THEORIES OF CONTINUITY AND INFINITESIMALS:
FOUR PHILOSOPHERS OF THE NINETEENTH CENTURY
Lisa Keele
Submitted to the faculty of the University Graduate School
in partial fulfillment of the requirements
for the degree
Doctor of Philosophy
in the Department of Philosophy,
Indiana University
May 2008
3319910
Copyright 2008 by
Keele, Lisa
All rights reserved
2008
3319910
Accepted by the Graduate Faculty, Indiana University, in partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
Doctoral Committee
___________________________
David C. McCarty, Ph.D., Chair
___________________________
Larry Moss, Ph.D.
___________________________
Timothy W. O'Connor, Ph.D.
___________________________
Frederick F. Schmitt, Ph.D.
May 7, 2008
ii
© 2008
Lisa Keele
ALL RIGHTS RESERVED
For Danforth J. Coonradt, who believed in a lot of hard work and a little magic.
iii
And for RPK. Cette thèse n'aurait pas pu se faire sans lui.
ACKNOWLEDGEMENTS
iv
This study could not have been done without the help of other people, whom I
have the pleasure now to thank:
To David C. McCarty I owe much indeed, including but not limited to gratitude
for his patience, his careful editing, his facility interpreting and explaining even the most
difficult proof, and his thorough knowledge of the history of mathematics. Without our
long conversations, this project would be substantially less rich; without his eye for detail,
this project would be riddled with many errors than it currently is.
I must thank Frederick Schmitt for suggesting the inclusion of C. S. Peirce in this
dissertation, and for his insightful comments on the manuscript; I also thank him for his
assistance with many practical aspects of completing this project.
I am grateful to Larry Moss for keeping me mathematically honest; simply
knowing he was to read it forced me to do my research on the most mathematical aspects
of this dissertation.
Thanks go to Tim O'Connor for his wise advice, including his advice, when I was
a young logic student, to write my dissertation on something philosophical. I have
thoroughly enjoyed doing so.
I am infinitely grateful to the people in my intellectual sangha, whose
encouragement, discussion, and advice have kept me going through the process of
creating this. There are too many wonderful and supportive people to list by name, but
special thanks must go to Erin Bolstad, Bill Gawne, Hanne Blank, C. J. Smith, Pragati
Jain, Jean D'Amato, and especially Joe Decker. Joe, you're wrong about the Axiom of
Choice.
To my sons Chris, Tim, and Jacob; I am grateful for the endless hours they spent
translating and typing my chicken-scratch handwriting, and for their patience in putting
up with a preoccupied mother. Most of all, I would like to thank my husband, Rondo,
v
for his philosophical and mathematical insight, for his scholarship, his help and
encouragement, and for simply being there. Thank you, sweetie. You can have your wife
back now.
ABSTRACT
vi
Theories of Continuity and Infinitesimals: Four Philosophers of the Nineteenth Century
Lisa Keele
The concept of continuity recurs in many different philosophical contexts.
Aristotle and Kant believed it to be an essential feature of space and time. Medieval
scholars believed it to be the key to unlock the mysteries of motion and change. Bertrand
Russell believed that, while everyone talked about continuity, no one quite knew what it
was they were talking about.
The subject of this dissertation is mathematical continuity in particular. By
mathematical continuity, I mean continuity as it applies to or is found in mathematical
systems such as sets of numbers. Mathematical continuity is a relatively recent concern.
The need to address whether numerical systems are continuous came about with the
creation of calculus, specifically, of limit theory.
The dissertation focuses on four mathematicians/philosophers from the late
nineteenth and early twentieth centuries who were concerned with mathematical
continuity. Richard Dedekind and Georg Cantor, in the 1870s and 1880s, developed the
concept of a 'point-continuum;' i.e. a continuum composed of discrete entities, such as a
collection of numbers arranged on a straight line. Paul du Bois-Reymond, in 1882, and
Charles S. Peirce, especially in his post-1906 essays, criticized this compositional pointcontinuum. Du Bois-Reymond believed infinitesimals were necessary for continuity;
Peirce believed no compositional continuum could ever satisfy our intuitions.
My ultimate conclusions are that (1) the concept of the mathematical pointcontinuum does suffer from philosophical difficulties, (2) the concept of the infinitesimal
vii
is neither as philosophically problematic nor as mathematically useless as is often
charged, but that (3) infinitesimals by themselves cannot solve the problems raised by a
compositional view of continuity.
TABLE OF CONTENTS
viii
Acknowledgements......................................................................................v
Chapter 1: Introduction ...............................................................................1
Chapter 2: History of Continuity...............................................................21
Chapter 3: Richard Dedekind....................................................................46
Chapter 4: Georg Cantor ...........................................................................75
Chapter 5: Paul du Bois-Reymond..........................................................102
Chapter 6: Charles Sanders Peirce ..........................................................133
Chapter 7: Infinitesimal Interlude ...........................................................162
Chapter 8: Conclusions ...........................................................................172
Appendix: A Partial Translation of the General Theory of Functions ...204
Bibliography.............................................................................................329
ix
And what are these fluxions? The velocities of evanescent
increments. And what are these same evanescent increments?
They are neither finite quantities, nor quantities infinitely small,
nor yet nothing. May we not call them the ghosts of departed
quantities?
Bishop George Berkeley, The Analyst, 1734.
Chapter 1
Introduction
1.1 The Historical Importance of Mathematical Continuity
In the last century, differential and integral calculus have been commonly referred
to as "the calculus," in much the same way as medieval scholars referred to Aristotle as
"the Philosopher" and ancient Romans referred to Virgil as "the Poet." It is as though
there is no other, or at least no better, way of calculating. It was not always so. Calculus
was born in conflict, with Gottfried Wilhelm von Leibniz (1646 – 1716) and Sir Isaac
Newton (1643 – 1727) both staking the claim of invention, and more importantly,
initiating a vehement argument about the philosophical details underpinning their largely
similar mathematical procedures. Calculus was invented in the seventeenth century but
did not gain prominence in the mathematical world until nearly two hundred years later,
early in the nineteenth century, largely because these same philosophical differences had
1
not yet been settled, or even addressed directly.
The philosophical difficulties which dogged the calculus in its early years were its
assumption of actual infinites, the assumption that the real numbers formed a kind of
continuity, and the philosophical difficulties surrounding the mysterious small part of the
equations. Calculus, as a means of calculating motion at an instant, necessitates some
method of referring to the distance between two numbers which are infinitely close.
Leibniz called this distance an infinitesimal, but claimed the infinitesimal was no more
than a useful fiction, not an actual number. Newton referred to this small distance
sometimes as a fluxion – the rate of change of flowing quantities over time – and
sometimes as an infinitesimal. While both mathematicians hinted at the possibility of
something like limit theory taking the place of fluxions or infinitesimals, limit theory was
not mathematically formalized until the early nineteenth century.
Limit theory, roughly, is the mathematical system that uses variables to allow one
quantity to approach infinitely closely to another but never reach it. Thus, limits rid
mathematics of the need for infinitely small entities: numbers no longer have to reach
infinitely small distances, as the gap between them (at all times finite) could decrease
infinitely. The introduction of this theory allowed calculus to move forward and gain
wider acceptance, but it was not itself universally accepted, some claiming it raised more
difficulties than it solved. In particular, it was limit theory which necessitated that the
real numbers themselves display a kind of continuity. If calculus was to be done
algebraically, without necessary recourse to geometry, the numbers themselves must be
able to make the continuous approach necessitated by limits, and thus, the continuity of
the real numbers must first be established.
2
The topic of this dissertation is this mathematical continuity, its philosophical
implications, and most importantly, the historical struggle to define and defend this
continuity, as the struggle took place in the late nineteenth century and on into the early
twentieth. By "mathematical continuity" I mean continuity as it is allegedly found in or
as it is supposed to apply to systems or sets of numbers. Mathematical continuity does
not therefore include the continuity of geometrical objects such as the straight line,
though geometry is of course a branch of mathematics (and though geometry will be
much considered in what follows). The following chapters will address such question as
whether mathematical continuity requires infinitesimals, whether limits require
mathematical continuity, and whether there is a definition of real numbers that respects
our intuitions and calculations, and also allows the real numbers to be collectively
continuous. The main question to be addressed, of course, is whether mathematical
continuity exists at all.
For most of the twentieth century, it was largely assumed that these questions
were settled, as far as calculus went. In the argument between mathematicians
calculating with infinitesimals and those calculating with limits, the limit theorists were
believed to have won. The infinitesimal theorists, it seemed, had failed to present a
coherent mathematical infinitesimal calculus, and meanwhile, limit theorists had grand
successes. However, in the 1960s, Abraham Robinson (1918 – 1974) presented a
mathematically coherent and powerful calculus which incorporated infinitesimals,
transfinite quantities, and real numbers into a single system, thereby reawakening the
sleeping dragons of the previous century. It is my hope that by reexamining the fertile
mathematical and philosophical arguments of the past we can better frame the current
3
debate about the philosophical underpinnings of the standard calculus and Abraham
Robinson's non-standard analysis.
The purpose of this introductory chapter is to define and discuss in a preliminary
fashion certain concepts which shall be referred to often throughout the course of this
dissertation, concepts relied upon in our investigation of continuity. These concepts are
infinity, infinitesimal quantities, and continuity itself. More precise and philosophical
discussions of continuity will be presented in subsequent chapters; this chapter will focus
instead on intuitive notions of continuity. However, we will follow our discussion of
intuitive continuity with some preliminary remarks on the concept of mathematical
continuity. The chapter will end with an outline of the structure of the rest of this
dissertation.
1.2 Infinity, Infinities, and Transfinites.
Infinity has had a long and troubled history, some viewing infinity as simply
impossible, others viewing it as simply factual, and still others viewing it as mystical or
mysterious. For example, one of René Descartes' (1596 – 1650) arguments for the
existence of God depends intimately on the concept of the infinite:
So from what has been said it must be concluded that God necessarily exists. It is
true that I have the idea of substance in me in virtue of the fact that I am a
substance; but this would not account for my having the idea of an infinite
substance, when I am finite, unless this idea proceeded from substance which
really was infinite.1
The infinite, for Descartes, is too large to be comprehended without the help of a
1
Descartes, Meditations on First Philosophy, Cambridge: Cambridge University Press, 1996, 31.
4
similarly infinite being – thus, without the existence of God, humans could never
comprehend the infinite. About twenty years after Descartes wrote his Meditations,
Baruch Spinoza (1632 – 1677) used the infinite to characterize his concept of God as
well. Proposition XXI of Spinoza's Ethics is, "all things which follow from the absolute
nature of any attribute of God must always exist and be infinite, or, in other words, are
eternal and infinite through the said attribute."2 It is not only necessary that God be
infinite, but also that everything that comes from God must be similarly infinite.
Not all discussions of the infinite are theological, of course. Aristotle (384 – 322
BCE) considered the infinite in Book I of the Physics, before he considered anything else
in the subject of science. When Aristotle laid out the principles of the physical world, he
wished to begin by discussing the most basic concepts, and for him the most basic
concept is that of number. When considering a thing, that thing must either be one or
more than one. If one, it must either be still or in motion; if more than one, it must be
either finite or infinite. Thus, number, motion, and the infinite must be addressed before
anything else in the physical world.3
Aristotle distinguished between potential and actual infinity. Potential infinity is
infinite in the sense that one can always find or create a bigger number, another instant, a
further corner of space, but at any given time, only finitely many such objects are
2
Spinoza, Ethics, Prometheus Books, 1989, 58.
3
Aristotle, Physics, Book I, 184b 15-22. Throughout the dissertation, all Aristotle quotations refer to The
Complete Works of Aristotle. Princeton: Princeton University Press, 1984. The pagination is the standard
Aristotelian pagination, however.
5
considered. Actual infinity is be an infinity of which it could be said that every one of
infinitely many objects exist, simultaneously. Aristotle himself believed that potential
infinity was a more coherent concept, and that actual infinity was problematic. He thus
argued that space is infinitely divisible, but only potentially so: we could always perform
another division on a spatial entity, no matter how many we have already performed, but
this infinite division could never be completed, one could never have an actually
infinitely divided section of space. For Aristotle, the infinite, in division, only exists
potentially.4
As for infinite magnitude, Aristotle argued that there is no such thing. Infinitely
many things do not exist in actuality, only finitely many. Infinite magnitude does not
even exist potentially; for "it is impossible to exceed every definite magnitude, for if it
were possible there would be something bigger than the heavens."5 However, Aristotle
did not want mathematicians to despair over the non-existence of the actual infinite, or
the lack of infinite magnitude. "In point of fact, they do not need the infinite and do not
use it."6 Thus, mathematicians would not miss infinite magnitude, and they have as much
infinite division as they need.
Georg Cantor (1845 – 1918), whose theory of continuity is discussed in Chapter 4,
would disagree. His transfinite theory pushed the idea of the infinite to new levels.
Cantor not only proved the mathematical existence of actual infinities, but he went on to
4
Ibid., Book III, 207b 10-15.
5
Ibid., Book III, 207b 20-21.
6
Ibid., Book III, 207b 30.
6
establish different magnitudes of infinity, and began to calculate with them.7 Transfinite
theory, the theory that these different magnitudes of infinity can be specified with
precision and then well ordered, is possibly the most controversial theory Cantor
forwarded, but classical set theory, his most popular, also relies on the idea of an actual
infinite.
Richard Dedekind (1831 – 1916), the subject of Chapter 3, also played an
important historical role in the development of the actual infinite, by solving a long
standing paradox of infinity. Ancient and medieval scholars puzzled over the fact that a
part of an infinite set is, in some cases, the same size as the infinite set itself, which they
viewed as paradoxical because a part must always be the smaller than the whole.8
Dedekind argued that rather than being paradoxical, this is the very definition of infinity.
“A system S is said to be infinite when it is similar to a proper part of itself; in the
contrary case S is said to be a finite system.”9 Thus, the fact that the even numbers can be
put into one-to-one correspondence with the set of all natural numbers is not a paradox,
7
See in particular Cantor, Contributions to the Founding of the Theory of Transfinite Numbers. It should
be noted that Cantor was not the first to prove the existence of the actual infinite, nor was he the first to
speculate that there was more than one type of infinity. He was, to my knowledge, the first who not only
proved that there is more than one magnitude of infinity, but who also proved the existence of infinitely
many magnitudes of infinity.
8
St. Bonaventure (d. 1274) and al-Ghazali (1058 – 1111) use this paradox as part of an argument against
the eternity of celestial motion, for example.
9
Dedekind, “The Nature and Meaning of Numbers,” 63. C.S. Peirce, as we'll see in Chapter 6, had a
similar proof of infinity, which involved what he called the "inference of transposed quantity."
7
but rather proof that the natural numbers are indeed infinite. This property does not hold
of finite sets. We will see to what use Dedekind puts this definition of the infinite in
Chapter 3.
1.3 Ghosts of Departed Quantities
The infinitely small is no less vexing or inspiring than the infinitely large. A
perfect example of the vexing nature of the infinitely small is given by George Berkeley
(1685 – 1753) when he suggested fluxions be defined as, "ghosts of departed
quantities."10 Though Berkeley was speaking of fluxions in particular, his jibe seems to
apply equally well to infinitesimal quantities and to limits. In all these cases, we attempt
to capture a measurement of a magnitude as its variable determinant shrinks beyond all
finite bounds. Calculus seems to require us to perform calculations using a quantity that
is smaller than any given quantity. Limits are not claimed to be "quantities," and thus
avoid these particular difficulties, though we are left with the question of what, exactly, a
limit is.
Though calculus places the infinitesimal in the realm of mathematics, the question
of the infinitely small predates Newton and Leibniz, appearing, for example, in
theological questions such as the old Medieval chestnut, "How many angels can dance on
the head of a pin?" The question is almost a joke today, but seven hundred years ago,
determining the answer to the question took some thought, as angels were thought to have
10
Berkeley, "The Analyst, A Discourse Addressed to an Infidel Mathematician," Works of George
Berkeley, ed. Alexander Campbell Fraser, Oxford: Clarendon Press, 1901, v. 3, paragraph XXXV.
8
no physical extension at all. The question, then, of how many things lacking physical
extension can fit in a particular space was a controversial one, and mathematics contains
an updated version of this puzzle when we ask whether it is possible for a line to be
composed of points. A similar medieval question wondered how angels could move
through space – whether it was logically possible for something non-extended to move
through extended space.11
Though philosophically troublesome, infinitesimal quantities are held by some to
be necessary elements of anything continuous. Paul du Bois-Reymond (1831 – 1889), the
subject of Chapter 5, believed infinitesimals to be necessary for continuity, and thus,
necessary for not only calculus, but for geometry and for any other branch of mathematics
or science that dealt with the continuous. As non-extended non-spatial entities, du BoisReymond was convinced that a straight line could not be composed merely of points.
Continuity required that the line contain intervals of infinitely small length (i.e.
infinitesimal intervals) in addition to points. Charles Sanders Peirce (1839 – 1914), the
subject of Chapter 6, came to a similar belief late in his career – the belief that points
were insufficient for continuity, but also that numbers were insufficient for similar
reasons. The solution Peirce gave is quite different than du Bois-Reymond's, but Peirce
11
See, for example, the first quodlibet of William of Ockham, question 5, "Can an angel move locally?"
William of Ockhams Quodlibetal Questions, tr. Alfred J. Freddoso and Francis E. Kelley. New Haven:
Yale University Press, 1991. The puzzle was briefly this: to say that x moved is equivalent to saying that x
was here, and is now there (with some important codicils). An angel, as a non-extended, non-spatial entity,
can never be here or there for any particular space. Similar concerns plague some explanations of
geometry.
9
also believed in the importance of infinitesimal quantities, and developed an infinitesimal
theory in stages throughout his philosophical career. While du Bois-Reymond argued
that the real number system of Cantor and Dedekind was inadequate, Peirce agreed but
went one step further, arguing that limit theory itself was philosophically confused. Both
men believed that infinitesimals helped solve these mathematical and metaphysical
problems.
Thus, the concept of the infinitesimal is one which will recur in every chapter in
this dissertation. Both Dedekind and Cantor believed infinitesimals were either
impossible or at least philosophically suspect; du Bois-Reymond and Peirce believed they
were integral and necessary parts of any actual continuity. Infinitesimals are one of the
issues over which our mathematicians divide, and any thorough treatment of the concept
of continuity must examine the role that infinitesimals do or do not play in this concept.
1.4 Intuitive Continuity
A necessary first step in a detailed investigation of the metaphysics and
epistemology of mathematical continuity is to define continuity itself. Definitions change
depending on which object is claimed as continuous. Often, mathematicians appeal to
our intuitions about continuity when forwarding a rigorous definition of continuity –
Peirce, for example, thought that the most important feature of a definition of continuity
was that it must match up with our intuitions. To attempt to examine these intuitions
about continuity, we will briefly look at the discussion a distinctly non-mathematical
continuum – that of the color-wheel – in Friedrich Waismann's Introduction to
10
Mathematical Thinking12.
Waismann asked, when you look at the color wheel, how many colors do you see?
Does each color nuance represent a new color? If it does, and the color spectrum is
continuous, then there should be infinitely many colors in a spectrum. But do you see
infinitely many colors? Waismann answers 'no.'
The color continuum has a structure entirely different from that of the number
continuum. In the case of two real numbers it is uniquely established whether
they are equal or different. No matter now close they may lie near one another on
the number axis, they are and always will be different numbers. A color,
however, runs into another imperceptibly, it blends with it; more accurately stated,
it is without meaning to speak of isolated elements out of which the continuum is
to be erected. The concept of number is not applicable to formations of this kind,
for the first assumption of counting is that the entities to be counted be clearly
distinguishable.13
Thus, colors are indistinct from one another, blending and merging in ways that are
predictable but not always perceptible. The color wheel, as a representation of this color
continuum, thus differs from crayons in a box, each of which represents a distinct,
identifiable color.
Though Waismann himself does not call the color wheel the epitome of intuitive
continuity (he discusses intuitive continuity directly, though he does not define it), the
changes on the color wheel do have features most people would associate with continuity
at an intuitive level. The lack of any clear distinction from one place on the wheel to the
next, the merging of one part into another, and the lack of gaps or spaces, all seem
12
Waismann, Introduction to Mathematical Thinking: the formation of concepts in modern mathematics,
Mineola: Dover Publications, 2003. .
13
Ibid., 212-213.
11
important features of continuity. For all their plurality, the real numbers, the nature of
which we shall consider in detail in the following chapters, are in some sense distinct.
They are a totally ordered set, and for any two non-identical real numbers, we can
distinguish them from each other – they do not blend into one another. Thus, the main
question with respect to the continuity of the real numbers is whether it is possible for a
continuum to be composed of distinct objects.
Typically, we view distinct objects, or even collections of distinct objects, as the
contrary of continuity. To continue with our investigation of our intuitions on the matter,
consider objects which are clearly non-continuous, such as such as tables, or a glass of
water. A table is not continuous, no matter how smooth, flat, or large it might seem; once
it is divided enough times, it becomes something different. Some tables, chopped in two,
might still function as tables, but chopped again and again, they cease to be tables, and
instead become firewood. Water will take finer-grained divisions before you get
something non-watery, such as individual hydrogen and oxygen atoms, but only a finite
number of divisions is needed to reach that level; water is no more continuous, at bottom,
than a table.
We may argue drawing on our above examples, that a color wheel is not
absolutely continuous either; color, after all, is a physical thing. We may take smaller and
smaller slices of our color wheel without seeing any breakdown of continuity; but that is
more a function of the limits of our eyes than it is true indication of the continuity of the
color wheel. If we could focus on a small enough wedge of the color wheel, we would
instead see a collection of atoms or sub-atoms, which are too small to reflect light and
12
therefore colorless.14 Our allegedly continuous color wheel breaks down. This raises two
important questions. First, is anything at all actually continuous? And second, what
would the nature of an actually continuous entity be?
Space, time, and motion have been considered actually continuous at least since
Aristotle. There are, however, those who hold that some or all of the above are noncontinuous – that time, for example, is composed of discrete atomic instants15 – but if
continuity ever exists in the physical world, these are the three top contenders for
continuous entities.16 Indeed, it seems as though no matter how short a segment of space
you have, it will resemble any other segment of space in important respects; you can
divide space into smaller and smaller parts and never reach anything qualifying as "nonspace." Also, one part of space merges into the next without identifiable borders,
distinctions, or gaps. Time, too, at least appears to flow by smoothly, without bits of nontime breaking things up. Motion is more complex, usually analyzed as an interaction
14
John Locke (1632 - 1704) posited such colorless particles in his theory of perception, in his Essay
Concerning Human Understanding, Oxford: Oxford University Press, 1979.
15
Of course, the acceptance of the existence of instants does not immediately lead to rejection of the
continuity of time.
16
As an interesting side note, Immanuel Kant (1724 – 1804) did not believe that space or time were part of
the physical world, but rather, that they existed as the very conditions of our perception of the external
world. However, he still held that they were continuous entities (see The Critique of Pure Reason,
A26/B42, and A33. See also A213/B260 and A170/B212 for Kant's discussion of the continuity of space
and time, respectively). This seems to suggest that for Kant, continuity itself was a condition of perception,
rather than a property of the external world.
13
between space and time, and therefore in some sense continuous; unlike space and time,
however, motion is usually limited to particular events, and even particular objects. In
Chapter 2, the continuity of space, time, and motion is discussed in further detail.
1.5 Preliminary Remarks on Mathematical Continuity
While continuity has been an essential part of geometry ever since Euclid (325 –
265 BCE), numbers have often been seen as the antithesis of continuity. Aristotle and
other ancient Greeks viewed geometry as the science of space (and thus naturally
continuous) and mathematics as the science of number. The concept of number, for
Aristotle, began with 1, and the collection of numbers included only multiples of 1.
Thus, there were only finitely many numbers (see Section 1.2 above), and numbers were
in no sense infinitely divisible; you could only divide numbers until you returned to the
smallest number: the unit. The set of natural numbers is clearly non-continuous; each
one is uniquely identifiable, there is no smooth connection between them, they are
discrete at least in the sense of being fully distinguishable, one from the other.
As we will see in Chapter 2, during the sixteenth and seventeenth centuries, the
concept of number broke out of this Aristotelian mold, and the division between
mathematics and geometry began to blur. Number systems grew to include the negatives
numbers, the rationals, and the reals. Cartesian advances in algebra and geometry
allowed geometrical shapes to be described completely numerically. Thus, as numerical
and mathematical systems grew, it became at least possible to view numbers as in some
sense continuous.
While it is clearly impossible to view the natural numbers as continuous, the same
14
difficulties do not automatically apply to, for example, the set of rational numbers. The
rational numbers are distinct in the sense of being uniquely identifiable and
distinguishable one from the other. However, the rationals have a feature the naturals
lack. The rationals are dense: between any two rational numbers, there exists another
rational. Furthermore, there exist infinitely many rationals between any two. Though 2/3
does not run and bleed into 3/4 the way red and orange subtly merge, it can be said that 2/3
and 3/4 are parts of a spectrum of their own sort, with the infinitely many rationals
between them resembling the infinitely many colors of the color wheel. Rational
numbers are distinguishable one from the other, but there is no next rational number – in
this sense, the rationals could be said to be smooth.
However, the rational numbers do not form a continuous set. The reason for this
failure highlights a third feature of our intuitions about continuity (the first two being
non-discreteness and smoothness): something which contains gaps is not continuous. A
span of time from which an hour has been removed is no longer a continuous span. The
time previous to this missing hour can be continuous, as can the time after – continuity
does not require all time to be present at once – but the time span with the gap in the
middle loses its cohesion, and therefore its continuity. The rational numbers have
identifiable gaps, at least in the sense that there are lengths which no rational number can
measure.
Bertrand Russell (1872 – 1970) discussed these gaps in his Principles of
Mathematics:
In the last chapter of part III, we agreed provisionally to call a series continuous if
it had a term between any two. ... [However,] ever since the discovery of
incommensurables in Geometry – a discovery of which is the proof set forth in the
15
tenth Book of Euclid – it was probable that space had continuity of a higher order
than that of the rational numbers.17
Russell believed that density was a type of continuity; yet, due to the incommensurability
(i.e. irrationality) of certain geometrical measurements, space is richer, more complete,
than the rational numbers. This incommensurability is easy to demonstrate. Consider a
right triangle with two sides of length 1.
Figure 1.1
Now take the three sides of the triangle and lay them end to end along a straight line,
comparing them to the rational number line.
Figure 1.2
The top line in Figure 1.2 is our deconstructed triangle; the bottom line is the rational
number line, attempting to measure the line above. A clearly measures 1, A and B
together are 2. However, our rational numbers contain no number corresponding exactly
to the sum of all three sides. We could find a rational that estimated the measurement of
17
Bertrand Russell, The Principles of Mathematics, London: George Allen & Unwin Ltd, 1903, 287.
16
this line with some precision – 3.4, for example, would be quite close, 3.41 would be
even closer – but exact measurement of this line is beyond the rational numbers. The
rationals have gaps when we attempt to compare them to geometrical structures. An
entity with obvious holes violates our intuitive sense of continuity.
The real numbers, unlike the rationals, are capable of measuring so-called
incommensurable geometrical forms. With the reals, we no longer find gaps of this kind
when we attempt to match our number line against the geometrical line, as our real
number system captures them all. Dedekind will argue convincingly that there are no
gaps at all in the real number system.18 However, while the real numbers have the
property of density, just as the rationals do, and while they have no obvious holes, there is
still some sense in which the set of real numbers is composed of discrete entities. While
it is harder to specify the irrational real numbers with exactitude, the real numbers are still
discrete in the following sense: given any two non-identical real numbers, it is possible
to distinguish them, one from the other. 2 and
distinguishable, but so are
2 are easily identified and unique, and
2 and 1.4142, though they are closer to each other.
As we will see in Chapter 2, Aristotle argued that a continuum could not be
18
Of course, this does not mean the set of real numbers is absolutely complete, in the sense of containing
every possible number or being able to perform every possible mathematical calculation; the reals do not
contain the imaginary numbers, for example, nor do they contain transfinites. They are gap-free in a very
specific sense, which will be discussed more thoroughly in later chapters. As we shall see in Chapter 6, the
fact that the set of real numbers does not contain every possible number was significant in Peirce's
determination that this set was not continuous.
17
composed of discrete entities, therefore, to the extent that
2 and the rest of the real
numbers are viewed as discrete entities, Aristotle would argue that the set of reals is not
continuous. These two issues – whether real numbers are discrete entities, and whether
continua can be composed of discrete entities – will become essential in much of what
follows. Dedekind did not address this difficulty directly, but followers of Dedekind have
taken various approaches, some arguing that Dedekind's real numbers are not discrete
entities, some arguing that Dedekind's continuum proves that continua can in fact be
composed of discrete objects. Cantor called the "point-continuum" (that is, a continuum
composed of discrete entities such as points or numbers) the very essence of continuity.
Du Bois-Reymond and Peirce believed that numbers (and points on a line, for that matter)
are indeed discrete entities, and they followed Aristotle in asserting that continua cannot
be composed of discrete entities. Thus, whether a continuous entity can be composed of
discrete elements is the very issue around which much of this dissertation turns.
1.6 Organization of this Dissertation
The debate over the discrete versus the continuous is just one of many issues
which will be examined carefully in the chapters that follow. The real number continuum
and its development in the late nineteenth and early twentieth centuries is, as we shall see,
rich with philosophical ramifications of all sorts. In order to give a background to the
debates of these four figures, as well as to highlight the philosophical concerns which
have followed continuity throughout history, the next chapter shall follow the history of
continua from Aristotle to Abraham Robinson, in brief outline. After setting this ground,
the next four chapters present in depth analyses of our four figures, starting with
18
Dedekind, and then moving on to Cantor, du Bois-Reymond, and finally Peirce.
Dedekind’s groundbreaking essay, “Continuity and Irrational Numbers,” which
explains the Dedekind-cut theory of real numbers, will be discussed in detail in Chapter
3, as will Dedekind’s philosophical view of the nature of numbers. Chapter 3 also
contains an argument that Dedekind's continuity is mathematically inconsistent with the
existence of infinitesimal quantities. Chapter 4 presents Cantor’s theory of continuity as
he developed it over time, beginning with his formulation of real numbers in 1872 and
ending with his specification of necessary and sufficient conditions for continuity from
his Gründlagen of 1883. Chapter 4 also treats Cantor’s argument that infinitesimals are
self-contradictory and therefore impossible.
In Chapter 5, we move toward mathematicians who believed infinitesimals are
consistent with continua, and consider du Bois-Reymond’s real number theory, theory of
continuity, and infinitesimal theory. Chapter 5 is mainly concerned with du BoisReymond's General Theory of Functions, in which he wished to thoroughly investigate
the philosophical groundwork and mathematical intuitions for functions themselves. This
chapter analyzes in detail du Bois-Reymond's theory of continuity as well as his argument
for necessity of infinitesimal magnitudes. Chapter 6 is dedicated to Peirce. Like Cantor,
Peirce developed his theory of continuity over time; unlike Cantor, Peirce changed his
view wildly. Chapter 6 thus follows this development in Peirce and analyzes the
ramifications of each of Peirce's theories of continuity. This chapter also contains
Peirce’s criticisms of Cantor and Dedekind, and considers the interesting relationship
between his theory of infinitesimals and his theories of continuity.
Chapter 7 is a short chapter meant to defend the idea that infinitesimals are not
19
prima facie useless as mathematical entities, thus attempting to overcome a criticism of
infinitesimals seen frequently; i.e. the charge that even if infinitesimals were consistently
definable, they serve no mathematical purpose. The chapter presents a few mathematical
examples in which the use of infinitesimals seems to add in a productive way to the
calculation. Chapter 8, the concluding chapter of this dissertation, will begin with a
summary of each of our four figures, and then analyze the theories of continuity presented
within. Chapter 8 will conclude that Cantor and Dedekind, by composing continua from
discrete elements, create problematic theories. Also, while du Bois-Reymond and Peirce
forward fair criticisms of Cantor and Dedekind, ultimately, Peirce's most mature theory of
continuity is even more problematic, leaving du Bois-Reymond's as the most promising
theory of the bunch. Finally, Chapter 8 indicates how the results of this dissertation
might be applied to the philosophical investigation of contemporary theories of
infinitesimals.
20
Chapter 2
History of Continuity
2.1 Historical Context
As was noted in Chapter 1, the concept of mathematical continuity is tied
intricately to particular mathematical developments and systems, and thus to a particular
point in history. Mathematical continuity is defined as continuity as it is meant to apply
to systems of numbers, and thus only becomes possible with the introduction of real
number theory in the sixteenth century, and only becomes relevant with the formal
introduction of limit theory in the early nineteenth century. However, just as intuitive
notions of continuity helped give our preliminary discussion of mathematical continuity a
framework for discussion, so too an overview of philosophical approaches to the concept
of continuity will prove highly useful in discussing mathematical continuity throughout
the rest of this dissertation. Toward that end, this chapter presents a brief overview of the
main philosophical approaches to continuity through the ages, as well as an overview of
some of the historical mathematical developments most relevant to this project.
The chapter begins with ancient Greece. Section 2.2 briefly looks at some of
Aristotle's more important contributions to our understanding of the concept of
continuity, especially his definitions of continuity and his argument that continua cannot
be composed of indivisible entities. Section 2.3 discusses Archimedes, whose startlingly
modern methods of calculation led him to a concept of continuity which differed starkly
21
from Aristotle's. Section 2.4 raises some of the more interesting medieval debates
concerning continua, especially as surrounds the discussion of how one ought to
understand motion and change. These debates on motion lead naturally to the earlymodern period, discussed in Section 2.5, which centers on the development of the
calculus – one goal of which is to be able to measure motion at an instant. Section 2.5
glances at the mathematically fertile century before calculus itself was invented, then
looks into how the idea of continuity was central to the ensuing controversy which
surrounded the calculus for centuries after its introduction. These historical
investigations should leave us, by the end of this chapter, with a thorough background of
the issues which concerned our four figures.
2.2 Aristotle's Continuity
Continuity was an important concept for Aristotle, and he spent much of the
Physics considering its various properties. Aristotle wished to understand continuity
thoroughly in order to understand space, time, and motion, all of which he saw as
necessarily continuous entities; he also wished to develop a theory of continuity which
could be used to overcome Zeno's paradoxes. In this section, however, we shall ignore
Aristotle's actual applications of his theory in favor of a look into the theory itself,
particularly focusing on his argument that a continuum cannot be composed of indivisible
entities. This argument, as well as the distinctions and definitions he uses to develop the
argument, will be directly relevant to material in later chapters.
Aristotle explicitly equated continuity with infinite divisibility. Thus, for
22
Aristotle, "the continuous is divisible ad infinitum;"1 and also the reverse, "what is
infinitely divisible is continuous."2 Similar statements appear throughout the Physics. As
we saw in Chapter 1, our intuitions on continuity agree that entities things are easily
destroyed by a few divisions do not seem continuous to us. If we divide a unit of space,
the result is still a unit of space; if we divide a table, the result is firewood. Tables are
not, therefore, continuous.
However, by emphasizing infinite divisibility, Aristotle was not simply invoking
the idea that space was not destroyed by division. He went quite beyond the simple
intuition expressed above when he elaborated further: "By continuous I mean that which
is divisible into divisibles that are always divisible."3 In other words, Aristotle believed
there can be no end to the division of a continuous entity – no smallest part can be
reached through division. Also, for Aristotle, the opposite holds: continua are not
composed of indivisibles any more than they are divided into them.4
Thus, if Aristotle is correct, time is not composed of instants, and continuous
magnitudes – including the geometric line – are not composed of points. This does not
mean that the line contains no points, only that successive divisions of the line never
produce points or anything else non-divisible, and that it is impossible for a line to be
composed exclusively of points.
1
Aristotle, Physics, Book I, 185b 10.
2
Ibid., Book III, 200b 18.
3
Ibid., Book VI, 232b 23.
4
Ibid., Book VI, 232a 24.
23
Aristotle's argument for the non-compositionality of the line relies on a three-part
distinction: things "next to" each other are either in succession, contiguous, or
continuous. Before we explain this distinction, note that this is an inclusive and not an
exclusive disjunction, as contiguous things are also in succession, and continuous things
are contiguous – the distinction is rather a matter of degree of closeness. Also note that
Aristotle means this to be an exhaustive definition; these are the only three ways in which
two things can be said to be next to each other.
Briefly, Aristotle's tripartite distinction is this: two things are in succession if one
immediately follows the other without something of the same kind in between. There
could be something of a different sort between them, such as air between two people
standing in succession in a line at the movies, but there is no third person between them.
Two things are contiguous if they are in succession, and the outer extremity of one
touches the outer extremity of the other. Two books next to each other on a tightly
packed shelf are contiguous. They are in succession because there is no third book
between them; they are contiguous because their covers physically touch each other.
Finally, two things are continuous if the outer extremities, rather than merely touching,
are in fact one – continuous things share an outer extremity. Two countries which share
a land mass, such as Egypt and Sudan, form a continuous entity: unlike books with
separate (but physically touching) covers, there is only one border between Egypt and
Sudan; the southern border of Egypt is also the northern border of Sudan, there are not
two distinct borders abutting each other.
With this distinction in mind, we turn now to Aristotle's main argument for the
24
conclusion that continua cannot be composed of indivisibles.5 Assume for reductio that
there is a continuum composed exclusively of indivisibles, such as points.6 It follows,
then, that there must exist at least two points which are next to each other – for if they
were not, then either there would be nothing in between them and thus the continuum
would not be continuous, or there would be some non-point thing in between them, and
thus the continuum would not be composed exclusively of points, but instead composed
of points and something else. Consider now, in what sense these two points could be
next to each other, given our previous distinction. They cannot be merely in succession,
as that would allow for gaps which would destroy the continuity. Thus they are either
contiguous or continuous. However, contiguity requires contiguous entities to have
extremities which touch each other, and points, being indivisible and thus without parts,
have no such extremities.7
Thus, there is only one option left; the two points must be next to each other in a
continuous manner, i.e., they must touch such that their extremities are one. However, as
we have just noted, indivisible entities have no extremities; in fact, since points have no
parts at all, no part of a point can touch a part of another point. The only manner in
which these neighboring points could be next to each other is to touch each other by
5
Ibid., Book VI, 231a 18 – 231b 16.
6
Aristotle states the argument in terms of points, but notes that the same reasoning applies to any type of
indivisible.
7
Just to make the reasoning here explicit: if a point had an outer extremity with which it could touch a
bordering point, then that point would be divisible into outer extremities and inner non-extremities – and
would therefore be divisible.
25
being in exactly the same spot as one another; i.e. they must be "in contact with one
another as whole with whole."8 However, if this were so, the extension of a single point
would be the entire continuum, which is impossible, as continua are infinitely divisible,
and points are indivisible. As we have exhausted all of the possible ways in which two
things can be next to each other, we must conclude that two points cannot be next to each
other in any way. As our assumption that there was a continuum composed exclusively
of points led to the conclusion that two points on the line must be next to each other, we
must conclude that continua cannot be composed of points.
There are several interesting assumptions and definitions contained within this
proof that will repay examination. First, the characterization of a point as an indivisible,
while not surprising, is quite important to later developments in the theory of continuity.
Second, note the first step of this argument, i.e. the claim that if a continuum is composed
of points, it follows that two points must be next to each other. Those who have some
experience with the real numbers, or even the rational numbers, will recognize that this
contradicts the property of density: between any two real numbers, there is always a
third, and the same holds true of the rationals. This third point does not constitute a gap
in the continuum, and it does not constitute a non-point entity, therefore, this third
alternative has not been ruled out by Aristotle. The ramifications of this third alternative
will be discussed in later chapters, since several of our authors, including Cantor and
Peirce, argue that density is a necessary condition of any continuous system.
In fact, we will see echoes of Aristotle's argument throughout this dissertation.
8
Ibid., Book VI, 231b 4.
26
Whether a line can be composed of points, or whether any continuum can be composed of
indivisibles, is a fundamental debate in many discussions of continua. His argument was
repeated and elaborated on throughout the medieval period, and in Chapter 5, we will see
that similar considerations led du Bois-Reymond to a conclusion that is very different
from Aristotle's. In Chapter 6, we will see that Peirce used a modified version of
Aristotle's tripartite distinction to define his own system of mathematical continuity, even
though he later came to agree with Aristotle that continua cannot be composed of
indivisibles.
Before leaving Aristotle, note that so far as we have seen, he has only argued that
continua cannot be composed of indivisible entities; the other side of the coin, that
continua cannot be divided into indivisible entities, is proved later on in the Physics, but
relying on the same principles as the first argument:
Moreover, it is plain that everything continuous is divisible into divisibles that are
always divisible: for if it were divisible into indivisibles, we should have an
indivisible in contact with an indivisible, since the extremities of things that are
continuous with one another are one and are in contact.9
Thus, imagine a division in a continuum that produced, instead of two parts that were
themselves continuous and infinitely divisible, two parts that were indivisible. These
indivisibles would be connected continuously (as they are the product of a division on a
continuum), but this implies that they share borders. As Aristotle demonstrated in the last
argument, indivisible entities have no borders to share, thus such a division is impossible,
and every division on a continuum results in things which are themselves divisible. As
9
Ibid., 231b 16-19.
27
we shall see in the next section, Archimedes made significant mathematical advances by
rejecting just this characterization of continuity, by instead viewing continua as being
divisible into indivisible entities.
2.3 Archimedes and the Infinitely Small
Archimedes of Syracuse (281 – 212 BCE) assumed that continuous things were in
fact composed of indivisible parts; and more importantly, that a continuum could be
divided into indivisibles. A plane, for example, could be viewed as composed of
infinitely many lines.10 While lines are certainly divisible along their length, they are not
divisible along their height, as they lack that dimension; in this sense, they are indivisible
elements. The assumption of an entity containing infinitely many indivisible objects was
key to Archimedes' "method of exhaustion," a startlingly accurate system he used to
measure seemingly immeasurable geometric figures and shapes. The method of
exhaustion is similar in many respects to the calculus of Newton and Leibniz, and in fact,
this method heavily influenced the mathematicians of the sixteenth and seventeenth
centuries.
The method of exhaustion involved first inscribing the figure to be measured with
infinitely many figures of a specified shape. For example, to measure a parabolic
segment, he first inscribed a triangle within it.
10
For a thorough description of the method of exhaustion, see Carl B. Boyer, The History of the Calculus
and its Conceptual Developments, New York: Dover Publications, 1959, 50 ff.
28
Next, he would inscribe smaller triangles, which had for their bases the sides of the
original triangle.
The sides of these triangles were then used as the bases for new inscribed triangles, and in
this manner, Archimedes obtained “a series of polygons with an ever-greater number of
sides. […] He then demonstrated that the area of the nth such polygon was given by the
series A (1 +
1 1
1
1 1
4
+ 2 + ... + n −1 + ⋅ n −1 ) = A where A is the area of the inscribed
4 4
3 4
3
4
triangle having the same vertex and base as the segment,”11 the segment being the
original parabolic segment which was to be measured.
Thus, we view the area inside the parabolic segment as composed of infinitely
many infinitely small triangles, creating a polygon of infinitely many sides. Similarly,
11
Ibid., 52. In the calculation, the result is estimated through the method of exhaustion. As Boyer explains,
"As the number of terms becomes greater, the series thus 'exhausts'
29
A only in the Greek sense that the
Archimedes would circumscribe a triangle around the outside of the polygon, and
circumscribe infinitely decreasing triangles between this larger triangle and the external
walls of the parabolic segment, creating another polygon of infinitely many sides. One
could use the equation in the Boyer quote above with the theory of limits to find the
length of the curve and the area of the parabolic segment, but Archimedes did not view
this simultaneous inscribing and circumscribing of polygons with smaller and smaller
sides as approaching a limit. Rather, after inscribing and circumscribing his two different
infinitely-many sided polygons, he argued by a double reductio ad absurdum that the
parabolic segment above “could be neither greater nor less than
4
A .”12
3
Archimedes did not operate with the limit concept, nor try to formulate a
generalized means of finding the area of any such figure, but rather restricted himself to
inscription and circumscription, using different inscribed figures as each shape
demanded. Thus, although he was highly influential on the mathematicians who helped
give rise to the calculus, he himself can not be viewed as having invented the calculus. It
is interesting, however, to note that a sort of infinitesimal calculation was in use as early
as Archimedes. In fact, had medieval philosophers and mathematicians been less
influenced by Aristotle to the exclusion of Archimedes, it is possible that calculus or
something like it may have been invented much earlier, building on these Archimedean
remainder,
A, can be made as small as desired. This is, of course, exactly the method of proof for
the existence of a limit, but Archimedes did not so interpret the argument."
12
Ibid., 52.
30
concepts.
2.4 The Medieval Debate on Motion, Change, and Continuity
Even before medieval scholars had access to Aristotle's Physics and its analysis of
motion as implying continuity, they were highly interested in the peculiar logical qualities
of the words "stop" and "start;" i.e. they saw philosophical problems precisely where
motion changes into non-motion, and the reverse.13 An analysis of this medieval debate
will help set the ground for the philosophical problems that arose with the introduction of
calculus itself.
To understand the difficulties inherent in the concepts of starting and stopping,
consider the sentence, "This ball begins to roll." It is perfectly straightforward, on the
face of it; a simple present-tense positive statement. Yet embedded within the sentence is
an implicit negation, as well as an implicit reference to either the past or the future (or
both), depending on how the sentence is analyzed. For "This ball begins to roll" implies
that an instant ago, the ball was not rolling (the implied negation), and also implies
something about both the past (that the ball was not rolling), and the future (that the ball
will, at least in the immediate future, continue to roll, as motion must take place in time).
"The ball ceases to roll" has parallel complexities; it implies that an instant ago, it was
13
Norman Kretzmann supports the thesis that beginning and ceasing were important to the medievals before
Aristotle's Physics was widely available throughout Western Europe. See Kretzmann, "Incipit/Desinit,"
Motion and Time, Space and Matter: Interrelations in the History of Philosophy and Science, Columbus:
Ohio State University Press, 1976.
31
rolling, and an instant from now, it will no longer do so, and thus we have an embedded
negation and reference to the past and the future, all contained in a seemingly simple
present-tense positive statement.
Thus, early medieval discussions dealt with the complex logical implications of
the notions of starting and stopping. Once the medieval philosophers became familiar
with Aristotle's Physics, the somewhat complex logical puzzle of starting and stopping
exploded into full-blown metaphysical conundra. For all beginnings and endings imply
change; and change, as analyzed by Aristotle and medieval philosophers, is a type of
motion. For Aristotle, and thus for later medieval philosophers, motion implied
continuity.
Medieval philosophers inherited Aristotle's analysis of continuity almost
wholesale when they gained access to the Physics. Thus, they inherited the ideas that
space, time, and motion were all continuous, and that continua were infinitely divisible.
The first idea meant that every discussion of natural philosophy had to include some
discussion of continuity, as natural philosophy almost always dealt with change of some
sort; thus, for example, when William of Ockham (1287 – 1347) tackles the question "Do
angels move?" he must not only reckon with the Bible, with the metaphysical status of
angels, and with the meaning of motion itself, he must also grapple with the continuity of
motion, time, and space.14 The second idea meant that the nature of infinite divisibility
needed to be understood thoroughly before natural philosophy could be understood fully.
14
Ockham, William of Ockham's Quodlibetal Questions, question 5. Recall, this puzzle is presented in
footnote 11 of Chapter 1.
32
To see how the concept of continuity influenced the medieval debate on motion,
let us return to the example of a rolling ball. Medieval scholars wished to know what
occurs during the very instant the ball begins to roll. For consider: the ball begins to roll
at time t. Thanks to the law of excluded middle, we know that the ball either is or is not
rolling at t. Furthermore, the time span when the ball is not rolling must be adjacent to
the time span when the ball is rolling (for if there were any time in between states, the
ball would neither be rolling nor not rolling at that time); and the two periods cannot
overlap (for if they did, then at some period would be both rolling and not rolling – a
contradiction).
Yet given the assumptions that (i) time is a continuum, (ii) a continuum cannot be
composed of indivisibles, and (iii) any indivisibles which appear in a continuum cannot
be next to each other,15 we seem to have reached a difficulty. For if the ball begins to roll
at time t, what happens immediately before time t with respect to the ball? There cannot
be both a first instant of rolling and a last instant of non-rolling, for that implies either
that instants are next to each other, which Aristotle argued can never happen, or that there
is a period of time when the ball is neither rolling nor non-rolling, which is a
contradiction.16
15
Recall from Section 2.2, Aristotle's statement that "two points must be next to each other" was not a
statement Aristotle affirmed, but was supposed to follow from the reductio assumption that the straight line
is composed of indivisible points. This statement was rejected in the course of the argument.
16
My discussion of this difficulty is influenced by the discussion in Paul Spade's "How to Start and Stop:
Walter Burley on the Instant of Transition," Journal of Philosophical Research, Volume XIX, 1994, 193–
221.
33
The difficulty was approached in various ways by various philosophers. Though
it was not philosophically fashionable to reject Aristotle's key views, a small group did
so, and argued that time and space were in fact composed of indivisibles. Walter Chatton
(d. 1344), for example, argued that a line was not only composed of points, but was
composed of finitely many points. Other philosophers, such as Walter Burley (d. 1344),
analyzed change by using something similar to limits, thus claiming that the rolling ball
can have a first instant of movement but not a last instant of non-movement. It is
interesting to see the concepts of continuity and limits philosophically connected
centuries before limits were formally introduced in mathematics. Ockham, on the other
hand, argued strongly against the existence of indivisibles that his contemporaries
supported, drawing not only on Aristotle's arguments against indivisibles, but on
mathematical and geometrical arguments first proposed by al-Ghazali (1058 – 1111) and
popularized in the West by John Duns Scotus (d. 1308).17
An issue related to stopping and starting is the question of whether or not motion
is possible at an instant. Aristotle himself discussed this question, and concluded in the
negative, but his discussion focused on whether a motion can take place entirely within
an instant. A more complicated issue is whether motion can happen at all in an instant.
Think, for example, of our rolling ball; we know it is indeed in motion while it rolls, but
is it therefore in motion in every particular instant throughout the duration of its motion?
It seems as though it should be the case that, if an object has a property throughout a
period of time, it should also have that property through any part of that period, even a
17
See Ockham's first quodlibet, question 9, "Is a line composed of points?"
34
part as short as an instant. Yet some accounts of motion prohibit such an understanding.
Ockham, for example, defined the motion of x as meaning that x is in some place at time
t, and at a different place at a later time, t1. Some codicils apply to ensure that the object x
did not somehow, for example, cease to exist during the time period, but the notable
feature of Ockham's definition is that it is only possible to say whether something has
moved after a span of time has elapsed. An instant is not a span of time, thus to say
something is in motion at an instant is therefore nonsense, even if that instant occurs
during the time period between t and t1, that is, during the time at which the object was in
motion.
Of course, the idea that an object cannot have motion at an instant is not only
counter-intuitive (what does a moving object do at such an instant? Stop?) but it is also
antithetical to the basic principles of modern analysis, where we not only claim that an
object in motion is also in motion at an instant, but we are able to calculate the object's
velocity at that instant. Fortunately, not every medieval philosopher agreed with
Ockham; Chatton, for example, (who disagreed with much of what Ockham wrote)
presented an analysis of motion which left open the possibility of instantaneous motion.
Thus, continuity was very much a live and lively issue for philosophers during the
medieval period. The assumption of time, space, and magnitude as continuous entities
influenced discussions about motion, starting, and stopping, and also appeared in
geometrical discussions, such as whether or not a line is composed of points. This period
is interesting not only in its own right, but because these discussions set the stage for
early modern mathematical developments, from Cartesian geometry to the development
35
of the calculus.18
2.5 Analysis
It is notable that up until this point in our history discussion, continuity was not in
the main a mathematical property. It was mostly a physical and metaphysical issue;
geometry certainly touched on such topics, but arithmetic never did. This was to change,
of course, with the introduction of real number theory, Cartesian graphs, and similar
mathematical advances; but no event or theory changed the face of mathematics quite so
much as the introduction of the calculus.
A fascinating feature of the century leading up to the invention of the calculus was
a change in attitude towards Aristotle. Elie Wiesel (b. 1928) has said many times of his
own religious faith, “With God or against God, but never without God.”19 Medieval
scholars had a similar approach to Aristotle, analyzing his works, arguing with him, but
almost always mentioning him in some form or fashion. This changed in the sixteenth
18
The discussions of Burley, Ockham, Chatton and others in the 14th century and earlier influenced a group
of logically and mathematically minded gentlemen often referred to as the "Oxford Calculators" or the
"Mertonian Calculators", who worked on similar issues of space, time, and motion in the late fifteenth and
early sixteenth centuries; the Calculators in their turn influenced the science of Leonardo da Vinci and
Galileo. These medieval ponderings where thus not inconsequential to the history of mathematics and
science. See Edith Dudley Sylla's article "The Oxford calculators" in The Cambridge History of Later
Medieval Philosophy, ed. Norman Kretzmann, Anthony Kenny, and Jan Pinborg, Cambridge: Cambridge
University Press, 1982.
19
As quoted in an interview in the U.S. News and World Report, October 27, 1986, 68.
36
century. While Aristotle was still studied in this century, a new and growing group
rejected both Aristotle and the Aristotelianism of the medieval age.20 Mathematicians of
this age rejected Aristotle’s argument that mathematics was the science of counting and
number began with 1. The door was thus opened for negative numbers, fractions,
irrational numbers, and imaginary numbers, all of which were accepted as types of
numbers by Western mathematicians of the period.21 This partial rejection of Aristotle
also opened the door for reappraisal of Archimedes, and it was Archimedes who burst
into prominence in certain intellectual circles, his works appearing in numerous editions,
his method of exhaustion thoroughly studied, repeated, and expanded upon.
Mathematicians in this period who did notable work on furthering the method of
exhaustion were Simon Steven (1548 – 1620), Luca Valerio (1552 – 1618), and Johannes
Kepler, (1571 – 1630). In each case, limit theory was approached, but never reached.
Kepler in particular innovated considerably on Archimedes’ method. In computing the
volume of various solids, Kepler assumed them to be composed of infinitesimal shapes of
various kinds, not always assuming these infinitesimals to be indivisibles.22 Bonaventura
Cavalieri (1598 – 1647) took Kepler’s methods even further (though he denied any
inspiration from Kepler’s work), writing the highly influential Geometria indivisibilibus
in 1635.
Thus, by the mid-seventeenth century, the time was ripe for analysis; so ripe that
20
Boyer, 96.
21
Ibid., 97-98.
22
Ibid., 109.
37
two apples fell almost simultaneously from the calculus tree. Newton and Leibniz,
apparently independently, introduced a method of mathematics that went by several
names – infinitesimal analysis, differential calculus, etc. – and which today is referred to
in college courses simply as "the calculus," as though there was never any other. Among
other things, this new calculus allowed mathematicians to calculate the tangent to any
curve with remarkable precision. Applied to science, the theory allowed the measurement
of trajectory at a point, and of motion at an instant. The continuity of time and space
became a feature of mathematics at large, rather than being a feature only of geometry, as
continuous functions were mapped, plotted, and calculated; and mathematics had
suddenly imported into it all the metaphysical and philosophical concerns of continuity.
Chief among the metaphysical concerns associated with calculus was "What,
precisely, are we measuring?" To demonstrate the problem explicitly, a concrete example
can help. The advances Descartes made in algebra allowed curves to be measured with
precision, and algebraic equations to be plotted as curves. However, calculating the slope
of a line was still only possible under certain circumstances. Imagine, for example, that
we have a curve on a graph, the equation of the curve, and a particular point on the curve
– call the point P1, with coordinates (a, b). We wish to find the slope of the tangent line
that intersects the curve at P1. If we had another point on the tangent line, we could easily
calculate its slope: if we had point (a, b) and also (c, d), the slope m is found with the
equation m =
d −b
. However, with only one point, this formula is useless.
c−a
One of the fundamental advances in calculus is that it allows us to find this slope,
given only the point P1 and the equation of the curve, in roughly the following manner.
38
Find a second point on the curve, relatively close to P1 – call it P2 – and draw a straight
line through these two points. We can use the above formula to figure the slope of this
line, but this is not the tangent line, it is rather a secant – it crosses our curve at two points
rather than intersecting it at only one. Now, we find a point on the curve closer to P1 –
call it P3 – and draw another secant line. The slope of this line is closer to the slope of the
tangent line than the previous secant, since P3 is nearer to P1, but it is still not the tangent
itself. We can keep choosing points closer and closer to P1, and with every step closer,
the slope of the resulting secant line becomes a more accurate estimation of the slope of
the tangent.
However, notice what happens if we choose the point P1 itself, rather than a point
near it – that is, notice what happens if the distance between the points on the curve
collapses. If we plug the coordinates of P1 into the slope formula we are left with the
equation m =
b−b
. Not only is this not informative, it is nonsensical; the equation
a−a
requires that we divide by zero, which we cannot do. Thus, ideally, the point we choose
to use as an estimation for the tangent line itself must be as close as possible to P1
without actually being equal to P1. The closer the point is, the better the estimation, but
some distance must be maintained to avoid division by zero. In fact, if we could find a
point Pn that was infinitely close to P1, the secant line would no longer give us an
estimation of the tangent line – rather, we could calculate the actual slope of the tangent.
Both Newton and Leibniz were sure this infinite closeness could be suitably formulated
and calculated with. Their original attempts to do so were met with philosophical and
mathematical skepticism, but notably, though the two men used very different means of
39
formulating this infinite closeness, their calculations produced the same results.
However, characterizing with exactitude this part of the calculation – the place
where P1 and Pn become infinitely close without becoming equivalent – proved a difficult
task, and one which created much ado among the mathematical community. What,
precisely, was this "vanishing point"? Of what did it consist? Was it indeed an infinitely
small ratio? Did the idea of an infinitely small ratio make any sense? Did it make any
more sense than dividing by zero? Were they, after all, no more than "ghosts of departed
quantities" as Berkeley charged? Infinitesimals had never been universally accepted
among the philosophical community, and infinitesimal quantities were no less
controversial.
The two founders of the calculus dealt with this open metaphysical and
mathematical question in different ways. Newton was, at bottom, a scientist in need of a
tool, and the tool was needed to analyze motion. He saw continua as created by motion,
and saw the difference between these two points (P1 and the very close point used to
calculate the tangent) as an infinitely small moment in time.23 Thus, imagine the curve as
generated by the motion of some entity; and imagine a very accurate stopwatch. We start
the watch when the entity reaches P1, and stop it a mere moment later, when it reaches Pn.
The distance traveled in that moment is, roughly speaking, a fluxion. However, Newton
ultimately became dissatisfied with this formulation, and especially with the explicit
reference to infinitesimal quantities; he eventually rejected it in favor of a theory of ratios,
23
See John Bell's "Continuity and Infinitesimals" article in the Stanford Encyclopedia of Philosophy (Fall
2005) Section 4.
40
keeping the references to motion, but more closely resembled modern limit theory.24
Leibniz thought of the difference between P1 and Pn as an infinitely small
distance,25 but it bothered him that he did not have a clear understanding of what an
infinitely small distance was. After much thought, he concluded that infinitesimals were
not actual numbers, but idealized fictions, useful for mathematics.26 An 'infinitesimal'
was no more than a way of speaking about the distance between P1 and Pn, a way of
referring to it and moving on with the calculation. However, Leibniz's subsequent
attempts to define this useful fiction rigorously were problematic. He presented an
elegant set of rules governing his calculus, and insisted that infinitesimal quantities were
to "obey precisely the same algebraic rules as finite quantities."27 This insistence led
directly to the conclusion that infinitesimal quantities must be treated "in the presence of
finite quantities, as if they were zeros."28 It is clear from our above example that if an
infinitesimal quantity is treated as if it were zero, then the introduction of an infinitesimal
24
Ibid, Section 4.
25
Leibniz does use the word "infinitesimal" (in Latin, infinitesima) as a noun, but more frequently uses the
term "infinitely small" (infinite parvum) as a substantive or an adjective.
26
Interestingly, the attempt to define infinitesimal quantities in mathematics ultimately led Leibniz to reject
their mathematical existence, but consideration of the same issues led him to postulate the actual existence
of infinitely small entities called "monads," on which his metaphysics is based. See Bell, SEP, Section 4.
For Leibniz's proof that infinitely small things are fictions, see Leibniz, The Labyrinth of the Continuum:
Writings on the continuum problem, 1672 – 1686, New Haven: Yale University Press, 2001, Aiii52.
27
Bell, SEP, Section 4.
28
Ibid.
41
difference between P1 and Pn no longer accomplishes what it was meant to – that is, no
longer keeps a large enough distance between the two points to avoid division by zero.
Perhaps because it was relatively well defined, Leibniz's calculus dominated the
continent for several decades, gaining faithful followers such as Guillaume de l'Hôpital
(1661 – 1704), who developed it further. Yet, the mathematical and philosophical unease
about infinitesimals led mathematicians to look for a new foundation. The proto-limits
mentioned in Newton's calculus held some appeal as an alternative, and more precise
definitions of them were sought. Great strides toward ridding the calculus of vague
definitions and useful fictions were made by Augustin Cauchy (1789 – 1857), who
developed a rigorously defined theory of limits. As Boyer wrote,
In the work of Cauchy, however, the limit concept became [...] clearly and
definitely arithmetical rather than geometrical.29
To return briefly to our tangent example above, Cauchy's limit theory allowed us to
calculate the tangent at P1 arithmetically, without reference to geometry at all, and rather
than positing a second number Pn existing an infinitesimal distance away from P1, we
could instead calculate a variable quantity Px which infinitely approached P1, but never
reached it. In other words, limit theory allowed the calculation of the tangent without the
necessity of positing an infinitely small magnitude.
Limit theory was further developed and refined, and calculus gained wider
acceptance. However, limit theory did not have universal approval among
mathematicians and philosophers. Some were suspicious of limits, and many were still
29
Boyer, 272-273.
42
convinced that infinitesimals had a place in calculus and in mathematics in general. Paul
du Bois-Reymond, Charles Sanders Peirce, Otto Stolz (1842 – 1905), and Charles
Dodgson (1832 – 1898) are some of the mathematicians who attempted to develop
infinitesimal theory further, even after limits had a wide-spread following.
2.6 Conclusions
Limit theory proved to be a mathematically powerful system, one which has been
used to great effect since its adoption: one need only think of the advances in physics,
astronomy, chemistry, and virtually every scientific field which uses calculus to be
convinced of this fact. Philosophically, limit theory solved the troublesome need to refer
to infinitesimals (though as was briefly noted above, and as shall be seen in more detail
below, not everyone believed infinitesimals to be troublesome or unnecessary), but
necessitated the extension of the concept of continuity into the realm of systems of
numbers. For, after Cauchy, calculus was freed from its necessary connection to
geometry, and one could calculate motion at an instant without reference to curves and
lines, but only if number systems themselves contained the property of continuity. If Px
(now a variable quantity, not a point) is to approach some number P1 infinitely and
continuously, it can only do so if the numbers themselves form a continuous set.
Though I hope this historical overview of various theories of continuity has been
interesting in its own rights, I also wish to focus for a moment on concepts introduced in
this chapter which apply directly to the difficulties inherent in mathematical continuity in
particular. Of primary importance is the debate between those who believe a continuum
can never be composed of indivisible elements, and those who think it can. At the dawn
43
of this debate, Aristotle argued that continua can never be composed of indivisibles,
while Archimedes simply treated continua as though they were composed of indivisibles,
and made mathematical advances based on that assumption. In the medieval period,
Aristotle was favored in many things, but indivisibilists still argued their case vigorously.
In the early-modern period, it was a rejection of Aristotle that led to the expansion of the
very concept of number, but he still had his influence: Leibniz himself was convinced
that Aristotle's argument was correct, and that continua could not be composed of
indivisibles.30
This ongoing debate is relevant to the subsequent chapters as it is one of the main
issues all four figures will address in one way or another. By discussing whether number
systems can be continuous, the debate itself changes, as it is not immediately clear
whether a number is an indivisible, or what it would even mean to refer to a number as
indivisible or not. However, those who argue in favor of a mathematical continuum are
compositionalists in the sense of believing that continua can be composed of atomic
elements. The nature of those elements, as well as the manner of composition, become
important issues in determining whether the continua can or cannot be composed of these
elements. Dedekind and Cantor believed that continua could indeed be composed of
atomic elements; both believed that the real numbers form a continuum without the
necessity of adding any other mathematical or non-mathematical element.
We shall also see, in the chapters that follow, what became of the debate about the
nature and mathematical formulation of infinitesimals after the introduction of limits. Du
30
Bell, SEP, Section 4.
44
Bois-Reymond believed that limit theory was only possible if the continuum over which
limit theory ranged contains both numbers (or points) and infinitesimal intervals. Peirce
believed that both mathematical and non-mathematical continuity required infinitesimals,
though these different continua used infinitesimals in different roles. Peirce also believed
that infinitesimals were useful mathematical entities in their own right, not only adding
continuity to otherwise discontinuous systems, but allowing for certain calculations and
fine-grained distinctions impossible with limit theory alone.
Thus, as our four figures are addressed one by one, the reader should keep in mind
the following historical lessons. (1) The definition of continuity itself, as presented by
Aristotle and medieval philosophers, must change as it applies to systems of numbers;
however, it must remain close enough to be identifiable as continuity. (2) Defining
continua as composed of atomic elements has always had its detractors, but has had many
supporters who have used the composition to great effect. (3) The role of infinitesimal
elements in continua has been an issue since before infinitesimals were mathematically
introduced, and their mathematical introduction allowed the birth of calculus, but almost
single-handedly destroyed it in its early years.
Though each of the four figures below approached the concept of continuity
differently, each addressed these three issues above: the definition of continuity
(sometimes distinguishing the definition of continuity in general from the definition of
mathematical continuity in particular), whether continuity could be composed of atomic
elements, and the role of infinitesimals in continuity. With a solid historical background
in the debates which framed these three issues, we are now ready to examine our first
figure: Richard Dedekind.
45
Chapter 3
Richard Dedekind
3.1 Biography and Introduction
Julius Wilhelm Richard Dedekind was born October 6, 1831, in Braunschweig, in
what is now Germany. Early in his career he was interested in science, particularly
physics and chemistry. He soon became disenchanted with the imprecise nature of
physical science, and his interest in mathematics increased. At the age of 16, he entered
the Collegium Carolinum, and two years later he attended the University of Göttingen,
where he studied physics with Wilhelm Weber (1804 – 1891) and mathematics with Carl
Friedrich Gauss (1777 – 1855); in fact, he was Gauss's last student. He earned his
doctorate in 1852, and spent the next two years learning the latest mathematical
developments.
In 1855, Dedekind began to take courses in number theory, definite integrals, and
partial differential equations from Lejeune Dirichlet (1805 – 1859). He associated with
Dirichlet almost daily, writing of this association that he was "for the first time beginning
to learn properly."1 Around this time he studied the works of Evarist Galois (1811 –
1832), and was among the first to lecture on Galois Theory. In 1858, Dedekind was
1
Written in a letter in 1856, as quoted in the University of St. Andrews biography; URL = <http://www-
groups.dcs.st-and.ac.uk/~history/Biographies/Dedekind.html>.
46
appointed to the Polytechnikum in Zürich, where he first taught differential and integral
calculus. In 1862, Dedekind moved to the Polytechnikum in Braunschweig, and
remained there until his retirement in 1894. He died in 1916, in the same city in which he
was born.
When he began to teach calculus in Zürich, he was troubled by the necessary
references to geometry that were standard in teaching methods. Recall from Chapter 2
that Cauchy had arithmetized the theory of limits; still, the introduction of these limits
depended on references to geometrical curves and tangents. Pedagogically, of course,
geometric figures and diagrams were most helpful, but Dedekind was worried that
calculus might be thought to depend essentially on these diagrams. At times it seemed to
him as though the whole of analysis itself might rest on such an unscientific and nonrigorous base as geometric intuitions. This was absolutely unacceptable to Dedekind, and
he became determined to fix it. In this connection, he wrote, "I made the fixed resolve to
keep meditating on the question till I should find a purely arithmetic and perfectly
rigorous foundation for the principles of infinitesimal analysis."2 He eventually achieved
this end to his own satisfaction, and published his result in his 1872 essay, Stetigkeit und
irrationale Zahlen.
The elements of calculus which most troubled him were those which involved the
assumed but not yet proven continuity of the real numbers, and the geometrical and
2
Richard Dedekind, "Continuity and Irrational Numbers." Essays on the Theory of Numbers, New York:
Dover Publications, 1963, 2. This small volume contains translations of both Stetigkeit und irrationale
Zahlen and Was sind und was sollen die Zahlen?
47
infinitesimal references often used in proving the theorem that "every variable magnitude
which approaches a limiting value finally changes by less than any given positive
magnitude."3 In fact, this theorem appears at the end of Stetigkeit und irrationale Zahlen,
and is proved without reference to infinitesimals or geometrical intuitions, relying instead
upon his newly established principle of continuity.
Dedekind is an important figure in this dissertation in several regards. Dedekind's
method of defining real numbers is closely related to other algebraic methods, but stand
out as remarkably self-contained and straightforward. Dedekind's real numbers are also
notable because he explicitly linked them to his principle of continuity, defining
continuity itself in terms of them. This chapter will first explain Dedekind's real numbers
and his principle of continuity in Section 3.2. Section 3.3 will investigate Dedekind's
conception of number itself, and Section 3.4 will contain with a fuller discussion of
continuity and infinitesimals in relation to his real number theory.
3.2 Dedekind cuts and the principle of continuity
Dedekind created real numbers through a method he called "cuts." In this section,
I shall first describe Dedekind's method of creating his cuts, and then how these cuts
relate to his principle of continuity. Cuts are a method of creating the real numbers in
terms of rational numbers, and are created as follows. First, notice that the rationals can
be totally ordered: for any two rationals a and b, either a ≤ b or a ≥ b. The rationals are
also dense; between any two rationals there is a third. Using the density and the order of
3
Dedekind, 26.
48
the rationals, we can divide them into two classes,4 A and B, such that every member of A
is less than every member of B. In fact, every rational number produces just such a
division: for any rational number n, every other rational number is either greater or less
than n; thus, by arbitrarily placing n into the lower class, we can form A = {x: x ≤ n} and
B = {x: x > n}. Every member of A under this assignment is clearly less than every
member of B. Any division of the rationals into two non-empty classes such that every
member of the one is less than every member of the other is called a "cut" on the
rationals.
For any cut (P, Q), if the set P has a greatest rational number, then we say this
number produces the cut. Thus, in our example above, the rational number n – the
greatest number of A – produces the cut (A, B). Note that we could just have easily
thrown n into the second class rather than the first, forming cut (A', B') such that A' = {x:
x < n} and B' = {x: x ≥ n}. In this case, n still produces the cut (A', B'), since it is also
true that for any cut (P, Q), if the class Q has a least number, then that least number
produces the cut. Notice, though, that as there are infinitely many numbers between any
two rationals, and as every rational must belong to either P or Q, it is not possible for P to
have a greatest member and Q to have a least member in the same cut.
However, it is perfectly possible for neither P nor Q to have a greatest or least
member, respectively, and thus, perfectly possible for a cut on the rationals to be
4
Modern presentation of Dedekind cuts often refer to two "sets." Dedekind himself had a theory of
systems, which is similar to the modern conception of sets, but differs from it in important respects. Thus, I
will use either "system" or the more neutral term "class" to avoid confusion.
49
produced by no rational number. As Dedekind demonstrates, there are infinitely many
cuts of this latter sort, infinitely many divisions of the rationals into cuts (P, Q) where
every member of P is less than every member of Q, yet P does not have a greatest, nor
does Q have a least, rational number.5 Thus:
Whenever then, we have to do with a cut (A1, A2) produced by no rational number,
we create a new, an irrational number a, which we regard as completely defined
by this cut (A1, A2); we shall say that the number a corresponds to this cut, or that
it produces this cut.6
Thus, the irrationals are born. Essentially, wherever a gap appears in the rationals, we fill
it with an irrational; this irrational is simply and completely determined by a cut.
Dedekind went on to argue that these irrationals have some of the properties we
would like them to have, and that we can calculate with these irrationals in just the ways
we need to. The basic method behind these proofs is simple:
To reduce any operation with two real numbers α, β to operations with rational
numbers, it is only necessary from the cuts (A1, A2)¸ (B1, B2) produced by the
numbers α and β in the system R to define the cut (C1, C2) which is to correspond
to the result of the operation, γ.7
In just this way, he proves that cuts work as we would expect them to in terms of
addition, and indicates how similar proofs could establish the ability of real numbers (as
defined by cuts) to perform "the other operations of the so-called elementary arithmetic
[...], differences, products, quotients, powers, roots, logarithms," and eventually, the
5
The mathematical proof of this occurs in Dedekind, 13-15.
6
Ibid., 15.
7
Ibid., 21.
50
proofs of theorems.8 In other words, these irrationals, determined by cuts in the rationals,
have many of the properties we desire.
Now that we have both the rational and irrational components of the reals, what,
precisely, is the continuity of the reals? Dedekind offered his definition of continuity
early in the essay, when considering the geometry he was trying to move away from.
First, in the section entitled "Continuity of the Straight Line," Dedekind wrote that "in the
straight line L there are infinitely many points which correspond to no rational number,"9
and thus that "the straight line L is infinitely richer in point-individuals than the domain R
of rational numbers in number-individuals."10 While continuity is not merely a matter of
size, the bigger somehow being the more 'continuous,' clearly this difference shows that
something about the rationals is lacking when we attempt to measure the line.11 Thus,
whether rationals are 'continuous' or not (Dedekind would say "not," of course), they are
definitely insufficient if, as Dedekind says, "we try to follow up arithmetically all
phenomena in the straight line."12 The rationals do not allow for complete arithmetic
analysis of the line, but we need a number system as least as complete, as least as
continuous, as the line.
8
Ibid., 22.
9
Ibid., 8.
10
Ibid., 9.
11
The reader may recall the example in Section 1.4, which demonstrated that a comparison between the
geometrical straight line and the rational number line could find gaps in the latter – places where there
simply is no number.
12
Ibid., 9.
51
And indeed, Dedekind finds the very essence of continuity in the straight line:
If all points of the straight line fall into two classes such that every point of the
first class lies to the left of every point of the second class, then there exists one
and only one point which produces this division of all points into two classes, this
severing of the straight line into two portions.13
Thus, claimed Dedekind, if you cut a geometrical straight line, you must necessarily cut it
at a point. It is impossible to cut a geometrical line between points.14 This, not infinite
divisibility or metaphysical smoothness, is what Dedekind took to be the feature that
distinguishes continuous from non-continuous things, or at least it is the feature that
distinguishes continuous from non-continuous geometrical objects.
The resemblance between this "essence of continuity" and Dedekind-defined cuts
is obvious. Replacing a few words, we could say that continuity of the reals happens on
the following condition: if all real numbers fall into two classes such that every number
of the first class is less than every number of the second class, then there exists one and
only one number which produces this division of all numbers into two classes, this
severing of the reals into two portions. Dedekind cuts establish half of this condition: the
creation of irrational numbers through Dedekind cuts ensures that every such division is
produced by a number (if there is no rational number at that cut, we create an irrational
one). If in addition we could prove that each cut is produced by only one number, this
condition would be fulfilled, and the real numbers would exhibit this essence of
continuity.
13
Ibid., 11.
14
As we shall see in Chapter 5, Paul du Bois-Reymond thought quite the opposite.
52
Dedekind argues that only one number produces each cut on the reals as follows.15
First, consider a cut (Φ, Ψ) on the reals – that is, consider a division of the real numbers
such that every real number belongs either to Φ or to Ψ, and that every member of Φ is
less than every member of Ψ. At the same time we produce cut (Φ, Ψ), we also produce
cut (P, Q) on the rationals, P containing all the rationals in class Φ and Q containing all
the rationals in the class Ψ. As we saw above, every cut on the rationals is produced by a
number; if there is no rational which produces it, we create an irrational and say that this
irrational produces the cut. Thus, let a be the "perfectly definite"16 real number which
produces cut (P, Q). It will become clear that a also produces cut (Φ,Ψ). Now, we
attempt to find a second real number, distinct from a, which also produces the cut (Φ,Ψ).
For any b ≠ a, either b < a or b > a. Since the rationals are dense, there are infinitely
many rationals c between b and a, and thus, if b < a, it is also the case that c < a. Since a
is the number that produces cut (P, Q), all numbers less than a belong to P, and therefore
c must belong to P; since all members of P are also members of Φ, c also belongs to
Φ. Since c belongs to Φ, and b < c, b also belongs to Φ. If, on the other hand, b > a,
then c > a, and c is therefore a member of Q, and thus of Ψ. And thus, every number b
different from the number which produces the cut a must either belong to Φ or Ψ, and
furthermore, b cannot be the greatest number of Φ or the least number of Ψ (because if b
is a member of Φ, then there are infinitely many numbers c greater than b which are also
15
Dedekind's proof appears on 20-21.
16
Ibid., 20.
53
members of Φ; and parallel reasoning holds if b is a member of Ψ), and therefore cannot
produce the cut (Φ,Ψ). Thus, only one number can make the cut; only one number
produces the separation between Φ and Ψ.
Thus, Dedekind proved that for every cut, there is one and only one real number
which produces the cut. These cuts correspond to Dedekind's essence of continuity, and
so, according to Dedekind, the real numbers thus produced form a continuum. But is this
Dedekind continuity related at all to our intuitive ideas of continuity or to the
philosophical notions of continuity we have already seen? Given that 'continuous' is a
word with many aspects, a word seemingly applicable in physics, metaphysics, algebra
and geometry, it is reasonable to expect it to take on different shades, or even different
meanings, in different fields. It is natural for us to need continuity to function differently
in different places, and for its definitions to change accordingly; not every use of the word
will fall directly out of the dictionary definition. Jules Henri Poincaré (1854 – 1912), for
one, thought that mathematical continuity to be a very different from our "ordinary
conception" of continuity:
The continuum thus conceived is nothing but a collection of individuals arranged
in a certain order, infinite in number, it is true, but external to each other. This is
not the ordinary conception, in which there is supposed to be, between the
elements of the continuum, a sort of intimate bond which makes a whole of them,
in which the point is not prior to the line, but the line to the point. Of the famous
formula, the continuum is unity in multiplicity, the multiplicity alone subsists, the
unity has disappeared.17
In other words, Poincaré is objecting to the compositional nature of mathematical
continua. As we shall see in later chapters, Cantor, du Bois-Reymond, and Peirce attempt
17
Henri Poincaré, as quoted in Russell, Principles of Mathematics, 347.
54
to formulate mathematical continua which characterize the "intimate bond which makes a
whole of" the elements of the continuum, but Dedekind himself has no such bond. He
simply characterizes continuity in terms of completeness: there are enough real numbers
such that anywhere you wish to divide them, you will divide at a number, there are no
gaps. Yet these numbers have no intimate or non-intimate connection between them;
they simply exist in this collection, in this order, in this completeness, and that is enough,
for Dedekind, to call the collection continuous.
The question now facing us is two-fold: (1) Is Poincaré's criticism just – is
Dedekind's mathematical continuum nothing more than discrete entities collected
together, and (2) if so, in what sense is this collection of the real numbers justly called a
"continuum"? What role does such a continuum play in mathematics, in geometry, in
calculus? In order to answer these questions, we must first examine Dedekind's theory of
the nature of numbers in general, so that we may better understand his theory of real
numbers in particular, and thus understand what it may mean for these real numbers to be
called "continuous."
3.3 Dedekind's Theory of Number
Dedekind's seminal work on the nature of numbers, Was sind und was sollen die
Zahlen?18 was written over twenty years after Stetigkeit und irrationale Zahlen, though
Dedekind took it to be merely an elaboration of the arguments forwarded in Stetigkeit. In
18
Translated as "The Nature and Meaning of Numbers;" page numbers refer to the Essays on the Theory of
Numbers volume.
55
Was sind und was sollen die Zahlen? he develops the notion of a chain, establishes the
existence of simply infinite chains, and uses these two ideas as a logical foundation for
the major tenets of arithmetic. This course of development is remarkably similar to other
nineteenth century projects, such as that of Guiseppe Peano (1858 – 1932); in fact, Peano
himself was inspired by Dedekind's essay. Although Dedekind did not axiomatize his
system as explicitly as Peano did, it is organized in such a manner that axioms can
essentially culled from his exposition.
Essential to this development is Dedekind's characterization of number, which is
based on a definition of the natural numbers in particular.
If in the consideration of a simply infinite system N set in order by a
transformation φ we entirely neglect the special character of the elements; simply
retaining their distinguishability and taking into account only the relations to one
another in which they are placed by the order-setting transformation φ, then are
these elements called natural numbers or ordinal numbers or simply numbers.19
Loosely speaking, the claim here is that that if a system is ordered in a certain way, if we
can simply ignore all unique features of the elements, the resulting generalization will be
the system of natural numbers, and each generalized element will itself be a number.
Before we define each aspect of this definition, first note that Dedekind does not focus on
some mysterious property held within each individual natural number – the "twoness"
contained within the number two, or any such nonsense. What gives a natural number its
character is its ability to be distinguished from other numbers, and the relations of order it
has to other numbers.
The first concept in this definition which needs to be understood is the concept of
19
Dedekind, 68.
56
element. The elements of Dedekind's systems, the things that gain their numberhood
from being organized in this fashion, are any things we can think of; literally they are
"every object of our thought."20 Anything thinkable counts as a possible element of one
of these systems, and thus counts as a possible number, as long it can be organized into a
simply infinite system. Not only does the number two not have some particular sense of
two-ness, it does not have a particular numerical property at all; it is simply an object of
thought bearing the proper relationship to other objects of thought in the proper type of
system.
To understand the proper type of system, that is, a simply infinite system, first one
must understand Dedekind's transformation, and his chain. A transformation is a law
which assigns every element in a system to a determinate thing, which may or may not
itself be a member of the same system.21 A chain is a system K such that there is a
transform K' of K that is a part of K itself. For example, take each of the natural numbers
n, and apply the 2x transform to each. It is clear that 2x is a transformation, because every
element in the system of natural numbers is assigned to a determinate thing, viz., to a
particular even number. The result of collecting all of these determinate things into their
own system would give us the even numbers, which are themselves part of the system of
natural numbers. Thus, 2x is a transformation which produces a chain. In modern set
theory, a chain is any set such that there is a function mapping the set into a subset of
itself.
20
Ibid., 44.
21
Ibid., 50.
57
A similar transformation is any two-way transformation – i.e. any one-to-one
correspondence. Dedekind then defined an infinite system as any system which can be
similarly transformed into a proper part of itself.22 This is notable because, as was
discussed in Chapter 1, for centuries the fact that the natural numbers could be mapped
onto a proper part of themselves was considered a paradox, sometimes called "the
paradox of different infinites." In fact, merely forty years before the publication of
Dedekind's Was sind und was sollen die Zahlen?, this feature of infinite sets was still
viewed as troublesome. Bernard Bolzano (1741 – 1848) noted that while there are
infinitely many rational numbers between zero and five, and infinitely many between zero
and twelve, the two sets could be put into one-to-one correspondence, though "the latter
set [is] greater than the former, seeing that the former constitutes a mere part of the
latter."23 Bolzano wrote of this correspondence, "As I am far from denying, an air of
paradox clings to these assertions," but he solved the paradox by insisting that the one-toone correspondence only proved that if one set was infinite, they both must be, and that
one-to-one correspondence only established equimultiplicity in the case of finite sets.
Thus, for Bolzano, two such sets are in fact both infinite, but not equally large.24
Dedekind retained the assertion that one-to-one correspondence established
equimultiplicity, but abandoned the belief that relative size can be determined by the
22
Ibid., 63.
23
Bolzano, Paradoxes of the Infinite, as quoted in Guillermina Waldegg, "Bolzano's Approach to the
Paradoxes of Infinity," Science & Education, 569.
58
part/whole relationship. Rather than viewing this one-to-one correspondence between a
collection and a proper part of that collection as a paradox, Dedekind uses this condition
as the very definition of an infinite collection.
Having defined chains and transformations, we can now define simply infinite
systems:
A system N is said to be simply infinite when there exists a similar transformation
φ of N in itself such that N appears as a chain of an element not contained in
φ(N).25
A simply infinite system is thus a system which has a transform that maps the system into
a chain which itself contains the entire system, except for one element. The element
lacking from this chain is the first element the transform is applied to, the 'base element'
of the chain. Succession is the perfect example of a similar transformation leading to a
simply infinite system, but many other simply infinite systems are possible, based on
many different similar transformations.
For a somewhat more humorous example of a simply infinite system, consider the
(possibly apocryphal) story of Bertrand Russell faced with a stubborn elderly woman at
one of his lectures. Reportedly, the woman argued that the world was supported by a
large turtle. When Russell asked what the turtle was standing on, she replied "another
turtle." After Russell asked what that turtle was standing on, she is supposed to have
24
Ibid., 570. It is interesting to note that Peirce had a rather similar attitude, as displayed by his often
repeated "syllogism of transposed quantity," as we will see in Chapter 6.
25
Dedekind, 67.
59
replied, "You can't fool me, young man! It's turtles all the way down."26 Supposing the
woman were correct, this infinite regression of turtles would itself form a simply infinite
system, with "x rests on the back of y" as the similar transformation. The collection of
every such x is our simply infinite system; the collection of every y forms a chain that
includes everything in the system but the base element – the earth itself.
With the key terminology in place, we are now in a position to return to
Dedekind's full definition of number. To review the terminology, we began with
elements, which are any object of our thought, and an element is "completely determined
by all that can be affirmed or thought concerning it."27 Elements can be collected into
systems, some of which have particular characteristics. Considering systems that are
simply infinite, we note that all simply infinite systems are similar to one another
(isomorphic), being organized in the same manner. Next,
We entirely neglect the special character of the elements; simply retaining their
distinguishability and taking into account only the relations to one another in
which they are placed by the order-setting transformation φ.28
The things, remember, were simply objects of our thought. We ignore any irrelevant
characteristics of these objects, such as the color or the size of the object of thought, and
focus merely on their distinguishability and their relationship to one another. These
elements, in these systems, simply are the natural numbers. Dedekind referred to the act
of stripping away irrelevant qualities from the elements as "freeing the elements from
26
Stephen Hawking opens his book, A Brief History of Time, with a slightly different form of this anecdote.
27
Dedekind, 44.
28
Ibid., 68.
60
every other content (abstraction)."29
Given that we began with random objects of thought, the question must arise, has
Dedekind defined one system of natural numbers, or instead many different ones? While
it is perfectly possible that there are many such simply infinite systems that fit the
appropriate requirements, one must assume that his method of abstraction saves us from
needless duplication of number systems. Consider again the system of infinitely
regressing turtles. Turtles are objects of our thought, as is the earth, and these objects can
be mentally arranged into a simply infinite system. Applying Dedekind's method of
abstraction would mean we could no longer distinguish between a turtle and the earth, but
only between the base element, the next element, etc. It also would mean that we can no
longer distinguish between turtles in an infinite regress and, say, notes in an infinite
melody, as the features of turtles that distinguish them from notes would be abstracted
away as irrelevant to their organization in the system and to their distinguishability from
each other within the system.
The relations between these elements "are always the same in all ordered simply
infinite systems, whatever names may happen to be given to the individual elements."30
Therefore, we can abstract away from any names the elements bear naturally and name
the elements of these systems what we will – calling the base element 1, as Dedekind
does. Hence, the initially many isomorphic systems are thus abstracted, one from the
next, so that in essence they collapse into one, which we can then call the number system.
29
Ibid., 68.
30
Ibid., 68.
61
The only possible hindrance to this tidy collapse into one system is when we
consider the transformation φ. Nowhere does Dedekind specify that the nature of φ itself
is abstracted away, and this makes a difference. Recall, in the case of turtles, the
transformation is one of regression, while with notes in a melody, it is one of succession.
There are differences in meaning between succession and regression, but also logical
differences: one transform runs forward, while the other runs backward. However, the
important feature of φ for Dedekind was that it gives an infinite system some order, and
thus, one can possibly argue that we can abstract away from φ all of the peculiarities other
than the sheer fact that it sets the elements of our system into an order.
Dedekind gives an example of a system thus organized, before abstracting away
from all its particularities; in fact, he gives this example as proof of the existence of
infinite systems themselves:
My own realm of thoughts, i.e., the totality S of all things which can be objects of
my thought, is infinite. For if s signifies an element of S, then is the thought s',
that s can be an object of my thought, itself an element of S.31
Thus, S is the collection of all the things Dedekind could think, and the transform φ
31
Ibid., 64. Interestingly, Bolzano uses a very similar argument to establish the existence of actual
infinities. However, rather than an object of thought serving as the base case and "can be thought" as the
transformation, Bolzano uses "any truth taken at random" as the base case, calls it A, and notes that the
proposition "A is true" is distinct from A itself, forming B, from which we can form C = "B is true", etc. To
complete the proof that the set so formed is in fact infinite, Bolzano subjects it to what he considers the
ultimate proof of infinity: he shows it is in one-to-one correspondence with the set of natural numbers. As
Dedekind has not yet proved the existence of numbers at the time he gives this proof, he must refrain from
using this last step to avoid circularity. See Waldegg, 567.
62
would be "x is an object of my thought." Beginning with a random thought s, we plug it
into φ and are rewarded with s' = "s is an object of my thought." A similar transform can
be made on s' itself, yielding s'', and so on, creating a simply infinite chain, the base of
which is s itself. Abstracting away from the particulars linking this system to Richard
Dedekind, to his thoughts in particular, and to the transform φ, it is obvious that the
system is organizationally isomorphic to any other system we could organize in this
manner, and thus, according to Dedekind, the general features such systems have in
common are the numbers themselves.
The numbers thus established – as any simply infinite system, with its
particularities abstracted away from it – are simply the natural numbers. Dedekind then
creates the rational numbers from the natural numbers. Or rather, the rationals follow
with "natural consequence" from the naturals:
I regard the whole of arithmetic as a necessary, or at least natural, consequence of
the simplest arithmetic act, that of counting. ... Addition is the combination of any
arbitrary repetitions of [counting]; in a similar way arises multiplication.... Thus,
negative and fractional numbers have been created by the human mind, and in the
system of all rational numbers there has been gained an instrument of infinitely
greater perfection.32
After the naturals are produced, all else follows logically; i.e. arithmetic, algebra, and
even analysis immediate results "from the laws of thought."33 So let us follow this
progression. After counting, which, as we have seen, is logically founded by chain
theory, one naturally wishes to perform addition – to speed up counting, so to speak. The
combination of acts of addition gives us multiplication. Wishing to decrease as well as
32
Dedekind, 4.
63
increase, one inverts addition to get subtraction. This act, however, is incomplete within
the natural numbers themselves, if we wish to apply subtraction to any two numbers in
our range; thus, negative numbers are needed to close the operation.
Similarly, the inverse of multiplication leads to the need for the naturals to be
expanded to the rationals.34 One might then take exponents as the combination of acts of
multiplication, and the inverse operation to exponentiation as the creator of irrational
numbers, but here we reach a problem. Taking the logical progression to the creation of
the irrationals does not yield all of the irrationals: not all real numbers are reachable
through taking the nth root of rational numbers. Nice, orderly irrationals such as the
square root of 2 appear in this progression, but wildcards such as π do not. This is one
reason Dedekind does not establish the irrationals through similar "operational" methods;
but rather, establishes them through the method of cuts.
It was clear, through our progression from the naturals to the rationals, that the
members of our new systems of numbers were a similar sort of thing as the members of
the naturals; they completed operations we wished to perform on naturals, and fulfilled
basic mathematical needs. Even the rationals, for example, were totally-ordered, and they
were derived from the naturals using a simple operation. The metaphysical status of the
new irrational numbers, however, is distinct. Our exposition reveals their nature: they
33
Ibid., 31.
34
It is interesting to compare this 'just-so story' of the creation of numbers with the actual history of the
inclusion of various mathematical entities in our numerical canon, particularly the rapid inclusion of
64
are in essence gap-fillers. Look at the rationals, find a gap, then fill it in; call the filling
an 'irrational.' It seems that the end result is a number system with different types of
numbers – those directly derived from natural numbers by various operations, and those
which fill the gaps between these derived numbers.
Now that we have a clear idea of what number means to Dedekind, we can
investigate his continuity and answer the following questions: whether Dedekind's
continuum is compositional, whether it is composed of discrete elements, and what role
Dedekind's continuum plays in calculus itself.
3.4 The Nature of Dedekind Continuity
To review briefly: Dedekind defined "natural number," or simply "number," as
the abstraction of any simply infinite system – any system that was sufficiently organized.
From the naturals, he follows a logical progression to generate the integers and the
rationals, and with cuts, he generates the reals from the rationals. The reals satisfy
Dedekind's principle of continuity, and thus, he judges the collection of reals a
continuum. Yet, looking over this progression of creation from the naturals to the reals,
one thing is clear: Dedekind's continuum is distinctly compositional in nature. In
Chapter 2 we reviewed Aristotle's proof that a continuum cannot be composed of
indivisibles, and cannot be divided into them either; while Dedekind's numbers are not
indivisibles in the classic Aristotelian sense, his continuum is composed of individual
negative numbers, fractions, irrationals, and imaginaries, all in the sixteenth century. Recall the brief
discussion of this explosion of the number system in Chapter 2.
65
elements, and nothing more.
The inherent nature of these elements vary: the elements of the natural numbers
are any thinkable object, organized and stripped of any particular features other than this
organization itself; the rationals are ratios of naturals; the irrationals are cuts on the
rationals. Yet, each number is an individual element, and Dedekind's principle of
continuity does not posit essential connections between the members of the continuity,
the principle only specifies that no cut can be made between elements. Dedekind's reals
are continuous, according to Dedekind, because the collection is full enough that no gaps
can be found, but the collection is still a collection of elements.
Dedekind is the only figure of this dissertation who was satisfied with such a
strictly compositional continuum. As we shall see in later chapters, Cantor believed that
in addition to a completeness of this sort, the elements of a continuum must also have an
essential connection between them. Du Bois-Reymond attacked the idea of
compositional continua such as this "on the ground that it was committed to the reduction
of the continuous to the discrete, a program whose philosophical cogency, and even
logical consistency, had been challenged many times over the centuries."35 Peirce came
to believe that compositional continua failed to satisfy our basic intuitions about the
nature of continuity.
As was briefly noted in Chapter 2, Aristotle's argument against compositional
continua does not succeed against Dedekind's principle of continuity. The thrust of
35
Ehrlic, "General Introduction," Real Numbers, Generalizations of the Reals, and Theories of Continua,
Dordrecht: Kluwer Academic Publishers, 1994, x.
66
Aristotle's argument was that compositionality required two elements to be next to each
other (which he proves to be impossible), because if they were not next to each other,
there would either be a gap between them (and thus a gap in the continuum), or there
would be something foreign between them, and thus the continuum would not be
composed entirely of these elements. Yet in Dedekind's system of real numbers, no two
real numbers are next to each other, and yet there is no gap, and there is nothing foreign
in the set; rather, there are always infinitely many real numbers between any two.
Aristotle's argument thus does not refute Dedekind's principle of continuity;
however, many philosophers and mathematicians believe that a continuum can never be
composed of discrete elements. The question before us can thus be framed: are the
numbers composing Dedekind's continuum discrete entities, and if so, does this prevent
them from being viewed as continuous?
In 1917, Edward V. Huntington (1874 – 1952) argues that Dedekind and Cantor's
real numbers form non-discrete series.36 Huntington calls a discrete series one which is:
(1) divisible into two parts K1 and K2, such that every element of K1 precedes every
element of K2, and that there is at least one x such that any element preceding x belongs to
K1 and every element following x belongs to K2; and (2) every element of the series
except the last has an immediate successor, and every element of the series except the
first has an immediate predecessor.37 Thus, the integers are a good example of a discrete
36
Huntington, Edward V, The Continuum and Other Types of Serial Order: with an introduction to
Cantor's transfinite numbers, Mineola: Dover Press, 1917.
37
Ibid., 19.
67
series, but the reals, failing the second condition, are not. Notably, the most important
feature of a discrete series for Huntington is that one can perform induction on the series.
However, Huntington himself never refers to the elements of any series as discrete or
non-discrete, only to the series themselves as discrete or not. Thus, while Huntington
quite clearly states that the real numbers form a continuous series, and hence not a
discrete series, nowhere does he state that therefore the real numbers are not discrete
elements.
Conversely, John L. Bell argues that all numbers are discrete, and also that
discreteness is commonly regarded as antithetical to continuous. First, whole numbers
are the essence of discreteness: "In mathematics it is the concept of whole number, later
elaborated into the set concept, that provides an embodiment of the idea of pure
discreteness, that is, of the idea of a collection of separate individual objects."38
Geometry, on the other hand, is the essence of continuity: "by their very nature geometric
figures are continuous; discreteness is injected into geometry, the realm of the
continuous, through the concept of a point, that is, a discrete entity marking the boundary
of a line."39 The difficulty with discrete numbers being used to analyze the continua of
geometry, Bell says, arose as early as Pythagoras and his discovery of incommensurable
magnitudes. "Here the realm of continuous geometric magnitudes resisted the
38
Bell, "Dissenting Voices: Divergent Conceptions of the Continuum in 19th and Early 20th Century
Mathematics and Philosophy." URL = <http://publish.uwo.ca/~jbell/Dissenting%20Voice1.pdf>, 1.
39
Ibid., 1-2.
68
Pythagorean attempt to reduce it to the discrete form of number."40
Bell separates late nineteenth century mathematicians into two categories: those
who believe the continuous is not reducible to discrete entities, such as du BoisReymond, Giuseppe Veronese (1854 – 1917), Hermann Weyl (1885 – 1955), L. E. J.
Brouwer (1881 – 1966), and Peirce; and those who not only believed in the reducibility of
the continuous to the discrete, but in some cases, believed they had accomplished just
such a reduction, like Dedekind and Cantor. Thus, a full analysis of whether Dedekind's
reduction of the continuous to the discrete by means of cuts actually works can come only
after we have examined the position and arguments of the other figures to be considered
in this dissertation: viz. Cantor, du Bois-Reymond, and Peirce.
Pending the fuller analysis, we can still judge Dedekind's continuum on his own
stated goals. Dedekind wished to create this real number continuum to free calculus from
necessary geometrical references. In this, he succeeds: he has defined a concept of
continuity which is completely arithmetical; a function can now approach a limit
continuously, numerically, consistently. No tangent lines or curves are necessary for this
analysis. However, one might ask if Dedekind's separation between geometry and
systems of numbers has now gone too far. In terms of logical foundations, it was
necessary to free calculus from geometrical references; in terms of application, however,
it is necessary to be able to apply calculus to realms outside of algebra and number
theory. Thus, now that Dedekind has succeeded in shoring up the algebraic foundations
of calculus, one might ask if it is still possible to apply this algebraic calculus to space
40
Ibid., 2.
69
and time.
One of the greatest difficulties in this application seems to stem from Dedekind's
insistence that mathematics is both mental, and logical. Thus:
In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that
I consider the number-concept entirely independent of the notions or intuitions of
space and time, that I consider it an immediate result from the laws of thought.41
And again:
Without any notion of measurable quantities and simply by a finite system of
simple thought-steps man can advance to the creation of the pure continuous
number-domain.42
Thus, if numbers are thought and logic, and Dedekind specifically denies any reference to
measurable quantities or anything non-mental, how can we trust that the resulting system
applies to our physics, or even to our carpentry? Further mysteries arise when, after
proving that his real numbers form a continuous system, Dedekind suggests that space
itself may not be continuous:
For a great part of the science of space the continuity of its configurations is not
even a necessary condition. [...] If any one should say that we cannot conceive of
space as anything else than continuous, I should venture to doubt it.43
Thus, if Dedekind's real number system is necessarily continuous, but space is not, in
what sense do they link up?
Dedekind did believe that space was in fact continuous, and it is clear that he
believed that geometry gives us our paradigm of continuity itself. However given that he
41
Dedekind, 31.
42
Ibid., 38.
43
Ibid., 37-38.
70
divorced numbers from geometry in the development of the calculus, their reunion
becomes something that is not quite so obvious. Perhaps we can reunite them along these
lines: given that cuts assure us that the real numbers have no gaps, one will no longer run
into the same difficulties of incommensurability as the Greeks did. That is, wherever we
need to measure, there will be a number there. We will not reach a situation where the
reals are not sufficient to measure a triangle, as the rational numbers proved to be. Thus,
while Dedekind has not proved a correspondence between the points on a line and the real
numbers, the assumption of such a correspondence is not immediately refuted by
incommensurable quantities.
3.5 The Relationship Between Dedekind's Continuity and Infinitesimals
Before we end our discussion of Dedekind and turn to Cantor, we can draw some
interesting conclusions about infinitesimals from Dedekind's system of cuts itself.
Though Dedekind did not discuss infinitesimals directly, his real number system creates a
calculus that does not need to rely upon infinitesimals, and his conception of continuity
mathematically excludes infinitesimals. One of the most intriguing features of an
infinitesimal quantity is that adding an infinitesimal to a finite quantity such as 1 simply
results in that same finite quantity; the quantity has not been moved, even by a very small
amount. Therefore, as Cantor points out (in an argument we will examine more closely in
the next chapter), infinitesimals violate the Archimedean principle. The Archimedean
principle states that "if a and b are any two positive numbers of [a] system and a < b, then
it should be possible to add a often enough that the sum
a+a+...+a
71
eventually surpasses b. Briefly stated, there should always exist a natural number n such
that na > b."44 It is clear that if infinitesimals are considered to be numbers, they must be
non-Archimedean numbers; if a in the above definition is an infinitesimal, no natural
number n, no matter how large,45 can take a past another number b.
Thus, infinitesimals are excluded in any Archimedean system, but Dedekind's
continuity logically implies the Archimedean principle. This can be proved as follows.
Assume that any cut of the reals is determined by a unique real number. (This is
the assumption of Dedekind continuity).
[Prove: For any positive reals a and b, there is a natural number n such that na ≥
b.]
Assume for reductio: There exist two positive reals a and b such that na < b for
any natural n.
From the series a, 2a, 3a ... we can form the set A = {x: x ≤ na, for any natural n}.
It is clear that b lies outside of A, and since b is a real number, there are infinitely
many numbers greater than b which also lie outside A. In fact, we can form
another set B = {x: x > na}.
A and B together encompass all the reals, and it is clear that every member of A is
less than every member of B, and thus (A, B) constitutes a cut. By our original
assumption, (A, B) must be determined by a unique real number; call it c. Now, c
must be the greatest member of A or the least member of B.
Assume c is the greatest member of A. Thus, by definition of A, c ≤ ma for some
m. But if c < ma, then since ma is itself a member of A (since A contains all {x: x
≤ na} for all n), c would not be the greatest member of A. Therefore, c = ma. Yet
this cannot be the case either, for if c = ma, then since (m + 1) a > ma, by
substitution, (m + 1) a > c. But (m + 1) a is a member of A (by definition of A,
since (m + 1) is a natural number), and thus c would still not be the greatest
member of A.
44
Waismann, 209.
45
And, Cantor says, no transfinite number either. More on that in Chapter 4.
72
Therefore, c must be the least member of B. By definition of B, c > na for all n.
Notice also, that for every real number r, if r < c, then r is a member of A (since
every real is either in A or B, and as c is the least member of B, for all x member
of B, x ≥ c).
Consider, then, (c – a). As c and a are both positive real numbers, (c – a) < c, and
therefore, (c – a) is a member of A. By definition of A, (c – a) ≤ ma for some
natural m.
But if (c – a) ≤ ma, then c ≤ ma +a, which is equivalent to c ≤ (m + 1) a.
However, m + 1 is a natural number, and thus by the definition of A, c must be a
member of A, which contradicts the above.
Thus, the existence of infinitesimals is inconsistent with Dedekind continuity. Since
Dedekind has become so influential, several proponents of infinitesimals have tried to
work with modified versions of Dedekind-continuity – versions which retain the main
advantages of the system without also insisting on the Archimedean principle. However,
as Dedekind defined continuity, though he wrote nothing in these pages directly against
infinitesimals, his system is confirmedly Archimedean.
We now have examined several features of Dedekind's continuity. (1) The most
important feature of a continuous collection is that it cannot contain gaps. (2) The real
numbers, when defined as cuts on the rationals, exhibit this principle of continuity. (3)
Dedekind's continuity, given his theory of number, is thus compositional in nature. (4)
Infinitesimals are inconsistent with Dedekind's principle of continuity, as the latter
implies the Archimedean principle.
As we shall see in the next chapter, Cantor created a definition of continuity that
shared some similarities with Dedekind's, but attempted to overcome some of the
problems Cantor saw in Dedekind's theory. Cantor similarly saw the real numbers as
created from the rationals, and saw the real numbers as continuous (and in fact as
73
exhibiting the very essence of continuity itself), but he wished to overcome the charge of
his continuity being a simple collection of discrete elements by mathematically defining
an essential connection between the elements.
74
Chapter 4
Georg Cantor
4.1 Biography and Introduction
Georg Ferdinand Ludwig Philipp Cantor was born in St. Petersburg, Russia, in
1845, the son of a relatively wealthy merchant. Cantor was taken to Germany at age 11,
and was sent to study at a trade school in Darmstadt at age 15, in preparation for a career
in engineering. When he finished at Darmstadt in 1862, he decided to commit himself to
a career in mathematics. As Dauben wrote, “From the very beginning, apparently, Cantor
had felt some inner compulsion to study mathematics."1 He began his mathematical
education at the Polytechnic of Zurich, though his studies were cut short by his father’s
death in 1963. He moved to the University of Berlin, attending lectures by Karl
Weierstrass (1815 – 1897) and Leopold Kronecker (1823 – 1891), and completing his
dissertation on number theory in 1867.
Cantor’s career was a vigorous one, his brilliance earning him acclaim (such as
his promotion to Extraordinary Professor at Halle in 1872, a mere three years after he was
appointed there), his innovation and enthusiasm earning him enemies and embroiling him
in mathematical controversies. Among the most controversial aspects of his work, and
1
Dauben, Joseph, Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton: Princeton
University Press, 1990, 277.
75
also among the most influential, must be numbered his set theory and his transfinite
number theory. Though this chapter will not deal directly with these two theories, they
are not unrelated to our project here; his set theory was developed in conjunction with his
real number theory, and his investigations of mathematical continuity and infinity played
no small part in his development of transfinite theory.2
Though Cantor’s real-number theory and treatment of mathematical continuity are
in many ways similar to those of Dedekind, Cantor goes beyond Dedekind in significant
ways, and thus a careful treatment of Cantor’s thoughts and developments in this arena is
both interesting in itself and necessary to understand fully Cantor-Dedekind continuity.
As Cantor often developed his mathematical theory as needed to prove the particular
theorems he was working on, the best means of understanding his theory of the
mathematical continuum is to follow his mathematical progression toward his eventual
definition of continuity. Thus, this chapter will proceed chronologically. Section 4.2 will
discuss Cantor’s 1872 work on trigonometric series and development of his real-number
theory, Section 4.3 will treat the papers in which he develops nondenumerability theory,
from 1872 to 1878, Section 4.4 discusses his early theory of continuity, as it appears in an
1878 paper, and Section 4.5 presents Cantor's most thorough and philosophical discussion
of continuity, contained in the 1883 work primarily concerned with a defense of
transfinite theory. Finally, in Section 4.6 we will consider in some detail his famous
2
For a more thorough biography of Georg Cantor, see the University of St. Andrew’s biography of Cantor,
URL = <http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cantor.html>, and Dauben’s Georg
Cantor, especially Chapter 12.
76
argument against the existence of infinitesimals, which he believed were selfcontradictory entities.
4.2 Real Numbers (1872)
Cantor developed his theory of real numbers in the 1872 article, "Über die
Ausdehnung eins Satzes aus der Theorie der trigonometrischen Reihen."3 This essay is
one in a series of Cantor's early papers (1870 – 1872) which were concerned with
particularly interesting trigonometrical series. As we shall see below, Cantor developed
his theory of real numbers as a tool to prove the uniqueness theorem he was working on
in 1872; thus, a brief overview of Cantor’s project in this paper would not be out of place.
This uniqueness project has its roots in the early nineteenth century, with the
introduction of Fourier series. Joseph Fourier (1768 – 1830), while studying the
conductivity of heat, “had established that arbitrarily given functions could be represented
by trigonometric series with coefficients of a specified type.”4 The term "Fourier series"
refers to just such expansions of functions as trigonometric series, and these series are a
powerful mathematical tool used in a variety of equations, proofs, and sciences. Many
mathematicians throughout the nineteenth century spent considerable time expanding on
the notion of Fourier series, making the transformations more general and applicable to a
3
Georg Cantor, "Über die Ausdehnung eins Satzes aus der Theorie der trigonometrischen Reihen,"
Mathematische Annalen v 5, 1872, 123-132. In this dissertation, pagination refers to that of the French
translation, "Extension d'un théorem de la theorie des series trigonometriques," Acta Mathematica 2, 1883,
336 – 348. Translations from the French are mine, unless otherwise noted.
77
wider variety of functions, generally making the tool more powerful.
Cantor’s 1872 paper is an attempt to establish uniqueness conditions for one such
transformation from certain functions to trigonometric series; i.e., he was working on the
problem of whether an arbitrary function could be represented by exactly one
trigonometric expansion. As Cantor wrote, "I would like to make known in this work an
extension of a theorem according to which a function can be developed by a
trigonometric series using exactly one method."5 He had earlier established uniqueness
conditions under certain restrictions, when he proved the theorem,
That two trigonometric series;
½ b0 + Σ (an sin nx + bn cos nx)
and
½ b10 + Σ (a1n sin nx + b1n cos nx)
which, for all values of x, converge and have the same sum, have the same
coefficients; I further showed ... that this theory stays true if, for a finite number of
values x, one renounces either the convergence, or the equality of the sums of
these two series.6
The purpose of the 1872 paper is to extend this result.
The extension I have in view here is this: that for an infinite number of values of
x in the interval [0 .... (2π)] one can renounce either the convergence or the
4
Dauben, Georg Cantor, 6.
5
Cantor 1872, 336.
6
Ibid., 336. Cantor originally established this theorem in “Beweis, dass eine für jeden reellen Wert von x
durch eine trigonometrische Reihe gegebene Funktion f(x) sich nur auf eine einzige Weise in dieser Form
darstellen lässt,” Journal für die reine und angewandte Mathematic (Crelle’s Journal) 72, 1870, 139.
78
agreement of the sums of the series, without the theorem ceasing to be true.7
Thus, Cantor’s project in 1872 was to expand his uniqueness results by extending the
range of values under which the conditions placed on uniqueness can be relaxed.
He does prove this extension in the last section of the paper, but the details of this
proof are beyond the scope of this chapter; rather, our focus is on the tool Cantor needs to
make the jump from finitude to limited (bounded) infinity – that is, a precise
mathematical understanding of the real numbers. Thus, Cantor developed a theory of real
numbers, and did so in a manner somewhat similar to that of Dedekind – by using
rational numbers as the foundation. Cantor's rationals included zero, and the set of
rationals is called A.
His construction of irrational numbers begins with an infinite series of rationals.
This infinite series must be "obtained by a law"8 and is represented thus:
(1)
a1, a2, .... an, ....
The series is constituted such that the difference an+m – an becomes "infinitely smaller as
n grows."9 Here, n is an arbitrary integer such that (an+m – an) < ε, where ε is a positive
rational and m is an arbitrary integer. In later papers, Cantor called this type of series a
"fundamental sequence," which is defined more concisely in Dauben:
The infinite sequence
a1, a2, ...., an, ....
7
Cantor 1872, 336 – 337.
8
Ibid., 337.
9
Ibid., 337.
79
is said to be a fundamental sequence if there exists an integer N such that for any
positive, rational value of ε, |an+m – an| < ε, for whatever m and for all n > N.10
Notice that while these two definitions pick out the same sequences, the emphasis in each
one is different. Cantor’s original definition draws attention to an important feature of
fundamental sequences – that the members of the sequence draw ever closer to each other
as the sequence progresses.
Such sequences had long been used to define sets that had limits without
assuming the existence of the limit itself, which is, no doubt, why Cantor chose to use
them to define his real numbers. For, as he wrote in 1883, the main mistake encountered
in previous attempts to establish the real numbers based on some collection of rationals is
the assumption of the existence of the very limit used to define the real number. Cantor
wrote,
"I believe that this logical mistake, which was first avoided by Herr Weierstrass,
was committed almost universally in previous times, and not noticed because it
belongs among those rare cases in which actual mistakes cannot cause any
significant damage to the calculus."11
The important point, which Cantor noticed, is that such sequences are a natural tool to use
if one wishes to develop real numbers as limits without previously assuming their
existence.
After thus defining fundamental sequences, Cantor associates a "determined limit
10
Dauben, 38. Though Cantor does not refer to them as such, fundamental sequences are usually called
“Cauchy sequences.” Bolzano named them as such after Augustin-Louis Cauchy (1789 – 1857).
80
b"12 with each sequence. These determined limits will eventually be established as real
numbers, but for the moment, they are merely symbols associated with particular
sequences. As Dauben writes, the phrase "determined limit" must be understood "as a
convention to express, not that the sequence {an} actually had the limit b, or that b was
presupposed as the limit, but merely that with each such sequence {an} a definite symbol
b was associated with it."13
Cantor then proves that such b's associated with such sequences are linearly
ordered. Let us call the limit of sequence {an} above b. Now consider another
fundamental sequence {a'n} and call its determined limit b'. One (and only one) of three
relationships must hold between the two sequences. Either: (a) an – a'n becomes
infinitely smaller as n grows, or (b) an – a'n, after an n of a certain size, stays always
larger than a positive rational magnitude ε, or (c), an – a'n, after an n of a certain size,
remains smaller than a negative rational –ε.14 If (a) is the case, then b = b'; if (b) is the
case, then b > b'; and if (c) is the case, then b < b'. Thus, the sequences (under the
equivalence relation) are linearly ordered, and so too are their associated limits.
Cantor next shows that the conjoined set of B (the set of all such b's) and A (the
11
Cantor, Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Leipzig: B. G. Teubner, 1883, 81. Here
and in all references to the Grundlagen, pagination refers to the English translation: "Foundations of the
Theory of Manifolds." Trans. U. Parpart. The Campaigner v. 9 1976, 69 – 96.
12
Cantor 1872, 338.
13
Dauben, 38. The original reads "...a definite symbol a was associated with it," but it seems obvious b is
intended here.
14
Cantor 1872, 338.
81
set of rational numbers) is itself linearly ordered, by showing the same three relationships
hold between any member of A and any member of B. Finally, he then indicates that
there is good reason to suppose that these symbols b, b', b'', .... are the actual limits of the
sequences with which they are associated.
From these definitions and from those which immediately follow, it results (and
can be rigorously demonstrated) that, b being the limit of series (1), b – an
becomes infinitely smaller as n grows, which justifies consequentially in a precise
manner the designation of "limit of series (1)" given to b.15
This is far from proof that b is an actual limit of its associated sequence; it is not even
proof that the sequences in question have limits. Cantor's real numbers, at least as far as
this article goes, are the fundamental sequences themselves; b or b' is simply an easy way
to refer to the sequence, and it seems helpful if we can think of b as a limit of {an}.
Thus, after proving that the elements of his set B can satisfy the operations of
addition, subtraction, multiplication and division, and after a small detour,16 Cantor went
on to demonstrate how this collection of b's could be used in the measurement of
distances. This demonstration is key to our thesis here, as it will be used later in his
15
Ibid., 338.
16
The detour is an interesting one historically, if not directly relevant to our current thesis. After
developing the set B (which is, recall, the collection of determined limits associated with fundamental
sequences from set A), Cantor claims we can take fundamental sequences of members of B and use them to
develop a set C, and reiterate the operation on C to produce D, etc. He here claims that while this process
could be iterated indefinitely, the only time the process results in a jump in magnitude is the movement from
set A to B. B is thus in one-to-one correspondence with all such sets save A itself. Thus, Cantor was
thinking about and working with magnitudes of infinity as early as this 1872 paper – a process, of course,
that will eventually culminate in transfinite set theory.
82
definition of continuity. Thus, in the next section we will examine this demonstration.
We will also discuss an important feature of the real numbers; i.e. their
nondenumerability.
4.3 Continuity and Denumerability (1872 - 1878)
After Cantor developed the set B as described above, (and indicated how further
sets, C, D, etc. could be generated), he began referring to the elements of these sets as
numerical magnitudes rather than as symbols. That is, he believed himself to have
established numbers of a sort, though perhaps a peculiar sort of number.
In the theory I have here described (according to which numerical magnitude,
having at first, in general, in itself, no objectivity, appears only as an element of
theorems which have a certain objectivity, of this theorem, for example, that the
numerical magnitude serves as the limit to a corresponding sequence) it is
essential to maintain the abstract distinction between B and C.17
Thus, though Cantor treated these elements as numbers, he restrained himself from
embracing their objectivity as numbers, instead accepting their tenuous nature as entities
merely associated with particular sequences.
In order to have them function fully as numerical magnitudes, however, it is not
enough to prove that they are linearly ordered and that they satisfy basic equations; he
must also establish that they can be used in measurement. This is the task he took up in
Section 2 of his 1872 paper, and eventually accomplished by means of a connecting
axiom.
It now follows without difficulty that numerical magnitudes from systems C, D, ...
17
Ibid., 340.
83
are also capable of determining known distances. But to achieve knowledge of
the bond that we observe between the systems of defined numerical magnitudes in
§1 and the geometrical straight line, we must still add an axiom, here enunciated
simply: to each numerical magnitude reciprocally belongs a determined point of
the line of which the coordinate is equal to this numerical magnitude in the sense
discussed in §1.18
That the connection is assumed and not proved is no accident. Cantor continues:
I call this theorem an axiom because it is in its nature that it cannot be proved in a
general fashion.19
As we will see a little further on, Cantor does indeed take this unprovable connection as
the essence of continuity itself, not just the essence of the real number continuum, thus its
status as an assumption is a notable one. In Section 4.5, when we discuss Cantor's
definition of continuity itself, it will be good to keep in mind that Cantor believes this
connection between the real numbers and points on the geometrical line to be an essential
but necessarily unproved connection.
But first, since we are discussing Cantor chronologically, it is necessary to
comment on denumerability, another important ingredient in his theory of the continuity
of the real numbers. As is generally well-known, Cantor proved that the rational numbers
can be put in one-to-one correspondence with the naturals, and are thus countable, or
denumerable. The "power" of the naturals is thus the same as the power of the rationals,
18
Ibid., 342.
19
Ibid., 342.
84
and Cantor calls the power of the naturals the smallest possible power of an infinite set.20
Obtaining a result remarkable even to Cantor himself, he next proved that, though
the algebraic irrational numbers are also denumerable, the irrationals themselves, and thus
the reals as a whole, are not.21 The jump from the denumerability of the rational numbers
and the nondenumerability of the reals is not simply an interesting feature of the number
system; it is often used as a partial justification for why the real numbers form a
continuous set and the rational numbers do not. Certainly, as Dedekind showed, the real
numbers have the property he termed the essence of continuity; that is, wherever one
wishes to divide the set, one always is able to do so at a number. The nondenumerability
of the real numbers adds a further intuitive reason to suppose them continuous. The reals
have a larger power; they are more extensive than a denumerable set. Of course,
nondenumerability alone does not suffice for continuity, but some do take it as evidence
that the real numbers are in a separate metaphysical category from the rationals.
4.4 Early Real-Number Continuity (1878)
Returning to Cantor's theory of continuity, we move now to his essay of 1878, Ein
Beitrag zur Mannigfaltigkeitslehr. In this essay, Cantor abandoned trigonometrical
20
Cantor, "Ein Beitrag zur Mannigfaltigkeitslehre," Journal für die rein und angewandte Mathematik. v
84, 1878. References to the Beitrag use the pagination of the French translation, "Une contribution à la
théorie des ensembles," Acta Mathematica v 2, 1878, 311 – 328.
21
Unfortunately, there is neither time nor excuse to discuss these denumerability and nondenumerability
proofs here in detail; though the proofs bring up many interesting points of discussion, they are outside the
domain of this dissertation.
85
series, and was working directly with the implications of set theory and various theorems
one could prove about the set of real numbers. Here, then, he directly discussed
continuity, as well as the relationship between continuous sets, and also the relationship
between continuous and non-continuous sets. The Beitrag thus gives us several
important insights into his early theory of continua.
The main point of the Beitrag is to prove theorem (A):
(A) Let x1, x2, .... xn be n real variable magnitudes, independent of each other,
such that each variable can take every value ≥ 0 and ≤ 1. Let t be another variable
with the same limits (0 ≤ t ≤ 1). We can make this magnitude t correspond to the
system of n magnitudes x1, x2, ...., xn such that to each determined value of t
belongs a system of determined values x1, x2, ...., xn and vice versa – to each
system of determined values x1, x2, ...., xn belongs a certain value of t.22
(A) is derived straightforwardly from three theorems Cantor wished to establish for
continuous sets. First, we can completely and uniquely relate a continuous set of n
dimensions to a continuous set of one dimension. Second, the elements of an ndimensional continuous set can be uniquely determined by a continuous and real
coordinate t. Third, the elements of an n-dimensional continuous set can also be uniquely
determined by a system of m continuous coordinates t1, t2, ...., tm. With these three
theorems, one can reduce continuous sets of any dimension to sets of one dimension.
After several intricately entwined lemmas and sub-proofs, Cantor finally arrived
at the (A) he desired – though not in an uncontroversial manner.23 Along the way,
22
Cantor 1878, 315.
23
Not surprisingly, Cantor's constant foe, Leopold Kronecker, led the critical charge against the arguments
in this article. See Dauben, Georg Cantor, 66 – 72 for a discussion of Kronecker's opposition to Cantor's
theories in general, and of his objections to the theories discussed here.
86
investigated more thoroughly the axiom introduced in his 1872 paper and discussed in
Section 4.3 above – the postulate that the numbers do indeed correspond to the
geometrical line. It is noteworthy that Cantor, like Dedekind, developed his real numbers
first and only then axiomatized their relationship to geometry. As Ferreiros wrote,
It had been customary to assume that the continuity of space or of the basic
domain of magnitudes induces, through the definition of real numbers as ratios,
the continuity of the number system. But now we find two mathematicians who
emphasize the point that it is possible to define abstractly a continuous number
system, while geometrical space is not necessarily continuous. One needs an
axiom, sometimes called the axiom of Cantor-Dedekind, to postulate that space is
continuous.24
Or, at the very least, one needs an axiom to link our abstractly defined real continuum
with geometry, and hence, one supposes, with space.25 Our first observation, then,
regarding the Beitrag and Cantor's continuity, is that he still was firmly of the belief that
the relationship between the real numbers and points on a geometrical line must be
assumed and not proven.
Another claim in the Beitrag which is interesting for us is that continuous sets
must be non-denumerable. After discussing various properties of sets of the “first power”
(i.e. sets that can be put into one-to-one correspondence with the natural numbers) Cantor
introduced the concept of non-denumerability to the discussion.
We will now in what follows examine the sets that we call continuous from the
point of view of their power. ... [I]t is certain that these sets do not belong to the
24
Ferreriós, José, "On the Relations between Georg Cantor and Richard Dedekind," Historia Mathematica,
20 (1993) 135.
25
Though, of course, whether geometry has any connection at all to actual space is a topic for another
dissertation.
87
first class, i.e. they have a power superior to that of the first.26
Thus, “from the point of view of their power,” continuous sets must be non-denumerable.
After this point in time, mathematicians will frequently list nondenumerability as a
necessary though not sufficient property of any continua, but it is particularly interesting
that Cantor specifies nondenumerability as a necessary condition so soon after proving
that the reals have this property. In the space of five years, nondenumerability rose in
status from a startling and non-obvious quality of the reals to a necessary property of any
continuum.
The last observation I wish to make on Cantor's continua in this 1878 paper
concerns the dimension-reducing project itself. When Cantor claimed that a continuum
of any dimension could be treated mathematically as a continuum of one dimension, i.e.
as a straight line, he did so by arguing that the members of a continuous set of n
dimensions can be put into one-to-one correspondence with the members of a continuous
set of one dimension. The philosophical implication of this is that any continuum,
whether a line, plane, or three-dimensional figure, contains nothing more than the sum of
its points. For Cantor, any continuum is thus completely defined by specifying the
members belonging to it as a continuous set. We shall see Cantor expand on this belief in
Section 4.5 below.
Before finally assembling all of these ingredients into Cantor's explicit definition
of continuity, I wish to note one element in the Beitrag of historical interest. Cantor ends
his 1878 paper with an early version of his continuum hypothesis:
26
Cantor 1878, 313.
88
By a procedure of induction, the description of which we shall not enter into here,
we are led to the theory that the number of classes of sets obtained after this style
of grouping [i.e. grouping them according to power] is a finite number, and is
equal to two.27
The claim here is that if we group our infinite sets by their powers we shall come up with
only two groups. Cantor identified these groups as, first, the sets sharing the power of the
positive whole numbers, and second, those sharing the power of the real numbers
between 0 and 1. He would later come to believe there were many more than two
powers, but he would hold firmly to the belief that there were no powers larger than that
of the natural numbers but smaller than that of the real numbers, and his search for
definite proof of this Continuum Hypothesis would continue for most of his life.
4.5 Infinity, and the Definition of Continuity (1883)
After first constructing real numbers (in his 1872 article), subsequently showing
that the set of real numbers is nondenumerable (1873), and comparing the real number
continuum to continua of many dimensions (1878), Cantor finally gives us an explicit
definition of continuity in his Grundlagen, in 1883. The Grundlagen is one of Cantor's
most explicitly philosophical works, and intentionally so. In the preface, he wrote that
this essay was intended primarily for two groups, "for philosophers who have followed
the development of mathematics up to the most recent period, and for mathematicians
who are familiar with the most important older writings and more recent works in
27
Cantor 1878, 327.
89
philosophy."28 Cantor was quite aware that his mathematics – particularly his treatment
of infinity – was causing a philosophical stir, and he wrote the Grundlagen particularly to
defend some of the metaphysical and epistemological assumptions behind the acceptance
of actual infinities implied by his transfinite theory. The Grundlagen is a complicated
and fascinating essay; here I shall focus on only a small part of it. In particular, I wish to
focus on Cantor's discussion of different types of infinity, his rejection of infinitesimals,
and finally, his necessary and sufficient conditions for continuity.
In addition to distinguishing various magnitudes of infinity, Cantor distinguished
between two different mathematical concepts of the infinite. The first concept of infinity,
and the one Cantor regarded as more historically common, is the potential infinity
contained the concept of variable magnitude, "either growing beyond all limits, or
diminishing to an arbitrary smallness, always, however, remaining finite."29 This type of
infinity Cantor refers to as das Uneigentlich-unendliche – the non-genuine infinite.
Cantor himself based his transfinite theory, and much of the mathematics of his
later years, on a different concept of infinity, the Eigentlich-Unendliches, or genuine
infinite.
According to this concept, in the investigation of an analytic function of a
complex variable, for example, it has become necessary and in fact common
practice to imagine in the plane representing the complex variable a single point at
infinity, i.e., an infinitely distant but determinate point.30
This genuine infinite, the assumption of a point at infinity or the comparison of two
28
Cantor 1883, 70.
29
Ibid., 70.
30
Ibid., 70.
90
actually infinite magnitudes, is of course the infinity Cantor was most concerned with, the
concept assumed by transfinite theory, and the concept that many people of his time
found philosophically objectionable. The main point of the Grundlagen is to defend this
genuine infinite.31
The primary philosophical objection to this actual infinite, and the one which
Cantor most wished to refute, is an argument found in Descartes, Spinoza, Leibniz, and
others: namely, that humans, as finite beings, can never comprehend infinity, since
infinity comprises the absolute; thus, only an infinite mind can truly comprehend this
infinity. Cantor's rebuttal is simple: there are levels of magnitude between finitude and a
true absolute; all of Cantor's magnitudes fall in these medium levels, and in fact, all
magnitudes short of God himself belong to these middle-range, comprehendible
infinitudes. Though these levels of infinity are, by definition, non-finite, they are
nevertheless definite and determined – they are not finite, but they are limited. Thus,
wrote Cantor, "All things, whether finite or infinite, are definite and, with the exception
of God, can be determined by the intellect."32 It is the absolute nature of God's infinity
that Cantor agreed was incomprehensible by human beings; regular infinity was always
bounded by something or another, always had limits and lacks (the infinity of the natural
numbers does not contain real numbers, the infinity of real numbers does not contain nonnumbers, etc.), and thus was comprehensible.
31
Paul du Bois-Reymond posited a similar distinction between types of infinities. See the General Theory
of Functions, 72 – 73.
32
Cantor, 1883, 76.
91
Just as infinity is comprehensible by humans, so too is continuity, according to
Cantor. He argued that comprehension of continuity did not require intuitive
understanding of continuous entities, such as space or time.
In my opinion, the enlistment of the concept of time or of the intuition of time in
the discussion of the much more fundamental and more general concept of the
continuum is not in order. It is my judgment that time is a notion which for its
clear explication presupposes the concept of continuity, which is independent of
it.33
The same holds true of space. An understanding of continuity, Cantor argued, does not
depend upon a prior intuition of continuity such as that of space or time. Rather, the
opposite holds; understanding space or time requires a prior understanding of continuity
itself, and understanding continuity itself requires "sober and exact mathematical
investigations."34
Cantor's sober and exact investigations lead him to define continuous sets as
having two conditions, which are individually necessary and jointly sufficient: a
continuous set must be (1) connected, and (2) perfect. A set P is connected when
between any two numbers t and t' at least one finite collection of fellow members {tν}
could be found such that the distance between tν and tν-1 are collectively smaller than ε, an
arbitrarily chosen positive number.35 From connectivity, everywhere-denseness follows.
A set P is perfect when it equals each of its derived sets P(γ). Cantor used the
'derived set' concept in his 1872 paper, where, for a set P, its first derived set P' was
33
Ibid., 85.
34
Ibid., 85.
35
Grattan-Guiness, 93.
92
defined as the set of limit points of P.36 Here he expands on the notion in concord with
his expanded number system: the derivative P(γ) is such that "γ can be any whole number
of one of the number-classes (I), (II), (III), etc."37 That is, γ can be a finite or transfinite
number, and thus the set of all of P's derived sets, P(γ), includes derived sets of any
magnitude. If P is infinite and perfect, then P is non-denumerable,38 and the set of real
numbers as Cantor defined it in 1872 is a perfect set; the collection of derived sets of R is
identical to R itself.
Thus, with these two conditions, Cantor believed himself to have established a
"purely arithmetical concept of a point-continuum,"39 rather than one based on intuitions
or experience, and believed that this notion could then be applied to our understanding of
non-mathematical continua such as space or time. Much like the continua of many
dimensions discussed in Section 4.3 above, which were analyzed completely in terms of
the points contained within them, this definition of continuity relies solely on the
connection between the points of the set in question. Thus, one supposes, when
analyzing the continuity of time, one would first have to identify the basic elements to be
analyzed – perhaps instants – and then we could determine whether the set of instants
under consideration was a perfect and connected set. It is of great consequence to note
that Cantor's continuity is necessarily one composed of individual elements.
36
Cantor 1872, 343.
37
Cantor, 1883, 86.
38
Dauben, 111.
39
Cantor, 1883, 85.
93
Cantor's theory of continuity is similar to Dedekind's in some respects.
Particularly, Dedekind's set of real numbers can be shown to have the property of
perfection. However, Cantor's mathematical concept of connectedness is precisely what
Dedekind's theory of continuity lacked, which led to objections that Dedekind had simply
presented a collection of distinct objects, and not a continuum at all. While Cantor's
connectedness does determine some sort of interlinking between the elements of his
continuous sets, his continuum is, like Dedekind's necessarily composed of elements; in
order for a set to be judged continuous, one must first have a set of elements to judge, and
Cantor's continuity is predicated upon the existence of infinite sets. As we shall see in
later chapters, du Bois-Reymond and Peirce were critical of this compositional theory of
continuity, and both strove to develop substantially different theories. It is notable that
both theories include infinitesimals in crucial roles. Thus, before ending this chapter on
Cantor, it behooves us to examine in some detail Cantor's objections to infinitesimal
magnitudes.
4.6 Infinitesimals (1883 and 1887)
Cantor was a staunch opponent of the inclusion of infinitesimals into our
mathematical systems. In the Grundlagen, in 1883, he argued that those who believed
infinitesimals to be real quantities were laboring under a confusion; in a letter to
Weierstrass in 1887, he formulated the sketch of an argument which was meant to prove a
stronger conclusion – that infinitesimals were self-contradictory entities, and thus could
not be consistently formulated. In this section, we will examine Cantor's antiinfinitesimal arguments in both of these works.
94
In the Grundlagen, while Cantor vigorously defended the addition of transfinites
to our concept of numbers, he took a moment to indicate that this should not imply his
acceptance of the existence of infinitesimals. In fact, he believed that those who posited
actual infinitesimal magnitudes were confusing the non-genuine infinite with the genuine
infinite.
The infinitely small magnitudes, to my knowledge, have so far been worked out
for useful purposes only in the form of the non-genuine infinite [...]. On the other
hand, all attempts to transform the infinitely small by force into a genuinely
infinitely small magnitude should finally be abandoned as purposeless. If
genuinely infinitely small magnitudes exist in any other form at all, i.e. are
definable, still they surely do not have any immediate connection with the
ordinary magnitudes that become infinitely small.40
Thus, Cantor here claimed that the infinitesimals which appeared in some works of
analysis and function theory at the time were not infinitesimals at all, but rather variable
magnitudes approaching the infinitely small but at all times actually finite – i.e. examples
of non-genuine infinity. Here in the Grundlagen, he hedged on whether infinitesimals
actually existed, however, writing not that they are impossible entities (as he later would
argue), but rather, that if they were somehow definable, they would have no connection to
ordinary magnitudes, and thus no connection to the variable magnitudes which are
actually useful to analysis.
As the charge of uselessness is a common criticism lodged against theories of
infinitesimals, it is worth taking a moment to consider Cantor's stance on the difference
between pure and applied mathematics, the criteria for admitting new numbers as actually
extant, and the role of use in mathematics – all of which he discusses in his defense of
40
Ibid., 74.
95
admitting transfinites in to the numerical canon. Cantor's explicit view on usefulness is
that pure (or, as Cantor preferred, 'free') mathematics should not be hampered by
considerations of whether mathematical creations were useful in the sciences or in any
sense whatsoever, and that the most creative and important mathematical advancements
were achieved by those who cared not one whit for the usefulness of their theories.41
While he firmly believed that those working in applied mathematics should indeed
consider the metaphysical ramifications of their theories, the criterion restricting the
addition of new entities to pure mathematics were looser.
In fact, he wrote that mathematics is "bound only by the self-evident concern that
its concepts be both internally without contradiction and stand in definite relations,
organized by means of definitions, to previously formed, already existing and proven
concepts."42 Thus, for the pure mathematician, usefulness is not a necessity when new
concepts are introduced; only internal self-consistency, and the relationships the new
concept would have with the wider system, need be considered. To meet the latter
criterion, Cantor thought that a new concept must be defined well enough to permit
distinction from the old concepts.
Cantor argued that transfinite magnitudes meet these criteria, and thus "can and
must [be regarded as] existant and real."43 However, many mathematicians, such as
41
Ibid., 79. Among the mathematicians so identified are Gauss, Cauchy, Dirichlet, Weierstrass, Poincaré,
and Bernhard Riemann (1826 – 1866).
42
Ibid., 79.
43
Ibid., 79.
96
Peirce and du Bois-Reymond have argued that infinitesimals also meet these criteria, and
thus, according to Cantor himself, also must be regarded as extant and real. Hence,
though Cantor in fact may be correct that genuine infinitesimal magnitudes are
"purposeless" and have no "immediate connection with the ordinary magnitudes,"
according to Cantor himself, neither of these complaints are enough to justify their
exclusion from pure mathematics.44 According to his own criteria, infinitesimals may
only be justly excluded if they are proved inconsistent, or if they are incapable of being
defined in such a way that they can easily be distinguished from existing mathematical
entities.45
Cantor possibly saw this flaw in his reasoning, and a few years later he took the
more direct approach of attempting to prove infinitesimals internally inconsistent. If he
had succeeded, and had shown that infinitesimals could not be consistently defined, then
he would have barred them from mathematical reality by the first condition listed above.
His argument sketch appeared in a letter to Weierstrass.
"Linear number magnitudes ζ different from zero (i.e. shortly put, such number
magnitudes as can be represented by bounded, straight, continuous segments)
which would be less than any ever so small finite number magnitude do not exist,
44
In Chapter 7, below, I shall myself argue that Cantor is in fact incorrect about this, and that infinitesimals
are mathematically useful.
45
It must be noted that Cantor believed useless mathematical entities, while they should not be excluded
from the canon on the basis of their uselessness, would eventually be abandoned in favor of more fruitful
concepts. However, he noted that many concepts had no apparent use when introduced, but soon became
indispensible to science; uselessness was, for Cantor, something to be determined over time, not at the
moment of mathematical invention. See ibid., 79.
97
i.e. they contradict the concept of a linear number magnitude." The train of
thought in my proof is simply the following: I begin from the supposition of a
linear magnitude ζ which is so small that its product by n, n · ζ, for every finite
whole number n however great is smaller than unity, and then prove, from the
concept of a linear magnitude and with the help of certain propositions from
transfinite number theory, that then ζ · ν is less than every finite magnitude
however small, where ν is an arbitrarily great transfinite ordinal number (i.e.
Anzahl or type of a well-ordered set) from any arbitrarily high number class. But
this means that ζ cannot be made finite by any actually infinite multiplication of
any power, and hence surely cannot be made an element of finite magnitudes. But
then the supposition made contradicts the concept of a linear magnitude, which is
such that every linear magnitude must be thought of as an integrated part of other
ones, and in particular of finite ones. Hence, there remains no alternative but to
1
drop the supposition that there is a magnitude ζ which is smaller than for every
n
finite whole number n, and hence our proposition is proved.46
The argument has obvious gaps (for example, Cantor does not here specify which
"propositions from transfinite number theory" would here help us), but the general idea is
clear. Infinitesimals, or linear number magnitudes different than zero but less than any
finite number magnitude, cannot be made finite by multiplication of any finite or infinite
number. Therefore, infinitesimals cannot be elements of finite magnitudes – they are off
the map, so to speak, and no road whatsoever leads to them. For, to re-quote an essential
element of the proof, "every linear magnitude must be thought of as an integrated part of
other ones, and in particular, of finite ones."47 This requirement reminds one of the
second condition for allowing new mathematical objects into the canon – that the
relationship to the existing objects be specified.
However, though infinitesimals are necessarily disconnected from finite
46
Cantor's letter to Karl Weierstrass, Mitteilungen zur Lehre vom Transfiniten (Cantor 1887, 407-408 in
1932 ed.), translated by W.D. Hart, in Moore, 306. The italics are Cantor's.
98
quantities, pointing out this feature does not prove that they are "impossible, i.e.,
intrinsically inconsistent imaginary things." Rather, it merely proves that infinitesimals
cannot be an integrated part of a system of magnitudes, since no finite or infinite
multiplication can help an infinitesimal reach the finite numbers of the system. This not
only does not show infinitesimals to be inherently, internally contradictory; it is rather a
well-known feature of infinitesimals, one that lends infinitesimals a unique character.
Further, Cantor himself states that transfinite numbers are similarly isolated from finite
quantities.48
Perhaps the key to understanding Cantor's intended contradiction is hidden in the
first sentence of his argument: "Linear number magnitudes ζ different from zero ... which
would be less than any ever so small finite number magnitude do not exist, i.e., they
contradict the concept of linear number magnitude." It is the notion of an infinitesimal
magnitude qua magnitude which is, Cantor believed, inconsistent. Infinitesimals cannot
be an integrated part of a collection of infinitesimal and finite linear magnitudes, but
more importantly, they cannot consistently be considered magnitudes at all. In what
seems to be an extension of the Archimedean Principle, Cantor defends this supposed
inconsistency by claiming that ζ cannot be made finite by any finite or infinite
47
Ibid.
48
See, for example, the Grundlagen, pg. 71, where Cantor wrote that "the point at infinity in the complex
number plane stands isolated vis a vis all finite points." He went on to show that there were infinitely many
distinct such points, and further, that it is their isolation from finite quantities which give them the
distinctness necessary to justify adding them to the mathematical canon; for were they entirely reducible to
finite numbers, the creation of new numbers is wholly unjustified.
99
multiplication. And indeed, in the same letter to Weierstrass, Cantor gave a definition of
the Archimedean Principle, and insisted that it should not be considered an axiom.
The so-called Archimedean Axiom is not an axiom at all but a proposition which
follows with logical necessity from the concept of linear magnitude."49
As we saw in Chapter 3 above, the inclusion of the Archimedean Principle does
conflict with the inclusion of infinitesimals. If the Archimedean Principle follows
necessarily from the concept of linear magnitude, then an infinitesimal linear magnitude
is indeed a contradiction. Thus, Cantor's argument revolves around one key point;
whether non-Archimedean magnitude is possible. This is not a point Cantor here
supports, non-Archimedean systems have since been proved possible. Veronese
proposed non-Archimedean geometries around 1890,50 and David Hilbert (1862 – 1943)
proved the consistency of a non-Archimedean geometry and corresponding number
system in his Grundlagen der Geometrie in 1899. In the twentieth century, Abraham
Robinson created a non-Archimedean calculus. It is noteworthy that Cantor believed that
the Archimedean Principle was an essential feature of magnitude; however, his belief was
false.
One further note on this proof. Possibly the most controversial point of this
argument is where he claims that no finite or infinite multiplication can integrate
infinitesimals with finite magnitudes, for soon after Cantor's Grundlagen, in which he
introduced transfinite magnitudes, mathematicians were defining infinitesimals as the
49
Ibid.
50
See the biography of Veronese at the University of St. Andrews, <url = http://www-groups.dcs.st-
and.ac.uk/~history/Biographies/Veronese.html>.
100
multiplicative inverses of transfinites.51 Abraham Robinson's non-standard analysis does
something similar, and incorporates both infinitesimal and infinitely large quantities into
the same system with finite quantities. Systems of magnitudes which include
infinitesimals – such as the systems of Paul du Bois-Reymond and of Charles Sanders
Peirce – are decidedly non-Archimedean. Thus, as we will soon see, these systems
require a theory of continuity and even a theory of real numbers substantially different
from those of Cantor and Dedekind.
51
Cantor, 1883.
101
Chapter 5
Paul du Bois-Reymond
5.1 Biography and Introduction
Paul du Bois-Reymond was born in 1831 in Berlin. His father was from
Neuchatêl, an area of modern-day Switzerland which was at the time part of Prussia. The
du Bois-Reymonds spoke French at home, and Paul and his elder brother Emil were
educated at the Collège in Neuchatêl. Emil was a noted scientist, making a name in
physiology, and Paul followed in his brother's footsteps, studying the physical sciences
and medicine. While studying in Könisberg, he changed his focus to add more of a
mathematical element, and wrote his doctoral thesis in mathematical physics, focusing
particularly on mathematical analyses of the movements of liquids.1
Paul Du Bois-Reymond's physical sciences background brings an element of
practicality to his mathematics and his philosophy, and many of his examples are drawn
from the physical world. Applied mathematics was the focus of much of his career, and
considerations of how the theory matches up with the application dominate much of his
philosophical works. He is an interesting addition to this dissertation for precisely this
reason: that his applied and even physical approach to mathematics is an important
1
Paul du Bois-Reymond, De aequilibrio fluidorum, 1859. URL = <http://gdz.sub.uni-
goettingen.de/dms/load/img/?IDDOC=42460>.
102
counter-balance to the more theory-based work of the other three mathematicians being
considered. Interestingly, though his mathematical and philosophical theory is well
grounded in application, many of his mathematical and philosophical contributions center
on things far removed from the physical world: infinite series, infinite functions, and
transfinite and infinitesimal quantities, things he himself admits are not grounded in or
reducible to physical experience.
It was his study of applied mathematics that led to his interest in partial
differential equations and functions with infinite domains. He was the first person to
give an example of a continuous function the Fourier series of which diverged at every
point. Much of his career was spent creating his Infinitärcalcül, or Infinitary Calculus, a
theory whereby functions with infinite domains and ranges were compared and ordered.
In particular, the ratio of two functions with infinite domains was subjected to the limit
operation, which produced interesting results; we shall examine this Infinitärcalcül further
on in the chapter as it has philosophical ramifications related to continuity.2
The work of du Bois-Reymond with which we are most directly concerned is the
1882 work, Die allgemeine Functiontheorie. Written late in his life (he died in 1889), it
is a philosophical attempt to show that mathematics is a science like any other.
According to the introduction of Die allgemeine Functiontheorie, fundamental
mathematical concepts are not as rigorously established as key concepts in other sciences,
instead left to be intuited; and our mathematical intuitions often conflict. He wrote,
2
For more information on Paul du Bois-Reymond's life, see the University of St. Andrews biography; URL
= <http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Du_Bois-Reymond.html>.
103
"What mathematician could deny that ... the concept of limit and its near-parents, those of
the unlimited, the infinitely large, the infinitely small, the irrationals, etc, still lack
solidity!"3 He argued that teachers and researchers alike gloss over these concepts rather
than defining them rigorously, and then make use of them freely in calculus as though
they had been proven. Du Bois-Reymond wished to solidify the basis upon which our
mathematics is built, by closely examining the intuitions behind all of these concepts.
In this chapter, we will briefly consider the details of the Infinitärcalcül and
suggest how this system inspired du Bois-Reymond to consider the foundations of
mathematics an important field of inquiry, then discuss in detail du Bois-Reymond's
attempt to rigorously define the concept of limit and its near-parents, in the General
Theory of Functions. Du Bois-Reymond believed that at bottom, our mathematical
concepts were founded on two competing intuitions, an Idealist and an Empiricist
intuition, and the first third of the book represents the argument between these two
camps. We will spend some time with the Idealist/Empiricist argument, as it gives us
great insight into what du Bois-Reymond saw as the fundamental philosophical problems
behind the basic concepts of mathematics. A final section will draw together du BoisReymond's own conclusions about these basic concepts, discuss how these concepts
interact in his theory of continuity, and briefly compare his theory to those of Cantor and
3
Du Bois-Reymond, General Theory of Functions, 21. The original text of this book was published in
1882, and was titled Die allgemeine Functiontheorie. In this chapter, references will be to the French
translation, Théorie générale des fonctions, Nice: Imprimerie Niçoise, 1887. The English is my own
104
Dedekind.
5.2 Infinitärcalcül
Du Bois-Reymond's Infinitärcalcül was in essence an attempt to order functions
with infinite domains and ranges using the limit theorem. He began this project in an
1870-1 paper,4 and developed it through many papers written over many years. The
mathematical basics of the Infinitärcalcül are as follows.5 First, let f and g be functions
with infinite domains and ranges; we can then take the limit of their ratio, thus:
lim f ( x) / g ( x)
x →∞
The results of computing this limit tell us something about how the two functions relate
to each other. If the limit approaches infinity, then we say that f(x) ≺ g(x). If the limit
approaches zero, then we say f(x) ≺ g(x). If the ratio has a finite, non-zero limit, then we
say f(x) ~ g(x). We could say that, if it was absolutely clear we were only speaking in
analogies, in the first case, g(x) is infinitely bigger than f(x), in the second case f(x) is
infinitely bigger, and in the third case they are equivalent. Du Bois-Reymond viewed this
result as producing a rough ordering of such functions, and attempted to prove a more
translation from the French unless otherwise noted. See the Appendix for a translation of the first third of
this book from the French.
4
Du Bois-Reymond, "Sur la grandeur relative des infinis des fonctions," Annali di matematica pura de
applicata 4, Series IIa, 338 – 353.
5
This summary is drawn largely from Gordon Fisher's article, "The Infinite and Infinitesimal Quantities of
du Bois-Reymond and their Reception," Archive for History of Exact Sciences, v 24, 1981, p 101-164.
105
rigorous ordering and to draw results from this organization of functions. As he wrote in
his first paper on the subject, "This new algorithm which shows some analogy with
ordinary inequalities, can be called one of infinitary inequalities."6
Thus, for example, take the function f(x) = x, and the function f(x) = x2 (or, to
abbreviate, the functions x and x2). "The student soon learns that as x tends to infinity (x
→ ∞), then also x2→ ∞, and moreover, that x2 tends to infinity more rapidly than x."7
The magnitudes of infinity8 that characterize du Bois-Reymond's infinitary inequalities
are just these: that some increasing functions approach infinity more rapidly than others,
and that some decreasing functions approach zero more rapidly than other decreasing
functions. Thus, applying our above definition of the ≺ relationship, the limit of
x
is
x2
zero, and thus, x ≺ x2. The inverse relationship also holds in this case; as x grows,
x2
approaches infinity, and thus x2 ≺ x.
x
The above example provides us with a way of generating many such magnitudes
of infinity, in fact infinitely many. Consider not only x and x2, but also x3, x4, etc.,
insuring as the exponents increase that the function approaches infinity even more rapidly
6
"Sur la grandeur relative des infinis des functions," 339, as quoted in Fisher. At the time, no algorithm
existed, but only notation. Fisher suggests du Bois-Reymond might have been anticipating algorithms he
intended to develop. See Fisher, 102.
7
G. H. Hardy, Orders of Infinity, Cambridge: Cambridge University Press, 1.
8
These magnitudes of infinity, then, have nothing to do with transfinite numbers, unlike Cantor's differing
magnitudes of infinity.
106
than the previous function. Noting that there are thus infinitely many magnitudes of
functions, du Bois-Reymond thus wished to define a series of infinitary inequalities
f1(x) ≺ f2(x) ≺ f3(x) ≺ … .
Du Bois-Reymond envisioned this infinite ordering of functions as closely
analogous with the real numbers, and wished to prove the density of the ordering to
further the resemblance:
Just as between two functions as close with respect to their infinities as one may
want, one can imagine an infinity of others forming a kind of passage from the
first function to the second, one can compare the sequence F [a scale of infinity]
to the sequence of real numbers, in which one can also pass from one number to a
number very little different from it by an infinity of other ones.9
Thus, in this 1870-1 article, du Bois-Reymond established a family of functions, ordered
them linearly, and made some indications on how density could be approximated. In
subsequent articles, he drew consequences from this ordering and developed theorems for
his calculus of functions. By 1875, he began to explicitly address philosophical
considerations surrounding the Infinitärcalcül. In this article, he finally committed
himself to accepting the concept of the actual infinite, and begins to speculate on the
relationships between different types of functions (increasing versus decreasing, for
example), as well as on the nature of functions and the concept of limit – essential tools
in his theory of infinitary functions.10
An 1877 paper, "Ueber die Paradoxen des Infinitärcalcüls," was dedicated to
investigating the similarities and differences between this arranged chain of functions and
9
Ibid, 343.
10
Fisher, 105.
107
the real numbers. Within this article, he claims that the continuity of his Infinitärcalcül is
comparable to the continuity of the real numbers, which gives us further insight into the
Infinitärcalcül itself, but more importantly for our current purposes, gives us insight into
his early ideas of mathematical continuity and the creation of the real numbers.
Thus through more precise consideration the rational numbers always approach
more closely to one another, yet in our minds gaps are always left between them,
which mathematical speculation then fills with the irrationals.11
Notably, in general outline this account is similar to Dedekind's definition of real
numbers and mathematical continuity, but the overtones are quite different. The density
of the rationals is reached through "precise consideration," while the gaps between them
are not mathematically proved, as in Dedekind, but rather the gaps occur "in our minds."
The real numbers do not arise as mathematical necessity as in Dedekind, but rather
through "mathematical speculation," clearly implying a lack of rigor and justifiability,
which is to be contrasted with the precision of the rationals.
Indeed, du Bois-Reymond went on to claim that irrationals were introduced only
to complete the comparison between numerical sequences and geometric ones, and
depend intimately on the limit concept, which itself needed further analysis and
investigation.12 As we shall soon see, this is precisely where the General Theory of
Functions begins: with an effort to comprehend these mathematical concepts at the most
basic level. This work is du Bois-Reymond's attempt to philosophically ground the
11
Du Bois-Reymond, "Ueber die Paradoxen des Infinitärcalcüls," 1877, 150. As quoted in G. Fisher, 107;
presumably, the translation from the German is also Fisher's.
12
Fisher, 108.
108
continuity of the real numbers, the concept of limit, the relationship between geometry
and number systems, and of course, the nature of functions themselves.
5.3 Idealist versus Empiricist: basic theories, and straight lines
Du Bois-Reymond began his General Theory of Functions by considering two
philosophical approaches to mathematics; that of the Idealist, and that of the Empiricist.
It is important to note that du Bois-Reymond was not here discussing actual theories of
actual philosophers or mathematicians. Rather, du Bois-Reymond believed that each of
us who thinks carefully about mathematics will find both empirical and idealistic
tendencies within ourselves, and that in fact, this split in intuitions is the very reason we
have such trouble elucidating precisely the most basic elements of our mathematics. As
he wrote in the introduction to the General Theory of Functions, "There is, in the mind,
two completely distinct manners of apprehending things, which have an equal right to be
taken for the fundamental intuition of exact science."13
Du Bois-Reymond spent much of this book drawing out the logical consequences
of the Idealist versus the Empiricist world view (if we may call it that) as concerns our
mathematical tendencies. He imagined fictional proponents of each stance and staged a
discussion between them, until each imaginary philosopher was pushed to specify both
the fundamental principles of their particular viewpoint and elaborate on the logical
consequences of these principles, that is, on how exactly mathematics would have to
proceed to be consistent with each set of principles. As we shall see in the next section,
13
General Theory of Functions, 22.
109
du Bois-Reymond eventually gave us a “neutral exposition”14 – a system of mathematics
that is objectionable to neither the Empiricist and Idealist, and as faithful as possible to
both of them, from which he builds his own system.
The Empiricist/Idealist split is interesting in itself to our project, as it contains
arguments for and against the existence of infinitesimals, and the two viewpoints imply
completely distinct conclusions regarding the continuity of the geometrical line, and
regarding the relationship between the geometrical line and the number line. Thus, we
shall look at each philosophical viewpoint in some detail in subsection 5.3.1. We shall
then apply these viewpoints to a particular case, that of the straight line, for three reasons:
first, this particular example demonstrates well how different the two positions are;
second, it is necessary to understand the different theories of the line to understand the
different theories of infinitesimals; and third, our discussion of du Bois-Reymond's theory
of continuity shall also depend largely on these intuitions about the line itself.
5.3.1 The basic viewpoints and their fundamental theses
In basic outline, the Idealist represents our belief in logic, idealized geometrical
figures, and precision that is beyond that which we can accomplish physically. The
Empiricist represents our belief in the fundamental connection between mathematics and
the world of sensation – of things that can be seen and felt and physically manipulated.
Thus,
Idealism believes that the truth of certain limited forms of our ideas is required by
14
Ibid., 128.
110
our understanding, though they may lie outside of all perception and sensory
representations. Empiricism is the system of complete abnegation; it admits only
as extant that which can be perceived or reduced to perception.15
The Empiricist, then, holds as central our tendency to connect mathematics to what can
be applied, and to base it on what we experience, much like Dedekind’s assertion that
mathematics begins when we begin to count; while the Idealist focuses on our tendency to
extend mathematics as far as logic and our imagination can take us, and to create
mathematical objects which help us overcome difficulties, whether or not these objects
have any basis in reality.
To illustrate the difference, for the Empiricist, geometry can contain shapes such
as circles because we experience round things in our daily life. This is reminiscent of
Plato’s definition of shape in the Meno; “A shape is that which limits a solid; in a word, a
shape is the limit of a solid.”16 Plato’s definition could apply to geometrical, idealized
solids, or it could apply to actual solid objects in the world; the shape of an object is the
limit of that particular object, for du Bois-Reymond’s Empiricist, and a shape in general,
such as a circle or a triangle, is a mental collection of all that is similar in the shapes of
circular or triangular objects I have experienced. We must only be careful how we
abstract away from our experience; the Empiricist believes the Idealist abstracts so far
away from experience that it is impossible to return to it.
For the Idealist, on the other hand, the allowable mathematical objects are those
that meet certain stringent logical criteria, whether or not they are connected in any way
15
General Theory of Functions, 22.
16
Plato, Five Dialogues, Indianapolis: Hackett Publishing Company, Inc., 2002, 65 (76a).
111
whatsoever to our experience. The Idealist's geometry is thus quite Euclidian, with
partless points, breadthless lengths, and the Idealist would most likely agree with Euclid's
assertion that "a circle is a plane figure contained by one line such that all the straight
lines falling upon it from one point among those lying within the figure are equal to one
another, and the point is called the centre of the circle."17 Thus, a circle, for an Idealist, is
not a generalization of the shape of all circular objects, abstracted away from particulars
such as color and location, and perhaps also extracted away from flaws, but it is rather an
idealized object, constructed from already idealized objects such as point, line, and plane.
This Idealist circle is not based on experience, nor is it reducible to experience; and
further, it cannot exist at all. Any particular representation of a circle fails to meet the
stringent logical criteria of this definition; any drawn circle is drawn of a curved line with
width, which the idealized circle does not have, is likely to not be precisely round, etc.
5.3.2 Two Theories of the Straight Line
An important difference in Idealist and Empiricist mathematics, from our point of
view, is how they each view the nature of the straight line itself. For the Empiricist, the
line is one drawn from experience; the idealized straight line does not exist in nature, and
is not deduced from or reducible to representations, therefore it does not exist.18 In
Idealist thought, a straight line is an idealized geometrical object, infinite in extent,
17
Euclid, The Elements, Vol 1., 153-154. In fact, the Idealist defines a sphere in a similar way to this; see
General Theory of Functions, 95.
18
General Theory of Functions, 90.
112
infinitely divisible, existing only in two dimensions.19 The ramifications for each
viewpoint are worth going into in detail. Du Bois-Reymond argued that the Idealist
theory of the straight line leads directly to the logical necessity of infinitesimals, as we
shall see in the next section. The Empiricist theory, on the other hand, leads to a
conception of the straight line which is quite unique.
First, let us consider the Idealist straight line. Beginning with the geometrical
idealization of a line as discussed in the above paragraph, the Idealist attempts to draw a
correspondence between the real numbers and the points on this line. He does this as he
recognizes that precise measurement of geometrical objects is "fundamental to the science
of magnitude."20 Points, it must be noted, are themselves idealized objects; as they lack
length, they therefore have no extension at all. For simplicity, the Idealist considers only
the line segment between the points 0 and 1 – the "unit length," as it is precisely one unit
in length.
The "rational lengths" – points on the line which correspond to rational numbers,
and which mark out lengths which are a particular percentage of the unit line – are easy to
find. Thus, we can easily locate "halves, thirds, quarters, etc. ... and also multiples of
19
The Idealist specifies two different ways of conceiving of the idealized line and point. One is to begin
with an idealized plane, and define the line as an intersection of two planes, and the point as the limit point
of a line segment. The other is to begin by defining the point as space which is infinitely contracted, such
that it has no extension in any direction, and then creating a line segment by following the motion of this
point through space in one direction. See General Theory of Functions, 96 – 97.
20
Ibid., 64.
113
these fractions of the unit length."21 We can also easily construct innumerable irrational
lengths, such as the roots of rational numbers and multiples of these roots, and "the unit
length is covered in our representation by a more and more dense collection of points."22
However, this process will never completely fill the unit length with points.
On the contrary, two neighboring points always remain separated by a segment of
straight line which, abstracting away from its length, completely resembles the
unit length. This is the image which always accompanies my representation of
magnitude.23
This Idealist representation of magnitude is thus that of a line segment populated by
points which correspond to certain numbers, and which are such that each and every pair
of points maintains a line segment between them. Further, no matter how many points
populate a line, it is never composed of them. "I reject the enlargement of the concept of
magnitude according to which the line must be composed of points, the surface of lines,
etc."24
This representation of a line segment is common enough, but there at least two
interesting features which should be noted. First, the only irrational numbers explicitly
corresponding to points were roots of rational numbers and their multiples. This is
because the Idealist believes, as does du Bois-Reymond himself, that most if not all
21
Ibid., 64.
22
Ibid., 64.
23
Ibid., 64.
24
Ibid., 70.
114
irrational numbers are themselves limits,25 and the Idealist is not guaranteed that points
corresponding to any mathematical limit actually exist.26 Second, it is notable that mere
density is not the essential feature of the Idealist's magnitude; rather, it is the existence of
a line segment between any two points that is precisely identical to every other line
segment, except in regards to their size. The importance of this feature is precisely what
leads the Idealist to also accept the existence of infinitesimals, as we shall later see.
First, though, let us look carefully at the straight line of the Empiricist. As we saw
in 5.3.1, du Bois-Reymond characterized the Empiricist viewpoint as one of "complete
abnegation,"27 thus characterizing it mainly by what it denies. To be precise, it denies the
existence of any mathematical object not connected to or reducible to our experience. In
fact, the title of the section in which the Empiricist begins to enunciate his own view is,
"Purification of the System of Concepts."28 However, for a system of complete
abnegation, the Empiricist's ontology is rather rich, and many positive claims are made.
For example, after denying the existence of idealized geometric objects,29 the
25
This is a long-held belief on du Bois-Reymond's part. See, for example, du Bois-Reymond, "Ueber die
Paradoxen des Infinitärcalcüls," 152, and Fisher, 108.
26
He will eventually prove the existence of these point-limits, but not until he has proven the existence of
infinitesimal quantities; thus, infinitesimals are necessary for the geometrical line and our number system to
be truly comparable.
27
General Theory of Function, 22.
28
Ibid., 102.
29
"It is absolutely forbidden to deduce whimsically from our representations the idea of the perfect straight
line." Ibid., 105.
115
Empiricist commits himself to the existence of non-idealized geometric objects. Thus,
points are not idealized bits of non-extended space; they have extension. Lines are not
idealized two-dimensional objects that exist only in the mind, but all lines have a
thickness as well as a length. In fact, the thickness of the line is related to the extension
of the point; the line can be as thin as you wish (as long as it is not infinitely thin, which
makes no sense to the Empiricist), but as a point is "the portion of space common to two
lines which intersect,"30 the point thus has precisely the width of the lines, only instead of
having that thickness in only one direction, it is just as tall as it is thick. In fact, as it is
possible for three lines to intersect in three-dimensional space, the point is also as wide as
it is tall and thick; it forms a very small sphere. Consequently, there is no contradiction
viewing the line as composed of points. Recall Aristotle's argument against the
compositionality of the line from Chapter 2: the main engine of the argument was the
assumption that points have no extension, no parts, and therefore it is impossible for them
to be next to each other. The Empiricist point has no such restrictions. Therefore, he
concludes, the line "is composed of points without gaps."31
It is important to emphasize that the line (and therefore the point) does not have a
specified thickness, but rather, it can be as thin as one wishes.32 Traveling toward infinity
is not a problem for the Empiricist; only reaching it is a problem. Thus, while the Idealist
admits any actually infinite set which has a logical rule describing it, the Empiricist only
30
Ibid., 105.
31
Ibid., 106.
32
In the French, à volonté. See, for example, General Theory of Functions, 105 and 106.
116
allows that, using the rule, we can continue to generate as many numbers as we wish.
Otherwise, we are at risk of postulating that the numbers, once we give them a law,
proceed "separated from human mind to continue all alone their route toward the
infinite," or that the rule itself is identical to an actually infinite set, both absurdities.33
Thus, instead of infinite, the Empiricist prefers 'as large as one wishes,' instead of
infinitesimal the Empiricist would substitute 'as small as one wishes,' and instead of
idealized geometric shape, the Empiricist would use 'as perfect as one wishes.'34
5.4 Infinitesimals, for and against
The Idealist, according to du Bois-Reymond, is logically committed to the
existence of infinitesimals, and so, in addition to arguing for their existence, the Idealist
provides us with an analysis of their properties and how they fit into the larger
mathematical universe. This section shall have three subsections: an analysis of the
Idealist's argument for the existence of infinitesimals, a discussion of the properties of
these infinitesimals, and the Empiricist's counter-argument.
5.4.1 The argument for infinitesimals
33
Ibid., 84.
34
This is reminiscent of Aristotle's view that the actual infinite did not exist, but that mathematicians need
not despair as they "they do not need the infinite and do not use it." As we saw in Chapter 1, however,
117
The argument is complex, but it turns on the Idealist's theory of magnitude already
discussed above, therefore, is relatively easy to untangle. I shall quote the argument in its
entirety before looking at each step in turn. 35
The proposition that the number of points of division of the unit length is
infinitely large produces with logical necessity the belief in the infinitely small.
In fact, we have established above that, according to the true concept of
magnitude, these points do not follow each other without an interval, thus they
cannot be united but are always separated by extensions, so that points alone can
never form extensions; therefore infinitely many points are separated by infinitely
many extensions. Thus, of these extensions none can be finite, which is to say
cannot be contained a finite number of times in the unit length, because the unit of
length being arbitrary, every extension as small as it may be must be organized
like the unit length, and similarly contain infinitely many points of division.
One thus sees that the unit extension is decomposed into an infinity of
partial extensions, of which none is finite. Thus the infinitely small really exists.36
Let us assume that the "true concept of magnitude" refers to the Idealist's "image which
Aristotle did promise mathematicians that they could have as large of a number as they did need, just not
actually infinitely many of them. Aristotle, Physics, Book III, 207b 30.
35
The original German of this proof is as follows:
Denn halten wir fest, was wir oben als correcten Grössenbegriff hinstellten, dass Puncte auf der Länge nicht
ohne Abstand aufeinander folgen, also nicht aneinander stossen können, sondern immer durch Strecken
getrennt sind, dass also blosse Puncte nie eine Strecke bilden können, so sind auch die unendlich vielen
Puncte durch unendlich viele Strecken getrennt, und von diesen Strecken kann keine endlich, d. i. in
endlicher Zahl in der Einheitsstrecke enthalten sein, weil bei der Willkürlichkeit der Längeneinheit jede
noch so kleine Strecke die nämliche Beschaffenheit, wie die Längeneinheit haben muss, so dass auf ihr
wieder unendlich viel Theilpuncte vorhanden sein müssten.
Es ergiebt sich also, dass die Einheitsstrecke in unendlich viele Theilstrecken zerfällt, von denen
keine endlich ist. Also existirt das Unendlichkleine wirklich. (pp. 71-72 of the German edition).
36
General Theory of Functions, 73.
118
always accompanies my representation of magnitude"37 discussed in the previous section.
Thus, recall, the essence of linear magnitude for the Idealist was that any two
"neighboring" points were separated by a segment, which itself was identical to the unit
segment in every respect except for size. We now have the first step of the argument: if
the unit length can be infinitely divided, then the points of division all have segments
separating them, and therefore there must be infinitely many segments.
So far this all seems relatively unobjectionable. For if the line is truly infinitely
divisible, then surely it is infinitely divisible into segments; that is, each division will
result not only in points of division, but also in smaller and smaller segments. The
Empiricist would agree to this, as long as it was understood that the infinite divisibility
was a process which had no end, rather than an infinite process that had somehow been
completed, and that the segments grew as small as we wish, but remained always finite.
However, the next step of the proof, which elaborates on the nature of these infinitely
many segments, takes the Idealist rather beyond what is acceptable to the Empiricist.
This important next step, though, is something of an interpretive puzzle. It
begins, recall, "thus, of these extensions none can be finite, which is to say cannot be
contained a finite number of times in the unit length."38 First, that none of these
extensions can be finite can only mean that they are infinitely small, as they certainly
cannot be infinitely large. That they "cannot be contained a finite number of times in the
37
Ibid., 64.
38
This is a rather literal translation from the French, which reads, "Donc, de ces étendues aucune ne peut
être finie, c'est-à-dire ne peut être contenue un nombre fini de fois dans l'unité de longueur." Ibid., 73.
119
unit length," is a denial of the Archimedean principle: a finite number of repetitions of
one of these segments cannot equal (or, for that matter, surpass) the unit length itself.
The heart of this argument for the existence of infinitesimals, now that we have
established that these segments are our infinitesimals, lies then in the next clause,
"because the unit of length being arbitrary, every extension as small as it may be must be
organized like the unit length, and similarly contain infinitely many points of division."
This echoes the Idealist's image which always accompanies his representation of
magnitude, but does not seem a straightforward explanation for why these segments must
be infinitesimal.
A clue to what du Bois-Reymond's Idealist might mean at this stage in the
argument comes later, when the Idealist is summing up the types of quantities that exist:
infinitesimals, the unlimited in smallness, the finite, the unlimitedly large, and the
infinite. The relationship between the finite, infinite, and infinitesimal is summed up in a
sentence: "Before the parts of a finite quantity can be infinite in number, each must be
infinitely small."39 If the segments created by infinite division were finite segments, or
merely as small as one would wish (i.e. unlimited in smallness), then they would not be
the result of an infinite division; rather, they would merely echo the original unit length,
themselves being infinitely divisible.
Thus, the Idealist does not claim that the infinite divisibility of the unit length
leads to proof of the existence of infinitesimal segments; rather, it is the fact that there are
actually infinitely many division points which leads to infinitely many segments, each of
39
Ibid., 83.
120
which infinitely small. Notice, too, the key role played by the Idealist's rejection of a line
composed of points; infinitely many points may imply infinitely many segments, in that
any two points can be joined by a segment, but this assumes that a segment is merely a
connection between points, and not a necessary element of the composition of the line.
On this analysis, "points alone can never form an extension" precisely because they are
always separated by extensions, making these extensions themselves metaphysically
important in the constitution of the line, not simple connections between parts of the line
(i.e. points). And indeed, the conclusion reiterates the important role played by these
infinitely small extensions: "One thus sees that the unit extension is decomposed into an
infinity of partial extensions, of which none is finite."
Notice one last feature of this argument. If these infinitely small extensions were
extensions precisely like any other, they would themselves permit of an infinite division,
and thus infinitely many division points. As we shall see in Chapter 6, Charles Peirce
held precisely this conception of infinitesimals; he believed they were themselves
infinitely divisible. The Idealist seems to distance himself from this conclusion in several
places in this argument, but particularly by the introduction of the word "partial" in the
conclusion. It seems fair to conclude that, whatever the features of these partial
extensions, none of which is finite, we should not immediately import all of our ideas of
finite extensions, or even extensions as small as we wish, into this new concept.
5.4.2 Some Features of Infinitesimals
Briefly, I wish to overview some of the features of the Idealist's infinitesimals, so
that we may better understand the nature of these infinitely small segments and their place
121
in mathematics.
Infinitesimals are non-zero.40 This follows immediately from the assumptions we
have already seen, as if they were zero, the Idealist would be left with a line composed
merely of points. Further, the Idealist believes that zero is not a magnitude, but rather, it
represents the lack of all magnitude, whereas infinitesimal segments have magnitude.
"A finite number of infinitesimal extensions added to each other do not equal a
finite extension, but an infinitesimal extension."41 Du Bois-Reymond calls this the
"principle property of the infinitely small,"42 and it implies either that there are different
sizes of infinitesimals, or that the addition of an infinitesimal to an infinitesimal does not
lead to an increase in magnitude. Thus, assume γ and δ are infinitesimals. According to
this principle, either γ + δ = γ, or γ + δ equals a third infinitesimal. If the first is the case,
then it is also likely that γ = δ, i.e. that there is only one infinitesimal. There is some
indication that the previous interpretation holds, as in the discussion of this property, du
Bois-Reymond indicates that it seems to violate our concept of equality, and thus he
limits the concept of equality to finite quantities: "I call equal two finite extensions when
there does not exist between them any finite difference."43 However, the Idealist does not
here provide us with conclusive evidence for one interpretation over another. The focus
of this property is that finitely many infinitesimals (or finite reiterations of the same
40
Ibid., 78-79.
41
Ibid., 73.
42
Ibid., 73.
43
Ibid., 74.
122
infinitesimal) will never reach a finite magnitude, just as finitely many finite magnitudes
will never reach beyond finitude.
"Two finite quantities differing by only an infinitesimal are equal."44 This is
related to the above definition of equality, but it is also closely related to the next
property, that "a finite quantity does not change if one adds to it or subtracts from it an
infinitesimal quantity."45 This is a common property of infinitesimals; that adding or
subtracting them to or from finite quantities does not increase or lessen the finite
quantities. The properties are particularly noteworthy here as they explicitly introduce the
notion of infinitesimal quantities, whereas before the Idealist limited his discussion to
infinitesimal magnitudes or extensions.46
"The infinitely small is a mathematical quantity and has in common with the finite
the set of its properties."47 This is the most intriguing of the Idealist's properties of
infinitesimals. First, the Idealist now explicitly defines infinitesimals as mathematical
quantities, rather than geometrical extensions. This is consistent with the Idealist's view
of number as measurement; if there is a geometrical magnitude to measure, there is a
corresponding mathematical quantity which measures it. (Recall, it was the other
relationship which did not necessarily hold – a limit of a mathematical series, such as an
infinite decimal expansion, did not necessarily correspond to a limit-point on the straight
44
Ibid., 74.
45
Ibid., 75.
46
Ibid., 74.
47
Ibid., 75.
123
line). Thus, it is notable, but not astonishing, that du Bois-Reymond's Idealist introduces
infinitesimal quantities as well as infinitesimal magnitudes.
It is, on the other hand, astonishing to claim that the infinitesimal quantity "has in
common with the finite the set of its properties," especially after we have enumerated the
properties infinitesimal magnitudes have which finite magnitudes do not: that two finite
quantities differing only by an infinitesimal are equal, and that a finite quantity cannot be
changed by the addition or subtraction of an infinitesimal quantity. One property of finite
quantities is that they change each other when added to or subtracted from each other;
clearly, this is one property the infinitesimal quantity does not hold in common with the
finite quantity.
However, one can safely assume that du Bois-Reymond's idealist does not mean
that infinitesimal quantities have in common with finite quantities every single property.
The properties discussed above, in particular, are properties which arise when quantities
of different types interact – infinitesimal versus finite. Similar properties come to light
when comparing finite quantities and transfinite ones. Du Bois-Reymond stated, in the
discussion of this last property of infinitesimals, that the science of comparing different
types of quantities to each other is called the "calcul infinitaire"48 – the Infinitärcalcül
discussed in Section 5.2 above. He notes that the addition of infinitesimal quantities to
our mathematics opens the door for this comparison between types of quantities, and in
fact, the Idealist believes that transfinite quantities can be proved from the existence of
infinitesimal ones (they are simply the inverse of each other), thus, different levels of
48
Ibid., 75.
124
infinity are now open to analysis.
Given that the peculiar properties of infinitesimal quantities were in regard to how
they relate to quantities of differing types, one can assume that by this last property of
infinitesimals, the Idealist means for infinitesimals to have properties within their type
similar to the properties of finite quantities, when finite quantities are not being compared
to or added to quantities of different types. Thus, we can assume that an infinitesimal
plus an infinitesimal equals a third, distinct infinitesimal, for example, and that there are
infinitely many distinct infinitesimals which exhibit trichotomy (therefore solving our
interpretive quandary over the Idealist's second property of infinitesimals).
5.4.3 The Empiricist's Response
The Empiricist does not accept infinitesimals; they are not drawn from
experience, they are not reducible to experientially-derived principles. As the Empiricist
states, "I believe that we are not authorized to admit to or to create in our rational
mathematics things of which we neither have nor could ever have representations."49
Furthermore, he does not accept the premises of the Idealist's argument. He does not
agree that "points alone can never form an extension", nor that there are infinitely many
division points on a line. There are as many division points on a line as we need, and no
more; it makes no sense to extend the division of a segment beyond practicality. We are
told, by the Empiricist:
We could represent equal parts [of the unit length] in innumerable quantity, each
49
Ibid., 83.
125
as small as one wishes, precisely because inexactitude diminishes as we are
permitted to push the division of an extension as far as we wish. But the pursuit
of this division, when we do not halt at any size, finishes by being lost in vague
hand waving. Thought can push this division as far as we want, but this act does
not prove the infinite small, but rather, fatigues and discourages the mind which
cannot see the end of its journey, hidden in a uniformly cloudy region.50
The Empiricist thus claims that mathematics has no need of exactitude of measurement at
an infinitesimal level, and further, infinite division serves no purpose other than to fatigue
and discourage us. Each measurement can be as exact as we need it to be, the division
can be carried on until we reach that useful exactitude, and while the mind can perhaps
imagine, in some sense, infinite division, nothing practical is to be gained by positing the
result of such infinite division. Thus, there are only finite divisions to be made, and only
finitely many points on a geometrical line. "[For the Empiricist,] the set of points as
dense as one would wish always remains a finite set."51 The Empiricist thus rejects the
original premise of the Idealist's argument – the infinite division of the unit length – and
rejects the existence of infinitesimals themselves.
5.5 Continuity and a Unified Mathematics
As the task of this chapter is to find out not only the foundations of mathematics
for Paul du Bois-Reymond, but specifically du Bois-Reymond's theory of continuity, we
must end this chapter by turning our attention to answering this question. This last
section will contain two subsections. In the first, I will present the conception of
continuity entailed by the Idealist and Empiricist systems (briefly, the Empiricist has no
50
Ibid., 92.
126
conception of continuity, while the Idealist does, but that his conception must differ
markedly from the continuity of Dedekind and Cantor). In the second, I will discuss du
Bois-Reymond's neutral system of mathematics, a system acceptable to both the Idealist
and Empiricist tendencies within ourselves; and then I shall briefly discuss the status of
the mathematical continuum in this neutral system.
5.5.1 Idealist and Empiricist Continua
Though in the General Theory of Functions and in this chapter the Idealist has
typically been given the podium first, it is useful here to begin with the Empiricist
conception of mathematical continuity, due to its simplicity. In a word, it does not exist.
There is no such thing as mathematical continuity. "For the Empiricist the continuum of
numbers does not exist, either as a limit, nor as the final end of an approximation. It
absolutely fails to exist for him."52 Given the finite nature of the set of points on any line
segment discussed above, it should not be surprising to realize that the Empiricist also
does not believe that the line itself is continuous. This is quite in line with the Empiricist
worldview; continuity, like infinity, is one of those idealized concepts of which we never
have direct experience, nor can we deduce it from or reduce it to our experience.
The Idealist does have a concept of mathematical continuity, as well as a
conception of a continuous straight line. This conception of mathematical continuity is
more or less the one we are used to seeing in mathematics, that is, a continuity which is
51
Ibid., 162.
52
Ibid., 163.
127
infinite, dense, and has no gaps, though it does have some unusual features. First of all,
the Idealist's continuity can never be reached through a foundational approach; that is, the
Idealist admits that we cannot begin with empirical facts, such stones which are counted
and organized, and eventually derive a mathematical continuum. For the Idealist, "the
continuum of numbers is not the actual limit of any empiricist series of representation."53
The continuum is divorced from empirical representation, even for the Idealist, who is
used to making leaps from representation to idealization.
The second unusual feature of the Idealist's numerical continuum is that does not
have the property of Dedekind continuity. As we proved in Chapter 3, Dedekind
continuity is inconsistent with the existence of infinitesimals, and, as we saw above, the
Idealist believes infinitesimals follow with necessity from the most basic assumptions
about the continuity of a straight line. Thus, while it looks as though the Idealist is
forwarding a type of Dedekind continuity at the beginning of his proof of infinitesimals,
the fact that he goes on to use that conception to argue for the existence of infinitesimals
cues us in to the non-Archimedean, and therefore non-Dedekind nature of the continuum.
Thus, recall that du Bois-Reymond's Idealist began his argument for the existence of
infinitesimals with the sentence, "The proposition that the number of points of division of
the unit length is infinitely large produces with logical necessity the belief in the infinitely
small."54 The nature of continuity, according to Dedekind, is that wherever I may divide
a set, I always do so at a member of the set, never in between members. Wherever a line
53
Ibid., 163.
54
Ibid., 73.
128
is divided, it is divided at a point; wherever the set of real numbers is divided, it is
divided at a real number.
As this is the very property of Dedekind's real numbers which led to our proof of
the Archimedean principle, it must be that the Idealist's infinite division points of the unit
length do not directly entail this property. Thus, while there are infinitely many points of
division – points marking actual divisions of the unit segment – we must conclude that
there is at least the possibility of divisions occurring which do not occur at particular
points. Though it seems at first glance that this type of continuity is no continuity at all –
that is, that these theoretical divisions which do not occur at division points must
therefore occur at gaps in the line or in the number line – this is not necessarily so. For a
gap in the line to be proven under such a scenario, the proof would depend upon the
assumption that there was nothing on the line but points, and thus, a place which lacked a
point would not be part of the line, but the Idealist clearly believes that there is more to a
line than simply points; in fact, he believes that the line decomposes into points and
intervals.
These two properties of the Idealist's continuum – that continuity can never be
reached as a limit of an empirical series, and that continuity is non-Archimedean and
lacks Dedekind continuity – are consistent with the Empiricist's indication that continuity
is an idealized property which goes beyond direct experience. This is one conclusion
both the Empiricist and the Idealist agree upon; they only differ in their willingness to
129
accept mathematical entities so divorced from empirical reality.55
5.5.2 A Neutral Mathematics
Du Bois-Reymond believed that these two world-views about foundational
matters were so diametrically opposed that they were separated as if by a large, chasm,
and this chasm "is too profound and too vast to be able to be filled by reciprocal
concessions. The counter-proposals are absolutely irreconcilable."56 Yet, both systems
are "equally authorized to serve as the base of the science [of mathematics]."57 Thus,
compromise is impossible, and deciding once and for all in favor of one worldview over
the other is also impossible. However, du Bois-Reymond did attempt to create a system
that was objectionable to neither the Idealist nor the Empiricist. The first step to creating
this system is to follow the Empiricist in ridding mathematics of any metaphysical entity
underivable from experience. Thus, any infinitely large or infinitely small magnitude will
be treated as if it were "an idealist fiction."58 Similarly, phrases such as "precise
measurement" will be replaced with phrases such as "as exact as we wish." The logic of
this neutral system, however, will follow that of the Idealist, for restricting ourselves to
prove that certain mathematical properties only hold for sets "as large as we wish" will
55
Do note, however, that neither the Idealist nor the Empiricist has claimed that reality itself is continuous,
or that it is discontinuous. They only agree in the limits of our ability to experience continuity empirically,
just as they agree that whether reality is infinite or not, we can never experience actual infinity.
56
Ibid., 128.
57
Ibid., 129.
58
Ibid., 129.
130
not satisfy the Idealist, and any proof that holds for infinite sets, whether or not these
infinities actually exist, will also hold for the weaker sets which are only as large as we
wish them to be. Thus, concludes du Bois Reymond, "empiricist language, idealist
proofs."59
It is surprising how much of mathematics can be done with a system such as this,
which is restricted on both ends; ontologically restricted to only objects related to direct
experience, proofs restricted to only those strong enough to capture actual infinity and
perfectly exact measurement. Walking the Idealist/Empiricist tightrope is perfectly
possible when we are dealing with calculations and procedures that do not address
troublesome issues. For example, we can perform the operations of differential calculus
without ever settling the question of whether differentials are actual infinitesimals, as
Leibniz was sometimes prone to believe, or whether they are simply as small as we would
wish. However, vast areas of mathematics are closed to us on this neutral system.
Infinitesimals and transfinites are denied to us, but so are actual infinities (and thus, one
would imagine, the Axiom of Choice among some other set theoretical results), as well as
actual continua.
Notably, du Bois-Reymond's Infinitärcalcül is similarly off bounds in both the
Empiricist and the neutral system. Recall from Section 5.2 above that the Infinitärcalcül
was a system open to us once we had access to the levels of infinity derived from Idealist
principles, varying magnitudes of the infinitely large as well as varying magnitudes of the
infinitely small. I do not believe that du Bois-Reymond wished to claim his own
59
Ibid., 130.
131
mathematical system was out of bounds; but I also do not believe that du Bois-Reymond
wholeheartedly accepted this neutral system. It is surprising how much mathematics can
be accomplished without the Empiricist/Idealist divide breaking into open warfare, and
the neutral system of mathematics ensures that as far as we are able, we can accomplish
just this sort of calculation without settling the larger foundational issues.
These larger issues, however, must sometimes be confronted, and unfortunately,
as we saw above, they are not issues which can be settled; the Idealist/Empiricist divide
remains. "Since no conclusive mathematical consideration will ever decide between the
two, such questions constitute undecidable problems whose solutions will remain forever
outside the range of our mathematical abilities."60 Among these undecidable issues are
the very problems this dissertation is concerned with: whether the real numbers are
continuous and whether infinitesimal quantities exist.
60
McCarty, David Charles, "David Hilbert and Paul du Bois-Reymond: Limits and Ideals," One Hundred
Years of Russell's Paradox, Berlin: De Gruyter, 2004, 526.
132
Chapter 6
Charles Sanders Peirce
6.1 Introduction
Charles Sanders Peirce was born in Cambridge, Massachusetts, in 1839. He
studied science at Harvard, and after receiving his Bachelor of Science degree in
chemistry at 1863, went to work for the U. S. Coast and Geodesic Survey program run by
his father. Though the extent of his academic appointments was limited to five years as
an instructor at Johns Hopkins University, throughout his life he applied himself to
studying and writing on a variety of subjects, notably philosophy, logic, mathematics,
psychology, and semiotics (the study of signs and signage).
Peirce was a prolific writer. Plagued by trigeminal neuralgia, a progressive
condition causing severe pain, he was driven to write with great fervor in an attempt to
record all of his thoughts before he was no longer capable of intellectual work. Peirce
also suffered from financial instability for much of his life; for several years he was
financially dependent on the charity of friends, especially William James (1842 – 1910).1
Peirce is known for the development of the American school of Pragmatism, and
1
For more details of Peirce's life, see Robert Burch's Peirce article, "Charles Sanders Peirce," in the
Stanford Encyclopedia of Philosophy, and Joseph Brent's book, Charles Sanders Peirce: a life,
Bloomington: Indiana University Press, 1993.
133
he was heavily influenced by Aristotle, Immanuel Kant, and Georg Cantor. His work
reflects a systematic, global approach, but also pays much attention to the mathematical
and logical implications of his philosophical ideas. Continuity was, for Peirce, an idea
which linked his global approach with his fondness of mathematics; it was essentially a
mathematical concept, and yet he believed it influenced every realm of life, from
psychology to history, from philosophy to biology.
Peirce's philosophy of continuity is an interesting addition to this dissertation for
two main reasons. First, like Paul du Bois-Reymond, he believed that infinitesimals were
essential to our understanding of continuity. Second, he developed a mathematical
definition of continuity that was quite similar to the definitions given by Cantor and
Dedekind. Yet he became frustrated with his own mathematical definition, and
ultimately rejected it, in favor of a definition which more closely satisfied his intuitions.
In fact, he formed three theories in all, which I shall call "early," "middle," and "late." In
this chapter, I will first briefly introduce Peirce's theory of synechism – the theory that
continuity is central to many intellectual disciplines – since this view supplied a
framework for Peirce's life-long study of continuity. I will next explain each of Peirce's
three theories of continuity in turn, including his reasons for rejecting the first two
theories. Lastly, I will discuss the interesting relationship between Peirce's infinitesimals
and his different theories of continuity.
6.2 Synechism
Though Peirce changed his mind about many things during his long
philosophical career, a consistent feature of his philosophy was his theory of
134
synechism. He defined synechism as, "That tendency of philosophical thought
which insists upon the idea of continuity as of prime importance in philosophy
and, in particular, upon the necessity of hypotheses involving true continuity."2
For Peirce then, continuity was not just important, but was "of prime importance,"
and synechism itself was the "keystone of the arch" of intellectual theories.3
Continuity was necessary to explain space, time, and motion, but he also believed
it could explain love, evolution, psychological development, chemistry, and more.
Synechism was the path not only to philosophical truth, but also to scientific truth
in all areas.
Thus, when Peirce wrote that he intended to, "Outline a theory so
comprehensive that, for a long time to come, the entire work of human reason ...
shall appear as the filling up of its details,"4 he proposed to start by finding
"simple concepts applicable to every subject."5 Though he does not mention
continuity and synechism directly in this paragraph, and in fact mentions no
theory or concept specifically, he clearly believes that continuity is the epitome of
an idea "applicable to every subject."
Synechism thus demanded that Peirce study multiple academic disciplines, but he
relied upon the tools of mathematics and logic to define continuity. Peirce agreed with
2
As is traditional, all references to the Collected Papers of Charles Sanders Peirce (CP) will be indicated in
terms of paragraph number, rather than page number. This quote is from CP 6.169.
3
CP 8.255-257.
4
CP 1.1
135
Aristotle about the prevalence of continuity in the physical world, and he agreed with
Dedekind about the ability of mathematics alone to provide us with the tools necessary to
build a theory of continuity. His early and middle definitions of continuity are quite
mathematical in nature. Even his final definition of continuity, which breaks away
substantially from continuity as a primarily mathematical or logical idea, depends upon
mathematical procedures and examples for its explication.
6.3 Early Definition of Continuity
In 1878, Peirce defined continuity as "the passage from one form to another by
insensible degrees."6 This definition was presented in the course of Peirce's theory of the
continuity of botany – how two very similar examples of the same botanical species differ
from each other by insensible degrees, by the slightly different shape of a particular leaf
or slightly different coloration in a marking.
Dissatisfied with the imprecision of this definition, in 1893 he amended it with a
footnote, writing that this account of continuity was meant to refer to "limitless
intermediation; i.e., of a series between every two members of which there is another
member of it."7 Though by 1893 Peirce had already developed his middle theory of
continuity, he understood his earlier theory to be equating continuity with density:
between any two members of a series, another member can be found. However, the set of
5
CP 1.1
6
CP 2.646.
7
CP 2.646, footnote added in 1893.
136
rational numbers has the property of density, and the rationals are clearly non-continuous;
thus, he realized, this first definition was inadequate. In fact, in his 1893 footnotes to this
passage, Peirce suggested that to save the integrity of this early essay, the word
"continuity" be replaced throughout in this essay with "limitless intermediation." It is
clear that by the time he added these explanatory footnotes, he had rejected density as a
sufficient condition for continuity.
6.4 Middle Definition of Continuity
Though density is not a sufficient condition of continuity, Peirce retained it as a
necessary condition. In 1893, in "The Logic of Quantity," he laid out what he then saw as
the three conditions for continuity.8 The first condition was non-denumerability: if a set
is continuous, it (i) must be infinite, and (ii) cannot be in one-to-one correspondence with
the natural numbers. The second condition he dubbed 'Kanticity,' in order to give credit
to Kant as having seen it as a necessary condition of continuity. Kanticity is the property
that "between any two points upon [the line] there are points."9 This, of course, is just the
density requirement of Peirce's early definition of continuity already rejected as
inadequate in itself. Peirce still recognized it as insufficient, writing that Kanticity holds
true of a line from which a closed segment has been removed – a line with a clear gap in
8
In section 6 of "The Logic of Quantity," entitled "The Continuum." CP 4.121-124. Peirce forwarded a
similar definition of continuity in 1892 (CP 6.121-124) and again in 1903 (CP 6.166).
9
CP 4.121.
137
it, which is certainly not continuous.10 Thus, "the union of the disjoint line segments (-∞,
a) and [b, ∞), a < b, is dense and, thus, continuous according to Kanticity."11 Clearly
another condition is needed to prevent such non-continuous sets from being captured by
our definition of continuity.
Peirce's third and final condition of continuity was called "Aristotlicity." As we
noted in Chapter 2, Aristotle equated continuity with infinite divisibility. Hence it might
seem more logical to name the density condition for Aristotle, as the notions of density
and infinite divisibility are similar.12 However, Peirce is not relying on Aristotle's
explicit definition of continuity, but rather on what he sees as the spirit of Aristotelian
continuity. For Peirce, then, a set has Aristotlicity when it is the type of set "whose parts
have a common limit."13 He claimed that Aristotle seemed to have had this idea
"obscurely in mind" when defining continuity. Before we examine the details of
Aristotlicity, it seems wise to inquire whether Aristotle really had something like this in
mind, obscurely or otherwise.
In a footnote immediately following this definition of continuity, Peirce cited the
Metaphysics, 1069a5, where Aristotle distinguished among various ways in which two
things can be considered together: they can touch, they can be contiguous, or they can be
continuous. Aristotle also made this distinction in the Physics, in his argument that
10
CP 4.121.
11
Spencer Gwartney-Gibbs, " Continuous Frustration: C.S. Peirce’s Mathematical Conception of
Continuity," section 1.
12
In fact, at this time in his life, Peirce equated density with infinite divisibility.
138
continua are not composed of indivisibles. 14 We discussed this distinction above in
Chapter 2. To recount briefly: two things of the same kind are in succession when no
third thing of the same kind is between them (but something of a different kind may).
Two things that are contiguous are in succession, and their external borders touch, so that
nothing is between them, either of the same kind or of a different kind. As for continuity,
The continuous is a subdivision of the contiguous: things are called
continuous when the touching limits of each become one and the same and
are, as the word implies, contained in each other: continuity is impossible
if these extremities are two. This definition makes it plain that continuity
belongs to the things that naturally in virtue of their mutual contact form a
unity.15
This version of Aristotle's continuity requires, in addition to infinite divisibility, that all
limits or borders are shared in common.
By Aristotlicity, however, Peirce did not literally mean continuous sets must
"touch" or "share borders." Vincent Potter and Paul Shields defined Peirce's Aristotlicity
as "the requirement that a continuum contain its limit-points."16 Peirce himself later, in
1903, defined Aristotlicity as "having every point that is a limit to an infinite series of
points ... belong to the system."17 And similarly, in 1892, he stated that Aristotlicity
13
CP 4.121.
14
Aristotle, Physics, 227a10-15.
15
Ibid. 227a10-15.
16
Potter and Shields, "Peirce's Definitions of Continuity," Transactions of the Charles S. Peirce Society, v
13 (Winter 1977), 25.
17
CP 6.166; written in 1903 in the margin to his 1889 article in his personal copy of the Century
Dictionary.
139
holds "if a series of points up to a limit is included in a continuum the limit is included."18
Thus, a set A has the property of Aristotlicity if and only if the limit of every subset of A
is included in the set A itself. This seems related to Aristotle's definition of continuity in
terms of parts sharing borders; the borders here would be analogous to the limits of the
subsets. However, the analogy can only be a loose one; if we were to take it literally, the
requirement of Aristotlicity would lead us to believe that every segment on a continuous
line must share end points with every other segment, which is clearly false.
Indeed, Peirce's requirement of Aristotlicity seems more closely related to
Cantor's notion of perfect sets than it does to Aristotle's definition of continuity, In fact,
Peirce was explicit about the extent to which Cantor influenced his definition,19 and
Potter and Shields compared Peirce's Aristotlicity to Cantor's requirement that a
continuous set must be perfect.20 Recall from Chapter 4 that for a set P to be perfect, it
must be equal to each of its derived sets P(γ). Thus, a perfect system is equal to its first
derivative, and "the first derivative of a system is simply its collection of limit-points."21
Cantor's perfection implies that every limit-point of a system is contained within the
system – which is Aristotlicity – but it also implies that "every point of the system is a
limit-point of the system."22 Aristotlicity is thus half of Cantor's perfection.
With all three conditions in place, we have that a set A is continuous if and only if:
18
CP 6.122; written in 1892.
19
CP 4.121.
20
Potter and Shields, p. 23.
21
Ibid., p. 24.
140
(a) it is infinite and non-denumerable, (b) it is dense, and (c) for every subset of A which
has a limit, that limit is itself a member of A. Sets with clear gaps, such as the union of
the disjoint line segments (-∞, a) and [b, ∞), a < b, considered above, are not continuous
according to this definition: the third condition would require that a itself, as a limit of
the subset (-∞, a), be included in the larger set. If a were included, then the set would
instead be the union of (-∞, a] and [b, ∞), but since a < b, the set would no longer be
dense, as a and b are both in this set yet there is nothing between them. The set of
rational numbers is also excluded from continuity by the third condition; as we saw
Chapter 3, Dedekind proved that some sets of rational numbers have non-rational limits.
In fact, Peirce's middle definition is rather like Dedekind's definition of continuity in
several respects, and as we shall prove in the section on infinitesimals below, Peirce's
definition is logically implied by Dedekind's.
This middle definition is adequate in the sense that it excludes clearly noncontinuous sets such as the set of rational numbers and discontinuous line segments.
Peirce was satisfied with it for several years, but he ultimately rejected it in favor of a
third and final definition. The reasons for this rejection are complex, but they stem from
a seemingly simple concern about density, Peirce's second requirement in the middle
definition.
22
Ibid., p. 24.
141
6.5 Later Definition of Continuity
By the late nineteenth century,23 Peirce had come to believe that he had
misunderstood Kant's definition of continuity, and sought to rectify this mistake.
Kanticity had been named after Kant because he supposedly defined continuity as density.
According to Peirce, Kant defined continuity as density in A169 (B212) of the Critique of
Pure Reason24. However, this is not strictly true. At this point in the Critique, Kant
wrote, "The property of magnitudes by which no part of them is the smallest possible, that
is, by which no part is simple, is called their continuity."25 It is possible to understand this
property of magnitudes as density, but in his later work, Peirce believed that by this
definition Kant did not refer to density at all. Rather, he thought Kant meant that a
continuum was something such that "All of [its] parts have parts of the same kind."26 For
Peirce, this no longer implied that between any two points there is a third point; rather,
"Kant's real definition implies that a continuous line contains no points.”27
Let us try to unpack the reasoning behind this audacious claim. It is true that a
point is not divisible into parts of the same kind; in fact a point is not divisible at all. If
23
The actual timing of Peirce's change of heart is somewhat mysterious. He forwarded something very
much like his third definition of continuity as early as 1895, but in 1903, he repeated his middle definition
of continuity. At any rate, it seems clear that by 1905, he had completely forsaken his middle definition in
favor of his later one.
24
Peirce, "Law of Mind," p. 320 f. 10.
25
Kant, A 169 (B211).
26
CP 6.168. This was written in 1903.
27
CP 6.168.
142
one believes that a continuum must be infinitely divisible in the sense that no part of it is
indivisible, then a point cannot be an essential part of the continuum. In fact, wherever a
point appears, according to this third definition of continuity, that point "interrupts the
continuity."28 Points conceived as indivisible atoms were now considered by Peirce to be
antithetical to continuity; thus, a continuum could no longer be composed of them, or
even contain them. Rather, as we shall shortly see, his late definition required that any
division of the continuum yielded parts that resemble precisely every other part, no matter
how big or small.29
Thus, Peirce abandoned his middle definition of the continuum in favor of one
that reflected his new-found understanding of Kant's definition. To fully understand
Peirce's new conception of continuity, a brief overview of his transfinite and
denumerability theories is in order. Even as early as 1893, Peirce was considering the
theory of non-denumerability and its implications. First, he defined three classes of
magnitude: those that are enumerable, i.e. finite; those that are "dinumerable," i.e.
countable and infinite; and those that are non-denumerable, i.e. infinite and uncountable.
He then wrote:
[These] three classes of multitude seem to form a closed system. Still,
nothing in those definitions prevents there being many grades of
multiplicity in the third class. I leave the question open, while inclining to
the belief that there are such grades.30
28
CP 6.168.
29
Notice that this has similarities to du Bois-Reymond's conception of continuity, as described in our
previous chapter.
30
CP 4.121.
143
He later proved that there were in fact infinitely many such grades, and this discovery
caused him trouble with his middle definition of continuity. In particular, he was
troubled by the idea that the set of real numbers was continuous, according to his middle
definition, even though there are infinitely many magnitudes greater than this set. By this
point in his life, Peirce firmly believed that a continuum must contain all possible points,
therefore, there could not possibly be a collection larger than one which is continuous.
Since this is such an important argument in the development of Peirce's
continuity, it is worth a close look. Peirce frequently referred to the set of real numbers
and a geometrical straight line as though they were interchangeable; for the sake of
clarity, let us focus on the number line in our argument sketch.
1.
A continuous set must contain all possible parts; as a continuous line must
contain all possible points, so too a continuous set of numbers must
contain every possible number. [assumption]
2.
The real numbers form a continuous set. [derived from Peirce's middle
definition of continuity]
3.
Therefore, the real numbers must contain every possible number. [from 1
and 2]
4.
However, the set of real numbers is not the largest possible set. [Peirce
and Cantor both proved that there are magnitudes of infinity that are
higher than the magnitude of the real numbers.]
5.
Therefore, if there is a set larger than the set of real numbers, there must be
numbers that are missing. [from 4]
Steps 5 and 3 contradict each other. If the argument were valid, this contradiction
would lead us to reject one of the two assumptions: either a continuous set of numbers
does not have to contain every possible number, or the real numbers do not form a
continuous set. Peirce rejected the second assumption, jettisoning the continuity of the
144
reals in favor of the view that continuous sets cannot lack anything. Thus, Peirce
believed, continuity is not found in the set of real numbers, but elsewhere:
I still think ... that there is room on a line for a collection of points of any
multitude whatsoever, and not merely for a multitude equal to that of the
different irrational values, which is, except one, the smallest of all infinite
multitudes, while there is a denumerable multitude of distinctly greater
multitudes, as is now, on all hands, admitted.31
Because Peirce believed that there are denumerably many magnitudes, and the real
numbers are among the smallest of transfinite magnitudes, he rejected the belief that the
real numbers formed a continuum. In fact, at this point he began referring to the set of
reals as a "pseudo-continuum."32 He believed that the most important feature of a
continuum was that it lacked gaps, and as Joseph Dauben succinctly phrased it, "Peirce
argued that if a continuum did not contain all the points that it possibly could, then there
would be gaps or discontinuities present."33
However, this argument as sketched (and as argued by Peirce) is not valid. First
of all, when we restrict ourselves to the number line, as we have in our proof sketch
above, rather than shifting from the number line to the geometrical straight line and back
again as Peirce did, the argument looks less convincing. No one, not Dedekind, not
Cantor, not du Bois-Reymond, argued that in order for a set of numbers to be continuous
it must contain all possible numbers. Imaginary numbers are not included in the real
31
Peirce, The New Elements of Mathematics by Charles S. Peirce, The Hague: Mouton Publishers, 1976,
(NE) 880, footnote 2.
32
Ibid, 880.
33
Dauben, "C.S. Peirce's Philosophy of Infinite Sets," p 129.
145
number continuum, for one. Also, the set of real numbers between zero and one is
usually taken to be a continuous set, and it does not contain all numbers; it does not
contain the number two, for example, nor the square root of two, nor negative five. Yet
the absence of negative five does not make this set of real numbers any more or less
continuous.
Peirce's argument begins by assuming that a geometrical straight line must contain
all points, and ends by concluding that the set of real numbers does not form a continuum
without showing an essential link between the two. Yet "containing all possible points"
does not seem to be a necessary requirement for the continuity of a line. A single line
never contains all possible points, e.g. it only contains one of the points in a straight line
perpendicular to it, and it does not contain any of the points of a line parallel to it. The
requirement that a continuous set be absolutely continuous in the sense of not lacking
anything is absurd for any set smaller than the universe itself. Furthermore, if Peirce is to
be believed, the complete continuity he seeks cannot have any points at all, for a point
represents a break in the continuity. How can a straight line simultaneously contain every
single point and no points at all?
Whatever the problems with this argument, it seems clear that the collection of
real numbers appeared to Peirce to be too small to be continuous, once he discovered that
there were magnitudes greater than the set of real numbers. To continue with Peirce’s
later definition, it is necessary for us to momentarily accept this argument, and let him
have his conclusion: assume, for the moment, that the real numbers are too small a
multitude to be continuous, because they do not form the largest possible multitude. If
we were able to determine the next greatest multitude that too would be too small to bear
146
the weight of continuity, as it also would not be the greatest possible multitude. So it
would go for every specific multitude possible.
Peirce concluded from this that continuity cannot be a multitude. Furthermore, he
argued, continuity cannot be composed of numbers, or even points. For if there was a set
made up of numbers or points, it would be a multitude and have a specific magnitude, and
there would be at least one magnitude greater than it. Therefore, "we must either hold
that there are not as many points upon a line as there might be, or else we must say that
points are in some sense fictions which are freely made up when and where they are
wanted."34 Peirce chose the latter option. He also argued that continuous lines could not
be made up of points because there will always be gaps between the points. Thus, "there
is no reason why the points of one line might not slip through between those of the other.
The very word continuity implies that the instants of time or the points of a line are
everywhere welded together."35 Dedekind in particular, Peirce wrote, presented a
continuum that was not actually continuous, and was deeply flawed by its insistence that
continua could be composed of individual points or numbers. At this point in his life
Peirce believed that "Kant's ideas about the continuum, and Dedekind's further
elaboration thereof, are faulty [...] because they use a foundationalist conception of a line
as consisting of points."36
34
NE, v 3, pp 58-59.
35
NE, v 3, pp 60-61.
36
Herron, Timothy, "C. S. Peirce's Theories of Infinitesimals," Transactions of the Charles S. Peirce
Society, v 23 n 3 (Summer 1997) 604.
147
This, then, was Peirce's solution and final definition of continuity – an intuitive,
“true” continuum must consist of points so close to each other that they are literally
welded together.37 There would have to be so many elements in this continuum that it is
larger than any given multitude – and thus would be too large itself to be a multitude.
Peirce called this a supermultitudinous collection, but do not be fooled by the inclusion of
the word "collection," for this continuum is no longer a collection of discrete elements:
A supermultitudinous collection, then, is no longer discrete; but it is
continuous. As such the term "multitude" ceases to be applicable to it.38
Supermultitudinous collections, Peirce wrote, are created by starting with a nondenumerable multitude, such as the real numbers. Then, between any two numbers, all of
the real numbers between zero and one (the collection of which is itself a nondenumerable multitude) are inserted. If one continues to insert non-denumerable
multitudes between all the numbers, Peirce claimed, the numbers themselves will
eventually lose their distinction and become a supermultitudinous collection:
I am careful not to call supermultitudinous collections multitudes.
Multitudes imply an independence of the individuals of one another which is
not found in supermultitudinous sets. Here the elements are cemented
together, they become indistinct.39
37
Peirce was not the first person to conceive of continuity in this manner. Galileo (1564-1642), who was
himself troubled by the inability of himself and his peers to mathematically define continuity, believed that
continuous magnitudes were composed of indivisibles, but that "the aggregation of these [indivisibles] is
not one resembling a very fine powder but rather a sort of merging of parts into unity, as in the case of
fluids." Boyer, The History of the Calculus, pg. 116.
38
NE, v 3, 87.
39
Ibid., v 3, 87-89.
148
Numbers would "cement together," as would points. Any distinct, individual point would
disrupt this continuity. As Hausman wrote: "A point marks a break in a continuum, and a
collection of points is as much a collection of gaps as a collection of units."40
Of course, if we actually inserted the set of real numbers between one and zero
between any two numbers on the line, the result would be equinumerous to the set of real
numbers that we began with, as that is what it means to say the real numbers are dense –
there are infinitely, and in fact uncountably, many numbers between any two numbers.
Adding uncountably more would not increase the magnitude of the set of real numbers,
nor would it force the numbers to collapse upon each other. There would simply still be
uncountably many.
Also disturbing is Peirce’s tendency to wish to compose this “true” continuum
from points (or numbers) added to each other until they weld together, and
simultaneously to claim the line has no points (or numbers) at all. Furthermore, while
accusing Dedekind of having a compositional view of continuity, he composed his own
continuity from points. The best we can do for Peirce at this point is to assume that this
“true” continuum is not formed from actually melding points, but that rather, the
compositional story he has told must be more of tool to help us think of what a line
without points would be like, in spite of Peirce’s own insistence that this procedure is a
creation of some sort.
40
Hausman, Carl, "Infinitesimals as Origins of Evolution: Comments Prompted by Timothy Herron and
Hilary Putnam on Peirce's Synechism and Infinitesimals," Transactions of the Charles S. Peirce Society, v
34 n 3 (Summer 1998), 630.
149
Even if we save Peirce from the hypocrisy of blaming Dedekind for having a
compositional continuum while at the same time composing his own continuum out of
points, this third definition of continuity has further problems, problems which Peirce
himself could see. We shall discuss some of these difficulties in the next section.
6.6 Advantages and Disadvantages of Peirce’s Late Continuity
Peirce believed that the greatest advantage of this third definition of continuity was
that it provided a continuum that lived up to his intuitive notions of continuity, i.e.
absolute smoothness. When one divides such continuity, every part resembles the whole,
and the whole resembles every part. Peirce disliked Dedekind's cuts because they violate
this latter intuitive idea about continuity. As Herron wrote, "Peirce is fond of producing
gaps in the continuum as one way to intuitively convince the reader that Dedekind's
definition of a continuous line is not sufficient to define a true continuum."41 Peirce also
thought the asymmetry of Dedekind's real numbers told against their continuity. When
one cuts the rationals at a place where no rational number creates the cut, one creates an
irrational number, and then adds it to one side or the other to complete the cut. This
means that every Dedekind cut is an asymmetry – one side has an end point while the
other does not.42
Peirce was determined to find a definition of continuity that satisfied his intuitions,
because intuitive continuity is important to the theory of synechism. Recall that
41
Herron, 606-607.
42
NE, v. 3, 95.
150
synechism is the theory that continuity can be found everywhere, in every physical object,
in every intellectual theory. The sort of continuity Peirce saw in the history of the Roman
Empire was not a continuity that could easily fit into a mathematical mold. He believed
firmly that continuity was smoothness that went beyond conceptions of number and point,
and he eventually created, with this final definition of continuity, a theory that lived up to
his intuitions.
The cost of retaining his intuitions was great. The main disadvantage of Peirce's
intuitive continuum is that one cannot use it in mathematics. In essence, by rejecting a
Cantorian continuum in favor of a philosophical model that satisfied his intuitions, Peirce
rejected mathematical continuity altogether. This new continuum, existing only in
potentiality, with no numbers and only potential points, is difficult to use in calculations.
Supermultitudinous collections themselves, assuming they could be consistently defined,
go beyond what Peirce thought was useful for mathematics. In speaking of the second
abnumeral multitude, which, if one assumes the Continuum Hypothesis to be true (as
Peirce does), would be the power set of the real numbers, Peirce claimed:
Mathematics affords no example of such multitude. Mathematics has no
occasion to consider multitudes as great as this.43
This meant, for Peirce, that continua such as space, time, and the straight line, were larger
and more complex than any system of numbers could reach. No matter how large, no set
of numbers could be used to measure space, time, lines, geometrical figures, or, one
supposes, objects that exist in space or time:
43
NE, v. 3, 85.
151
When the scale of numbers, rational and irrational, is applied to a line, the
numbers are insufficient for exactitude.44
Peirce then described points as the "hazy outlined part of the line whereon is placed a
single number," not precise determinations of exact measurement.45 However, if
measurement fails, so too does every part of mathematics where numbers and continua
supposedly mix, such as calculus, topology, and all of geometry.
Of course, Peirce's ultimate goal was not to create a mathematical continuity. His
goal was to explain the mysterious force that he saw linking all branches of science and
physical experience of the world. As Dauben wrote,
Peirce's interests had never been inclined towards analysis. From the very
beginning he had been inspired by the purely logical implications of the
syllogism of transposed quantity, and the logic of relations. Thus, unlike
Cantor, he was not concerned to develop the arithmetic properties of his
ideas [...]. He was interested in illuminating a deep philosophical problem
of long standing, namely that of the continuum, and he felt that
conceptually he had found an approach to the subject that was the most
satisfying of all.46
Thus, in some sense, comparing Peirce's final definition of continuity to the mathematical
continuum of the other three figures studied in this dissertation is like comparing apples
and oranges. Peirce's answer to the question, "How can a mathematical continuum match
up to geometry and to the real world" is, quite clearly, "It can't." However, he did provide
us with a mathematical continuum in his middle definition of continuity, and this
definition, while not representing his fully developed philosophical opinion on the matter,
44
NE, v. 3. p 127.
45
NE, v. 3, 127.
46
Dauben on Peirce, 131.
152
can be compared to those of Cantor, Dedekind, and du Bois-Reymond. In fact, since
Peirce himself did not believe mathematics could be done with his “true” continuum, it is
possible that he himself would have agreed to the use of his middle definition of
continuity for the purposes of doing mathematics. For, unlike Dauben, I do not believe
Peirce was unconcerned with mathematics. Even when considering his philosophical
continuum, late in his career and in his life, Peirce used the tools of mathematics and
logic to develop his theory of continuity, and considered carefully whether we could
create a continuum based on a system of numbers.47
Thus, in the last part of this chapter, it is essential to consider the theory of
infinitesimals that corresponds with Peirce's middle, mathematical definition of
continuity, although the theory of infinitesimals which corresponds with his final
definition of continuity will also be considered briefly.
6.7 Peirce's Infinitesimals and his Mathematical Continuity
Peirce stated several times that the theory of infinitesimals was preferable to the
"cumbrous" theory of limits.48 In 1892, he gave a definition of infinitesimals designed to
47
Recall, his answer was that we can indeed begin with a system of reals, and by adding non-denumerably
many reals over and over again, eventually reach a continuum – but by the time we have continuity the
numbers themselves have welded together and can no longer be distinguished one from another. Even this
non-mathematical continuum is mathematical in its origins.
48
For example, he wrote, "Men are afraid of infinitesimals, and resort to the cumbrous method of limits.
This timidity is a psychological phenomenon which history explains. But I will not occupy space with that
here." CP 4.151. Also see the footnote to CP 4.118: "The doctrine of limits should be understood to rest
153
demonstrate their superiority to limits. This definition appeared in "The Law of Mind,"49
along with a discussion of his mathematical continuity. At the time, he still believed that
Kant's view "confounds [continuity] with infinite divisibility"50 and at that time he knew
that infinite divisibility (i.e. density) was not sufficient for continuity. Peirce also
believed at this time that real numbers and points on a line were more or less
interchangeable, and substituted one for the other quite freely. It was during the
discussion of the continuity of real numbers in particular that he gave his definition of
infinitesimals.
Every number whose expression in decimals requires but a finite number
of places of decimals is commensurable51. Therefore, incommensurable
numbers suppose an infinitieth place of decimals. The word infinitesimal
is simply the Latin form of infinitieth; that is, it is an ordinal formed from
infinitum, as centesimal from centum. Thus, continuity supposes
infinitesimal quantities. There is nothing contradictory about the idea of
such quantities. In adding and multiplying them the continuity must not be
broken up, and consequently they are precisely like any other quantities,
except that neither the syllogism of transposed quantity nor the Fermatian
inference applies to them.52
This is a remarkable claim, one worth dissecting carefully. Starting at the end of the
argument, Peirce claimed that neither the "Fermatian inference" nor the "syllogism of
upon the general principle that every proposition must be interpreted as referring to a possible experience,"
which was a mathematical position Peirce found inferior to the simple introduction of infinitesimals.
49
The Essential Peirce, 312-333.
50
Ibid. 320.
51
I.e. is rational. Remember, the Greeks first discovered irrational quantities by discovering geometric
lengths that were incommensurable with a unit length.
52
Ibid., 322.
154
transposed quantity" apply to infinitesimals. The Fermatian inference refers to complete
mathematical induction, according to Stephen Levy;53 thus one cannot argue by
mathematical induction when one is dealing with a system that includes infinitesimal
quantities. The syllogism of transposed quantity was a favorite of Peirce's and he
discussed it in many different places, including this 1892 article in the Monist, where he
explained it in terms of Frenchmen:
Every young Frenchman boasts of having seduced some Frenchwoman.
Now, as a woman can only be seduced once, and there are no more
Frenchwomen than Frenchmen, it follows, if these boasts are true, that no
French women escape seduction. 54
For this inference to hold it is not necessary to have two distinct classes (i.e. Frenchmen
and Frenchwomen); elsewhere his inference was stated in terms of Texans killing Texans,
or Hottentots killing Hottentots.55 However, no matter how many classes are involved,
the inference only holds if the set or sets being referred to are finite. If the sets are
infinite, the conclusion does not follow, and this fact is precisely what made the syllogism
so interesting to Peirce. Thus, if the inference of transposed quantity fails to apply to
infinitesimals, it seems that there are infinitely many of them.
To return to Peirce's original argument, it is difficult to comprehend what Peirce
53
See Stephen Levy, "Charles S. Peirce's Theory of Infinitesimals," International Philosophical Quarterly,
v 31 (1991) 131.
54
Peirce, The Essential Peirce, 316.
55
See NE v. 3, pg 772, for the Hottentot formulation, and CP 3.288 for the Texan one. We might wonder
why Texans and Hottentots are so violent; however, we might also wonder if the French would agree that
each Frenchwoman can only be seduced once.
155
might mean by the infinitieth place. Fortunately, he provided us with an example. It is
well known that the infinite repeating decimal expansion 0.999... is equivalent to the
number 1.56 Peirce disagrees with this equivalence. He argues that they do indeed differ,
but only at the infinitieth place. Thus, 0.999... is infinitesimally smaller than 1.57 One
might imagine that the expression 1 - 0.999... would be equal to 0.000...1, with an infinite
number of zeros replacing the ellipses, and thus that 0.000...1 represents a mathematical
infinitesimal. One also assumes that 0.333... would differ from 1/3 by an infinitesimal
amount, though Peirce did not address this question directly.
Thus his infinitesimal is similar to the mathematical differential. In calculus, a
differential is related to linear approximations. Consider a point a on the graph of a
known function. Linear approximations (using the derivative of the known point and the
increment along the x-axis between the known point and the unknown point) can help us
approximate the secant line, connecting the known point a with an unknown point x. Of
course, the closer a is to x, the closer our approximation can become. If the two points
are the same, the calculation falls apart, and growing distance makes it unreliable; the
ideal distance is an infinitesimal one.
Peirce thus seems to be claiming that infinitesimals exist as quantities defined by
56
Though it hardly seems necessary, let me briefly prove this. One divided by three is 1/3, which is
precisely equivalent to 0.333... . If we add 1/3 + 1/3 + 1/3 we get 1; if we add 0.333... + 0.333... + 0.333...
we get 0.999... . Since 1/3 and 0.333... are equivalent, so too must 0.999... be equivalent to 1.
57
See Levy, 130; also see NE, v. 3, 597: "although the difference, being infinitesimal, is less than any
number can express[,] the difference exists all the same, and sometimes takes a quite easily intelligible
form."
156
such a differential. Leibniz himself conceived of a differential as a mathematical entity
related to the infinitesimal. The main problem with Peirce's infinitesimal is that he
introduced it too soon; it was defined by Peirce in 1892, when he still believed his middle
definition of continuity was the correct one. However, this definition logically implies
the Archimedean principle, which, as we saw in Chapter 3, is mathematically inconsistent
with the existence of infinitesimals.
The implication is easy to prove: Peirce's mathematical continuity implies
Dedekind's continuity, and we proved in Chapter 3 that Dedekind's continuity implies the
Archimedean principle.
Assume: There is a set P which is non-denumerably large, has Aristotlicity
and Kanticity. Assume for reductio that P is not Dedekind-continuous; i.e.
there is a cut in P for which there is no member of P at which the cut takes
place.
Proof: Since set P is not Dedekind-continuous, there is a division in P
such that every member of subset A is less than every member of subset B,
and the union of A and B equals the set P. Furthermore, this set does not
occur at a point, which means that there is no greatest member of A nor
least member of B. Consider for a moment the subset A. This set clearly
approaches the cut, but does not go beyond it; therefore, A has a limit.
However, because the cut does not occur at a specific member of P, the
limit itself is not a member of P. This violates Aristotlicity, and thus, we
have our contradiction.
Thus, either Peirce's infinitesimals are inconsistent with continuity as he understood it in
this period, or they are consistent with the Archimedean principle. Yet it is simple to
show that Peirce's infinitesimals violate the Archimedean principle. In Chapter 3, we
used Waismann's definition of the Archimedean principle:
If a and b are any two positive numbers of [a] system and a < b, then it
should be possible to add a often enough that the sum a + a + ... + a
157
eventually surpasses b. Briefly stated, there should always exist a natural
number n such that na >b.58
Consider our infinitesimal 0.000...1 as a and 1 as b. Is it possible to multiply 0.000...1 by
a number large enough that it surpasses 1? The answer is yes, but only if that number is
itself infinitely large. However, the Archimedean principle traditionally requires n to be
finite; it requires a finite addition a + a + ... + a, not an infinite one. Therefore, Peirce's
infinitesimals are inconsistent with his middle definition of continuity. However, this
inconsistency only holds if one insists upon viewing infinitesimals as numbers; these
infinitesimals can still be a part of mathematics in the same way that differentials can be
used in calculations without being themselves members of the set of real numbers.
In 1900, Peirce wrote a letter to the editor of Science, defending his position that
differentials could be considered to be true infinitesimals.59 Strangely, rather than defend
this claim, he did not discuss infinitesimals at all, but rather ended up introducing his
third and final definition of continuity: a continuum which does not contain and is not
constituted by points. Perhaps Peirce believed that the introduction of this later definition
was in itself a proof of the existence of infinitesimals. By presenting a continuum that
has no points at all, he again asserted his belief that infinitesimals are an integral feature
of continuity. However, any infinitesimals which are a part of Peirce's later conception of
continuity must be very different from the infinitesimals defined above as differentials,
and Peirce did not answer the charge that infinitesimals resembling differentials were
inconsistent with his middle definition.
58
Waismann, 209.
158
It is possible that Peirce himself would answer my charge of inconsistency by
explaining that the failure is not in infinitesimals themselves, but in his middle definition
of continuity, which he completely abandoned in his later career. After all, in his final
theory continuity cannot be Archimedean in nature, as the Archimedean principle requires
numbers or points, and his final continuum has neither. Therefore there is no
contradiction between this continuity and the existence of infinitesimals. However,
abandoning his middle definition and siding with his later definition would leave us in the
position we were in at the beginning of this section; since we cannot calculate with this
unwieldy, largely potential continuum, Peirce has ultimately not given us a theory of
infinitesimals which can be used in mathematics.
6.8 Peirce's Infinitesimals and his Final Definition of Continuity
All that remains in this chapter is to describe how infinitesimals themselves
underwent changes when Peirce's continuity changed, and to outline the role they play in
his final theory of continuity, even if that role is mathematically useless. Remember that,
according to his third definition of continuity, traditional non-extensional points are
impossible, as they interrupt continuity by introducing a part of the continuum that does
not mirror every other part. The discontinuity of a point comes from the fact that it does
not share the infinite divisibility60 of the rest of the continuum. However, infinitesimals,
59
CP 3.563-570.
60
Though early in Peirce's career he conflates infinite divisibility and density, at this point, infinite
divisibility means just that – the ability to literally divide the continuum or any part of the continuum
159
for Peirce, are infinitely divisible. As John Bell wrote:
Now the "coherence" of a continuum entails that each of its (connected)
parts is also a continuum, and, accordingly, divisible.61
Of course, when an infinitesimal is divided, the result is another infinitesimal. Peirce
clearly thought there were different magnitudes of infinitesimals; how many magnitudes,
however, is a mystery.
Thus, numbers and points cannot be used to measure a continuum, as the very
definition of this final continuum prohibits any of its parts from being indivisible.
However, infinitesimal-points are not forbidden, as they themselves are infinitely
divisible, and thus, they can be used to estimate measurements on this type of continuum,
although, as we mentioned above, "it [is] intrinsically doubtful precisely where each
number is placed."62 We can, however, identify the near neighborhood of the number as
an infinitesimal-point, and thus can approximate exactitude. "Thus a point is the hazily
outlined part of the line whereon is placed a single number."63 This gives us a clue of
how we might do something resembling mathematical calculation and measurement using
Peirce's unwieldy third continuum: by substituting infinitesimals for points or numbers.
As intriguing as this pseudo-calculation might have been, Peirce did not develop it
further. According to Dauben, "Peirce did not undertake a careful arithmetic
infinitely many times. As density implies that between any two points there is another point, density clearly
does not apply to a continuum without any points.
61
Bell, "Continuity and Infinitesimals."
62
NE, v. 3, 127.
63
Ibid., 127.
160
investigation of the properties of his infinitesimals, nor did he undertake any investigation
of non-Archimedean systems in general."64 And, further, "Ultimately Peirce's
infinitesimals remained vague rather than rigorously defined mathematical entities. He
never suggested how they might be useful in analysis."65 Like Paul du Bois-Reymond,
Charles Peirce believed in the existence and usefulness of infinitesimals, but
unfortunately he did not provide us with a coherent and consistent calculus which
included them.
Thus, Peirce's middle and late definitions of continuity have distinctly different
characters. His middle definition is internally consistent and mathematically useful, but
disallows the infinitesimals which he wished to use in place of limit theory. His final
definition is consistent with the existence of infinitesimals, and consistent with his
intuitions of continuity as an idealized smoothness, but it is not mathematically useful.
Perhaps the most important lesson for us in all this is that Peirce's intuitions led him to be
severely dissatisfied with Cantor's and Dedekind's theories of the continuum. Paul du
Bois-Reymond expressed a similar dissatisfaction. Peirce, however, did not provide us
with a system capable of satisfying both his intuitions and the needs of mathematics. In
Chapter 8, I shall attempt to characterize the root of this dissatisfaction.
64
Dauben on Peirce, 131.
65
Ibid. 131.
161
Chapter 7
Infinitesimal Interlude
This chapter will not be as short as the title seems to suggest; however, it will not
be lengthy. Its purpose is limited: to defend the mathematical usefulness of infinitesimal
quantities. As we have seen several times in this dissertation, many have claimed that
even if infinitesimals were consistently definable, given their non-additive nature (an
infinitesimal plus a finite number equals that same finite number), such entities are
mathematically useless. We saw in Chapter 4 how Cantor, in the midst of making this
charge against infinitesimals, also preached patience, attempting to persuade people to
refrain from judging a new mathematical concept as illegitimate simply because it does
not seem to have an immediate use. I would like to go further than Cantor did, and
demonstrate that some mathematical concepts are best understood with the use of
infinitesimal quantities. I will not prove their existence, their status as numbers, or even
the consistency of their definition; I simply argue that if we can consistently define them,
there are mathematical concepts to which they apply quite well.
In this chapter, I wish to present two examples of cases where the use of
infinitesimals preserves a meaningful difference that limit theory overlooks, one example
geometrical, and one from probability theory. Next, I wish to briefly indicate how
Abraham Robinson creates the conservative extension of the real numbers which yields a
non-standard system of numbers including both infinitesimals and transfinites, as some
162
indication of what such a system would look like. Finally, I end with a sketch of a proof
by Euler in which he uses infinitesimals to great effect.
Our first example is drawn from geometry.1 Imagine a circle with a line drawn
tangent to it, like so:
By definition of "tangent," the line touches the circle at precisely one point. Now imagine
that we wished to measure the angle formed by the tangent and the arc of the circle. We
can approach a measurement by fitting a rectilinear angle around our curvaceous one, like
so:
r
1
This example appears in Waismann, Friedrich, Introduction to Mathematical Thinking, 221-222.
163
Let us call the rectilinear angle r, as above. It is clearly much bigger than the curving
angle, but were we to make it smaller, the curved angle would be still smaller:
And so on. In fact, no matter how small our r angle gets, the angle of the curve will
always be smaller. Yet the curved angle is clearly not zero, as that would imply that it
was identical with the line itself, and the arc of the circle clearly does make an angle
distinct from the line. It is simply an angle which is smaller than any finite positive
rectilinear angle, no matter how small – and thus, the angle is infinitesimally small.2
Other circular and circular-to-linear angles can be constructed whose measurements are
infinitesimal quantities, and in fact, the study of circular angles possibly depends on
infinitesimals.
2
Waismann not only used this proof to demonstrate what an infinitesimal might look like, but also to define
an "actual infinitesimal quantity" η, in the following manner: η is defined "as the angle which the circle
of radius 1 forms with its own tangent" (ibid, 222) Despite the geometrical argument, the definition of an
"actual-infinitesimal," and the insistence that one could construct a geometry that lacks the Archimedean
Principle, Waismann did not believe there was a future for infinitesimals. In fact, he suggested that dealing
with infinitesimal systems "is nothing but idle play" (ibid).
164
Of course, we can analyze the angle in terms of limit theory rather than using our
visual intuition: as the straight angle decreases, we see if it is approaching a limit, which
it is in this case. The limit is zero, and thus limit theory tells us that the curved angle is
also zero. Yet we can see that this is false; we can see that the angle is greater than zero,
that it differs from the line. If one wishes to numerically express the difference between a
straight line and a curved angle, an infinitesimal is a helpful means of doing so.
We draw our second example from probability theory.3 Imagine someone tossing
a quarter. After one toss, the odds that the quarter will come up heads are
tosses, the odds of the coin coming up heads both times is
1
. After two
2
1
. We can generalize this
4
progression: for n tosses, the odds that every single toss will be heads is
1
. Imagine,
2n
that a coin could be tossed infinitely many times – let's say that God decides to while
away some time with a quarter. What are the odds that God would turn up heads every
single time -- that is, infinitely many times out of infinitely many tosses? Following the
formula above the odds would be
1
-- one over two to infinity. As two to infinity
2∞
results in an infinite number, the odds are one over an infinite number – which is an
3
This example is taken from Charles Dodgson's infinitesimal theory. See F. F. Abeles, "The enigma of the
infinitesimal: toward Charles L Dodgson's theory of infinitesimals," Mod. Log. 8, no. 3-4 (2000-01) 7-19.
165
infinitesimal.4
Some would say that the chances of infinitely many coin tosses turning up heads
every time is not an infinitesimal, but rather it is an impossibility, because in infinity,
every possibility is played out, and clearly the quarter turning up tails is a possibility
which must be actualized at some point. This is a mistaken view of infinity. Though
some philosophers, such as Hegel and Spinoza, have argued that ultimate infinity does
indeed have this property of every possibility, of necessity, being actualized, in limited,
normal infinities it does not hold. For example, the positive integers are infinite, and yet
they do not include every possibility. They exclude fractions, for one. The integers are
not an all-inclusive sort of infinity, and neither is an infinite coin toss. Even if we limit
the infinite coin toss to relevant possibilities, excluding the possibility that it turns up
pink elephants sometimes, it is impossible for all relevant possibilities to be played out.
For example, both the scenario where the coin turns up tails every time but one and the
scenario where the coin turns up heads every time but one are possibilities; it is logically
impossible for them both to occur. Even in an infinite series, not every possibility can be
expressed.
If we determine the odds using limit theory rather than calculating the odds as
as Charles Dodgson does, our equation becomes lim
n→∞
4
1
2∞
1
. As n grows, the fraction
2n
Charles Dodgson believed that this demonstration proves the existence of infinitesimals. Perhaps his
argument is not as strong as he believed; but at the least, it shows one place where infinitesimal quantities
could be usefully applied: in calculating infinitary probabilities.
166
shrinks, and approaches the limit 0. Thus, limit theory calculates the odds of an infinite
coin toss turning up heads every single time is zero, but since in probability theory, zero
means "absolute impossibility," one should be careful before applying the limit. Though
it is highly unlikely that a coin thus tossed would turn up heads every single time, there is
no logical impossibility to the event. If in probability theory we wish to distinguish
between something being highly unlikely and something being impossible, preserving the
formulation of the odds as
1
and admitting infinitesimals into probability theory is one
2∞
way to do so. Thus, one possible use for infinitesimals is to mathematically preserve the
distinction between an event being highly or even infinitely unlikely, and that event being
impossible.
The most developed non-Archimedean system is that of Abraham Robinson's
nonstandard analysis, and I wish here to present an outline of how he creates the set of
numbers he uses in his analysis. I will follow Robinson's own exposition.5 He begins
with standard first order logic, and proves the theorem
Let K be a set of sentences in a language Λ [where Λ is a higher-order
language than first order logic, and thus our relationship symbol can range
over sets as well as individuals]. Suppose that every finite subset of K is
consistent. Then K is consistent.6
Robinson uses this theorem to define the enlargement of K in such a way that if K
is consistent then its enlargement is consistent as well. Furthermore, such a conservative
extension retains many of the properties of the original set; thus, the nonstandard
5
Robinson, Non-Standard Analysis, Princeton: Princeton University Press, 1996.
6
Ibid., 27.
167
enlargement of the natural numbers *N has many properties in common with the natural
numbers themselves:
(i) Every mathematical notion which is meaningful for the system of
natural numbers is meaningful also for *N. In particular, addition,
multiplication, and order are definied for *N.
(ii) Every mathematical statement which is meaningful and true for the
system of natural numbers is meaningful and true also for *N: provided
that we interpret any reference to entities of any given type, e.g., sets, or
relations, or functions, in *N not in terms of the totality of entities of that
type, but in terms of a certain subset.7
Thus Robinson is able to create a larger mathematical system while preserving the
functions necessary to mathematics. It is unfortunately beyond the scope of this
dissertation to investigate the philosophical or mathematical definition of continuity
implied by Robinson's non-standard analysis, however, one imagines that a first step
would be to show that the conservative extension of the real numbers necessary for
calculus does not imply Dedekind continuity, i.e. that not every cut in Robinson's nonstandard system is determined by a unique real number. This seems intuitively correct,
since not all of the elements in Robinson's universe are in fact real numbers, as they are in
Dedekind's mathematical system, but further investigation into Robinson could settle the
matter definitively.
In closing, it is worthwhile to exemplify the potential usefulness of infinitesimals
in calculation by alluding to a proof of Leonhard Euler (1707-1783). Euler suggested a
system of infinitesimals in his book, Introduction to Analysis of the Infinite. In chapter
VII, Euler attempted to obtain infinite series expansions for exponential and logarithmic
7
Ibid., 49.
168
functions without using differentiation or integration. In doing so, he defines ω as an
"infinitely small number, or a fraction so small that, although not equal to zero, still aω =
1 + ψ, where ψ is also an infinitely small number."8 He let ψ = kω, so that aω = 1 + kω.
He then presented an example of this function at work using a finite ω:
He let a = 10 and ω = 0.000001, so that 100.000001 = 1 + k(0.000001). It
follows (from a table of logarithms) that k = 2.3026. On the other hand,
for a = 5 and ω = 0.000001, he found that k = 1.60944. 'We see,'
concluded Euler, 'that k is a finite number that depends on the value of the
base a.'9
Euler goes on to expand ax for a finite x by using the transfinite number j, defined
as j = x/ω. As such, Euler completely bypasses the need to refer to limits of any sort,
instead operating with defined infinitesimal and infinitely large variables. Such a proof is
ingenious, but of course controversial. In sketching the proof for us in his book, Euler:
the Master of us All, Dunham hails the cleverness of the logarithm proofs, but
simultaneously slanders the use of infinitesimals
Such reasoning hails from a pre-rigorous era. This is not to say, however,
that it should be casually dismissed. On the contrary, it accurately reflects
the standards of its day and, in that context, is both clever and
compelling.10
By excusing the use of infinitesimals as "reflecting the standards of the day," Dunham
echoes the many people who excuse racism and sexism of times past by calling the
perpetrators products of their time. Yet many today do not see infinitesimals as shameful
8
Euler, as quoted in Dunham, Euler: The Master of Us All, The Mathematical Association of America,
1999, p. 24.
9
Dunham's Euler, 24-25.
10
Ibid., 29.
169
or embarrassing mistakes made by our ancestors who were unfortunately born in an
unenlightened era; some view infinitesimals as useful tools in calculation, adequate and
sometimes preferable substitutes for limit theory.
The preceding examples do not prove the existence or consistency of
infinitesimals; they do not even prove beyond a shadow of a doubt that infinitesimals are
mathematically useful. What they do accomplish, however, is to show that infinitesimals
are not prima facie useless simply because they are non-additive, that there may be some
situations in which infinitesimals work just as well as, or perhaps even better than, limit
theory.
We must also keep in mind that mathematics frequently proceeds without
deciding beforehand what mathematical systems will be of use to later generations.
While Newton developed the calculus with uses firmly in mind, Leibniz did not. One
suspects that analyzing the square root of negative numbers was not first done in order to
find an application for imaginary numbers, but rather in order to see what would happen
if one analyzed imaginary numbers.11 As Dunham puts it:
Mathematics, of course, has been spectacularly successful in such applied
undertakings [as determining planetary orbits and balancing checkbooks].
But it was not its worldly utility that led Euclid or Archimedes or Georg
Cantor to devote so much of their energy and genius to mathematics.
These individuals did not feel compelled to justify their work with
utilitarian applications any more than Shakespeare had to apologize for
writing love sonnets instead of cookbooks or Van Gogh had to apologize
11
Cardan developed imaginary quantities as a way of dealing with equations involving the root of negative
numbers, but he thought that actually solving such equations was impossible. See Smith, A Source Book in
Mathematics. New York: McGraw Hill Book Company, Inc., 1929, 201-202.
170
for painting canvases instead of billboards.12
While "uselessness" may be a cutting charge in feminist political theory or in sociology,
mathematics has a habit of discovering mathematical entities first, and finding uses for
these theories later, if at all.
The next chapter is dedicated to an analysis of continuity itself. However, the
analysis presented will suggest a natural place for a system of mathematics which
contains infinitesimal quantities.
12
Dunham, Journey Through Genius: The Great Theorems of Mathematics, New York: Penguin Books,
1991, vi.
171
Chapter 8
Conclusions
8.1 Introduction
We have seen four different theories of continuity; here we shall compare them
directly, and draw some philosophical conclusions about continuity in general and
mathematical continuity in particular. In this chapter I wish to define the concept of
"compositional continuity," discuss the viability of compositional theories, and apply this
analysis to each of our four figures. I will conclude that, viewed in this manner, Cantor
and Dedekind present flawed theories of continuity, and that, despite his criticisms of
Cantorian continuity, so does Peirce. The most promising theory of the four is that of du
Bois-Reymond's Idealist. The analysis of continuity I will present here, together with the
Chapter 7 defense of infinitesimals as mathematically useful, will make apparent a logical
place for infinitesimals in relation to theories of continuity.
This chapter will thus consist of four main parts. First, drawing on the work in
Chapters 3 through 6, Section 8.2 will summarize the key philosophical positions of each
of our four figures on continuity and infinitesimals. Next, in Section 8.3, I shall define
"compositional continuity," address Aristotle's argument against composing continua
from points, and discuss Aristotle's definition of continuity by expanding on the
exposition in Chapter 2. From there the remainder of the chapter is philosophical
analysis. Section 8.4 will show that Cantor and Dedekind both have compositional
172
theories of continuity, and both suffer from certain philosophical problems as a result.
Section 8.5 will discuss the complicated way in which Peirce, despite his criticisms of
compositional continua, fell into a certain type of compositionality himself, and Section
8.6 will address and evaluate du Bois-Reymond's theory of continuity, and discuss the
role of infinitesimals in continuity itself. Finally, in Section 8.7, I will end with a few
concluding remarks.
8.2 Summary of Our Four Figures
In the following summaries, I pay particular attention to each man's theory of
continuity, his particular view on infinitesimals, and the ways in which continua and
infinitesimals interact in each theory.
8.2.1 Richard Dedekind
As we saw in Chapter 3, the impetus behind Dedekind's theory of real numbers
and theory of continuity was his dissatisfaction with the geometrical nature of the
calculus. He wished to create a truly arithmetized calculus, one that did not rely upon
geometrical intuitions about curves and tangents. His theory of real numbers and
continua reflects this essential dissatisfaction; indeed, Dedekind gave a purely arithmetic
interpretation of these mathematical concepts. Thus, when we combine Dedekind's
general theory of number with his theory of real numbers in particular, we get a picture of
human beings beginning with counting, and gradually adding complexity to their
mathematical system as needed for completeness and calculation, until the real numbers
are created. The irrational numbers are created from the rationals, through Dedekind cuts,
173
and Dedekind argued that the resulting set of real numbers exhibit the essence of
continuity.
For Dedekind, perhaps ironically, the essence of continuity was found in geometry
itself, in the geometrical straight line. As I quoted in Section 3.2,
If all points of the straight line fall into two classes such that every point of
the first class lies to the left of every point of the second class, then there
exists one and only one point which produces this division of all points into
two classes, this severing of the straight line into two portions.1
Thus, the essence of continuity, for Dedekind, is the ability to cut a continuous entity
anywhere you like, and have the division necessarily fall at an element of the entity,
and not between entities. For the straight line, the elements in question are points; for
the set of real numbers, the elements in question are numbers. While Dedekind does
not directly address the issue of whether the line is composed of points, or if the line
contains nothing but points, he is quite clear that the real number continuum is
composed of numbers and contains only numbers. Further, he is able to prove that
this property, this essence of continuity, holds for the real numbers created by
Dedekind cuts. That is, once the full set of real numbers is created, there must be a
number wherever you cut the set.
Of course, due to Dedekind's definition of irrational numbers, this concept of
continuity follows for the reals almost trivially. Recall that for Dedekind, an
irrational is created when one finds a cut in the rationals which does not occur at a
number; having found such a cut, we create a number at its location, and term it an
1
Dedekind, 11.
174
irrational. Thus, it follows easily from this definition that wherever there is a cut on
the reals, the cut happens at a number. Dedekind's principle of continuity clearly
holds of the reals created in this manner. Also, it follows that this real number
continuum contains nothing other than the real numbers themselves, as we were able
to follow the creation of this set from the counting numbers to the inclusion of the
irrationals, and nothing was added at any stage except numbers. Further, nothing
more can be added, since wherever a gap in the system occurs, an irrational number is
created to fill the gap; there is no opportunity for non-real numbers to be included in
such a composition. Intuitively, this is the deep reason why Dedekind's real number
system implies the Archimedean Principle, and is thus incompatible with
infinitesimal quantities.
In sum, Dedekind's real number system is built from the foundation of the
counting numbers, and is itself complete enough to satisfy our principle of continuity.
The system itself contains nothing but rational and irrational numbers; it does not contain
segments or intervals as metaphysical atoms (the way du Bois-Reymond's Idealist system
does), and is mathematically incompatible with infinitesimal quantities at its heart. While
Dedekind did not attempt to prove that the points on the geometrical straight line are in
any sort of correspondence with the numbers in the real number system, he did, at least,
create a number system which shares the same principle of continuity found in the
straight line, and, unlike the rational numbers alone, can be argued to be as rich in
"number-individuals" as the straight line is in "point-individuals."2
2
Dedekind, 9.
175
8.2.2 Georg Cantor
Cantor's theory of numerical continuity is similar to Dedekind's in many ways.
He, too, built a real number system based on rational numbers, and he, too, believed that
calculus could be founded upon purely arithmetical principles. He also was vehemently
opposed to the idea of infinitesimal magnitudes; however, his system of numbers was
much more open to the inclusion of non-real numbers than was Dedekind's. For example,
he firmly believed that irrational and complex numbers were an important part of our
mathematical systems; and, of course, he wished to extend our concept of number to
include the transfinite numbers, which are not a part of the real number system (though an
important tool, Cantor felt, to analyzing and understanding this system).
Though Cantor's irrational numbers are, like Dedekind's, built from sets of
rational numbers, he does not have quite the same foundational approach as Dedekind.
His chief anxiety was not to eliminate references to geometry, but rather, to make
mathematics more elegant, more functional, and better grounded. Thus, while Cantor's
real number theory, definition of continuity, and attitude toward infinitesimals are all
similar to those of Dedekind, his philosophical approach is markedly different.
His irrational numbers are based on sets of rational numbers, but there the
similarity to Dedekind's irrationals ends. Cantor's primary philosophical concern in
creating irrational numbers was to avoid a circular definition; he believed many, if not all,
mathematicians before Weierstrass assumed the existence of particular limits in creating
their irrational numbers, and then defining irrational numbers in terms of these limits.
Thus, Cantor's sets of rational numbers – Cauchy sequences, or, as Cantor called them,
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fundamental sequences – were associated with particular symbols, but Cantor was careful
not to assume these symbols were the limits of their associated sets, and even more
careful not to assume that these symbols were, in fact, numbers. Rather, he provided
evidence for the limit-like nature of these associated symbols, and went to great lengths to
prove mathematical functions such as addition, subtraction, and equality, held for these
symbols before he allowed himself to refer to them as magnitudes or as numbers.
Cantor's definition of continuity is just as carefully constructed. Rather than
intuiting the essence of continuity and then arguing that his real numbers lived up to our
expectations, he instead provided necessary and sufficient conditions for the continuity of
any set, and demonstrated that his real numbers met these conditions. Again, rather than
arguing that his mathematical continuity was consistent with other continua, such as that
of space, time, or the geometrical straight line, he argued that his mathematical definition
was logically prior to any other intuition of continuity, and that we should understand the
continuity of space and time in terms of his necessary and sufficient conditions, rather
than the reverse. Thus, these point continua of Cantor's provided the true essence of
continuity, not by developing our intuitions, but by grounding them in rigorous theory.
Despite these differences, and Cantor's openness to non-real numbers such as the
transfinites, his theory of continuity is still firmly committed to the idea that there are no
logical atoms in a continuum other than the point-elements – such as numbers, in the case
of mathematical continua, and points, in the case of geometrical continua. He even
referred to continua as "point continua", and believed that it is sufficient to show one
continua reducible to another to show that the elements (points or numbers as the case
may be) are in one-to-one correspondence. Thus, the essence of continuity, for Cantor, is
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contained entirely in his two necessary and sufficient conditions for point continua –
connectedness and perfection of sets. The mathematical definitions of these terms were
spelled out in Chapter 4; for our purposes here it sufficed to indicate that perfection of a
set means that every derived set is contained in the set itself (thus, the collection of limit
points of a set does not extend beyond the set itself). The connection of a set can be
variously interpreted (see, for example, Peirce's attempts to analyze Cantor's
connectedness), but at the very least, it implies everywhere denseness of the set.
As for infinitesimals, Cantor argued in the Grundlagen that even if infinitesimal
magnitudes could be consistently defined, they were useless (though he argued in the
same work that pure mathematicians should not concern themselves with the usefulness
of their mathematical creations). Later, however, he argued that they could not be
consistently defined. The later Cantor believed that infinitesimals were inconsistent with
the very notion of linear magnitude, and thus, an infinitesimal magnitude was a
contradictory concept. The argument Cantor put forward to try to show infinitesimals
contradictory relied heavily upon the Archimedean Principle, though his own beloved
transfinite numbers also fail to meet the criteria of the Archimedean Principle.
8.2.3. Paul du Bois-Reymond
Like Cantor, du Bois-Reymond was led to his interest in the philosophical
underpinnings of things like infinity and continuity through his sometimes controversial
mathematical systems. In du Bois-Reymond's case, his work with infinitary functions
(his Infinitärcalcül) raised foundational concerns; by using the limit operation to organize
functions with infinite ranges and domains, his system brought into sharp relief questions
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about infinity, as well as questions about the nature of limits themselves. Like Dedekind,
du Bois-Reymond was openly dissatisfied with the non-rigorous approach to these
difficult concepts found in pedagogical situations, and du Bois-Reymond's dissatisfaction
led him to attempt a systematic development of the intuitive underpinnings of our
mathematical concepts.
He found two intuitions at the ground level, and claimed each of us, if we thought
carefully enough about mathematics, would find both of these intuitions equally
compelling. First was the Empiricist intuition, by which we feel that mathematics must
spring from our experience of the world. According to our Empiricist intuitions, counting
must spring from our need to count objects, number sprang from the notion of counting,
and all mathematical concepts could and should be either traced back or reduced to our
experience of the world around us. Second was our Idealist intuition, from which we get
our sense that mathematics can be developed as far as logical organization can take it,
regardless of whether it retains its tether to the empirical world. He believed that these
intuitions were not only competitors, but in open conflict, with virtually no shared ground
between them.
These competing intuitions gave rise to different philosophical attitudes toward
mathematics, which, in their turn, gave rise to differing views on what mathematical
entities are acceptable, and what mathematical processes should be followed. Thus, a
mathematics which respected our Empiricist intuitions would proceed slowly, checking
with empirical reality at each step of its development, whereas a mathematics developed
along Idealist lines would proceed more rapidly, avoiding contradiction or chaos but
adding any other elements that seemed helpful and logically satisfactory. Crucially for us,
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an Idealist mathematics contains geometrically idealized objects and procedures; it also
contains transfinites and infinitesimals. An Empiricist mathematics, on the other hand,
forbids objects so idealized that they can no longer be said to be drawn from experience.
Extensionless points and one-dimensional lines are impossibilities, and obviously, so are
actually infinitely large or small magnitudes.
Thus, according to du Bois-Reymond, the Empiricist intuition in mathematics
leads to the conclusion that there is no such thing as continuity. Perhaps there is
continuity in the physical world, but if so, it is beyond the direct experience of human
beings, and thus, our mathematics and geometry contain no such thing. The geometrical
line is, for the Empiricist, as long as we want, but not actually infinitely long; as thin as
we want, but not actually only one dimension; as divisible into as many parts as we want,
but not actually infinitely divisible. The Empiricist may believe that we can find a point
wherever we wish one on the geometrical line, but that claiming actual continuity of the
line is entirely different, and unwarranted. What holds of the geometrical line, according
to Empiricist intuitions, holds even more strongly for the number line. The numbers are
not infinitely dense, nor are there infinitely many, and certainly they do not form a
continuous set. However, du Bois-Reymond's Empiricist believes, with Aristotle, that "as
many as we wish" is quite enough for mathematics to function well.
A mathematics built on Idealist intuitions, on the other hand, contains continua.
Idealist mathematics has no problem with the infinitely large, infinitely small, or
infinitely smooth. The geometrical line is continuous, as is the collection of real
numbers. A continuous set of numbers, in Idealist mathematics, is infinite, dense, and
has no gaps, but it is not built from the ground up – it is not built from empirically
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established phenomena. Even the Idealist admits that continuity is quite beyond our
experience.
Further, the Idealist continuum necessarily contains infinitesimal quantities. The
belief that the geometrical line is infinitely divisible leads, du Bois-Reymond argued, to
the conclusion that the geometrical line contains more than points; it also must contain
infinitesimally small intervals. If the number line is to measure the geometrical line, and
other infinitely divisible and continuous entities, it must therefore have an infinitesimal
quantity with which we can measure these infinitesimal segments, and a thorough
understanding of how these infinitesimal quantities interact with each other and with
finite quantities is necessary (and thus is developed by du Bois-Reymond).
Du Bois-Reymond believed that the internal struggle between our Empiricist and
Idealist intuitions was an insoluble one, one that we must continue to grapple with as we
create new mathematical systems and analyze the philosophical foundations of our
current ones. It is quite notable, however, that both the mathematics developed strictly
according to Empirical intuitions and that developed according to Idealist intuitions are
radically different from the mathematical systems of Dedekind and Cantor.
8.2.4. Charles Sanders Peirce
Peirce believed that continuity was one of the key concepts to understanding the
nature of the world, and was important to sciences as diverse as botany, history, and
psychology. He defined continuity early in his career as, "the passage from one form to
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another by insensible degrees,"3 but found this definition too imprecise and sought to
formalize it. This imprecise first attempt to pin down the concept of continuity is
instructive, however, as it reveals his early intuitions on the subject. He would spend
much time and ink over the course of his career attempting to find a precise philosophical
understanding of continuity that lived up to his intuitions on the subject. He first tried to
formalize this intuition by defining continuity simply as density, but soon judged this
inadequate, and went on to define a mathematical continuity that much resembled those
of Dedekind and Cantor; in fact, he was explicitly influenced by Cantor in his thinking on
continuity, as he was in many of his mathematical views.
Eventually he set out a definition of continuity containing three conditions, and
judged them to be necessary and sufficient for continuity. A continuous set, according to
Peirce at this stage of his thought, must have the properties of (a) infinity and
nondenumerability, (b) density, and finally the set must (c) contain all of its limit points.
In (c) we recognize a concept quite similar to the perfect sets of Cantor. This definition
shares many properties with Cantor's definition; in addition to the property of perfection,
Peirce's continuity is meant to hold of sets with members, such as numbers or points, or
perhaps instants.
Oddly enough, though Peirce accepted this mathematical continuity, which is
similar enough to Dedekind and Cantor's that we can actually derive Dedekind's from
Peirce's, Peirce believed at the time that infinitesimals were an important part of any
continuum. Whether Peirce saw this contradiction (recall that Dedekind-continuity
3
CP 2.646.
182
implies the Archimedean Principle and thus is inconsistent with the existence of
infinitesimals) is not clear; what is clear is that he soon became dissatisfied with this
definition as well. While his first attempt at a definition characterized his intuitions, it
lacked precision; this revised definition, although quite precise in the sense that it allowed
one definitively to determine whether any given set was or was not continuous, no longer
satisfied Peirce's intuitions.
His third and final definition of continuity was an attempt to characterize his
intuitions with precision. Under this final definition, Peirce abandoned density as a
requirement, replacing it instead with the requirement that, for an entity to be continuous,
"all of its parts must have parts of the same kind."4 This, he judged, was the essence of
continuity – a precise characterization of the "insensible degrees" intuition he expressed
so much earlier – and this essence of continuity had vast ramifications for the nature of a
continuum. No longer could Peirce accept the idea that continuity might hold of sets of
individuals such as points, for a point would be a part of that continuum that lacked parts
of the same sort. In fact, Peirce (unlike du Bois-Reymond's Empiricist) believed that
points had no parts at all.
A continuum, Peirce came to believe, was an entity which could be infinitely
divided, and every division would produce a part which was isomorphically identical to
every other part. Thus, this part would itself be infinitely divisible, and each of these
divisions would result in isomorphically identical parts, etc. Peirce believed that in so far
as these continuous entities could be measured, one could create points to aid our
4
CP 6.168.
183
measurement, but the continuum itself was not composed of points, and in fact, he
believed that wherever we place a point for purposes of measurement, we disrupt the
continuity of the entity we are measuring.
This final definition of continuity also featured infinitesimals in an important role.
Infinitesimals, he argued, could themselves be infinitely divisible. Thus, they could be
used as points of measurement without essentially disrupting the continuity of the object
to be measured. Thus, for Peirce, the elements of a continuum varied depending on what
we did with the continuum. All continua were divisible, with each division producing
continuous parts; one could find points on a continuum, at the cost of disrupting
continuity, or one could instead use infinitesimals as a continuity-preserving method of
measurement.
However, Peirce did believe that continuity thus defined could be built out of
discrete elements. One simply continues to add enough discrete elements so that the
discreteness disappears. Thus, in the case of the numbers, Peirce believed that if one took
every number in a set and replaced it with all of the real numbers between zero and one,
and then took that set and replaced all of those numbers with the real numbers between
zero and one, etc., eventually the numbers would lose their distinct nature and 'weld'
together, forming a continuous mass of undistinguished elements. Peirce himself
believed this "supermultitudinous" collection completely satisfied our intuitions about the
nature of continuity; however, he also believed that the collection was too unwieldy to be
useful to the science of mathematics.
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8.3 Compositional Continua and Aristotle
In this section, I would like to accomplish three things. First, I would like to
define precisely what I mean by "compositional continuity". Second, I would like to
analyze Aristotle's argument against composing continua from indivisibles, and show
why it is not applicable to continuity composed from points or numbers. Third, I wish to
argue that despite the failure of Aristotle's argument, Aristotle's analysis does succeed in
capturing our intuition of continuity itself.
By "compositional," I simply mean composed of non-continuous elements. A set
of five apples is composed of five discrete apples, and thus is compositional; an apple pie,
before it is sliced, cannot be said to be composed of five slices of pie (though in the case
of finite entities, once we create the divisions, we can then recompose the entity out of
those divided entities). An entity containing infinitely many elements can be said to be
composed of those elements if (a) the elements contained within the entity are necessary
constituents of the entity as a whole, and (b) any non-elemental part of the entity is itself
reducible to these elements. By "element" is understood any object, mathematical,
geometrical, or otherwise, which is not itself a continuous entity. By this definition, one
can compose a continuous segment of a line from shorter segments of a line, each of
which is itself continuous, without viewing the larger segment as thus compositional in
nature. If, however, each segment is itself viewed as composed from infinitely many
points, this would be a compositional continuity.
Using these notions, we can see that Dedekind's theory of continuity is clearly a
compositional one. His theory of number is foundational. We begin with the building
blocks of the natural numbers, given by the human need to count. We add numbers of
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different types as needed, but we always define them in terms of the numbers we already
have, building a system one layer at a time. Cantor has less of a foundational approach to
his continuity. He builds his irrational numbers out of rational numbers, but not with the
same levels of dependency. However, he too has a compositional continuity in at least
the following sense: his necessary and sufficient conditions for continuity assume that
the continuous entity is an actually infinite set of elements. Thus, though his irrational
numbers are merely associated with sequences of rational numbers, continuity, in its
essential form, is a property which applies to actually infinite sets; and thus, the elements
of the set are logically prior to the property which holds of the set itself.
As was discussed in Chapter 2, Aristotle believed a continuum could not be
composed from indivisible elements. His explicit argument against compositional
continua does not hold against the real number continua of Dedekind and Cantor; but it is
in his general definition of continuity that we find the intuitive properties of continuity so
well expressed, and thus, it is to Aristotle we turn for inspiration expressing what is
troublesome about Cantor and Dedekind's point-continua.
Recall, the essence of Aristotle's argument against compositional continuity relies
on a reductio: if we assume the existence of a continuum composed of elements, it
immediately follows that two points must be next to each other. An exhaustive
examination of the concept of 'next to' reveals that no two indivisible elements can be
next to each other, for any type of 'next to.' The problem with this argument is not in the
concept of 'next to', but in very first step – the notion that compositional continuity leads
directly to two elements being next to each other. Notice that this is not the case in either
Cantor or Dedekind's real number continua, nor is it the case in a line conceived of as
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composed from points; the property Aristotle misses is density. Though Cantor and
Dedekind view the set of real numbers as a continuum composed of elements, none of
these elements are next to another, for density guarantees that between any two elements
there is another. Thus, Aristotle's argument does not apply to them.
His idea of continuity, however, does, when he defined the continuous as "that
which is divisible into divisibles that are always divisible."5 This is the intuition of
continuity Peirce emphasizes when he rejects his second, mathematical definition of
continuity in favor of one which respects these intuitions, writing of continuity that "all of
[its] parts have parts of the same kind."6 The intuition that all essential parts of a
continuum must themselves be continuous is one which is persistent throughout literature
on continuity, and is echoed by Kant, when he claims that, "the property of magnitudes by
which no part of them is the smallest possible, that is, by which no part is simple, is
called their continuity."7 Du Bois-Reymond forwarded similar criticism against
compositional continua when he wrote
Points are devoid of size, and hence no matter how dense a series of points may
be, it can never become an interval, which always must be regarded as the sum of
intervals between points.8
Thus, while Peirce focuses on the difficulty of non-continuous parts forming a
continuum, du Bois-Reymond emphasizes the inadequacy of points themselves as
5
Aristotle's Physics, Book VI, 232b 23.
6
Peirce, CP 6.168.
7
Kant, Critique of Pure Reason, A 169 (B211).
8
Du Bois-Reymond, 1882, 66; translated from the German by Philip Ehrlich.
187
building blocks of continua. Philip Ehrlich adds that du Bois-Reymond "was not alone
among late 19th-century thinkers in believing that, if a continuous line is to be regarded as
composed of elements, these elements must themselves be extended."9 Of course, these
elements must themselves also be continuous, for if a continuum were to be formed of
discontinuous extended elements, the result would be a discontinuous entity.
Aristotle's definition of continua as that which is divisible into parts which are
themselves divisible fits with our intuitive sense that continua are not destroyed by mere
division, and that continuity ensures a particular kind of sameness. Recall the examples
from Chapter 1 of the division of tables and of water; physically dividing a table produces
something not at all table-like, and while dividing a glass of water into two results in
smaller amounts of water, it is possible to divide water enough times to break down the
molecules and even atoms themselves, proving water to be non-continuous.
Compositional continua are anti-intuitive precisely because they violate our sense that
continuity, at the very least, must guarantee that small parts of the continuum must
resemble the larger parts in every manner but size. Small areas of space must resemble
large areas of space; small units of time must flow in a manner similar to large units of
time; if these things failed to be generally true, we would judge space and time as
noncontinuous entities.
9
Ehrlic, "General Introduction," Real Numbers, Generalizations of the Reals, and Theories of Continua,
Dordrecht: Kluwer Academic Publishers, 1994, x.
188
8.4 Continuity, Cantor, and Dedekind
Hence, the problem with Cantor's and Dedekind's definitions of continuity is not
that they contain non-continuous entities. A stretch of time may contain a discrete event;
an area of space may contain a discrete entity. It is the definition of continuity as
logically dependent upon the discrete that violates our intuitions. The difficultly comes
only when continuity, which invokes an entity whose parts resemble each other no matter
how small they become, is defined as necessarily, logically dependent upon entities which
do not resemble the larger entity at all. Both Cantor and Dedekind define continuity in
such a manner, and both are troubled for similar reasons. I shall here address Cantor's
"point continuity" first, and then analyze Dedekind's essence of continuity.
Defining continuity as point continuity – as logically composed of discrete
elements – and then claiming that this is not only a good definition of continuity but the
essence of continuity itself, is to make a category error. Indeed, Cantor goes further than
to claim that a continuum can be formed from numbers whenever a set of numbers has
the correct properties; he claims that this definition of continuity is necessary to
comprehend all continua, even those such as space and time. If by 'to comprehend'
Cantor meant 'to quantify and measure', then he is correct. In order to analyze space,
time, or any continua mathematically, we must have a tool of measurement that helps us
understand continuity in discrete terms, as measurement is necessarily discrete. A line
can only be measured if we posit points upon the line, and units that correspond to those
points. However, if by 'to comprehend' Cantor meant that to understand the deeper nature
of the continuous, one must first understand the properties of the sets he defines as
continuous, then he has conflated the properties of the tool used to measure continuity
189
with the essential properties of continuity itself. Simply because discrete elements such
as numbers and points are necessary to measure, quantify, or manipulate continuous
entities does not mean that these continuous entities are necessarily composed of discrete
elements.
Dedekind showed a similar conflation when he defined the essence of continuity
with the geometrical line.
If all points of the straight line fall into two classes such that every point of
the first class lies to the left of every point of the second class, then there
exists one and only one point which produces this division of all points into
two classes, this severing of the straight line into two portions.10
On the face of it, this seems a reasonable approach to continuity, and it certainly captures
our intuition that continuity contains no gaps, or essential elements that are
metaphysically different from the continuous entity itself. However, by focusing on this
aspect of the geometrical line as the essence of continuity itself, Dedekind makes explicit
the requirement that all continua contain an infinite plenum of discrete elements. Were
we to attempt to apply this idea to time, for example, the first thing we would do is divide
time – a normal enough occurrence. Suppose at precisely noon today I divide time into
the past and the future; for any time in the past, it is clear it exists before any time in the
future. The point of division, the present, is only useful as a tool of analysis if it is itself
without time rather than an interval of time. So far, so good; time fits Dedekind's analysis
in so far as it is divisible into two such classes, and the division happens at a point;
neither at an interval of time, nor at a place somehow outside of time itself. However, if
10
Dedekind, 11.
190
we apply Dedekind's essence of continuity more literally to time, it requires time to
consist of infinite instants, and that the division between the past and the present consist
of a division of these instants, one of which is singled out as the divisor for the others.
I wish to avoid veering too deeply into the philosophy of time, and thus wish to
avoid pronouncing on the metaphysical status of instants, but it seems clear that the
continuity of time does not itself require an infinite plenum of instants. It requires only
that time take no breaks on its path, no gaps or jumps or other discontinuities. Instants
are only required if we wish to analyze time, if we wish to understand time as the sort of
thing we could measure, and separate into the past and the future. Thus, while one may
wish to posit or even prove the existence of infinitely many instants in time, time as a
plenum of instants does not follow from its continuity, even though division and analysis
of time does sometimes require a unit of analysis which is itself non-continuous. This is
what I mean by claiming that Dedekind conflates the means of measuring continuity with
the features of continuity itself: in order to analyze and quantify a continuum such as
time, non-continuous elements such as instants are desirable. However, our use of such
elements in no way necessitates their constitutive inclusion in the continuity itself. All we
must assure ourselves is that there are enough instants for us to measure time effectively,
that is, that 'instant' is a useful tool for analyzing time.
Historically, proving that numbers were up to the task of measurement was a
long-fought battle. Due to the incommensurability of the unit, it seemed as though
numbers were a particularly bad tool of measurement when one wished to measure space;
there were identifiable discrete points which could be found using the tools of geometry
to which no number corresponded. The inclusion of irrationals in our numerical canon
191
assures us that we can create a system of numbers of which Dedekind's essence of
continuity holds – that is, that there are no longer identifiable gaps in the real number
system, and each division of these numbers must occur at exactly one number. The
correct conclusion to draw from this fact is not, however, that sets of numbers themselves
are continuous, but rather that numbers so fashioned are in fact a good means of
measuring space. Dedekind's system of real numbers makes the Cantor-Dedekind
postulate feasible. We can in fact postulate without contradiction or incommensurability
the correspondence of points on a line with numbers in our real number system.
However, points themselves are not an essential feature of the continuity of the line, but
rather only an essential feature of our capacity to measure and quantify intervals of the
line.
Thus, neither Cantor nor Dedekind produce an intuitively satisfying theory of
continuity, and the primary reason behind this failing is the conflation of a property of a
measurable entity with the tools for measuring and comprehending the entity which has
that property. By defining continuity itself in terms of the points and numbers necessary
to break continuity down into analyzable pieces, Cantor and Dedekind have, as Philip
Ehrlich put it, reduced the continuous to the discrete.11
8.5 Continuity and Peirce
Both Peirce and du Bois-Reymond reject the Cantor-Dedekind style construction
of continuity, and both criticize this type of construction in ways similar to the criticisms
11
Ehrlich, "General Introduction," x.
192
above. However, the alternative theories of continuity they present are substantially
different from each other, and only one theory, that of du Bois-Reymond's Idealist,
overcomes the criticisms raised against Cantor and Dedekind. In this section, I shall
analyze Peirce's supermultitudinous continuity, and show how it retains some of the
assumptions which led Cantor and Dedekind to trouble.
Recall, Peirce first created a theory of continuity that was quite similar to that of
Cantor's and Dedekind's, and was in fact directly influenced by his study of Cantor,
before rejecting it as not satisfying to Peirce's intuitions about continuity. He then based a
new theory of continuity on his new understanding of Kant's definition of continuity as
that property of magnitudes "by which no part of them is the smallest possible."12 This is
quite similar to the Aristotelian insight into continuity we quoted above, whereby
Aristotle defined the continuous as that which can be divided into parts which are
themselves divisible. Peirce took this intuition seriously, creating a theory of continuity
which respected it.
Peirce's final theory of continuity, which tried to unify these disparate insights,
does in fact present us with a continuum of which no part is the smallest possible, and
which can always be divided into parts which are in turn infinitely divisible themselves.
However, the resulting supermultitudinous collection does not seem like a formalization
of our intuitions, but rather like a system in which nothing any longer is intuitive. It is
difficult to comprehend Peirce's construction procedure, the resulting supermultitudinous
collection (which he is careful to note is not a collection at all), or how this non-collection
12
Kant, 204 (A 169, B 211).
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relates to an entity such as space or time. Further, if Peirce himself is to be believed, then
this true continuum, this infinite plenum and supermultitudinity, is utterly useless for
mathematics.
I believe that Peirce's theory of continuity and the troubles contained within are
the result of Peirce attempting to model this Aristotelian insight into continuity within a
Dedekind-esque foundational framework. In so doing, Peirce has not only failed to avoid
the many difficulties with compositional continua, but he has also conflated the property
of continuity with the tools used to measure and analyze continuity, and in the process,
paradoxically created a system that is entirely useless in mathematics. Peirce creates his
continuity in much the same way as Dedekind creates his, by beginning with numbers and
adding enough to reach continuity. Rather than stopping at the collection of real numbers
as Dedekind did, however, Peirce believed more is necessary for true continuity, and
continues to add more and more numbers.
By creating continuity from elements such as points or numbers, Peirce retained
the philosophical problems that plagued Cantor and Dedekind's models, and this
supermultitudinous collection is itself a compositional continuity by our definition above.
By insisting that this set of discrete elements must be expanded until the elements
themselves meld together, Peirce introduced a new host of difficulties. First and
foremost, his creation process makes no sense. At precisely what point would numbers or
points find themselves so close to one another that they must bind? With a physical
entity, extended in space, such a picture may make sense; a small space may be filled
with increasingly many tiny particles until there is no longer room for them all and they
must somehow fuse or burst the boundaries which contain them. Yet points and numbers
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are not physical entities extended in space; the former is defined (from Euclid on down)
as that which lacks space, and numbers, whatever they may be, certainly contain no
physical breadth themselves. Without physical extension, these elements would never
run out of room, and thus would never be forced to meld; furthermore, there is no
physical extension with which to meld. If we were to do as Peirce directs, that is, take
the set of real numbers and, between every two reals, insert the entire collection of real
numbers between zero and one, we know that the result would be equinumerous to the
collection of real numbers itself, no matter how many times the process is repeated. The
creation process thus fails miserably.
Assume, for a moment, that the creation of this supermultitudinous collection
actually succeeded. By using points and/or numbers to create continuity, these tools
necessary to measure parts of this continuity are the very things that are melded away into
continuity. If numbers melt into one another in the process that forms a continuum, no
method of quantifying parts of this continuum remains; if points are merged into a
continuous entity, the points themselves are no longer available to us to mark distance on
that entity. Peirce has, in fact, retained Cantor and Dedekind's conflation between the
property of continuity and the elements used to measure continuous entities, and then he
proceeded to destroy the ability of this collection to serve mathematical purposes.
Moreover, this defect cannot be repaired, given his theory. His main criticism of a
real number continuum was that the real numbers are not the largest possible set, and true
continuity must contain all possible elements. Thus, he cannot restore the mathematical
viability of his continuum by introducing new points or numbers as tools of measurement.
Were we to admit the existence of such points, we would also have to admit that the
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supermultitudinous collection did not contain all possible points to begin with.
Thus, Peirce accurately criticizes Cantor and Dedekind for attempting to create
continuity from discrete elements, but ultimately falls into the same trap as they did. By
attempting to overcome the flaws in their system while retaining a foundational
mathematical approach to continuity, Peirce creates new problems and solves few; and
the resulting theory is an unintuitive, unwieldy, ill-defined, and mathematically useless
system, helpful neither for understanding the true nature of continuity nor for calculating
and comprehending continuity in a mathematical sense.
8.6 Continuity and du Bois-Reymond
All that remains is to discuss the theories of continuity presented by du BoisReymond, under the assumptions of the Idealist. Unlike the Empiricist, the Idealist
embraces abstract entities such as idealized shapes and lines, and draws logical
conclusions from them. In doing so, he forms a distinct theory of continuity, one which
applies specifically to the straight line but has consequences for number systems meant to
measure this straight line. The key to his theory of continuity is contained in his
argument for the existence of infinitesimals. The assumption on which the argument is
based expresses a clear concept of continuity; if the line is continuous, then "the number
of points of division of the unit length is infinitely large."13 This clearly echoes
Aristotle's idea of continuity; however, while Aristotle stated that continuous entities
were divided into parts themselves divisible, it seems at first blush as though du Bois13
Du Bois-Reymond, General Theory of Functions, 73.
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Reymond's definition of continuity would require division into points – themselves nondivisible entities.
This is not the case, and in fact the next sentence of this proof clarifies the
Idealist's stance. "According to the true concept of magnitude, these points do not follow
each other without an interval." A continuous line segment, for the Idealist, is not divided
into infinitely many points; rather, the segment is divided by points into infinitely many
intervals. Points are the means of division, not the result of it, and these points must
always maintain intervals between them. This theory of magnitude, and thus of
continuity, is not a compositional one; this continuum is not composed of points. This is
made clearest by the statement, "points alone can never form extensions."14 If points
cannot form extensions, they can never form a continuum. The commitment of the
Idealist to an Aristotelian continuity is made clear when he argues, "every extension as
small as it may be must be organized like the unit length, and similarly contain infinitely
many points of division."15 The Idealist further insists that if there are actually infinitely
many points of division on a line segment, then these intervals must eventually become
infinitesimal in length, but they never become extensionless; they are always intervals,
which, one assumes, can themselves be divided.
It may be tempting at this point to claim that the problems which plagued the
compositional theories of continua can be solved by composing our continuum from
points and infinitesimal intervals; in other words, we may wish to argue that
14
Ibid., 73.
15
Ibid., 73.
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compositionality of continua is not itself a problem, but rather that points or numbers are
particularly bad entities from which to compose continua. The Idealist’s infinitesimal
interval seems like a far better candidate for such a job, since composing a continuum
from finite intervals would do nothing to explain the nature of continuity itself. Think,
for example, of composing a long continuous line segment from shorter continuous linesegments; this would be in some sense compositional, but continuity would not be the
property thereby constructed, it existed previously to our construction, and so would only
be preserved.
However, composing a continuum from infinitesimal intervals does not solve all
of our problems. First of all, if the infinitesimal interval is itself infinitely divisible, as
Peirce’s infinitesimals are, one could make the argument that these intervals were
themselves continuous, thus, it is not continuity which is composed from infinitesimals,
but rather, we would merely be building a larger continuity from a collection of smaller
continuous elements. That du Bois-Reymond’s Idealist believes these infinitesimal
intervals are at least themselves infinitely divisible is evident from his assertion that
“every extension as small as it may be must be organized like the unit length, and
similarly contain infinitely many points of division.” This clearly is meant to include
infinitesimal extensions as well as finite ones.
Another difficulty plagues the proposed composition of continuity from
infinitesimal magnitudes; such a construction would commit the same conflation error
seen in Dedekind, Cantor, and Peirce. Infinitesimal magnitudes are, like the unit interval,
artifacts of measuring and quantifying, and there is no reason to suppose that they are
inherently contained in all continuous entities, assuming any actually exist. Composing a
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continuum from infinitesimals and points is no better than composing a continuum from
points alone. As an analogy, consider the unit segment which we discuss so frequently in
this dissertation; let us specify the unit segment as a line segment of precisely an inch in
length. Because one could theoretically construct an infinite line from an infinite number
of inch-long line segments is no reason to suppose that the inch itself is an inherent
component of any line, and a fortiori, that the inch is an inherent component of any
continuum.
It must be noted, however, that du Bois-Reymond’s Idealist does not make this
mistake; he does not speak in terms of composing a continuum from infinitesimal
intervals. Rather, the conclusion of his argument is “the unit extension is decomposed
into an infinity of partial extensions, of which none is finite. Thus the infinitely small
really exists.”16 The infinitesimal in the Idealist’s system is not a necessary element of a
composed continuum. Rather, it is a by-product of decomposing a continuous extension.
An infinitesimal interval results when a finite continuous extension is divided into
infinitely many parts, as "before the parts of a finite quantity can be infinite in number,
each must be infinitely small."17 Du Bois-Reymond's argument for the existence of
infinitesimals is not that they are a necessary element of continuity, but rather that they
are a necessary result from a continuous entity which is finite in extension but infinitely
divided.
It is possible that du Bois-Reymond’s Idealist does conflate the property of
16
Ibid., 73.
17
Ibid., 83.
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continuity with the tools for measuring continuous elements; one piece of evidence for
such a conflation is the seeming interchangeability of the concepts of point and number in
du Bois-Reymond’s writing. This is not necessarily a conclusive indication of this
conflation, however, as points and numbers are themselves both ways of quantifying and
measuring magnitude. His non-Archimedean continuity, which can be decomposed into
points and intervals, is an interesting one. Of the four theories of continuity examined in
this dissertation, it is the one which is the most compatible with the Aristotelian and
Kantian insight that any part into which a continuum is divided must itself exhibit the
basic properties of continuity, and which is still useful for mathematics.
8.7 Conclusion
In this last chapter I have argued that a true understanding of continuity must
make a sharp distinction between the property of continuity itself and the tools for
measuring continuity. Though tools of analysis are necessary in one sense to comprehend
and manipulate continuous entities (numbers, for example, are extremely useful for
measuring the passage of time), one must not confuse comprehending a continuum in this
manner with comprehending the essential features of continuity itself. Cantor, Dedekind,
and Peirce all conflate the property with our tools of measurement, and all run afoul of
philosophical difficulties as a result. Du Bois-Reymond’s Idealist system has
philosophical difficulties of its own, which even du Bois-Reymond himself admitted (for
example, it is wholly divorced from any empirical experience of the world), but by not
making the category error of the other systems, it is the most promising of the four
developments of the concept of continuity. Rather than simply ending with a summary of
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my main argument, however, I would like to end by permitting myself some remarks on
three larger issues, beliefs I have formed in the course of writing this dissertation but
which I do not intend to argue strenuously for here, as they are, strictly speaking, outside
of its range.
First, continuity itself. Notice that while I have insisted that continua are not
composed of the things we use to measure continuous entities, but rather that continuity is
a property entirely distinct from mathematics, I have not said much in they way of what
this property itself consists of. I do believe that Aristotle’s insight, that continuous things
must be divisible into parts which themselves are divisible, holds the key to continuity;
however, beyond that, continuity remains to me something of a mystery. Du BoisReymond’s Empiricist very well may be on the right track when he claims that continuity
is beyond our direct experience. However, it seems more likely to me that a Kantian
evaluation would be more accurate; that continuity, like space and time itself, is a
necessary feature of the way in which we experience the world. Whatever the true nature
of continuity itself, it seems obvious that if such a thing exists, in space, or time, or
elsewhere, it is a property quite independent of our means of quantifying and analyzing
continuous things.
As for infinitesimals, I firmly believe they should not be rejected as useless, since
they could contribute much of value to mathematics (I have argued to this effect in
Chapter 7, above). However, while they are not useless, neither are they necessary
elements of continuous systems. Infinitesimal quantities, like a variety of mathematical
systems and entities, are tools with very specific uses. It may well be that infinitesimals
are indispensable tools for the most comprehensive analysis of continuous entities, but
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just as continua are not dependent upon numbers or points, neither are they dependent
upon infinitesimally small magnitudes. A non-standard analysis, such as the one
presented by Abraham Robinson, may well turn out to be the system of calculus that is
maximally useful for thorough analysis of continuous phenomenon, but even this would
not prove infinitesimal quantities were necessary elements of these continuous
phenomena themselves.
Finally, Cantor and Dedekind were mistaken about the metaphysical importance
of their mathematical systems; they both failed to produce necessary and sufficient
conditions for defining continuous phenomenon. However, their mathematical systems
are far from useless. By providing a system of numbers which guarantees that wherever
one cuts the number system, one does so at a point, Dedekind gave mathematicians a
system of real numbers which was both defined independently of geometry, and uniquely
suited to be wed back to geometry, with the use of the Cantor-Dedekind postulate. By
providing necessary and sufficient conditions for a “continuous” real number system,
Cantor did the same – he too created a system of real numbers which is uniquely suited to
be used to quantify continuous things. Rather than creating continua from numbers,
Cantor and Dedekind accomplished something much more mathematically interesting and
important – they created systems of real numbers which guarantee that no matter how
fine-grained we get in our analysis of continuous phenomenon, our numbers will not fail
us. By using a Cantorian or Dedekindesque system of real numbers, we are guaranteed
that we can measure with as much accuracy as we wish without running into the
incommensurability problems which plagued mathematics before the sixteenth century.
We are also guaranteed that mathematics can stand on arithmetic feet, without constant
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reference to geometry – which was, after all, Dedekind’s original goal in creating his real
number system.
This does not mean that Cantor or Dedekind provided the one and only system of
numbers appropriate for analysis of continuous entities. A Cantor/Dedekind system of
real numbers, and a non-standard system of numbers, can both be useful tools in our quest
to understand the wider world. Both should be developed, as Cantor recommends,
without constant reference to applied mathematics, without constant reference to
precisely what they are best at measuring, as such a creative approach to mathematics is
not only interesting in itself, but provides the intellectual community with a variety of
mathematical tools, some of which may be better suited than others to particular jobs.
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APPENDIX
In 1882, Paul du Bois-Reymond published his philosophical masterpiece, Die
allgemeine Functiontheorie. Shortly thereafter, the work was translated into French. Du
Bois-Reymond took the opportunity of this new translation to make some corrections to
the original. Though he assured the reader that none of the core theses or arguments had
been substantially altered, he did attempt to clarify some things, and fix some others. To
my knowledge, no one has yet discovered precisely what alterations have been made to
this translation.
What follows is my own translation of the first one hundred pages of the French
edition. I present it for two main reasons. First, as a convenience to the English reader,
as most of the references in this dissertation are drawn from this first third of du BoisReymond's book, and no English translation has yet been published. But second, in the
hopes that someone may find this useful, should they wish to compare this with the
German in order to discover the mysterious alterations. This is only a rough translation; I
apologize in advance for any errors which remain. The numbers in square brackets refer
to the pagination of the French translation. Thanks go to Christopher, Timothy, and
Jacob Keele for their help in typing the corrections to my second draft of this translation.
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GENERAL THEORY OF FUNCTIONS
Part One: Metaphysics and Theory of Fundamental Mathematical Concepts:
Magnitude, Limit, Argument, and Function.
Tr. 1887
Preface to this translation
When Messieurs Milhaud and Girot had made clear to me their intention to
translate my General Theory of Functions which, published in 1882, had been drafted just
five years previously, I wished to take advantage of such a great occasion to revise some
passages, to insist upon several points which today have acquired more interest than
before, and finally, to make some additions. These changes do not at all effect the
foundation of the matters discussed in my book. On the contrary, the numerous criticisms
which have appeared, more often contradictory among themselves than in opposition to
my views, have convinced me more than ever that I have succeeded in discovering the
true nature of exact knowledge and the metaphysical concepts on which it is based.
Paul du Bois-Reymond
Berlin, 15 May 1887
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Translator’s Preface
The overly-exclusive care in rigor gives to the teaching of mathematics an often
dogmatic form. Those who have received this teaching in schools or in colleges
frequently do not comprehend that there could be concerning these sciences questions
capable of dividing mathematical thinkers, and all philosophical discussion on essential
notions of mathematics is often poorly received for the simple reason that it is difficult
feel the necessity of it.
For, here is precisely such a book, written by an eminent geometer from Berlin,
full of scholarly discussions on the fundamental concepts of magnitude, limit, function.
Whether the manner of the author’s observation is exact, or inexact, is of little
importance: it raises questions about philosophical problems of the highest interest,
touching on what mathematicians consider too ordinary, avoiding treating it like
forbidden ground. This is why it has seemed to us useful to publish a translation.
We will not enter into the debates stirred up by this book. We only wish to
prepare the reader who has drawn their mathematical instruction from certain courses or
tracts, to make them begin to understand how mathematics can give way to interesting
discussions on the origin and formation of their concepts. We would like to make clear
that the rigorous chain of deductions which stretches through the teaching of mathematics
is in reality posterior to the normal development of these sciences, and that it only attains
this ideal of rigor by becoming more and more formal and, at the same time, more and
more subjective.
When one opens a mathematical treatise, one is struck by the importance of the
role played by definitions. Each definition serves as the basis for a development which
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entirely forms a new chapter. These are, in other words, vital elements of mathematics:
the power of deduction of the mind seems used up by the first objective of its studies; a
definition is dropped in and carries new nourishment to its activity. This is how the
definitions seem each time to assure a prolongation of the life of mathematics in a manner
that pulls the boundaries back to infinity. But what is a definition comprised of? Under
what conditions is it acceptable?
A definition has as its object to construe a new thing or fact not already known, by
aid of certain reliable properties and the only condition imposed on these properties
appears to be that they do not present any logical contradiction. The sole point we judge
it useful to insist upon that is, in mathematics, we feel the need to justify a definition. But
then the first impression the reader of a certain treatise gets is the most strange. It seems
that the mind can be given free reign. Doesn’t it have, in creating definitions, an
unbounded scope? And not only does one feel that mathematical science does not have
known boundaries, but still we wonder if there couldn’t exist an infinity of distinct
mathematical systems, other than those which are taught, if our systems aren’t ultimately
due to a caprice of human intelligence -- which wished to follow one voice among so
many equally accessible ones? In geometry, the same situation! One feels vaguely
guided by bodies and forms similar to those that show us the exterior world, but to speak
of analysis, now more than ever, thanks to the reconstruction work of Lagrange, Cauchy,
Abel, etc., it is easy to make one’s way to the most elevated notions without bringing in
other experiential facts of the number.1
1
See the Introduction to the Study of variable functions of M.J. Tannery
207
To be reassured, and to see the capricious and arbitrary character of mathematics
disappear, we must go back to the genesis of the notions that we study, and look beneath
this perfect arrangement we are presented with today.
One ordinarily distinguishes pure and applied mathematics, the first being
geometry and analysis, the others being applications of the first. It is not well understood
that a single question of degree justifies this distinction. Geometry and analysis are the
primary and most simple applications of mathematics; that is to say, of the special sets of
proofs, of particular logical methods which belong to all mathematical sciences,
abstracted from their objects. We know that the first given facts of geometry and analysis
are drawn from the external world. Geometry borrows extension; it also borrows the point
and the straight line, with all their intuitive properties. Analysis is founded on number,
the concept and properties of which are supplied to us by experience. These are the naïve
truths on which insistence is futile. But the points of departure thus fixed, the
indetermination of the road to follow would not exist if geometry or analysis were left to
be guided by given exterior facts, and it is precisely this, in spite of appearances, that they
make ceaselessly. We do not wish to speak here only of Euclid’s postulate, which, far
from being a logical axiom, is clearly already the affirmation of an experiential fact. We
can add it to initial given facts. These are so complex that it matters little whether or not
we think that this new fact is implied by intuitive notions on which geometry is founded.
Otherwise, there would be no arrangement of any type that could conceal the completely
bare fact, and there would be little point to denouncing this loan from experience.
But there is more: all the elements new to the study of geometry, angle, right
angle, circle, length of circumference, etc., are suggested only by the external world. It is
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the same in analysis of fractional number, of incommensurable number, of limit, etc.
Imaginary numbers themselves are furnished by experience, although this appears
paradoxical. This experience is so refined, so to speak, that it has become the verification
of the result of a calculation or of an algebraic transformation. But all this is far from
evident. Each time a new object of study is suggested, mathematics assimilates it to the
point of concealing its origin, or rather, at the occasion of this element, they logically
construe a new being, they are created from all pieces, and if the unique care that guides
them appears to be the non-contradiction of the properties with which they dress it up,
and the possibility of expressing them with the aide of ancient elements, in reality the
primary preoccupation has been that this logical creation corresponds exactly to the
concrete object. This preoccupation is hardly visible, because it carries little of the rigor
of rationality, but, if we do not wish to leave mathematics as a purely platonic beauty, if it
is to merit its title of science, all these logical developments are destined to be used in the
general knowledge of the physical world, and then, for the simplest solution of the
problem, we should admit the identity of the exterior object and of the purely logical
concept: at precisely this instant we find a denunciation of the experiential origin of any
definition. If we have been able to hide it, it is due to the sole condition of substituting an
unprovable proposition for the instant of application, a true postulate, by which we affirm
that our logical theories can give explication to an objective fact.
Some simple examples will help to clarify these ideas. After the study of several
properties of straight lines considered together, geometry used these properties to present
a new element: the circle. The definition which serves, so to speak, as its passport, is the
following: the circumference of the circle is the geometric bond of points situated at a
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given distance from a determined point. Translated: by the determined point we draw
any line whatsoever, then we take this line, starting from this point, to a length equal to
the given distance. The extremity of this length is that which we call a point of the circle.
The possibility of thus constructing just the points of the circle that we wished contains
the notion of geometric space which enters into this definition. We deduce from this all
sorts of properties; for example, a line which has a point in common with a circle has a
second; a diameter splits the circle into two symmetrical parts, in other words, at every
point on the circle one find a corresponding second point, symmetrical by relationship to
any diameter whatsoever; etc. The consideration of inscribed polygons, of which the
summits are points of the circle, permits the definition of the length of the circumference;
this would be the limit of the perimeters of the inscribed polygons of which the number of
sides grows indefinitely.
In all this development, the idea of the form of the circle, of the perfect round that
we draw by abstracting from experience, does not enter in any fashion. This round is
formed by a continuous contour, it divides the plane into two parts, the one it limits and
that we call interior, and the other we call exterior; all these notions are absolutely distinct
from the geometrical definition and deductions. Likewise, the purely logical concept of
the length of the circumference is essentially distinct from that which by intuition we
understand as the length of a round, the circumference of a wheel, for example, or the
length of a thread applied exactly to the circumference of the wheel, then unrolled. It is
this that in mathematics seems to be the closest neighbors of concrete objects of
experiential intuition; they are developed parallel to these objects, and without ever
dissipating the duality offered by the given facts of the senses, even refined by
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abstraction, and the logical constructions of the mind. But here at least this parallelism is
perfect enough that we feel strongly how geometry has proceeded. Experience has first of
all furnished not only the notion of circle, but a multitude of its properties; from among
them an époque of many inferred properties, the geometer has chosen those which,
completely expressed only with the use of lines, points, lengths, those which appear as the
best characterization of the concrete circle, and the theory of the circle has been
constructed from these properties. But we only wonder, for example, what is the
circumference of a wheel of which we are given the radius? How do we solve the
problem with the aide of geometry, if we don’t admit the identity between the limit of the
perimeters of inscribed polygons and the circumference of the wheel? The method by
which we had expelled experience is betrayed at this instant by a postulate, which is
precisely what in reality had led us to the logical definition.
The preceding example carries us back to a previous epoch in the history of
mathematics; one which, on the contrary, borrowed from elementary arithmetic to
respond to these actual tendencies. At the beginning of a chapter on fractions on
arithmetic, we ordinarily accept a fact of experience: the division of a unity into equal
parts. The equality of two fractions or the superiority of one fraction over another is
explained by the consideration of two lengths, for example, measured by fractions. The
experiential given fact which breaks the chain of deductions in arithmetic has not
therefore disappeared. But meanwhile nothing is easier than to conceal here the origin of
the new development. We call “fraction” the formal symbol composed of two whole
a
numbers written one above the other ( ). We agree to say that two fractions are equal
b
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when the terms of the one are equimultiples of those of the other; we define sum,
difference, product, and quotient of fractions, the results of which we are led to by the
consideration of concrete quantities, etc. The definitions always conform to what results
we set in the experiential given facts, nothing will be changed in the collection of rules in
the chapter on fractions. It is probable that sooner or later we will thus readily expose
this chapter. But when we find the solution x = 2/3 to the simplest problem, where the
unknown is a length, it will be necessary to admit, for interpretation, that 2/3 represents
two times the third of unity, and, in sum, we will thus reestablish all that we have
appeared to suppress.
These examples will perhaps suffice to show the march of the mathematical
sciences: they are developed naturally under the impulse of experience, but sooner or
later they hide the origin of their concepts under logical constructions. Here is a last and
interesting configuration drawn from higher analysis.
We suppose that at any instant in the history of mathematics, someone has needed
to use a new property of concrete quantities: it would not always be easy to simply
express it, as explication by reducing it to other known elements, or unraveling the
complex elements that it implies. It is possible, meanwhile, that the elevated minds
conjecture as by instinct that there is a fruitful notion, and that a development founded on
it will lead to great progress in the general knowledge of things. A priori, we do not have
to go to the trouble of admitting the existence of long mathematical chapter constructed
on notions that are not explained: don’t geometry and arithmetic have as initial givens
some concepts impossible to define? If we suppose finally if, while researching applied
to new ideas, struck by interesting results, we were above all preoccupied with enriching
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the list, the logical reconstruction of experientially given entities will strongly risk staying
forever unachieved; for a long time, the analytic branch that was thus brusquely pushed
seems not at all attached to the primitive trunk. It is precisely this that has passed for
superior analysis: this part of analysis which treats incommensurables, limits, infinite
products and series, infinitesimals, differentials, etc. The experiential fact which is the
point of departure, and the introduction of which into analysis gives rise to an epoch
impossible to determine, is the notion of becoming a concrete quantity, of the continuous
passage from one state to the other. After having suggested these works, this notion
manages its last and complete consecration in the methods Newton and Leibniz
inaugurated. But the exposé of these methods is far from always having presented the
tranquil chain spectacle of a chain as presented in mathematical tracts today. This chain
only appears after large of fault effort that has definitively been settled here, since Cauchy
and Duhamel exposed it to infinitesimal analysis. Is the problem of logical
reconstruction, which has become a rigorous collection of anterior chapters, completely
resolved? We will show that it is not in education in any fashion, except in showing how
a pure formalism can furnish this solution.
The exposé that today presents the course of analysis, and which has for its point
of departure the mathematical definition of limit, reduces the fact of experience that we
have described to a simple indemonstrable postulate, the principle of limits, which is thus
described: if a variable quantity grows ceaselessly, always remaining smaller than a
determined quantity, it has a limit.
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Duhamel gives the following proof:2
Let A be the value under which the variable always stays, and let B be one of
those values of the variable:. Split the interval between B and A into equal parts as
small as we wish: the variable could pass every point of division, but cannot go to
the extremity. Suppose the variable could not pass every point of division; then
there would be one that was the last that it passes; It would thus always remain
between this last point and what follows it, that is to say in an interval as small as
one wishes, and within which it would always grow. In subdividing this interval
into equal parts as one wishes, we recognize just as the magnitude can only be
found in a new interval fixed between B and A, and length as close to zero as one
wishes. There thus exists a certain fixed value between B and A, which the
variable approaches indefinitely; it thus has a limit.
The reasoning is rigorous up to the conclusion: as for passing from that interval
which includes the magnitude which can become as small as one wishes to the existence
of the limit, there isn’t the least difficulty directly admitting the conclusion, without any
proof at all. In the same way, as Paul du Bois-Reymond clearly shows, the diminution of
the interval that includes the magnitude does not diminish the difficulty that there is in
conceiving the limit. The reasoning of Duhamel hides an illusion; and it is so true that
often, on the contrary, to explain that a magnitude bound in an ever smaller interval has a
limit, one depends on the values which include those forming two series, one growing,
2
Duhamel – Methods in the Sciences of Reason. 2nd part, p. 413.
214
one shrinking, admitting to each a limit after the principle in question, and from this it
sufficiently follows that the limits are the same.
M. Bertrand, speaking of the series of positive terms said simply, “It is clear that
if, in the sum u0 + u1 + … + un one takes an ever increasing number of terms, the results
obtained would grow and if they cannot pass every limit, they necessarily approach it
such that one wants the smallest number that cannot be passed. It’s the same manner of
reasoning as M. Briot used. But we have only felt the existence of this smallest number
that the variable cannot pass and can never be proved, and that in sum, this reasoning
changes, without explication into the enunciation of an intuitive fact.
These proofs, and all those that one can encounter, return at bottom to the two
things that the author of this book developed, showing each time the illusion that they
hide. For on the other hand, if we reject every proof, will we accept the principle of
limits of the same style as a proposition of this type: “The same thing can not both be and
not be at the same time?” It is quite clear, on the contrary, that the principle of limits
introduces a new fact, a particular property of concrete and continuous quantity. The
character of the evidence available to us is due only to experience. It is a true postulate
that teachers introduce at the beginning of upper analysis, reducing it thus to the truth that
it expresses as necessary given the facts based upon it. Is it possible after all to hide this
postulate? It suffices, to convince us, to read the “Introduction to the Study of the
Functions of a Variable” of M. J. Tannery. The rigorous chain of deductions by which
the author leads us insensibly from the whole numbers to all the most elevated notions of
analysis decidedly resolves the problem of logical reconstruction that begins to make all
lacunae between the ancient views and the new disappear in the formal exposition of
215
mathematics. To understand the simple mechanism by which it is possible to attain this
result, recall the example of fractions mentioned already. On the condition of being
resolved to a pure formalism, nothing is easier than applying an identical method here.
We are convinced that a collection of values growing, though not indefinitely, has a limit,
when under these values we see successive states of a concrete quantity: we leave this
last consideration and produce, in virtue of a definition, the limit of the collection of
values. This will not be a concrete thing, known by a sensible intuition; it will be a pure
symbol. We will agree to say that it is superior to every value attained or passed by the
collection that it serves to define – inferior to every value that it never attains – we will
agree upon the circumstances when two symbols satisfying to the new definition are
equal, thus agreeing upon the results of operations accomplished on them, and, thanks to
the constant care of making these definitions or conventions correspond to the truths
which result for us from the existence of the limit, the results will never change in the
deductions which follow. The principle of limits will have disappeared from analysis as a
proportion to be established – and, with it, the last trace of all experiential given facts
other than the initial fact of arithmetic, of number. But, to apply the ideas strongly
indicated in this preface to this last example, it is understood that as soon as, draw near to
the tool thus refined enough to resolve the most childish of problems, before treating
lengths, for example, the solution can only be justified and interpreted thanks to the
opinion that the symbols correspond to realities.
The tendency to eliminate all experiential facts other than these given initially —
is this a simple fancy of mathematicians? A dangerous fancy in this case, has the
consequence of giving their science a capricious air and the appearance of a simple mind
216
game. However, this tendency rather corresponds to a high philosophical necessity. The
elements of our knowledge, whether their origin is experiential or rational, are combined
in our mind to such a degree and nature that the certitude that they admit of is often very
difficult to specify. For, the logical reconstruction of mathematical facts has as its result
the clear separation of the part which, in itself, presents the character of certitude — if
not absolute certainty at least the highest to which we can attain — from the other part
which is only inductive truth. In other words, it effectively creates a mathematical ideal
gliding above all experience; it is this that mathematicians know above all. Behind their
logical scaffolding, they are really sheltered from all objections; and it is with reason that
the evidence for their conclusions is taken for the most complete sort of evidence that
there is. Stuart Mill pretends that the certitude of mathematical truth is inductive. This
thesis only attains objective mathematical truth, for it is the expression, if possible, of
being concrete that suggests experience. It appears to us in this case absolutely just, for
it returns to denouncing the eternal hidden postulate, hidden behind all creations of
geometry or analysis which reappears at the moment of application. But it is not in order
to know how to carry on these logical truths of this mathematical ideal that we will
define. This seems, it is true, to not be free of initial given facts: if we look at it closely,
we see that it affirms nothing about the given facts themselves, and shows only what are
the consequences if they are accepted as hypotheses. Finally, the construction of this
completely formed mathematics gives nevertheless, a precise indication of the things
themselves: it teaches us what is the minimum number of properties that it suffices to
assume in these things to justify the application of logical truths. All the geometric
properties of the circle, for example, will extend to concrete rounds, if they exist in it,
217
from which all points are at the same distance from a center. Higher analysis will be
applied to quantities which exist from states corresponding to all symbols it has created,
etc. And thus we know how to understand the minimum of characteristic properties by
which a concrete fact can enter into the gears of deductions of pure mathematics.
Here is where this draws its raison d’etre. But is interesting is that it proves, we
believe we have sufficiently shown, by its purely formal essence, mathematics will not
give the solution to any problem of concrete knowledge. This is why treatises of
mathematics, as rigorous as they are, and precisely due to their extreme rigor, will never
be able to render superfluous the study of problems of knowledge such as the treatise of
this book.
Le Havre, 1st April 1887
G. Milhaud
218
Preface
[-17-] The need to see clearly into the deep nature of analysis, from equations to
partial differentials of the second order, led me to study the general formulae which, like
that of Fourier, serve to express arbitrary functions. These studies obliged me to next
consider the essence of functions and their integrals independent from every hypothesis.
On this point, I did not reach and could not reach perfect knowledge, which would require
subjecting to the trial of an epistemological critique the fundamental analytic concepts of
magnitude and limit. I wish now, in the work contained in the first part of this book, to
traverse the path I have followed myself in the opposite direction but in less time, I hope.
This path is, in effect, the question a set of theories which deserve to be well and
solidly established in a special exposition and which make evident the fundamental idea
which can be seen in the division of medical sciences. General theories are here opposed
to specialized theories, like special anatomy, special pathology, etc. are opposed to the
general sciences of the same name. Similarly it would appear proper to distinguish in
analysis a special theory of functions from a general theory.
The first special theory in studying the functions of complex variables in a very
general manner has as its end to represent the functions of determined properties and to
study the nature of grand classes of transcendents, in particular those which have
relationships to algebraic functions.
The general theory of functions embraces, in my opinion, all that is connected
with the most general idea of function – at the head I place the metaphysics of the
concepts of magnitude and limit, which serve as the base of the theory of argument, of
219
function, and of the common condition of convergence and [-18-] divergence of the
different infinite operations. These, by the way, are the questions treated in the first part
of my work. Then one finds the general theory of series, the study of integrability and
differentiability of the functions and the general propositions relating to the definite
integral; and next the theory of so-called arbitrary functions expressed with the help of
integrals and series, but in particular the expressions of Fourier which further specify the
concept of function; finally, different parts which return in the theory of equations to
partial differentials of the second order.
Thus, in a few words, what is contained in the general theory of function is the
theory of the relations of magnitudes and of operations in general, unless one essentially
has in view the representation of particular dependencies.
Since, I first published a study on this subject in 1872 under a different title,
works which have pursued the same goal have appeared. It could occur to me that their
borrowing maintains things of value, as I did for when I borrowed some important
propositions on the nondifferentiability of certain functions from M. U. Dini. He will be
seen, however, that my work on the project, just as for the very foundation of the subject,
is absolutely original. This is assuredly true in what concerns this first part, but also in
what follows, such as, the next part which treats the general theory of series, and where,
outside of what is already found in the treatises, I bring more than a new insight.
As I have said, this first part contains a survey, from the point of view of the
theory of knowledge, of the concepts of magnitude and of limit, and the application of the
results of this study to the theories of argument and of function. The problem which
touches on the theory of knowledge, I believe to have resolved perfectly, in a way which
220
guarantees, among other things, a rigorous and solution natural to the paradoxes which
have appeared recently in analysis.
The concept of analytic limit, or (for after all it is only a question of this) of the
limit of the decimal fraction, originates from certain successions of representations,
which have, as I will show elsewhere, a rather wider meaning. As we divide, after certain
principals, these collections of representations or as we can make their ideal term
apparent, we give rise to a double form of intuition ascending to which we can follow the
line of demarcation through all speculations of the theory of knowledge.
[-19-] These representations, simple elements of thought, and whose succession
accompanies and governs every mental act, being the primitive material of every study of
the theory of knowledge, must to be preceded, as in our study on fundamental analytic
concepts, by a sufficient definition of the word “representation” (Vorstellung).
In fact, the sense of this word, despite its frequent usage, does not appear to be
established with the rigor demanded by its fundamental importance in the same way as
the multitude and immense variety of the individuals that it embraces are established. For
we are concerned here not only with visual representations, with those of hearing and
other senses, but also, for example, with representations of sadness, of gaiety, of color, of
will, etc. Briefly, one can say that every psychic phenomenon, that we call perception,
sensation, volition, is equally a representation or, to speak more exactly, furnishes the
mind with a representation of itself. For, how can we specify and delimit these concepts?
Here is my opinion:
A representation is any perception which can become an object of memory, and
under the same form it assumes at the instant when the memory receives it. And I
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naturally understand, in the totality of representations, every memory which, by the work
of thought, is offered to the consciousness.
Thus, are representations reunited under the same simple and well defined
principle that we can name directly; that is they are the immediate impressions of our
senses, and indirect representations drawn from memory.
Tubingen, February 1882.
222
On the concepts of magnitude and limit
Preliminary remarks and the simplest enunciation of the principle problem.
[-21-] By ‘concept of limit,’ we mean a certain mode of reasoning in virtue of
which, from the nature of a collection of values susceptible to being measured or
observed, we infer the existence of values which escape all perception, and whose
existence can never be demonstrated in the ordinary sense of the word. Despite all this,
however, we are used to being satisfied without flinching from this conclusions that
which we apply constantly.
This manner of inferring the actual existence of objects that cannot be obtained
through any mediate or immediate perception is, as we know, a method familiar to certain
sciences, where an appeal is made as a manner common to all men to conceive by
intuition and feeling. But we should not be astonished that one form of thought, which is
hardly rigorous at all, must serve to consolidate the most indispensable and the most
fruitful fundamental notions of mathematics, precisely that science which, more than any
other, is prided for having the cleanest and most meticulous rigor, and which each day is
shown incontestably more worthy of its reputation?
What mathematician could deny that (chiefly in the ordinary idea of it) the
concept of limit and its near-parents, those of the unlimited, the infinitely large, the
infinitely small, the irrationals, etc, still lack solidity! The professor, in his writings and
speeches, is accustomed to running through this perilous introduction to analysis rapidly,
in order to walk more easily on these comfortable paths of the calculus.
223
[-22-] In truth, we rarely forge a winding path when we try to search for the most
curious things without following the crowd. Anyway, we ourselves propose to travel
precisely that path that others avoid.
As was presumed, we will soon rediscover that the inherent difficulties of the
concepts I mentioned above are not of a mathematical nature: otherwise, they would
have been smoothed out before long! They rather have their origin in the simple elements
of our understanding, in representations.
The solution to this enigma, if I am not mistaken, is that it is and always will be an
enigma, only it seems to me that this enigma returns to its simplest psychological
expression. The most tenacious observation of our thought, and of its correspondence to
perception, do not lead us beyond the following statement: there is, for the mind, two
completely distinct manners of apprehending things, which have an equal right to be
taken for the fundamental intuition of exact science, because neither of the two bring
absurd results, at least as far as pure mathematics is concerned. And when, in the other
sciences, one of these two forms of thought seems to result in contradictions, the majority
of thinkers before the present time have preferred to support this inconvenience rather
than give up the intuition corresponding to the world.
It is always quite surprising that when all that can hide the truth has been
eliminated and we can finally look forward to contemplating the image clearly and
cleanly, it appears to us in a double form. The first follows the double image of the
simple object transparently, which proves it’s fellows with more emotion than I have
myself at this instant where, arriving at the term of the most scrupulous and most
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indefatigable examination, I had to resolve to show the reader the double mode of the
intuition of the fundamental principles of our science.
These two modes of representations as I mentioned, are connected to familiar
concepts, Idealism and Empiricism. To characterize both in a few words, Idealism
believes that the truth of certain limited forms of our ideas is required by our
understanding, though they lie outside of all perception and sensory representation.
Empiricism is the system of complete abnegation; it admits only as extant or as
corresponding to existence of that which can be perceived; it is not to be confused in any
fashion with classic pyrrhonism.
[-23-] We have indicated in the concept of limit the problem which reappears in
our present research; we must now show this problem more clearly.
In trying to form the question in the most general and abstract manner in the sense
of a rule of calculus which is continued indefinitely or of a so-called infinite operation,
we immediately also perceive the simplest form of argument of which the varied
formulae affected by it in analysis have not always been safe from the reproach of the
lack of rigor. And, in this simple form, it will be necessary to analyze this conclusion to
the complete satisfaction of our thirst for knowledge, which, on its side, must be calm in
the presence of an effective limit of our power of knowing.
225
Some principles and methods of analysis
where the concept of limit forms the essential hypothesis
We will designate by f(x) a collection of representations tied to another collection
of representations x such that to each particular representation x corresponds a particular
representation f(x). To this most general form of the concept of function or uniform
dependence also corresponds the most general manner of arriving at the mathematical
concept of limit: it is sufficient to imagine, under the representations, some states
numerically comparable to others of the same type, in such a manner that we can
substitute for these representations or these states their numerical measurements, which
we call their values. Then x designates a collection of magnitudes that can be expressed
in numbers. We then imagine, to explain the concept of limit, that they grow until they
finally pass all boundaries, a little like how the time which passes from a determined
instant to the present is extended indefinitely. There is however a difference, that is the
growth of our magnitudes that has no need of passing through all degrees, they can
proceed in bounds, like the jumping hands of certain clocks, or proceed by collections of
values succeeding each other without rules and in absolutely any manner whatsoever. If
there now exists a determined magnitude Y, which, x growing, f(x) approaches
indefinitely as x grows, in such a manner that there always exist values sufficiently large,
such that from these values the difference Y – f(x) becomes and stays smaller than [-24-]
any magnitude as small as you wish, Y is called the limit of f(x) and we write:
Y = lim f(x).
226
Thus x grows by any values whatever, 1 is the limit of
values, 2 is the limit of 1 +
x −1
, and for x taking all whole
x
1
1
1
cos x
+ +…+
. Finally, zero is the limit of
. On
2
4
2x
x
the other hand, there is no limit, for example, to the series of whole numbers 1, 2, 3, …
No limit, consequently, to the collection of all numbers x, when this symbol represents
every whole number or every fraction. Periodic phenomena, such as sidereal revolutions,
do not have a limit, assuming that the motion of the planets will remain in the future how
it has been in the past. The general concept of function also includes so-called infinite
operations, for example, infinite series and infinite products, continuous fractions,
integrals, etc. … In infinite series and infinite products, x designates, as in the example
cited above, the order of the last element to which we provisionally limit the operation.
At nearly every step in analysis we find ourselves facing this question: For some
function f (x), does it have a limit or not, when x clearly tends towards a determined
value or when it clearly grows indefinitely? Analysis poses this question for us so often
that, to resolve it, it has amassed over the course of years an extremely abundant treasure
of propositions, of rules and theories, of which several are so familiar to us that we apply
them unconsciously as rules of elementary calculation. In infinite operations, of which I
will speak further on, we call the rules which decide the existence or the non-existence of
a limit the (criterion of convergence or divergence).
Now, all the propositions, rules, or theories seem as though they could be
considered as transformations and purely mathematical reasoning, thanks to those in even
the most complicated cases, which always seem to have a certain extremely simple
227
criterion which immediate solve the question of the existence of a limit. This criterion is
provided by the following proposition which is certainly quite acceptable even when left
unproven:
If the difference f(x1) – (fx) (x1 > x), beginning with a sufficiently large x1, and for
arbitrary values of the difference x1 – x, stays below a given number as small as one
would wish, the function f(x) has a determined limit. [-25-] But if, for some value of x as
large as one would wish, we find for f(x1) – f(x) (with x1 > x) some values superior to any
small number whatsoever but independent from the variation of x, f(x) does not have a
limit.
This proposition is thus only an extension of the general rule for the convergence
of series given by Cauchy: For the series u1 + u2 + …. to converge it is necessary and
sufficient that um + um+1 + … un have the limit zero, when m and n grow indefinitely.
I call the proposition concerning f(x) the general principle of convergence and
divergence, because in fact I find that in all conclusive reasoning touching on limit that I
am aware of, we always end by requiring that in this rule resides the force of the proof.
One can affirm that this principle, which agrees with the definition of limit
according to which we must conceive of a magnitude Y such that y – f(x) falls below
every quantity when x grows, psychologically represents a large part of the mathematical
domain, at the very least the part of this domain that treats the inherent difficulties of the
concepts. In this sense, we can say that our research on the fundamental concepts of
mathematics accounts to the complete understanding of this principle.
This principle includes two parts, the one affirming the existence of a limit and
the other excluding it under a determined condition. The concept of limit is thus each
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time implied. In reality, the first part is more important because purely mathematical
paths lead from the first to the second. Now we can prove the proposition, as well the
fundamental rule of Cauchy, easily in several ways and certainly with the degree of rigor
can smooth out the proofs of other known principles. But if, in similar proofs, we return
more and more to the elementary ideas from which they were formed, and at each
conclusion we wonder again if it is satisfying enough, we then discover in each of them at
least one point which corresponds to a hole in the collection of our representations, where
without transition something appears whose existence we had never tried to prove. This
“something” is the limit. As close to us as it can be placed, an unfathomable abyss
always separates us from it; the continuity of representations is interrupted somewhere.
We mitigate this fact in saying: in being considered as a limit, by expressing the jump
that one makes over this abyss, the word limit indicates the point at which we are
conscious of a hole.
In fact, if we do not already know the limit of a fraction, [-26-] as in the simple
examples given above, but the condition posed by the general principle is fulfilled, and
we affirm that the function must have a limit this is the precise sense of what we have
affirmed: the symbol f(x) created or produced a magnitude which we never knew of
before and, which, without the operation f(x),would have no further need of existing. But
what is a limited magnitude limit thus produced? How must we conceive of it?
If, for example, the series 1 +
1
1
1
+
+
+ … has the number 2 for its
2
2 x2
2 x2 x2
limit, as we can easily establish in making known that this number satisfies the condition
229
of a limit – which is to say that the difference 2 – (1 +
1
1
+ … + n ), as n grows
2
2
indefinitely, becomes as small as we would wish – no man can still see the number or the
decimal fraction or any other thing which represents the limit of the following: 1 +
1
+
2
1
1
+
+ … No one sees the numerical result of the extraction of the square root
2 x3
2 x3 x 4
of 2, supposedly indefinitely prolonged, but we designate this result by
2 and perform
on this symbol all operations as naturally, for example, as on the number 2 itself. If,
however, we affirm that these limits exist, or that this statement is in error, or in need of
an explication, we vainly search for it in the works of special mathematics, and for a
stronger reason in the philosophical writings that will relate to them.
Nevertheless, the formation of the limit by the absolutely undetermined variation
of a function f(x) (this is what we have supported up until now) is evidently like f(x) itself
a very vast set of representations, which we will later strive to draw from the order of the
study, of the general principles of convergence and divergence. There is a particular case
of the general principle, ascending to which a function that always varies in the same
sense3, and which neither [-27-] grows nor diminishes beyond all limit, must necessarily
3
This is a simplified expression, designating a function which either only grows or only diminishes each
time it varies. This does not exclude it being constantly in whole intervals, or everywhere the same. It is
this that Lejeune-Dirichlet wished to designate in saying: The function never diminishes and never grows.
(Dore’s Report. BD 1, Dorstellung Willkührlichen Functionen durch Reihen). M. Newmann proposed not
long ago to call them monotones, functions submitted to the restrictions in question, and we will also make
use of this term where it will appear convenient.
230
have a determined limit. This case, as the simplest representation would already agree the
best with the search for the origin of the concept of limit, and at length presents itself as
an entirely special case as a an even simpler mode of constructing the limit, thanks to
which as I will also show in the chapter cited, the gap that I have revealed in deduction of
the general propositions is found to be diminished. I have in view the customary
expression of that which we call the irrationals in terms of unlimited decimal fractions. It
is thus for us tantamount to the simplest expression of these things which still appear
definitively positive in the analysis of infinite operations.
As soon as the origin of the limit of the decimal fraction has no more obscurity for
us, the charm will be broken and analysis will be its master. It will then govern as easily
and surely in the immense variety of accounts of magnitudes as the theory of whole
numbers always has, in its narrower domain.
As we see it, the reflections that give rise to the manner of reasoning founded on
the concept of limit could already be solved by the most usual arithmetic functions, such
as the development of an unlimited decimal fraction. Only, in elementary mathematics,
unlimited decimal fractions serve at the most only for calculus, could this not awaken the
suspicion that all is not absolutely clear? The examination of the problem of limit which
will follow has rather been led by certain new and fearless combinations which arise
when we transport the infinitely small differences of the “continuum” of numbers into the
variables of functions, and it is precisely in these combinations that have appeared to give
rise to insurmountable difficulties.
In fact, Fourier, who, in giving to geometers numerous examples of discontinuous
functions, has put us on the path of the modern notion of the analytic function, perhaps
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knowing only the continuous functions like those which had been studied before him, and
only presenting interruptions of continuity at isolated points, to study the brusque changes
of value. This too restrictive representation of functions, as before, with the exception of
several isolated points, changes in a continuous manner like algebraic functions and the
simplest transcendentals; it is Lejeune Dirichlet, as far as I know, who first made this feel
insufficient. But these preconceived ideas on the path of functions did not disappear
before [-28-] continuous functions which did not admit of derivations, and the
everywhere discontinuous functions and nevertheless integrable functions which have
been saved for science, from the treasure of ideas that brought on the premature fall of
one of the most profound unfortunately are of the most laconic researchers of the century
by the worthy students of his teachings, and the others, by a happy circumstance.
Quick overview of the research which will follow
An inquiry as serious as that which we will make into the demonstration of the
concept of limit, reasonably obliges us to ask before all else: of what does a
mathematical proof consist? According to what criteria would it be declared satisfactory
or defective? However, we will leave these intricate questions aside. I believe that to
judge the value of a proof, today we rely on logical judgment, which started to become
some time ago much more delicate, from a mathematician’s point of view. The scientific
definition of mathematical proof will perhaps one day succeed by a view analogous to
that which began with the logical calculus of Boole, which Mr. Schroeder, in recasting it
has rendered it more accessible.
232
It is perhaps quite evident that, to prove or to conceive of a representation, we
must rely on an initial representation already in existence, or on a concept (that is to say,
on the set of qualities common to a class of representations), of the final representation,
which is precisely the new object to be proved or conceived of, and this is helped by a
chain of representations of which the successive and continuous generation never
surprises, never troubles the quietude of our attentive consciousness. For conception, the
mind return back from the new representation to the initial representation; for proof, we
follow the path in reverse. In any case, our study of (numerical values) requires that we
first of all place in full light the first representation or initial concept, which can only be
the concept of magnitude in a general and entirely particular fashion, or more precisely,
mathematical magnitude considered chiefly in relation to number.
After having solidly established the concept of mathematical magnitude, we can
attack our principle problem, the concept of limit, or simply, the study of the limit of a
decimal fraction. There, our thought must, as we have said, follow two different views.
We [-29-] will try to describe the two absolutely opposed models of intuition, though we
will not prefer one to the other. To develop them both in the eyes of the reader as entirely
independent from one another, with their irreconcilable propositions, I have adopted a
particular form of exposition.
What the reader would correctly wish is supposed by the thought in the following
case. Suppose I had a chance conversation with two friends knowledgeable about on the
fundamental intuitions of analysis, and that I had begged them to write down their ideas
for me in great detail. Suppose that they gracefully promised to do so. Then suppose I
had, with their permission, submitted to each of them the communications of the other,
233
and I had thus provoked this written exchange of opinions which I will take the liberty of
submitting to the reader, after having analyzed the concept of magnitude within these
communications. I will endeavor to make my two interlocutors argue with the same rigor,
and thus the reader will not have to depend on my good will in order to see the logic
behind the arguments of the Idealist and the Empiricist. The Idealist will have the first
word.
The idea, according to which following two modes of essentially distinct
intuitions for the fundamental concepts of analysis are not only indicated in what follows,
but methodically realized, can be understood in a general manner thusly: with regard to
limits, or hypothetical terms of our collection of representations that are only estimated
more or less, we have the custom of taking sides sorts of different sorts following our
natural dispositions, or our education; no logical reason fetters the liberty of our choice.
This idea also seems to be an excellent guide in the fundamental principles of other
domains in human knowledge, as I propose to show a little further on in a special work.
It is in the limit of the decimal number which the multitude neither sees nor
wishes to entertain a shadow of a doubt, as in the famous axiom of Euclid.
Such efforts have obliged us to confine ourselves to the most distinguished
thoughts to prove each of these truths, the evidence of which spontaneously appears to a
carpenter’s assistant! But the end and the benefit of this research will not only be a pure
geometric certitude, but a more profound understanding of the mechanism of human
thought. For, to find the origin of these natural geometric intuitions, we must analyze the
representation of space that the human mind holds, and to return to the manner in which it
has been acquired.
234
[-30-] The conclusion of the following considerations, which must serve as the
introduction to the general theory of functions, will be provided by the mathematical
theory of the concept of limit and of the general principle of convergence. It is only after
a sufficiently long journey on a philosophical sea that we will tread for the first time with
this question on mathematical ground, and the question will no longer occur to us
hereafter. This will be the result of our study of the concepts: to feel the soil we lack
under our feet.
235
CHAPTER I
On Magnitudes or Mathematical Quantities
INTRODUCTION
[-31-] By magnitude, or mathematical quantity (quantum, quantitas, Grösse), we
mean a quality common to objects of different types, in relation to which they are
numerically comparable, such as their length or their weight. Accordingly, all the
collections of representations which can be submitted to mathematical operations are far
from being included in this definition. Generally speaking, we must understand by
magnitude or mathematical quantity the collection of a suite of representations submitted
at the least to the following conditions. 1st: each isolated representation holds in this suite
a sufficiently determined place; 2nd: between the magnitudes of the suite or between
these magnitudes and those from equally ordered suites, there exist relationships which
can be combined to give birth to new relationships.
However, we hasten to say, with these general definitions, which we formulate in
such a manner as to not let any particular case escape, we advance hardly at all. For, to
make one’s way from this to a clear and precise idea of the concept of mathematical
magnitude in the ordinary sense which is, and which will remain, the fundamental notion
of geometry, mechanics, and without doubt, also of abstract analysis, we would need to
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restrain the general definition to that which is adapted to the desired concept. This
manner of proceeding, which is found in several authors’ works, has a primary defect. It
is this, at bottom: that to know where to stop in restraining the general definition, we
must already be in possession of the final concept. Also the results obtained in this
manner appear to me not very satisfying, somewhat vague and even contradictory as far
as they are based on different preconceived ideas.
[-32-] To understand to the bottom the powerful concepts which control all
thoughts like those of space and time and also concepts less vast though still well
bounded, such as those we are occupied with, it appears more natural and even more
interesting to proceed in the opposite manner, and try to return to the origin of the
concept, to examine attentively by which abstractions it can be formed, to pursue it in the
different domains of knowledge where it is manifest, and to finally solidly establish the
common characters of these different manifestations. It is only like this that such a
concept of rich and delicate ramifications, like that of mathematical magnitude, can
finally be freed of all accessories, and this must be our goal. For the grand concepts
present in general two distinct states of development of which the first; is common to all
men, and the other is of a scientific nature which tends to an exact determination of a
common concept. This separation can be quite difficult, it is true; such as, for example,
the concept of the organized subject-matter, or that of the vegetable or animal realm:
whatever the difference is between a lion and an apple tree, science has still not
succeeded in tracing the line of separation between the animal kingdom and the vegetable
kingdom.
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We give in this work two examples of the method indicated: that which will
follow on the concept of magnitude, and further the examination of the concept of limit.
But we intend to submit elsewhere to a similar analysis other concepts, to know those of
space and time and the mechanical concepts of force, etc.
As for the present work, we will follow soon enough the path to a fundamental
form of the mathematical concept of magnitude which dominates not only the exterior
world, but also the interior life of the soul.
Linear mathematical quantities – this is the name I give to this fundamental form
– are the proper roots of analysis, by which analysis continually draws new nourishment
from its native soil, the study of nature. But, on the other side, we will know more clearly
the essence of mathematical magnitudes of another species if we compare them to linear
magnitudes, and if we find by which properties they differ.
Mathematical magnitude can, by its nature, take only discontinuous values, such
as the number of objects (Anzahl), unless it corresponds to a type of representations
passing in a continuous manner from one to the other, as takes place with lengths. This
traces for us, it is true, a [-33-] line of separations, we will soon carry out our
observations from each side. This line, however, as we will soon recognize, does not
realize a proper division in the concept of magnitude, because the continuous
mathematical quantity essentially can only serve to measure after the introduction of the
concept of number, and because the concept of continuous magnitude in its scientific
development assumes the concept of discontinuous magnitude.
1. – Discontinuous mathematical quantities. – Now, to come immediately to
discontinuous mathematical quantities, the concept of the number of objects ( I wish to
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restrict our present considerations) to this type of discontinuous magnitude has its origins
in the representation of the isolated state of the objects of perception; and we clean up this
concept with the help of words or signs by which we express the numbers of objects and
which are called the numbers.
The concept of the number of objects is entirely independent from the type of
object. Raphaël, a theorem, and a cannon are together three objects. The number of
objects is thus so to speak that which subsists in our thought, all that is distinguished
when the things disappear, and there is nothing more in the mind but a representation,
which knows that the things are distinct; it measures how many times our consciousness
has received a distinct impression. It follows immediately from this that separate objects,
which are identical or when they scarcely differ at all from each other, immediately
awaken the representation of number, for the impression of differences does not need to
be obliterated, it is not even formed. This supposes however that the attention is not
absorbed by the geometrical configuration of the objects, a stronger reason that they are
not at first blush understood by their order.
2. – Elucidations on the formation of the concept. – The type of origin, which
will be indicated, of the concept of the number of objects, is common to nearly all
concepts. Nevertheless, it seems proper to commence from a series of determinations of
concepts, to say how I represent to myself the formation of the concept. Concepts begin
where traits common to a group of representations arouse and capture our attention. The
variety of these representations is erased or, as Jean Müller wrote, is obscured before our
consciousness. When the ensuing concept (i.e. the collection of these common
characteristics) persists in our thoughts, it is fixed in its essential traits, in our
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representations and in our memory, to the word or sign by which language or science
designates it. If we wish to return to the real content
[-34-] of the concept, we make reappear in thought one or several particular
representations of what was abstracted from the common characteristics. Also, we can
observe for ourselves which concepts exist for which neither a word nor a sign are
attached to a certain representation which emerges in a particular fashion from the group
to which it appeared, representations which serve as signs. It is here where, in a general
fashion, the formation of the concept and the truest form of the concept debut in those
who think without the faculty of speech.
The characteristics common to a collection of concepts are reunited together to
form a new, more general concept, and by this an even poorer one, and so forth.
3. – Discontinuous mathematical quantities (continued). – The concept of
number expressed by figures or by words belongs by nature to those concepts which are
the most detached from the real representations which engender them. This is related to
the fact that a real representation cannot be made from every small number. There are
few men who, at first glance, can recognize a number of suitably chosen objects (for
example, balls of the same form and color), if there are more than five or six. I admit
here that the geometric order in which the objects are found is important in different
collections of objects, whether for example they are arranged neatly in a straight line. I
have heard that Dahse could at first glance count, a remarkably smaller number of objects
when they were arranged in a uniformly straight line, or even in a circle, that lacks a clear
point of departure, than the number of objects he could count when they were distributed
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on a plane surface, forming irregular geometric figures, which would render them easier
to count.
Thus we admit we have a clear representation of numbers up to around seven, we
can then, up to this, make the concept of the number of objects correspond to a collection
of particular representations, as we can make the concept of oak tree correspond to the
representation of all oak trees that we have in our memory. Thus this figure we have
from real representations is of larger or smaller pluralities, while the representations of
numbers a little larger are attached instead to the representation of enumeration; 31, for
example, is not a representation pulled directly from perception, but it assumes the
preliminary fact of enumeration.
[-35-] But the enumeration itself, notably often one passes the number 20, which
is the number of digits on hand and foot, requires a system of notation and
consequentially a certain degree of scientific development.
The concept of number thus shows clearly two stages of development; the second
of which carries the beginnings of science, and is followed by a third completely
scientific stage.
At the first stage of development, we find the most basic beginnings of the
measure of pluralities with the help of numbers; at this stage, number is connected to
representations pulled directly from the perception of all small numbers, perceptions even
animals can have when they must defend themselves against several enemies instead of
only one. According to similar reports on non-civilized people in the current epoch, who
are similar to people from all epochs previous to civilization, this primitive numeration
simply consists of the comparison with the numbers of digits on the hands and feet. We
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still have evidence of this detestable heritage of our fathers in our decimal system of
numeration. The subsequent stage of the development of the concept of number furnishes
the unlimited set of whole numbers, determined by the circumstance that each number
represents a collection containing one object more than the preceding one, and has
established in this set points of location which permit the measuring of every plurality.
Finally, the scientific development of the concept of number is marked in the
research of relations between whole numbers, and has finally led from the simplest rules
of calculation to the theory of numbers. In truth, however, the path is far from being
direct. Much to the contrary, the beginnings of the scientific development of the concept
of number proceed like the development of the concept of continuous quantity; both are
first developed and formed by the requirements that they have shown – in regards to each
other – until a more profound thought is granted as agreeable to the good properties of
numbers in themselves, and which, over the course of years, has given rise to the multiple
seeds of a science which, in the last two centuries, has become so richly developed.
4. – On continuous mathematical quantities. – Here are some examples of
continuous mathematical magnitudes: length, surface, volume, weight, time, speed,
force, temperature, intensity of light or of the sun, electric voltage, force of electric
current, etc.
[-36-] When we speak of mathematical quantities, we first think only of
geometric quantities, and, particularly of the straight line limited in length, to which we
seek to relate other quantities because it is incontestably the simplest representation of its
type, the most invariable, and the most open. It is not a representation, in the sense
already given above in the discussion of the number of objects; it is more properly a
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concept but which, here, unlike to number, is most often close to one of these particular
representations from which it the thought this representation supplied has been drawn
from. Thus sometimes thought of the line represents the end or the limit of a plane;
sometimes, the thought of the line belongs to the representation of a thread or an arrow or
a spoke. Finally, one also sees it as the trajectory of a traveling point, such as shooting
stars.
The continuous mathematical quantities I have cited have first of all this in
common: that their measure and their comparison depend on visual perceptions,
ascending to which their comparable or measurable quality always becomes a rectilinear
expanse, and they are left, like this, to be separated and combined by addition. We
observe, to explain this, the measurable representations of geometry.
5. – Characteristics common to the quantities already cited and to geometric
quantities. – In the mosaic of a visual field appear images such that each morsel that we
would single out somewhere possesses the same properties as the complete image. The
most immediate intuition of such properties is offered to us by the image of the limited
straight line. But it would be an error to want to affirm this also of an arc of a circle that
hasn't been straightened, because the relationship of the cord to the arc does not remain
the same when the arc becomes smaller and smaller; rather, in a prolonged division of the
arc of a circle, the representation ceases bit by bit to be that of an arc and is changed into
that of a straight line.
A plane, on the other hand, uniformly colored and illuminated, has this property,
like the straight line does; only this is no longer a fact of direct intuition. The analogy
between measurable properties of straight lines and the representation of measure that
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offers such plane surfaces is rendered possible by the intervention of the new concept of
area, i.e. of the exhaustion of a plane limited in some fashion and of the conveyance of
these portions of surface within the frame of a rectangle, for example, with a fixed base,
such that the height in its variation would determine at each instant the area of the
rectangle. By this [-37-] "exhaustion", the comparison and the measurement of the
surfaces are seemingly related entirely to those of the lengths. If we then imagine that our
visual representation is made up by the representation of space, then the concept of
volume appears, the measurement of which we express with the help of a length, exactly
as we have done for the area. This concept, like length, has this property: that the parts
are of the same nature as the whole to which they belong, and as any other thing of the
same type. In the other mathematical quantities already cited we equally and easily
discover the length which supplies the measurement. To cite several familiar examples:
the arc of the circle that describes the hand of the clock, once adjusted, makes the time
depend on a length; the gradation of the scale which see-saws to measure weight; the
force growing proportionally to the energy of the phenomena of pressure and movement,
which for their part can be immediately expressed in length, etc.
Now, in the case where our perceptions and observations bring us a type of
representation, only differing among themselves by which is greatest or the least, we feel
provisionally satisfied when we have posited the diverse degrees of this representation in
determined relationships with measurable representations of geometry, when,
consequently, we have found the length giving the measurement. This appears to us, in
fact, as the first step made towards the mechanical knowledge of things. We only do this
to follow our tendency to turn from what is new and complicated, these things which
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disturb the quietude of our soul, back to vulgar and familiar things, among which must be
counted first in line the measurable geometrical representations. For, comparing and
dividing extensions is for us so natural that these facts, like the first stage of the concept
of the number of objects, are perhaps not unique to man. The concepts of area and
volume truly appear to the first acquisitions of the human mind. The original form could
be abstract away from observations without numbers. The need of fabric for clothes and
for tent-covers, the capacities of different vases or other objects of the same nature, the
seed necessary for the field, all have different forms and magnitudes; thus, by the
progress of the state of the society of man, the concepts of the landowner and of
conventional measurements and of a pile of similar things have made area and volume the
most fundamental forms of the intuition of our mind. The child acquires these concepts,
if they are not innate, by, for example, [-38-] tearing up paper and by playing with vases
that the child fills with sand or liquid.
The measurable representations of geometry thus form the point of departure to
which our thought returns ceaselessly in its rigorous deductions. This assertion will
certainly not encounter serious contradictions.
6. – Introduction of the concept of linear quantity. – The quantities cited up to
the present thus have a remarkable common property; they can be related to lengths; their
differences, their divisions and their multiples are again quantities of the same type, as
lengths; they are, as lengths are, susceptible to becoming very small or very large; like
lengths, they are comparable, measurable. I will name the mathematical quantities of this
type: linear mathematical quantities. But it appears suitable to develop this concept by
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relating it to the case in which we make linear quantity correspond to a length that we are
never obliged to consider as unlimited.
Thus suppose we establish a relationship between a certain quantity and one of
these linear quantities described below, for example, an undetermined length, a
relationship such that this magnitude would be a continuous function of the linear
quantity (such that this relation could be observed or that it would be the consequence of
an argument, or finally, that it would be an example imagined by us for some purpose).
If this magnitude grew or diminished at the same time as the length to which it
corresponds changed by a magnitude of the same type, and if it is supposed continuous,
that is to say growing by degrees as small as one wishes, and it begins with zero, it
follows that it must then be considered as a linear quantity, and it must not grow
indefinitely. We could, in fact, make it correspond point by point at least to the limited
straight line and, as such, with the exception of the extension, it will possess the
properties of linear magnitude.
The concentration, i.e., the relationship of a quantity of a substance dissolving in a
fixed quantity of a liquid, furnishes us with an example of limited linear magnitudes. It
can vary between zero and the infinite, but also between zero and a finite value. It can
also, as with ether and water, be necessary to consider the two relationships separately, of
water to a fixed quantity of ether, and of ether to a fixed quantity of water, to make the
relationships of these quantities linear according to our definition. Another example of a
limited linear magnitude is the probability of a sharpshooter hitting his goal from a given
distance [-39-]. If we consider as the relationship the number of shots that miss at this
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distance to the very large number of shots fired, the difference of the two similar
probabilities can again be envisaged as one probability.
7. – That which can increase gradually following the extension or the
intensity belonging to linear quantities. – All the types of quantities that we can
consider as mathematical quantities are far from being linear, so thus our job is first of all
to acquire at least an approximate idea from the scattering of linear quantities in the
different domains of human knowledge. This task is facilitated by the following remark:
We are accustomed, we know, to dividing types of quantities into two categories
according to whether they increase due to extension or intensity. The multiplicity is
certainly not thus exhausted, but what is important is that all collections of quantities
distinguished by extent or intensity can be considered as relating to linear quantities.
I say relating to linear quantities, for it is quantities varying according to
extension, we cannot, for example, consider as linear a continuous collection of similar
triangles, their differences between them being, something undetermined, instead of the
differences being other similar triangles. But that which distinguishes and determines
triangles, perimeters, heights, surfaces, etc. … is of a linear nature. Thus we can say in
general that which changes only according to extension can be reestablished as linear
measurement.
On the other hand, the collections of continuous quantities gradated according to
intensity, when they can be conceived of as mathematical quantities, are always linear.
The condition that a collection of quantities is mathematical requires only that the
individuals of the collection can be sufficiently defined themselves and that we could not
actually determine them by making them correspond to a linear measure, or that we could
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not even suspect how we could ever restore them to linear quantity. This granted, we
must first seek to clearly show what is meant by "difference of intensity." This concept is
not in truth as easy to analyze the difference of extent. But this is precisely where we do
not have the right to forcibly create that which is not even familiar to us. Thanks to the
following explanation of this concept, I have never found myself at a loss.
[-40-] We use the expression "differing in intensity" when we understand the
variations in magnitude (growing or shrinking) as magnitudes of the same type; when
consequentially the differences in a type of magnitude appear to us themselves as
magnitudes of the same nature.
But, it is one of the principle properties of geometrical linear quantities that, we
do not distinguish them by intensity, but by extension. This designation differs from the
relations between quantities of the same type, of which some are said to be gradated
according to extension, and others according to intensity, and therefore does not
constitute a difference in the mathematical character of these magnitudes, and what we
have shown plainly proves the linear nature of mathematical quantities which differ
according to intensity, when, of course, we suppose them sufficiently determined and
continuous. In fact, when the variable changes by differences, so to speak, which have
the same nature as the variable, it must, if it is continuous, begin with zero; further, being
formed by increases as small as one would wish, it also necessarily admits of multiples
and of divisions of the same type, and this is precisely our concept of linear quantities.
To elucidate what precedes with examples, one says of temperature that it is
gradated by intensity. Accordingly, we must here distinguish temperature sensed
physiologically, that is, the sensation of heat as it relates to the senses which we will
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discuss further on, and the physical temperature which is the expression of the intensity of
calorific movement, and which is the linear quantity properly so called.
Finally, we again mention hardness which consists of a resistance, still not
rigorously analyzed from the physical point of view, which solid bodies present when we
want to make them go through variations of form which go beyond the limits of elasticity
(either by unraveling them, or by condensing them). If this is a question of difference of
intensity, we clearly understand this as forces of resistance, of which hardness is the
manifestation of their variation of one material to another or variations in the same
material (for example, when affected by a change of temperature) these forces of
resistance undergo changes which are themselves forces of the same type and to which
correspond certain degrees of hardness. Briefly, hardness is to the forces of cohesion
what temperature is to caloric movement.
I have chosen these examples because the precision of the explication that we
have given to the concept is perhaps not [-41-] as evident in itself as in the cases which
are ordinarily presented.
Thus linear nature of collections of quantities increasing according to extent or
intensity is sufficiently founded; we now seek to frame a general view of mathematical
quantities belonging to the different domains of thought.
8. – Quantities of the external world. – First, the world of perception which we
call the external world. On the first line are found the mysterious first causes that we call
forces and, in putting them first, we cross the limits of the domain of perception and we
thus find ourselves in the middle of the empire of the creations of human thought. Under
the influence of an interior growth, we conclude from the phenomena to the existence of
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primary forces, and then we save nothing from them, we must see in these phenomena
their complete effects. The effects are thus in their representation equivalent in quantity
to the force which produces them. For these effects, pressures, tensions, movements, are
linear magnitudes, and are thus from the same forces and because we have cited them as
such.
As far as we are, and always are, from being able to represent mathematically all
the phenomena of the external world, no type of magnitude seems to be able to be
offered, which could not probably one day be comprised in the concept of linear
mathematical quantities, or better yet, which one day could not be revealed as a linear
quantity. For wherever we penetrate, every variation appears graduated according to
extent or intensity, and we will consider as parallel quantities when they are susceptible to
a precise definition (in the same way that this has been more closely examined), as
essentially linear, even if we have not already reestablished them, as for the duration of
their last geometrically and mechanically linear variables.
9. – The quantities of the world of internal perception. – Sensations gradated
according to intensity. – We study several phenomena of the internal life of the soul as
follows by considering them as quantities from the point of view of their properties. The
sensations by which stimulus or irritation are revealed to us are quite the simplest
phenomena which must be presented here, and offer a multitude of very instructive types
of quantities.
In the first place we find sensations gradated according [-42-] to intensity, such as
sadness, the sensations of the skin, and the sensations of intensity which come to us
through the senses.
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In applying to the sensations our relative proposition on the quantities which
differ in intensity, we recognize that these are also linear quantities, when, properly
understood, they can be considered before all as mathematical quantities. They are
represented well, in fact, as if stimuli of the same type were increasing, a sum of
sensations produced respectively by the different forces of stimulation, formed equally
from a single and even more intense sensation. Thus, it is imagined that an impression of
livelier heat which corresponds to an elevation of temperature is produced by an increase
of sensation, which is itself a sensation of heat, of the sort that the impression of heat
becomes like an interior image of temperature.
There remains nothing but to ask if the intensity of the sensation is a mathematical
quantity, if it admits of particular values sufficiently enough determined to be accepted as
a defined function of measurable linear quantities. The difference of the forces of
sensation which correspond in different circumstances to the same stimulus, the brevity of
the time during which the intensity of the force of the sensation produced by a stimulus
stays constantly the same, the influence of fatigue, and above all things, the lack of
methodical observation of the self, are principally ascribed to the sensations of odor and
taste an appearance of inappreciable instability which can draw us to immediately give a
negative response to this question.
The scientific investigation is far from being jammed by such an appearance of
instability; it searches so much the more to fix the instability. There exists without doubt
in the control organ a state corresponding precisely to the stimulus, a state that our
conscience must train to evaluate and distinguish from neighboring influences, and can
certainly learn to know it bit by bit. Besides the sensations of intensity the sense of
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hearing and of sight, which yield to a longer observation and to easier comparison than
those of taste and of smell, and offer in general more certainty and stability. It seems thus
by no means improbable that, in all the types of sensation or in any of those which are
gradated according to intensity, we could find the conditions under which it is possible
for us to stir up at each instant intensities of particular sensations which would
correspond to equal quantities of stimuli and which appear to us absolutely equal. It [-43] suffices to recognize in the sensations mathematical functions of the quantities of
stimuli, especially if we think of the sometimes astonishing finesse of our senses when
distinguishing degrees of stimuli by the intensity of sensations, finesse which, as is
apparent in numerous examples, if not already naturally given, can be carried to a high
degree through exercise.
After this we could be authorized to consider the sensations gradated ascending to
intensity as linear quantities, that is to say, as functions of quantities of stimulus which
begin at zero and of which the differences, the divisions and the multiples are in their
turn, however non-expressible, forces of sensation. But how do we clarify their linear
essence, if there would exist some principle which enabled us, given two sensations, not
only to recognize the stronger or the weaker, but even to numerically calculate their
relation?
For sensations which vary ascending to intensity, we do not possess a precise unit
of transportable measurement, which would itself be a similar sensation that could be
perceived at the same time as a sensation in order to measure it; we can only compare the
intensities of sensations by recollection. In the theory of the relationship of measurement,
this is essential difference between vision and, for example, taste. For, suppose our
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perception of vision had the same restrictions as our perception of taste, we would not be
able to see two lengths at the same time, a single one would appear in the uniform field of
our vision: surely we would be as incapable of doubling an idea of length as we are in
reality incapable of representing a double sweetness. We cannot however conclude from
this incapacity that the cerebral phenomenon which corresponds to our sensation of
sweetness does not furnish a double idea of its energy.
We will push the opposition of two types of sensations still a little further. The
perception of a line of limited length will be transported from the retina to the brain via a
nerve without losing anything. What is added to the visual perception of the limited
length is the simultaneous vision of several lengths, or the possibility of their respective
replacements in the visual field; and this will furnish for us the intuition of double lengths
and more generally of their numerical relationships. Finally, the sentiment that this
intuition corresponding to reality comes only from the concept of space, that is from the
subsequent apparition of displacement according to the third dimension.
[-44-] In the sensations that differ in intensity here is how things happen: on the
path of nerves which goes from the mucus membrane of the mouth, for example, to the
brain, a fraction of the energy of the stimulus is likely lost; that which arriving in the
brain matter is resolved in a movement in which energy is in its turn partially transformed
into something else, while the last part of this energy comes entirely and completely to be
revealed to the consciousness as a sensation and to measure in proportion as we feel, want
it or turn our attention to it, it is found in the middle of the wheels of our thought.
This part of the energy is the linear quantity and it will always take (cœteris
paribus) from equal quantities of stimulus the same specific sufficient value in such a
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way as to be able to be assimilated to a mathematical quantity. Could we ever measure
it?
The problem that life ceaselessly presents us is that of the estimation and
calculation of the stimulus by the sensation, and in this, as we create it for our perceptions
of dimensions thanks to the intuition of space, we judge that our calculation of quantities
of stimuli founded on varied experiences are also in agreement with that which appears to
us to be reality.
However, Théodore Fechner had a bold idea the day when he set up the inverse
problem of expressing the intensity of the sensation as a function of the intensity of the
stimulus. If his result cannot yet appear very certain, it incontestably restores to him the
honor of having the first scientific research into the measurement of sensations.4
4
The reasoning which leads to his psychological law basically consists of this: If a stimulus begins at a
value β and grows beyond it, we will not perceive the variation, in spite of all our attention; before the
stimulus has attained a certain force β + ∆ β where ∆ β depends, following experience, on the force of the
stimulus β which already exists. As for weights set against the skin, and taken as stimulus, E. M. Weber has
found that ∆ β is proportional to β. In applying this law, beginning with the value β = o, there corresponds a
value ∆ β = ∆ βo which is called the threshold of the sensation. The important fact from the point of view of
the theory of magnitudes is that a parallel law can be adopted at least between certain limits, of a sort that
we will not occupy ourselves with here, from the extension in which the law of M. Weber responds to facts.
M.Fechner assumes, and this point is the foundation of his theory, that the growth ∆ γ of the force
of sensation γ which is precisely perceived, is independent from γ; so that γ is formed by the addition of the
equal growths ∆ γ of sensation. But since
∆β
β
∆β
β
is constant, he can then conclude that ∆ γ is proportional to
. It thus appears entirely admissible to treat ∆ γ and ∆ β as differentials, and he concludes that γ is
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[-45-] 10. – Sets of sensations not gradated according to intensity which are
actually mathematical quantities but are not linear. – Outside of the kinds of
β
proportional to log ∆β , which is his psychological law. The quantity ∆ γ is hypothetical; this is why,
o
lacking any measurement of sensation, we can only affirm that we would be in error finding more probably
the proportionality between the forces of stimulus and those of sensation. The theory of Bernstein appears
to me most particularly spoken in favor of the law of Fechner though in an indirect fashion. According to
all probability there is a correspondence in the central organ between the peripheral terminations of the
nerves which receive the stimulus and a halted system of functions which can be considered as the image of
the external world coming in direct relation with our body. By word "external world" we must here
understand all that exists outside of the brain.
M. J. Bernstein has tried to give us an idea of a central function of this type which would
immediately measure the force of the stimulus and explain the phenomenon of localized sensation.
(Untersuchungen über den Erregung'svorgang im Nervem und Muskelsysteme, p. 163 ff). He has
subsequently undertaken to explain through experiences the localization of sensations of the skin by a
certain cerebral mechanism, and his ingenious constructions have filled out Fechner's relationship between
stimulus and sensation. The experiential confirmation of the conclusions that J. Bernstein has drawn from
his theory can thus in fact be considered as a confirmation of the law of Fechner; although it is nearly
useless to penetrate further into the phenomenon which separate the phenomena generator from the
sensation in the brain and the apperception itself, I would make the following remark: I represent my
sentient self as a psychic phenomenon existing in isolation which either, following the need of thought,
collects this and that in the mechanism which gives the image of the exterior world, information which is
presented to itself, or, under the influence of particular and more powerful stimuli, exerts itself on the
mechanism, involuntarily closing it. Thus, the state of the mechanism which corresponds to the stimulus
must go beyond the threshold of Fechner, that is, it must reach a certain degree to be able to penetrate into
255
sensations growing according to intensity, we know there is a class of sensations
composed of individuals which furnish impressions similar to one another in a certain
relation, but of which the difference is not clear, as it is in the sensations studied up until
now; the spontaneous representation of each individual sensation is a sum of elemental
increases of the same type.
[-46-] Like the set of sensations not gradated according to force, we can list
among others: the pitch of a sound, the color of light, the timbre of a sound.
The pitch of a sound constitutes a sort of transition. We can never consider it as a
linear mathematical quantity, because for that, it would have to be a question of a
difference between two pitches of sound, one of which would be a pitch of a sound and
the other a fraction of the pitch of a sound. In truth the physical pitch which is given by
the duration of the vibration, and which is thus a simple duration of time, must
incontestably be treated like a linear quantity, if we see thing no other than a simple
duration of time or a number rendering it equal to the number of vibrations in a second.
The difference between two pitches is thus the difference between two numbers; but the
physiological pitch, i.e. that which is felt, even when it is susceptible to a more precise
and more constant determination than the intensities of sensation, does not admit of
parallel differences, and is thus not a linear quantity. Only, given the fact that these
impressions are only distinguished here by the difference in pitch that the majority of men
with any degree of culture know well, it has, a certain relationship (with the linear
the consciousness, a degree which depends entirely on the fatigue of attention, on natural dispositions, on
exercise.
256
mathematical quantity) which is missing in the other quantities cited, color and of light
and timbre of sound.
The colors of light are impressions of a completely varied nature, and before
Newton we had a presentiment of a necessary order in them; at least the natural
phenomena of dispersion had not to any knowledge been thus explicated. But in any case
the colors can be put in a determined order with the help of the fact that each of them
correspond to a particular value of a linear quantity, the duration of the vibration or even
an abscise of the specter that we will conveniently fix. Among the colors we establish
relationships in making each color correspond to its complementary one; these are
certainly mathematical quantities but they aren't linear.
In the sensations envisaged above the stimulus was directly given as a linear
quantity. This was either a mechanical phenomenon producing, for example, a sensation
of pressure or of sadness, or even, as with the sensation of heat, pitch, and color, etc., it
was a physical phenomenon eventually resolved into gradations of linear quantities.
In the sensations that follow we must consider the question from the point of view
of the essential nature of the [-47-] stimulus taken as quantity. For that of the timbre of a
sound, M. Helmholz studied more rigorously the nature of the physical phenomena which
take place in the air or in the mouth, and to which correspond a certain sensed timbre of
sound; however, by the confused nature of the phenomena, the relationship between the
timbre of sound and the type of vibration in the air is not yet ascribed to as simple and
precise a determination as in the pitch of the sound. My father established for vowels an
order similar to that of colors of the spectrum, which has been approved by the
257
phonologists.5 In any case, this is a first step in the view which can lead to a
representation of the sensation of timbre as a mathematical quantity.
Finally we must return to the sensations of the organs of smell and taste, as they
are different in nature. In a general fashion we have not yet discovered for them a
principle which can serve to classify either these stimuli, or these sensations. The
sensations of the organ of taste seem limited to a few types, whereas the sensations of the
organ of smell are of very numerous types; so that here a principle of ordination would
incontestably be real progress.
The quantities of sensation of the first type, which are distinguished by intensity,
are related to the quantities of sensation the second type, which are classified after their
nature, in the following relationship: each sensation of the second type represents a
particular collection of sensations gradated according to intensity. Every color, pitch,
timbre, every odor and every taste can represent a type of sensation, gradated according to
intensity.
12. – The dispositions6 of the soul, which can be understood as mathematical
quantities, are certainly linear. – We lay some stress on the phenomena of the soul.
After sensations come dispositions, if I can use this expression to designate the states of
the soul which engender desires and volitions, and which consequently determine acts, in
short, as Jean Müller said, and thus which also give birth to tendencies in a general
fashion. We cannot typically fail to recognize that joy and sadness present constant
5
Cadmus oder allgemeine Alphabetik von Fèlix Henry du Bois Reymond, page 153.
6
Stimmungen.
258
gradations which follow intensity, and that, in the oscillation of the disposition of the
soul, the same state is reproduced an incalculable number of times, so that in dispositions
one can speak of a certain equality. Everything called [-48-] passion, or affection, can be
more or less differentiated more or less; for example, fear, anguish, terror, anger, rage;
and, further, pleasure, sensual pleasure, etc. … The sensations and perceptions are in truth
sometimes to the dispositions in nearly the same relationship as the stimuli are to the
variable sensations, in that they directly produce a certain disposition: but in general
there is between two stimuli of the soul, the judgment that that makes it beyond our
power to make dispositions correspond to linear quantities. Nevertheless, they possess in
part some of their characteristic properties to a surprising degree. This is thus how
sadness or joy can increase by collections of small successive incidents, each of which
alone could bring a little sadness or joy. I am thus convinced that we are authorized to
properly speak of double joy, of doubly bad moods. However, the conception of
dispositions as linear quantities could be still more easily attacked than the mathematical
nature of sensations.
13. – The quantities which appear in mathematics itself are in part nonlinear. – The mathematical quantities cited above have been found completely in the
external world or in the internal world of the soul. We call similar quantities real*~~~.
But there are quantities created by human thought which are outside every direct
relationship with the world of perception. The logical process of the combination of
quantities, one of the favorite operations of the human spirit, leads to certain symbols
which in mathematics are called quantities, and which not only serve, like mathematical
signs in general, to summarize in a sign conclusions which reappear constantly, such that
259
the sight of an equal sign spares us the recapitulation of a series of conclusions, but which
also represents a discontinuous or an infinitely condensed set of mathematical signs of a
certain type.
These are of particular interest, these quantities which arise from the application
of mathematical operations beyond those cases where they are naturally valid,7 or else
they arise from the pursuit [-49-]of certain analogies, and those quantities which agree
with no numerical signification so that they can never be taken for linear quantities.
Here, it is clear to everyone that they are called quantities precisely because their origin
and their definition permits of calculation with them as with real quantities; most
frequently, however, on the condition that they restrain or alter certain operations to
which we can subject linear quantities.
In complex quantities, for example, the distinction between the greatest and least
vanishing point has recourse to the unit of measurement. "In the infinity of functions"
that which makes a function the greatest or the smallest it is not the difference but the
7
Hankel has called this the principle of permanence of formal laws. It is perhaps quite risky to create a
principle of something which is astonishingly successful, but by chance when, for the moment, we have only
been able to penetrate incompletely into the essence of the phenomenon. There is a multitude of cases
where the formal laws cannot be applied beyond the domain of validity, without leading to absolutely
useless results. For example, the formal law:
∞ du
∂ ∞
p
Σup = Σ
1 dx
∂x 1
is valuable in an enormous number of cases, but outside of these it gives nothing but a false formula.
260
quotient.8 Similarly all branches of mathematics are enriched each day by quantities
which, to distinguish them from those above, we can call analytic quantities.
14. – Finally, some mathematical quantities appear in human relations which
have nothing in common with those cited above. -- But in the main, analytic quantities
appear to also belong to the world of perception: the problems and methods of
mathematics satisfy all our tendencies to mechanically comprehend the world. Even the
most abstract doctrines that analysis and geometry have imagined eventually follow the
path traced by a physical problem, eventually spring from their own fountains, possess for
the intelligent man at least as much as methods which have practicality as their immediate
end.
When a combination of quantities appears new, it suffices to accord it a value, for
who can know if tomorrow it won't lavish upon us the most desired explications on the
very points we attend to the least? On the scientific needs of the future we are entirely
unable to judge with precision because, compared to the provision of unexplored things,
either within us, or out of us, [-50-] our knowledge and our perspicacity are an infinitely
small quantity in the most proper sense of the word.
On this remark, which will serve as a transition from real quantities to
mathematical quantities, it is not necessary to conclude that we must overturn the limits
8
We will show further on that these are infinitary quantities. It is principally the property of divisibility
which they do not possess in the same sense as linear quantities; considering that divisibility immediately
results in us being able to approach as close as we wish to every given linear quantity – like we approach 1
with the help of the set 1/2, 3/4, 7/8, … Leipz. Ann. XI Bd., Uber die Paradoxen des Infinitarcalcüls. Art. 9.
261
which separate scholarly games from mathematics, and that the combinations of the game
of chess should be considered as occupying the same scientific rank as the combinations
of algebra, for example, because no one can know what purpose the former will serve one
day. The inequality of the rank, that feel in general between combinations of chess and
those of mathematics, it appears to me, abstracts from practical utility, as follows:
We can only deny that the problem of the horseman's leap, and particularly that
which we call the endgame of chess, with its conclusion that is necessarily deduced from
the initial position of several figures, presents the proper character of truly mathematical
questions, in the limits of a field of very restrained combinations. But in this endgame is
mixed an element as necessary as it is, nonmathematical, which is to know the mistakes
of thought. The endgame properly so called does not appear to the mathematician
because it assumes players whose judgment is humanly imperfect. If we set aside these
errors, the game would be furnished of problems of combinations that would be serious
beyond a doubt, but of unprecedented complication. We would already have great
difficulty if we wished to explain what is understood by a perfect chess move.9 The field
of combinations is so restrained that it does not reward research of extraordinary
difficulty – no more than the similarly small number of new results, which are in the
distance. It is the same as an expert game of cards where one leaves to chance as small an
influence as possible.
9
Between rational creatures with perfect logic ability every game would amount to aiming at this type of
opening move.
262
The quantities which appear in games, distinguished by their values, have their
non-reality in common with the analytic quantities mentioned above, but, unlike those
with reality, they do not have a relationship as limited as many real quantities do; chiefly
they correspond to fields of an extraordinarily impoverished number of combinations; we
will give them the name quantities of games, and we will end here this review of the
different species of quantities in the world of perception.
[-51-] 15. – All that precedes shows that we have chiefly to study with as
much rigor as possible the concept of linear magnitude. – Here ends our review. After
having examined which paths necessitude the comprehension of mathematical magnitude,
we have learned to delimit this concept more closely. To summarize in few words our
acquired results, the rigorous science of nature strains without cessation to recognize the
same type of magnitude in everything which is variable, to know linear mathematical
quantities, joined to the concept of number, and the essence of things does not present any
invincible obstacle opposed to this tendency.
The fields of study, which uncover the obscurity of psychic phenomena, show that
we are near to several species of magnitudes graduated in the manner of linear quantities,
of the sort that one day could be clothed in the uniform of the measurement of length,
though they would be of an entirely different nature. The other magnitudes that we
encounter in the world of human thought are either the creations of our methods of
scientific reasoning, or else they serve as our diversion.
We have not yet been able to give a more precise specialized definition of the
concept of linear quantity. Their principle property is, we have shown, just as with length,
they yield to comparisons of measurement, and finally, they can be known as a length, -263
we have not yet indicated more precisely that which characterizes the measurement of
lengths themselves. We have supposed above that a certain homogeneity acts as an
indication of linear quantities a certain homogeneity, in virtue of which their differences,
parts, and multiples are still quantities of the same type. Finally, there has been a
question of the fusion of the concepts of quantity and number.
All this must be now explained with more care for the concept of magnitude is the
unique and necessary key to gaining the knowledge of other fundamental concepts of
analysis.
The concept of quantity such as we wish to describe now is it presented with three
different degrees of abstraction. First is a rough concept, implying a vague idea of
comparison, which very well might not be limited to humans. In view of measurement, it
is refined and receives the contents that we can consider as the origin of mathematics.
When, founded on the habitual and natural intuition of things, we have analyzed the best
possible refinement of the concept, it will seem, and this is at the moment our goal, it will
seem, [-52-] I say, that we will immediately attain the precise delimitation of the concept
of quantity and the last word on this will have been said.
But we will then discover that this has not simplified for us the difficulty of the
concept of limit. We will thus be obligated once again to show the birth of the concept
from linear quantity, and to pursue it this time until we find its most slender [most basic]
roots. We thus see that genuine penetration into the essence of the two concepts of
magnitude and limit is indivisible, or more exactly, the two things are equivalent, as I
264
have already declared some time ago.10 This third degree of the abstraction of the
concept of magnitude thus also helps us illuminate this remarkable opposition of two
fundamental intuitions belonging to man which we questioned in the preface and to
which we will return later on under the name of idealism and empiricism.
16. – A more precise definition of the concept of linear quantity. – Since linear
mathematical quantities the extension of which does not have a limit, behave as much as
magnitudes do, as for lengths, it is not necessary to constantly think of every multiplicity,
it suffices to think simply of lengths. However, we maintain the designation of quantity,
only employing the more explicit "linear mathematical quantity", when this character of
magnitudes must be emphasized.
The first degree of the abstraction of linear quantity is thus characterized:
Comparison without the representation of measurement.
I. – Linear mathematical quantities are either equal or unequal. They are equal if
their sensible manifestations produce the same impression in the same conditions. The
one is larger than the other if its sensible image could be diminished by exhaustion in
such a manner that it could exactly coincide with the other, the inverse thing being
otherwise impossible.
II. -- Among linear quantities of one type, for example, for all possible scopes,
none in particular enjoy a prerogative, and we thus do not present as necessary any limit
of the smallness or the largeness of a quantity.
10
I have commenced this study in the introduction to the article already cited, Ueber die paradoxen des
infinitarcalculs.
265
[-53-] III. – Two or several quantities of the same type reunited produce a new
quantity of the same type, larger than its composing elements. On the other side every
quantity can be divided into a number as large as we would want of quantities of the
same type, each of which is smaller than the non-divided quantity.
Accordingly the first part of property III gives the representation of the sum, the
second incontestably contains the germ of the comparison of measurement; for that which
occurs here is not a casual assembly of quantities, but an assembly which furnishes a
certain result and gives rise to the concept of difference in which we distinctly find the
character of measurement – we arrive thus at the Comparison with the representation of
measurement:
IV. – When one quantity is larger than a second, there always exists a third of the
same type as these two, which, united to the second, reproduces the first.
Supposing the second and the third are equal, we are then conducted by
proposition IV to division with the property of measurement. Since we here attaining
division into two equal parts, we could, being given a quantity, find a second smaller one,
such that the difference would make the half the smaller one, etc… Nevertheless, it is not
at all the case that the human race has acquired intuitions common to every man, which
we will now reach. But it suffices for our research to be able to suppose them common to
all men, and it is completely non-useful for us to be lost in conjectures on the fashion in
which they have been acquired.
If we unite equal quantities, choosing randomly the magnitude and the number of
these quantities, the quantity which represents the sum is equally arbitrary. It strangely
occurs to us that, as large as a quantity is supposed to be, we could compose one which
266
surpasses it in magnitude by adding equal quantities to it, among which the magnitude or
the number could be arbitrary. We then have the representation that any quantity
whatsoever could be rigorously composed with the help of smaller quantities equal to
each other, from which we can arbitrarily choose a number and which, as this number
grows, the smallness of the parts falls underneath any limit.
We possess these representations. They are as natural and familiar to us as those
which concern our physical needs. Whether this form of thought is innate or implanted in
the individual by transmission, or acquired by a personal observation, [-54-] is as
indifferent to us as the manner in which it produces the simplest representations.
The properties of addition and division of the quantities are fundamental
properties of analysis, but in particular this is true of unlimited divisibility of linear
mathematical quantities, which contain in nuce the concept of limit. Let us add now to
the properties given above:
V. – We can always add up equal or unequal quantities, of which the smallest
must not fall below a given quantity as small as one wishes, in sufficient number to attain
a quantity which is not less than to any other quantity of the same type, given in advance.
VI. – A quantity can be divided into smaller parts in a multitude of ways, and
among these divisions is distinguished those made in two or three, from those made in
one larger number of equal parts. The division of a quantity can be continued until all the
parts become smaller than a quantity of the same type assumed to be as small as one
wishes. But as far as one can suppose the division can be pushed, the parts are always
of quantities of the same type.
267
I wish moreover to remark expressly that in the preceding list, I am not proposing
to restore the properties of the concept to their necessary measurement and to thereby
acquire the smallest possible number of perceptions, under which our thought authorizes
its production to furnish us with the representation of the linear mathematical quantity, as
we doubtless drew upon varied and numberless perceptions. – This would exaggerate a
study of an extraordinarily delicate nature, which would mislead us. We could moreover
only respond to the question concerning the smallest necessary number of properties that
a deep and complete study on the concept of magnitude. My goal has only been to give a
definition such that each recognizes the concept that he possesses of the linear
mathematical quantities of geometry, and, following the extension of these ideas, the
innumerably many properties of other linear quantities which are offered in the most
diverse branches of science.
As far the first rough representations of linear mathematical quantity, that is to say
probably the oldest and most widely prevalent fundamental representation of things,
differing only in largeness or smallness, they are immediately to a more rigorous
measurement, which is incontestably joined to the concept of the unity of measurement,
and the unity has been [-55-] followed by fractional numbers. The fact of measuring with
unities has produced the science of measurement, and this, ceaselessly invoked in
research and observations in the sky and on the earth, has generalized its problems,
aggrandized its views, perfected its methods and always augmented the set of facts and
theories, from which mathematics began in the last century as an independent science.
We thus consider again the union of concepts of quantity and of number: these
preliminary explications on the concept of quantity will find their conclusion here.
268
Quantity, number, and literal formalism
17. – The unity and the one. – To compare more than two mathematical
quantities, and also to be able to express with perfect precision the variation of the
quantities, it is necessary to fix the unity of magnitude, or simply the unity, i.e. a
determined quantity among them of a certain type to which we compare the others and
which serves to measure them. It is not only the divisibility of magnitudes which
produces the concept of unity, it is also, as I have observed, the attention fixed
simultaneously on several quantities.
A certain degree of development of the science of measurement will require one
day the indefinite11 division of the unity to be able to make comparisons in a rigorous
fashion with the help of the unity of different quantities. This is evidently the origin of
fractions which, in the representation, in effect always returns to a certain division. We
will have assuredly distinguished here at least two degrees of development of the concept.
The simplest fractions probably correspond to a degree of culture scarcely more advanced
than the whole numbers; the daily need to divide the food or the treasure or any other
thing must have led to them; while the fractions which have the larger numerators and
denominators, and above all the concept developed from the fraction, would already
belong to the science of measurement. It is instructive to oppose the concept of quantity
in its relation to fractions, and the concept of whole numbers.
The thing which serves as the basis of the concept of whole number is one, which
is profoundly distinct from unity. As one is the point of departure for the whole numbers,
11
[Translator's note: "indefinite" could equally mean undefined and unlimited.]
269
unity is that of the fractions. Every unity is a one, but not everyone is a unity. And it is
happy that for these two absolutely distinct concepts we have at the same time two words
at our disposal. Moreover, these two sorts of [-56-] numbers, whole numbers and
fractions, are completed by forming one and only one set through the medium of the
concept of measurement. For although the idea of whole number comes into being
independently from the concept of quantity, and has nothing directly in common with it,
the concept of quantity as we have already described already assumes the concept of
number of objects and completes this concept of number with the help of fractions.
We are led by that which precedes, by the way, to see in the theory of numbers, in
insofar as it is limited to the study of relationships and properties of whole numbers, a
science existing theoretically in the middle of analysis, which sooner or later will form an
entirely distinct science. For if numbers that can be the points of contact with analysis, as
useful as this can still be – (it gives as hope of a rich compensation some day), we are
nevertheless strongly tempted to believe that the theory of whole numbers can attain a
term12 which appears to us today inaccessible to analysis, and that it will one day succeed
in drawing from itself the proof of all its theorems. I naturally do not dream of drawing
limits for the relationships of whole numbers and analysis. In every variable phenomenon
it can happen that whole numbers come to pull our attention and acquire a decisive
importance.
18. – Numbers as signs and quantities, and formalism. – As fractions must
have been born from division of the unity that measurement required, we also see, when
12
[Translator's note: "terme" can also mean end, limit, or boundary.]
270
we wish to attach it to them sense, that they are inseparable from the representation of
divided objects. After that it must appear against nature to wish to construe analysis by
means of pure numbers, without taking account of their source: the concepts of the
quantity of objects and of magnitude. How artificial a parallel separation is can be seen
clearly in the irrational limit of a set of simple numbers. The numbers 1, 2, 3, … 1/2, 1/3,
…, 2/3, … are either signs of real quantities, or else simply figures which can certainly be
used as quantities of a game (article 14). For it is true that, when we are plunged into a
numerical calculation or in an algebraic problem, the idea of seeing constantly real
quantities as numbers or letters which represent undetermined numbers, does not occur to
us. Nevertheless, in the development of fundamental analytic concepts, we are not
authorized to draw further abstractions about their true origin, as the philosophy [-57-] of
language does not know, for similar reasons, how to satisfactorily prove the
authentication of a completely formed language. The representation of the numerical
figure comes from the rapid work of thought, instead of from a relationship to the unity of
magnitude. It is in reality similar to language, in that the words have become independent
representations. When we write or speak, instead of real representations marching past
are largely representations of the words themselves, though during these acts, as we see in
observing even so little itself themselves, such and such corresponding representations
stored in the memory escape and go beyond the threshold of the conscience.
Of these two questions, the study of the texture of completely formed language
and the research on the origin of words and grammatical forms in the formation of human
concepts, we could believe ourselves to be well-founded to resolve the last, for example,
in supposing language to be innate in humans, because we find precisely that completely
271
formed language is partially disengaged from representations that the words designate:
assuredly this would spare us half the trouble. It is this which nearly makes the analyst
who would study the fundamentals of science, break away from the concept of limit, and
wish to never in analysis any numerical and literary symbols.
The insufficiency of formalism for conceiving the sense of the simplest
arithmetical operations arises among other places from the multiplication of fractions.
How do we arrive at the rule of the multiplication of fractions, according to which, for
2
4
and ? The detour thus made Euler conceal the
3
5
example, 8/15 is the product of
difficulty,13 for this is not how the rule is created. We vainly search also, it seems, in the
works of the most ancient authors, for an explication of the primitive sense of the rule.14
13
Euler defined
2
3
x
4
5
as 2/3rds of
4
5
, without doubt in this fashion the result is established. (See Algèbre
d'Euler, page 106).
14
To know how the most ancient treatises of calculation present the rule of multiplication, I invited M.
Cantor, at Heidelberg, to show me. He had the kindness to respond to me thusly: "The most ancient
treaties of calculation that we have, from around 1700 BC (a papyrus of Ahmos, translated by August
Eisenlohr) is comprised of multiplications of fractions of all sorts, and most often, in truth, of abstract
numbers. We have not discovered the foundation of the equality
conducted such that
a
b
=
1
b1
a
b
× αβ =
aα
bβ
. The calculation is
& αβ are equally split up into partial fractions of the numerator 1.
+ b12 + ... + b1m , and
[
[
a
b
α
β
1 1
b1 β1
+
1
β2
+ ... +
1
βµ
1 1
b2 β1
+
1
β2
+ ... +
1
βµ
=
]=
]=
1
β1
+
1
β2
+ ... +
1
βµ
. We then have:
1
b1β1
+ b11β 2 + ... + b11β µ
1
b2 β1
+ b21β 2 + ... + b21β µ
272
[-58-] But if we give up drawing an idea from pure symbols, and we see in the fractions
i.e. the parts of unities of linear magnitudes that they really are, the probable origin of this
rule appears then to the eyes of everyone with the greatest clarity. Near the unity of
length, the unity of the surface appears also necessary to measure planes, and it is known
that the square is the most convenient for this end, as the square reduces intuitively to the
measurement of lengths. The area of the simplest form after the square – the rectangle –
when the sides are multiples of the unit square, gives the direct image of the
multiplication of whole numbers. Rectangles smaller than this square, would require
division, and would thus lead to the multiplication of fractions. In regarding this
backwards, we assume, it only occurs in rational intervals of length. – The more
ingenious minds divide the sides of a rectangle by
squaring the side by
2
2
and in terms of the unit square by
3
5
1
to render the one comparable to the other, and they recognize
15
immediately that the rectangle is equal to
8
of the square. One believes one's eyes when
15
hunters who would establish a home make a truce and henceforth divide between them
their property following this rule. In any case, the multiplication of fractions becomes a
1
bm
[
1
β1
+
1
β2
+ ... +
1
βµ
]=
1
bm β1
+ bm1β 2 + ... + bm1β µ .
One reunites in a single expression all the partial products, either simply writing the ones after the
other elements of all the ordinal products following their magnitude, or in effecting the addition of some, for
example, by replacing
naturally obtain
aα
δβ
1
3
+ 16 with
1
2
, -- or finally in reducing them to the same denominator. We thus
. [Translator's note: I believe this last fraction is supposed to be
273
aα
bβ
.]
completely natural and extremely clear continuation of the multiplication of whole
numbers, and we could very well have found in the first epochs from the prehistoric
science of measurement.
For the rest, for that which I come to advance in favor of the true signification of
mathematical symbols, it is not necessary to conclude that I disregard the utility of
research on what we can call the mechanism of calculation, research by which Hamilton,
Grassmann, and others, notably our contemporary M. Weierstrass, have thrown light
upon the nature of composed operations and complex quantities. But to comprehend the
extent of this research we must glance backward onto the first historical developments of
the science. The simplest analytic operations on one or two quantities, addition,
subtraction, and division, which probably already has as its origin repeated subtraction,
have lead by repetition and extension of them to several quantities, to algebra, to the
calculation of powers, to logarithms, to the imaginary symbol. On the other hand,
trigonometry has come to be joined to these theories, blending entirely with them. We
add the creation of infinite operations and infinitesimal analysis and we have enumerated
the most important acquisitions that been have made over the course of centuries by the
analytic genius of humans. For all this has not been construed a priori. On the contrary,
we have no doubt, and we can conceive of it historically, that analysis is an experimental
science insofar as it has realized its progress tentatively, by slow steps, and it tries and
tries, but never succeeds. Ordinarily, simple problems on real quantities would suggest
operations, enlarged by induction, until they formed methods that we believe we can take
as certain, without even having dreamt of the necessity of a proof, as, for example, in the
decomposition of polynomials into simple factors. This is how things elapsed in algebra
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over the centuries, which we have nearly lost from view, and also recently in the domain
of infinite operations, where only since Gauss, Cauchy, Abel and Lè jeune Dirichlet, have
we felt the need of justifying the methods furnished by induction. The fact is that we
have found ourselves suddenly faced with the stupefying spectacle of completely formed
analysis, with its marvelous methods making partial use of purely literal symbols, but
always leading to the correct results. We learn to use this science, whose the primordial
genesis escapes us completely, only seeking to prove [-60-] isolated facts, proofs that are
founded themselves on the preexisting fact of analysis.
And here arises again a question: Is the actual form of analysis necessary? Could
we not invent other signs representing other combinations of conclusions and other literal
symbols, which would perhaps be even more profitable than the actual combinations and
symbols? For, when we reflect on this collection of questions that presents to us the fact
of completely formed analysis, we must evidently distinguish two things: the simple
operations the repetition of which has led to analytic calculus and this calculus itself. As
for the first, we must add the essence of the simplest infinite operations, especially the
concept of limit, which is the principle object of this book.
To not embrace too much at a time, it is convenient to treat these two types of
questions separately. In fact, the research on simple operations and on the concept of
limit revolve around the most abstract questions of knowledge, whereas the mechanism of
calculation could be treated apart from purely literal ideas. Here is how we can define
the problem that presents the mechanism of calculation: reduce the simple rules between
the letters or signs introduced to represent the indeterminates of calculation to the
smallest number possible; these rules are supposed to permit the execution of all
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operations which provide explication, under the condition that if we suppose certain
relations between letters or signs as given, fixed by these equations, these relationships
subsist during every calculation (the transformation of original equations by means of
simple or composed operations), and can be verified for each transformation that appears
in the calculation. In thus conceiving the mechanism of calculation, one can also divide
simple operations which no longer flow from the concept of the linear mathematic
quantity, but which, in satisfying the second condition, can serve as the base of a
completely legitimate calculation, and necessarily lead to correct results like that which
takes place for the interior multiplication of Grassmann. In the large number of treatises
and memoirs which have appeared on this subject, the authors have often confounded the
two problems -- that of the metaphysics of simple operations, and that of the mechanism
of calculation -- and the clarity of the ideas has not been gained. In all this literature it is
only ever a question of calculation between quantities supposed to be finite. One can
reduce, as we will show in what follows the calculation of differentials to this, which has
so intrigued geometers of the past century. But we will stay a bit longer with the analysis
of infinite operations and continuous variables. Here again we have [-61-] tried to
separate a purely literal analysis of the concepts of magnitude and limit. It is Heine who,
in his elements of teaching the properties of functions15 introduced a letter to designate
the limit of an established operation as convergent by the general principle of
convergence; a letter which figures like the other letters in the calculus. By itself it is
placed outside of every discussion of the metaphysics of limit, etc. But if, as I have
15
Die Elemente der Functionenlehre, J. de Borchardt. – Vol 74.
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discussed above, a parallel separation could be used for algebraic analysis, since the
elementary operations form a problem for knowledge apart from it, which can be easily
detached from the literal problem on the mechanism of calculation, a type of logical game
between pure symbols, it appears at least to be so for the calculation of functions in their
most general conception. A spirit which loves to thoroughly examine things does not
know how to be satisfied. This is not to say, that for the needs of ordinary practice, for
elementary education, where one is only concerned about of regular functions, it would be
necessary to first address the subtle questions relative to the continuum and to the limit.
A professor charged with teaching English to a typical student would not begin by giving
him a course on general linguistics and on the origin of the English language in particular.
But when it comes to treating all the questions which present the nature of the theory of
functions, for example, certain questions which address the solutions of equations to
partial differentials, we could hardly dispense with the return to fundamental concepts,
and when we cannot we have not reached our hopes, we would feel that we have not
penetrated to the bottom of the matter. In summation, if, for algebraic analysis, the
problem of the mechanism of calculation perhaps offers more attractions, for infinitesimal
analysis it is truly a problem of knowledge which presents the principle interest, and the
decisive importance. And here is a point spouting from the question.
A purely formal and literal skeleton of analysis, which amounts to the separation
of the analytic sign from magnitude, would in effect bring about the decline of this
science, which in truth is a natural science which carries its research to the most general
properties of perceived things, into a simple game of symbols, where the written signs
would bring arbitrary significations like the pieces of a game of chess or the game board.
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As interesting as such a game can be, as profitable and even necessary as it would be to
the end of the analysis, to follow the rules of the calculus of symbols to its last literal
consequences, created from the representation [-62-] of quantity, this literal mathematics,
when it would be completely extracted from the soil in which it has grown, would not
delay in wilting to a barren belief, whereas this science, if Gauss has correctly called this
a science of magnitudes, has, in the natural domain of human perception whose extension
will always grow, an endless source of new objects of study, and fruitful impulses.
Without any doubt, with the help of so-called axioms, of conventions, of philosophies
imagined ad hoc, of inconceivable enlargements given to concepts which are quite clear
in their origins, we could construct a supplementary system of arithmetic, similar in every
point to that which is born from the concept of magnitude to guarantee in this way
mathematical calculations as by a strand of dogmas and defensive definitions of the
psychological domain. A hardly ordinary shrewdness would be demonstrated in these
parallel constructions. Only we could also in this manner invent other arithmetical
systems, as that which is fact. Ordinary arithmetic is precisely the only one which
responds to the concept of linear magnitude; it is, so to speak, the first manifestation,
whereas analysis which already has the concept of limit forms the most elevated
development. Also, we would not have succeeded in smoothing over, with the help of
symbolism, these difficulties with the concept of limit, which we already will attack from
the front without fear. For every analyst who hides something there is a person who
creates combinations would wish to restore it to the origin of the game of symbols, and
then would find himself facing anew of the problems which it has turned up.
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We must therefore return to this matter when the Idealist and the Empiricist have
explained their systems, to show their relationship to the famous arithmetical system of
M. Kronecker who also appeared to respond to the views of M. de Helmholtz, such as
this classification into of two accounts, which eminent geometers are beginning to
publish.
Men who aren't mathematicians have often enough sought to enlighten
mathematicians on the fundamental concepts of their own science. We have seen in
geometry that it is definitively the serious and well managed work of mathematicians
themselves who have cleared the path leading from the perceptions of the senses to the
propositions of science. Similarly the most scrupulous research alone can abolish every
discontinuity between the fundamental concepts of analysis and our system of immediate
representations. This is a scientific question like any other, which necessitates, as
sufficient knowledge of things does, every faculty of work of which we are capable.
[-63-] 19. – How we must understand definitively the problem which has for
its object the concept of limit. – The considerations which precede thus finally lead us to
understand our problem as follows. – We have demonstrated the proposition:
Every decimal number o. α1 α2 … αn =
α
10
+
α2
10
2
+ ...
αn
10 n
approaches a limit value
as closely as one would wish when n grows sufficiently. The fractions o. α1; o. α1 α2; …
must be understood as multiples of tenths, of hundredths, … of a unit length of which the
extremes would be 0 and 1. If we place the lengths o. α1; o. α1 α2; … on the unity of
length 0 … 1, departing from the point 0, their extremities turn from the side of the point
1 and draw closer and closer to each other, and the limit point closes the set which
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compresses indefinitely in such a way that all the extremities are situated by relation to
this limit point on the same side as zero, the distances from the points o. α1; o. α1 α2; … to
the limit point falls below every value.
This geometrical conception of the problem must now be neither a hypothesis, nor
imply any restriction; but in treating in this chapter the concept of magnitude, I have had
for my goal precisely to show that in the perfectly unlimited field of thought we can
imagine nothing under the hypothetical limit other than a representative linear quantity,
thus, that which truly attains this end, a length. An abstract numerical limit is a type of
speaking devoid of any sense; we do not do the least harm to the generality of our
research, when under the hypothetical limit we imagine a linear mathematical quantity
and give to the problem the enunciation which precedes.
And now I open the discussion of the Idealist and the Empiricist on the proposed
problem. [-64-]
The two conceptions – Idealist and Empiricist – of limit and of magnitude.
Idealist system
THE IDEALIST
20. – Concept of limit. – The existence of the limit is in need of proof. – I place
exact length as the foundation of the science of magnitudes, and I represent unity as a
segment of the straight line which is perfectly delimited, whose extremities I will
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designate as 0 and 1. On this understanding, we know an innumerable amount of points
determined in different manners; among them must particularly be considered the sort of
points we can find representatives of in every portion of the unit length, as small as one
wants. These are rational lengths, i.e. the halves, the thirds, the quarters, etc., or multiples
of these fractions of the unity of the length. There are also lengths in innumerable
quantities susceptible of being thus construed which aren't rational fractions of the unity,
but rather roots of rational numbers, and multiples of these roots. The mind can
indefinitely multiply the number of these types of points, and the unit length is covered in
our representation by a pile of points of greater and greater density. We only have to
make the number of these points grow, our attentive thought never sees two points
coincide; on the contrary, two neighboring points always remain separated by a segment
of the line which, abstracted from its length, completely resembles the unit length. This
is the image which always accompanies my representation of magnitude.
I have only cited as distinct points of the extension 0 … 1 the rational fractions of
the unity, in the same way as lines can be theoretically construed.
In what follows, we will consider these points as given and as already known, and
we must immediately describe [-65-] and extend the manner in which new points are
connected to these givens. The origin of the points in question corresponds to the
consideration of the analytic transition to the limit. The new point determines on the
unity an extension which is the intended limit, i.e. the end of a collection of extensions
which differ less and less from each other.
Let α1, α2, … be numbers taken from the collection 0, 1, 2, …9 and for the series
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o. α1, α2 … =
α1
10
+
α2
10 2
+ ...
let there be a law which permits it to continue indefinitely. When the αp from a
determined range periodically reappear, the sum o. α1, α2 … αp, as p grows, approaches a
certain rational number that we learn in elementary mathematics. If we imagine now
placed on the unity of length 0 … 1, starting from 0, the extensions o. α1; o, α1 α2; etc., …
o. α1 α2 … αp; …, they are themselves rationals, because they are multiples of the tenth, of
the hundredth, etc, of the unit length; and in the case of periodicity of the unlimited
decimal number, the preceding series possesses a limit L, which is also represented by a
rational point on the extension 0 … 1, and which completes this condition that when p
grows the difference L – o. α1 α2 … αp becomes as small as one would wish.
But if the series of αp is not periodic, and no type of point whose existence we
have admitted contains the point L which completes the condition of the relative limit to
L – o. α1 α2 … αp, we then have the proposition which has occupied us here and which is
as follows: The series o. α1; o, α1 α2 … αp; etc., determines a new point on the unity of
length, i.e. this series forcibly introduces it into the domain of our representations; the
point is thus called the limit of this suite. It does not belong, by hypothesis, to the types
of points considered as known; it is thus beyond doubt that its existence must be
demonstrated before all else, and its existence is and will remain a hypothesis, if this
proof does not succeed; it is this that we must now examine.
21. – The concept of limit. Some proofs of the existence of the limit. – I have
searched thoroughly, and I have seen only two principle points of view for proving the
existence of the limit L of o. α1 α2 … αp. In the first place the limit point L is offered as
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the limit of an extension, of which the length indefinitely diminished and falls beneath
any quantity. In the second place is it known by the mind as the point of separation of
two extensions.
[-66-] The first proof can be explained as follows: First of all we have, for every
value of p and of q,
o. α1 α2 … αq < o. α1 α2 … (αp +1)
because:
1
9
9
= p +1 + p + 2 + ...
10 p 10
10
and, after the hypothesis that o. α1 α2 … is not a
rational number, the α will not begin at any line equal to 9.
If we thus construct on the unit length an extension sp of length
1
, connecting
10 p
the points o. α1 α2 … αp and o. α1 α2 … (αp +1), all the extensions o. α1 α2 … αn, as large
as n can be, it will terminate in the points situated on sp, so that the hypothetical limit L
(defined by the fact that L – o. α1 α2 … αn, for increasing values of n, can become as small
as one would wish) must fall as well on this extension sp. We now imagine that we give
to p larger and larger values, sp will become as small as one wishes and each new sp will
be contained in all those that precede. In this way, the hypothetical point is captured
between limits that are more and more narrow. Now let us stretch the extension sp
towards zero, its extremities will end by being identified with the same point and this will
be the hypothetical point L, of which the existence will thus be found to be proved.
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This is nevertheless no proof at all, but rather a surprise. In effect, the extension
sp is and stays indefinitely limited by points belonging to this type of supposedly given
facts. Our hypothesis authorizes us without doubt to diminish indefinitely the extension
sp, but this is an operation of thought which changes nothing in the nature of our
representation. Large or small, the extension sp always remains an extension contained
between two rational points. If now suddenly and without logical foundation we
substitute point for extension, it is an act by which we clearly and gratuitously bring in a
new representation; we let it immediately follow the first and anticipate precisely that
which must be demonstrated. A gradual meeting of two points, the expression of which
is felt and which very much seems to translate the fact, is absolutely nonsensical. Either
we have several points separated by an extension, or we have only one point, there is
nothing in the middle.
The other method of proof is as follows: The points that we imagine as given on
the unit extension are divided into 3 categories: 1st those which will eventually be
included by the extensions that we successively place on the [-67-] length 0 … 1, that is,
o. α1, o, α1 α2, etc.; 2nd the points for which this will never happen; 3rd, the hypothetical
points of such a nature that they belong to neither of these two first categories. It is first
of all easy to see that the third category contains either no points or one point at the most;
for we can make the two following remarks: 1st as far a point belonging to the first
category it is similar to all those which are found between zero and itself, and as far a
point belonging to the second, it is similar to from all those which are found between
itself and one. 2nd all the decimal points belong to one of the first two categories.
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It follows that there will not be two points of the third category, for we could
imagine decimal points in the interval which separates them; these points would not be
included in the series of constructions o. α1, o. α1 α2 … and it would likewise be so for the
two points in question, which it is against the hypothesis. Thus the 3rd category can only
contain one or no points.
From remark (1) it follows that the extensions on which are found the points of
the first and the second categories are without interruption so that the point of the 3rd type,
if it exists, can only be the extremity common to both extensions. But if the untied
extension is divided into two equal portions such that the points of the first would
eventually be included, and the points of the second never will be their common
extremity must be equally the hypothetical limit L, for this point satisfies precisely the
condition:
Lim (L – o. α1 α2 … αp) = o;
from which it follows that a point of the 3rd type is impossible because the point limit is
never attained.
As convincing as all this explication appears to us, it still hides a surprise, though
slightly less visible than the supposed first proof; I even acknowledge that at a certain
time that I had imagined to have really discovered the proof of the limit.
To characterize in a word the weak point of the proof, it would be absolutely
rigorous if we enlarged again the concept of geometrical quantity by this complimentary
proposition: "that the length is formed of points." This is going to become immediately
clear.
285
I first remark that up until the phrase which began by these words: "from the
remark (1) now results that…" the reasoning did not suffer from any objection. It is only
the conclusion of the proof which is no longer satisfying.
[-68-] We take our place at a point where we could judge the value of parallel
proofs, even without examining the conclusions separately, by committing ourselves to
that which could be proved in a general fashion from the foundation of our representation
of magnitude, such as it has been given at the beginning of the exposition of the idealist
system.
The assumed concept of quantity never furnishes us with more than a precise
length 0 … 1, and, innumerable points that we can without doubt imagine to be as dense
as we wish them to be; but, as I have said, in such effort of imagination, which
corresponds to a determined image passing before the consciousness without the least
cloud, we always represent between two neighboring points a segment of the straight line
whose extension coincides exactly with the unit length. That which we have said of
decimal numbers, knowing that they must belong to either category 1 or category 2, is
true of all types of points which we assume to compose the given on the unit length.
Thus we can say that the given points belong to category 1 up to the point N'16 inclusively
and to the category 2 after the point N' inclusively, so that, further, N and N' are in as
much agreement as one would wish, and that finally between them there cannot be any
points considered as given. There it is, in all its rigor, absolutely all that one can prove
and that which has proved the preceding reasoning.
16
[Translator's note: I believe N is meant here rather than N'.]
286
The surprise in this evidently is that we admit extensions determined by the points
which we assumed as known, without the extremities having been given. An extension is
only determined by its two extremities. The two extensions into which the unity 0 … 1 is
divided certainly have the extremities 0 and 1, but the common extremity is not
determined by the first part of the proof. One of the extensions is only precisely
determined up to a point N of the series of represented points, and the other starting at an
analogous point N'. On the part of these extensions situated between these two points, we
can discover nothing. We can augment at will by the imagination the density of the series
of points. Instead of reaching the extensions N … N', it will reach other extensions N1 …
N1' which are always contained in the former and fall in smallness below every limit. But
there is never anything but these extensions; and, as we have expressly raised in the first
type of proof, this fashion of diminishing makes no change to the essential nature of
representation; on the contrary, the fact of annulling the extension or of transforming it
into [-69-] a point is an arbitrary act which does not suffice as a real proof.
But if we essentially change the representation of length, and we conceive it not
only that which supports already known points, but as an aggregate of points -- if we
abandon this representation according to which line and point are completely different
things -- the critique of the second proof evidently collapses, and it must be recognized
that this is a rigorous whole. We perhaps cannot imagine all the points of which the
extension is composed; but it suffices that they actually exist. They are always broken
down into two categories following whether they will be reached once or never; and for
the third category, it could only have one; but which, given that a point does not have
length, does not enter into consideration and must necessarily coincide with the extremity
287
common to the two extensions. For the extremity of the extension of attained points
would never itself be attained, which is to say will never touch its limit.
We feel besides that in considering a length as formed of points the existence of
any limited length is first assumed, and that this representation must give rise to a series
of ideas completely different from that which occupies us here.
But this representation is neither clear nor neat. We could, to explain it, cite the
image of a pencil whose point traced a fine arrow. But there is in this the idea of
movement; the arrow is the trajectory of the mobile point. The representation of space,
immobile and fixed, never elicits the image of a true and uniform line from a series of
points, as dense as it would be, for the points do not have dimensions, and consequently a
series of points as dense as one would wish can never become a distance; much to the
contrary, we will always consider distance as the sum of the intervals of points.
The conception of the line as series of points still responds to the same lacuna in
the series of ideas that we have already critiqued twice. In fact the series of points on the
rectilinear extension, with their intervals as small as one wishes, is a representation whose
essence is not changed by diminution pushed as far as the infinity of intervals of points.
When suddenly we suppose these intervals to be null, we briskly jump to a limit,
following our fantasy and without transition, which this time is not only not
representable, but also is completely absurd or at the least paradoxical. [-70-] I thus reject
the enlargement of the concept of magnitude, according to which the line must be
composed of points, the surface of lines, etc.
I must thus give upon a proof of the limit L of the decimal number o. α1 α2 …
founded on the hypotheses made at the beginning.
288
We also in fact ask the impossible, when we wish a series of points composed
exclusively of points belonging to a given category, to elicit (bring about) a point that
doesn't belong to this category. For me the thing is so unimaginable that I affirm that no
intellectual work would ever extract a proof of this genre from a human mind, without
supposing united into one: the gift of divination of Newton, the clarity of Euler and the
overwhelming power of the mind of Gauss.
It has always conformed to general intuition that the infinitely prolonged decimal
fraction o.α1α1 … represents a value, to which we can give a sensible form, with the help
of a length or else with another mathematical quantity, such as a weight. In fact, an
innumerable collection of decimal fractions has a limit which can be represented by a
length, and nothing in the decimal fractions nor in the lengths can awaken the least
suspicion that, assuming the general enough property that the decimal fraction and the
limit coexist, a difference between them is possible. Finally we can, in any case,
approach the hypothetical limit as closely as we wish; that is we can find a quantity L1
such that L1 – o.α1α2 … αp … for every value of p stays beneath a length as small as one
would choose, by confining, as I have made clear, the hypothetical limit to a suitable
interval.
22. – The object of the preceding considerations is going to be completed in
the sense of the Idealist. – If we have not been able to draw necessary conclusions about
our representations of the point limit L, it is either because in reality it does not exist, or
because we have not yet drawn everything possible from our representations. It still
includes the relationships that we have not considered, for when our representation of
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magnitude is complete, and when these liaisons with the rest of our system of
representations are clearly shown, the limit points, if they exist, will necessarily proceed.
There is, in fact, an essential question that we have not yet posed. We would
study one more time the formation of the limit L; we have as our goal to immediately
approach it [-71-] in full light, by examining things from a more physical point of view.
Like the idealist, I believe in the reality of my ideas, I believe in the objective
reality of my ideas followed to the extreme limits of my thought, although, in truth, like
the infinite, they are not representable. I believe in the infinite and in the objective
existence of precise geometric images, or more correctly in their possibility; for this
possibility signifies that only secondary and subordinate circumstances can be opposed to
their existence, as we think it is possible that two watches could tick perfectly together.
And every conclusion that is built on parallel hypotheses must fall in the domain of
possibility, or else, following their nature, the domain of reality.
Now let us consider a precise length, not only imagined, but existing somewhere.
The lengths o.α1; o.α1α2 … will likewise exist, and then either their limit point is an
objective reality, or else there is no such point. All this is independent of the fact that in
the world there is or there isn't thinking minds; we thus do not ask:
Can make any way with necessity, by any path whatsoever, from our initial
representations to the final representation of a limit point?
But rather: Does there exist, independently from a thinking being, a limit point,
when the extensions o.α1; o.α1α2; … exist?
This fashion of envisioning the question evidently stipulates a route other than
that we first followed. We no longer have the right of posing in principle the points that
290
we can by thought place on the unit length as we have begun to see in the Idealist system,
but rather those which really are.
We are thus first presented with the question of the number of points of division
which must be distinguished on the unit extension. We can represent them as we wish,
but how many must be regarded as distinct? – We can still ask this from the physical
point of view: how many extensions less than a given extension exist or are possible; that
is, if they do not exist, must they not exist only by chance?
To give to this question a satisfying response, I will first establish a distinction of
the highest importance between two concepts.
[-72-] 23. – Concepts of the Unlimited and of the Infinite. -- The quantity of
points of division of the unit length or of the partial extensions into which the unit
extension is divided is called unlimited or infinite. When we have established the exact
sense of these two expressions, the related questions of the number of parts of the unity
and of the existence of the limit will immediately find their solution.
An example will clarify for us the sense of these two words. We wonder what is
the number of celestial bodies scattered in space. When we regard the starry sky with the
naked eye, several brilliant stars appear immediately, similarly to how with the points of
division of the united extension, our thought sees fractions first. A more exact
observation made with the help of a telescope can reveal to us, due to the intensity of the
light and of the magnification which we have prepared, wherever a uniform obscurity
seemed to spread out, masses of more and more dense stars. We thus have the impression
that there is no limit to the number of stars; and we say: The celestial bodies appear
unlimited in number. But to this question, "What, therefore, is the number?" – unlimited
291
would certainly not be a satisfactory response, because we always imagine a man who
counts and who does not arrive at the end of his calculation, either because there does not
exist an end to what he counts, or because he cannot reach the end which exists.
By the word infinite the majority of educated men mean that the extension or the
number thus qualified surpasses in reality and independently from the existence of
thinking beings everything which with the help of a determined unit can be measured or
counted. One says, "infinite space" and if one represents it as spreading out on all sides
with the celestial bodies which it contains, one also will say the number of celestial
bodies is infinite. The exactitude of this proposition can well be contested, but with
regard to the expression itself no one has an objection.
The expression "I represent space as unlimited" is beyond attack; but it would be
improper to say, "I represent space as infinite," because we cannot precisely represent
infinity, which is beyond the frame of every representation. We have to say: "I believe
that it is infinite."
All this taken into consideration, this is how we can summarize the relationship of
the unlimited to the infinite: We name unlimited in magnitude that which, while being
finite, cannot be surpassed by multiples of fixed measurements, because we picture it as
growing and leaving the confines of every sphere.
When we represent a measurement as large as we wish, [-73-] we picture the
unlimited quantity as going beyond it. Should the unlimited quantity should be set free of
the ideal which accompanies it in our mind, should we give up on the need that it has a
representation, then from the unlimited departs the concept of that which is superior to
every quantity and cannot be further represented, the infinite.
292
It is of the highest importance to well establish that the unlimited is always finite,
and that the infinite is applied to that which follows the finite in the direction of what we
had supposed unlimited, that is to say, to what is not finite.
The question, "What is the number of points of division lying in the unit
extension?" thus receives from what precedes the following solution: In representation,
this number is unlimited, and consequentially, in objective reality, it is infinite.
24. – The infinitely small and its principle properties. -- The proposition that
the number of points of division of the unit extension is infinitely large engenders with
logical necessity the belief in the infinitely small.
In fact, we have established above that, according to the true concept of
magnitude, points do not follow each other without interval on a length, that they thus
cannot be reunited but are always separated by extensions, so that simple points can never
form extensions; in their turn an infinite number of points are separated by an infinite
number of extensions. Thus, of these extensions nothing can be finite, which is to say
nothing can be contained a finite number of times in the unit of length, because the unit of
length being arbitrary, every extension as small as one requires must be constituted as the
unit of length, and contain also an infinity of points of division.
One thus sees that the unit extension is decomposed into an infinity of partial
extensions, of which nothing is finite. Thus the infinitely small really exists.
We see now more closely the properties of this new type of magnitude. The
principle property of the infinitely small is the following:
A finite number of infinitely small extensions added to each other does not
produce a finite extension, but a new infinitely small extension.
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It is true that we cannot create an idea of the domain of the infinitely small out of
the finite. We cannot fix an upper bound either to the finite or to the infinitely small; but
what is certain is that as large as one can represent a finite extension, a finite number of
parallel extensions [-74-] arranged in a series following each other always furnish anew a
finite extension. This property also is in accordance with the infinitely small, following
its origin. For if a finite number of infinitely small extensions would produce a finite
extension, they could not all be infinitely small, because this would necessitate that they
were infinite in number on a finite extension – from which results the proposition.
It is not without pain that the belief in the infinitely small is imposed.
Accordingly, if one seizes the idea boldly and without prejudging, if one does not recoil
timidly before the difficulties but rather one accepts the properties of this new type of
magnitude, as logic imposes them with necessity, as something where there is nothing to
change nor to interpret, then the first suspicion would already give way to a generous
security thanks to which one feels immediately at one’s ease in the fundamentals of exact
science, as obscure and as contested as they are, and one can take positions following
fixed principles.
The conception developed here of the infinitely small will be again found thorny,
concerning the extension that it implies of the concept of equality. In fact, I call equal
two finite extensions, when there does not exist between them any finite difference. This
is not prevented, and this is the signaled difficulty, that we can suppose that they differ by
infinitesimals. If we see serious difficulties in this extension of the concept of equality, it
is that we have not yet a clear view of logical reasoning which leads to the infinitely
small. Without doubt this concept of equality is contrary to the idea of equality that we
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ordinarily have, according to which the difference is absolutely nothing and cannot be
represented. But the proper essence of the infinitesimal is precisely that it resides and
must reside in the middle of our representations and consequentially we can only require
that according to one of its properties, which can appear to us strange, it is immediately
offered to us in a representation. We thus conclude:
Two finite quantities whose difference is infinitely small are equal.
In proportion to its importance, I will show that this proposition does not
contradict the concept of mathematical quantity, such as we have given in the
introduction (article 16). When two quantities of the same type are unequal, their
difference is in its turn a mathematical property of the same type (2nd property). But the
infinitely small is not a mathematical quantity of the same type. For if it was, in virtue of
the 3rd property, we could, with the help of a [-75-] finite multiplication of this infinitely
small difference, go beyond every mathematical quantity: which excludes the concept of
the infinitely small. Thus there is between two quantities no difference of the same type,
they can thus not be unequal, which is to say they are equal, in virtue of property I,
because mathematical quantities are either equal or unequal. We give the proposition
again in the following form:
A finite quantity does not change if one adds to it or deducts from it an
infinitesimal.
These relationships once and for all clearly established between the finite and the
infinitely small, we now ask: How must we imagine the infinitesimal? – for if its relation
to the finite is not at all representable, in any case the infinitely small can in itself, if we
abstract from this relation, correspond to certain representations. Naturally one must
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represent it such that it takes its beginning under our eyes, or rather (in so far as we
consider length as a type of quantity) as a portion of extension, consequently as
something identical to the finite, as an extension.
And generalizing naturally, thus we apply all that we know on the division and
addition of finite extensions to the measurement of infinitely small lengths, because in the
ideas acquired above nothing obliges us to admit for infinitely small lengths properties
removing it from the finite. Thus:
The infinitely small is a mathematical quantity and has in common with the finite
the set of its properties.
Once we have placed the infinitely small beside the finite in our concepts of
reality, it opens the door to numerous and novel inquiries.
It presents, for example, first of all the following question: With the help of the
preceding conclusions, how do we study the infinitely small in its relationships with the
infinitely small, drawing new conclusions and so forth. In this manner begins a series of
types of quantity succeeding each other such that a finite number of quantities of one type
never supplies a quantity from the preceding types.
When we extend to the quantities of one of these types the properties of ordinary
mathematical quantities, and consequently the same rules of calculation as for the finite,
the comparison of different types of quantities to each other is the object of that which
has been called the infinitary calculus. This calculus operates on the relationships of
infinitely small quantities or of infinitesimals from type to type, and these types show
between them relationships which escape from the ordinary concept of [-76-] quantity.
The transitions from one type to another does not show, for example, the continuity of the
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variation of mathematical quantities, although they do not change brusque by; here we
will depart briefly from the science of magnitudes, and enter onto a new path where one
does not study the relationships that quantities of the same type have, but the relationships
of the types of quantities which are distinguished from each other with the help of the
infinite.
In ordinary analysis, we distinguish orders of quantities called infinitesimals, and
we understand by this their whole powers. – The fractional powers have the same ability
to be used for orders, but the different possible types of infinitesimals form, as infinitary
calculus teaches, a series of degrees bound closer together, it seems, than that of numbers
exhausted already by the collection of powers of the quantity which becomes infinite.
Also from the point of view of the possibility of approaching any determined estimation
whatsoever, the series of degrees of orders of the infinite posses other properties than the
collection of linear quantities.
25. – The concept of limit and the numbers. – Just as we substitute numbers for
lengths, we similarly substitute numbers for partial extensions of the unit.
When we have completed by the infinitesimal the concept of quantity, the
existence of the limit point o.α1α2 … is immediately beyond all doubt. Each of the two
proofs given above thus acquires an absolute rigor. Let us consider for example the
second: the points susceptible of being covered go up to a certain point N; those which
will not be covered begin with a point N’. But the extension NN' is now infinitely small,
while previously it was considered to be growing without limit. We thus accept the point
N as the limit point, as the extremity of the length L: it must in reality satisfy the
condition that L – o.α1α1 … αp for increasing values of p fall beneath every limit, for the
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difference cannot stay above NN', and, as this extension is infinitely small, it adds nothing
to N or L
QED.
Inversely to each extension corresponds a number o.α1α2 … if L is not a decimal
fraction of the unit extension, α1 is then the biggest multiple of tenths of the unit
extension, such that it is inferior to L, α2 the greatest multiple of a hundredth, which
would be inferior to L – o.α1, etc… this gives rise to an unlimited decimal fraction, of
which the limit point is L.
[-77-] The points of division of the unit extension are thus completely equivalent
to numbers, they either comprise the limited decimal fractions or not. To each point
corresponds a number, to each number a decimal fraction of the unit, so that we say:
The set of all numbers is infinite.
Now let's go further in our conclusions. Here as when it was a question of the set
of celestial bodies, intervenes an indefinitely large set. Sets are measured by whole
numbers. There is therefore an infinity of whole numbers. In truth one would be
disposed at first to hold the set of whole numbers to be only unlimited, because one
thinks of counting; but it is again necessary to separate the mind which counts from
number itself. We read sometimes in Gauss: Numerus infinite magnus, and his
expressions are ordinarily carefully considered.
If one imagines the whole number not as it is written in the decimal system, but
under the form of an ordinal series following increasing powers of 10,
βo + β1 10 + β2 102 + … βp = 0, 1, 2, … 9
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the infinite series thus gives the image of the infinite whole number. The curious fact is
produced that the last terms of the second member, which we lost from view when the
number left the domain of representation, have precisely the largest influence on its value.
This circumstance presents a particular interest. If in effect with the concept of the
infinitely large we naturally admit, as we have done in our system of concepts, that of the
infinitely small, we must present it numerically by this fraction:
a finite number
_________________
an infinite number
From this fashion of conceiving the infinite number now results in the particular form of
the numerator and denominator completely disappearing from this fraction. For the
numbers in the finite are without influence in comparison to that which they have in the
infinite; and the numbers in the infinite so beyond perception, and cannot be indicated, so
that in fact the numerical value of each term is completely left a side and the infinitely
small appears as going beyond every numerical relationship with the finite.
Thus if I think that it is absolutely necessary to conceive the set of whole numbers
as infinite, the rational numbers on the contrary (we still suppose them comprised
between zero and one) form an unlimited set and is not infinite. This is implicated in
very definition that presents them as simple fractions having a finite numerator and
denominator. [-78-] The restriction that the numerator and denominator cannot become
infinite means that the set of rational numbers can, in truth, be supposed to be as large as
one wants, but always remains finite, that is to say that, as large as one supposes a set to
be during an instant, it can always be passed by another finite set.
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The set of all numbers is infinite, because it does not assume the existence of
thinking beings, whereas the set of rational numbers is bound to the person who thinks
which is already implied in the arbitrary character of the magnitude which the numerator
and the denominator attain.
26. – Zero and the infinitely small. – The property of the infinitesimal that,
added to the finite, does not change it, makes that sometimes inconvenient figure, the true
zero, which designates the absence of every quantity appear completely useless in
analysis. When a variable x is submitted to go through in entirety the domain of real
quantities, one most often thinks that this domain includes pure zero; I believe that this is
an error. In many cases, it is true, it matters little whether we accept or reject zero. This
can however, when one loses sight of the true sense of the symbols, obscure the
signification of the formulae, and many confusions have already resulted from a purely
formal zero being treated like a quantity.
The forms
1 0
, for example, that we meet so often, are absolutely to be
0 0
condemned, for they lack exactitude, or they lack sense. They are in all cases
incomprehensible from the start. If zero must be an infinitesimal, they are inexact; if zero
must signify "nothing", a division by nothing is evidently one, and nothing divided by
nothing stays nothing. Happily the tendency to purify mathematics of parallel
combinations of symbols which lack sense, something so desirable for the student, is
earning every day new partisans.
If we assume a positive variable, we stretch it towards zero as is prescribed in
questions of limit, and if we make it transverse every degree of finite magnitude, each
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degree being as augmented or diminished as one wishes with the help of an infinitesimal,
we consequently make it run through every degree of infinitesimal of every order, x
exhausts the domain of positive quantities. If we add even the supposedly negative
values, x would have exhausted the entire domain of real quantities. The true zero does
not appear in this domain, it is not a quantity.
[-79-] When however we approach the question from the psychological point of
view, it appears to be a violent act of the imagination to annihilate an undetermined
mathematical quantity or an arbitrary variation, and to create it anew, as one would relight
an extinguished candle. A variable quantity designates a series of representations of the
same type differing only by their numerical measurements, and annuls that which is
supposed as a variable, it evidently introduces, by a special act of will, a new
representation in the concept of quantity.
The Idealist does not doubt that the true zero is completely useless to analysis. It
is however necessary to insist on this point. We would not be able to say, for example:
x2 – 1 becomes nothing when x = 1, but rather it becomes infinitely small when x = 1.
We would say: lx becomes 1 when x becomes infinitely small,
1
becomes positive
x
infinity, when x becomes a positive infinitesimal, and infinitely negative, when x becomes
a negative infinitesimal;
1
is only well determined for a finite value of x or a value of
x −1
x which differs by an infinitely small quantity from 1. Finally we can see in every
equality U= 0 the fact that U is an infinitesimal of a higher order than the positive or
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negative terms which intervene in the equality U = 0, and which can by the addition or
subtraction of themselves in the two members of U = 0 be transported into a one.
The preceding expressions are far from being new, they have been often used, on
the contrary, but not always with clear understanding of their range.
27. – A few more words on the terminology of the Idealist. – We follow again
the language of the Idealist in the two most well known expressions of limit:
x
 1
1st Our conception responds to this fashion of speaking. The value of 1 +  ,
x

1
when x becomes infinitely large, is transformed into the irrational e. Similarly (1 + x ) x ,
when x becomes infinitely small, is transformed into e. Each expression must be taken to
1
the letter. This does not have the sense of saying that (1 + x ) x tends towards e when x
1
becomes null, because for x = 0, (1 + x ) x is a symbol devoid of sense. Let us designate
[-80-] by ε an infinitesimal from any order and of any value (its order being given), we
have then in all rigor:
1
(1 + ε )ε
=e
2nd If ∆y designates the growth of the function y = f(x), which results from the
growth ∆x ascribed to the argument, and f'(x) is the differential quotient of f(x) assumed
expressible with the help of preceding facts, so that in the equality
∆y
= f (x) + δ , δ falls
∆x
below every limit with ∆x, we say then that δ becomes infinitely small, when ∆x becomes
[infinitesimal] itself, and, with infinitesimals designated by dx, dy, and ε, we write:
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dy
= f ' ( x) + ε ,
dx
the absolutely rigorous equality. No less rigorous is this one:
dy
= f ' ( x)
dx
after the proposition on the addition of infinitesimals. Similarly:
dy =f'(x)lx
and this equality, following that which precedes, is equivalent to this one: dy –f'(x)dx =
an infinitesimal of an order superior to dx.
These examples suffice to characterize the conceptions and the language of the
Idealist and put us in a position to use them correctly in all occasions.
28. – The different conceptions of infinitesimals in mathematics. – The
infinitesimal has acquired the right of being cited in mathematics since Leibniz, who
called it quantitas infinitésima, but only as a scientific term. But Leibniz did not appear
to have clearly apprehended the sense of the infinitesimal, and since the apparition of
these famous six and a half pages in the acta eruditorum, 168417, the work of
mathematicians [-81-] essentially applied to the methods and results of differential
17
Page 467-473. Introducing the differential as early as the first lines of this memoir, which is still difficult
to understand today, he said: Jam recta aliqua pro arbitrio assumta vocetur dx … so that (and it is this
which renders the figure probable) here, lines seem first of all to have been supposed proportional to
differential growths, later becoming an everyday usage. We still read on this subject the preliminary
remarks of M. C. Gerhard, Leibniz, and Pertz, III and following, Volume V, page 215.
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calculus have not clarified any more the concept of the differential. This incertitude gives
little integrity to our science which is ordinarily given as a model to others.
Some, notably the mathematicians of the last century, have taken the infinitesimal
completely seriously; they wish to understand by it a sort of quantity exterior to the
common domain of quantities, which possesses a real existence. This is rather, in truth,
an indication of a logical deduction, like that which I have tried above, which has led
them in this manner of understanding.
Later analysts, who represent an intermediary opinion, considered the
infinitesimal, not as an ordinary quantity, but on the contrary as a quantity which runs, in
movement towards zero, possessed by the vanishing point, and consequently something
like that which we have called unlimited in smallness.
A third sort of representation absolutely rejects the infinitesimal, taking it as an
improper expression put in place of 'as small as one would wish;' the quantity as small as
one would wish can be assumed small enough to give to relations the degree necessary of
rigor. The relations carrying on quantities thus made infinitely small are inexact, pretty
close to those of mathematical physics, with this single difference; that they are also as
inexact as one wishes. Accordingly I regret in our literature the absence of a logical
development of this idea, which would completely disperse in its fashion the relative
difficulties of calculus with the help of differentials.
Finally, we do not lack mathematicians who have not introduced in their
relationships the hypothesis of the infinitely small as a literal expression, and certainly
here we must at least consider Newton and Lagrange. The efforts of Lagrange have
perhaps only served to celebrate the triumph of the differential which rather penetrated
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immediately across every pore of science in a fashion so irresistible that the edition of his
works which appear now adopt differential notations.
The Idealist makes room at the beginning for these four conceptions; however it is
not based on these presentiments, but rather on the conclusions that he must accept.
Many experts are disposed to place the infinite below the unlimited in space and time,
many among them have with difficultly decided to believe in the infinitesimal, even
though it possesses the same right of existence as the infinitely large, and even though it
must be deduced with logical necessity.
[-82-] One voluntarily swerves very little and with difficulty from the large tracks
of the domain of representation traced by the general education of the mind. The view of
the starry sky could be withheld from man, the human genius could begin and develop in
the fashion of troglodytes, in closed spaces; instead of traversing the remote regions of
the universe with the help of the telescope, these learned men would be habituated to only
examining the smallest elements of the body with the help of the microscope and would
understand the meaning of smallness beyond all measurement. Who would doubt then
that the infinitesimal would occupy in their system of concepts the same place that the
infinitely large does in ours? -- Moreover, hasn't the tendency to return mechanically to
the smallest active elements recently introduced into science the atom, which is the
infinitesimal made physical? And the efforts that we have made to render it useless to
physics, does this not mirror Lagrange’s struggle against the differential?
29. – Quick glance backwards on the system of the Idealist. – To finally
summarize in a few words my fundamental mathematic representations, I will say that
they give way to this series of quantities.
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Infinitesimal, unlimited smallness, finite, unlimited in magnitude, infinitely large.
Zero is not included, it is not a quantity. Some of these quantities are real and
independent from the existence of thinking beings: the infinitely small, the finite, the
infinitely large. The representation of the unlimited is bound to the existence of a being
who thinks, it is not the representation of a fixed immobile quantity; rather, it captures the
idea of a movement beyond all limit, in which our thought inseparably follows the
quantity which increases or decreases, this is thus always finite.
The idealist thought assumes a world which, not only corresponds in some
manner to our representations, and indeed even our intuitions and our most abstract
concepts, but which also holds a real continuum that is beyond our representations and of
which we are profoundly aware, despite the impossibility of representing it to ourselves.
Thus the idea which most men have of the universe is precisely the idea of a space; it is
true that it has dimensions as space does, but its extension escapes every representation.
The Idealist, to designate the true dimensions of the universe, as for the number of the
parts of a mathematical quantity, uses a word which does not mean that we [-83-] cannot
measure these dimensions or count this number, but that these dimensions and this
number really exist, however non-measurable with finite measurements: this word is
infinite. Before the parts of the finite quantity can be infinite in number, each must be
infinitely small.
The primary results of this is that every unlimited decimal fraction o.α1α2 … has
for its fixed limit a mathematical magnitude; the second result is that there is an infinity
of numerical quantities o.α1α2 …
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As for the infinitesimal, it gives rise to infinitely varied orders. If we stay in the
same order, it is composed like an ordinary mathematical quantity, and an infinitesimal
added to an infinitesimal of the least order does not change it. The finite corresponds to
the order zero.
The fundamental conception of the Idealist is thus the real existence of not only
that which is the object of our representations but also of intuitions which result
involuntarily from our representations. When we take them as existing, it is permitted for
us to wonder what is the essence of that which is hidden from our perception, and whose
existence is as indispensable for us as the real dimensions of the universe; and this
question cannot receive any other response than that which we have given.
Critique of the System of the Idealist made by the Empiricist
THE EMPIRICIST
30. – Of the unlimited decimal fraction which is developed following a certain
law, and of that whose development has no law. – My manner of understanding the
fundamental concepts of mathematics is distinguished from that of the Idealist by the fact
that it is prohibited to pass the limits of the natural domain of representations. I believe
that we are not authorized to admit into mathematical reasoning things which we neither
have nor are able to have a representation. Consequently, I will combat several essential
hypotheses of the idealist system, but first I would like to oblige the Idealist to pursue his
views on the parts into which the limited extension is divided until we draw the
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consequences that have not yet been seen. This is why I will for now admit the value of
the concepts of the unlimited [-84-] and of the infinite such as they have been defined by
the Idealist, and study the nature of a decimal fraction which continues without end, from
the point of view of its unlimited development.
The Idealist concludes in his manner that already the set of whole numbers, and a
fortiori the set of all numbers, is infinite. This logical conclusion has a singular result.
Let us consider a decimal fraction < 1, of the form:
o.α1α2 … =
α1
10
+
α2
10 2
+ ...
A formula, a law, according to which we can develop it, is necessary so that we
can continue [the decimal expansion] towards infinity; the unlimited series and its law are
in some sense identical. After all, an infinite operation without a formula which
determines how to continue it is a contradiction in adjecto. For, if by infinite one means
that the development is separate from the human mind and continues its route towards the
infinite all alone, it is quite necessary, if the development must be made in a determined
manner, that we give a formula with which it makes its path: this correspondence of the
series and the law also implies that the law is determined by the series, although, by its
nature, this relationship is not imposed as necessarily as the inverse relationship. Choose
any number at our inclination from the terms of the set o.α1α2 …, we can imagine that a
mind of sufficient penetration could always deduce the law of succession, because it is
necessarily determined by the unlimited series of α’s.
But what could be the nature of parallel laws? -- A particular law will contain any
number whatsoever from arbitrary quantities. For example the number Z = o.α1α1… must
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be calculated by the formula Z = α1µ1 + α2µ2 + … where the α's designate arbitrary
numbers in unlimited quantity, and the µ's designate given numbers. If thus with the
Idealist I inquire into the number of parallel laws, I am obligated to respond that it is
infinite; for the form of the law is given in advance, it is thus henceforth independent
from thinking beings, and consequently, after the manner of reasoning of the Idealist, it
does not admit of another possible response. But if the number of arbitrary quantities α,
unlimited in its representation, is in reality infinite, it follows that it is similar to the
quantities α, and in a word, the entire series o.α1α2 … is arbitrary and, as far as infinity, is
not summarized by any law. Even if some number of these digits were exposed to the
eyes of the most penetrating man, their succession could never be expressed for certain.
Briefly, these would not be numbers. It is only permitted, in fact, that human
thought would make use of this word for parts [-85-] of the unity expressible in numbers
and for parts which, without being thus expressible, are nevertheless determined by the
law of the succession of its digits.
Therefore the series eternally deprived of laws are not numbers. Would they be
able to correspond to quantities? Yes, without doubt, in the idealist sense. – The Idealist
supposes them continuous up to the most distant terms
α∞
10 ∞
, which are infinitesimals.
Those who follow the Idealist’s definitions do not change the sum of the preceding terms,
and consequently a segment of the limit corresponds to the infinite decimal fraction,
though without law. He must then conclude that once the unit is well fixed, there exist
innumerable portions of this unit which are not expressible in numbers.
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THE IDEALIST
I am armed against parallel consequences drawn from my fundamental
hypotheses; it embarrasses me not at all. The empiricist concludes rightly that I must
admit series of numbers which can never furnish a law for their succession and,
accordingly, march towards the infinite. This certainly is not representable, no more than
the "infinite" and the "eternal." These non-expressible numbers are a necessary
consequence of the conception which most men have on the essence of things, drawn
equally from the infinitude of space and the eternity of time.
The limitation of the number of possible decimal fractions by the laws that require
infinite operations is not evidently contained at first in the concept of all the infinite series
formed from such numbers as o, 1, 2, … 9 written one following the other, this has rather
been contrarily added a posteriori by speculation. If we stray from the representation of
quantity, every length smaller than the unit is, in the representation, of the same nature.
Let the unit and a smaller length be given: if we determine by measurements the
relationship of this length to the unit, we will admit to the expression one decimal digit
after the other; we could, under the hypothesis of sufficient instruments, follow this
occupation as long as we wish, and empirically develop an unlimited decimal fraction.
Nothing in this formation of a decimal fraction requires a law which rules the order of
digits. By the rigorous relationship of two lengths, the infinite decimal fraction is
perfectly determined without [-86-] any rule needed to exist which would permit it, for
example, to be given N first digits, to calculate anything else apart from it.
We could also imagine the following mode of formation of a number which
continues to infinity without any law. Each digit is simply determined by the roll of a die.
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As we can imagine that we have thrown the dice for all eternity and that we will throw
them for the eternity to come, so too we can think of a number without law. However,
the consideration of nature furnishes many examples.
Incontestably many constants in nature are determined for all eternity by the state
of the universe. The temperature of space, and its optical constants, certainly exist before
its entire potential plays out. Of potential and of similar quantities one can say that they
are determined by the totality of masses expanding in space; under the controversial
hypothesis, it is true, that forces beginning from a point produce their effects at the same
time. But the other constants of space are engendered by all that is and has been in space.
For example, the temperature of space is the result of all the states of the infinitude of
space and the eternity of time. Let us suppose matter finite, limited somewhere or other
in each direction, and let its state be given at any instant, the expression in numbers of
these natural constants must lead to a law. But suppose matter infinite: a constant like
the temperature of space depends then on the actions which must necessarily be limited
by no decimal range. If we would prolong the series of digits by a law of formation, this
law would contain the history and the image of the eternity of time and of the infinitude
of space. As we have seen, physical considerations of this space have already furnished
the existence of irrational numbers without laws. From the point of view of the limits of
parallel numbers, my considerations apply without changes, then in my proof of the
existence of limit, I have not made use of any law. I could, having in view the preceding
examples, call “empirical” the irrational numbers without laws, to distinguish them from
those whose development has a law and which could be called analytic.
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THE EMPIRICIST
I well appreciate the solidity of this development of idealist principles. I have
wished only to show, by one example among a thousand, how the concept of the Idealist,
from [-87-] the first elements, has already gone beyond all boundaries of human
understanding, instead of restricting and simplifying the principles of the science. Such
exuberance of thought is perhaps for the Idealist a laudable quality, but for me it would be
worrisome. I could not feel at ease despite all the rigor of these conclusions, in a science
in which the branches penetrate so often into the monstrous and unknowable.
31. – The fundamental hypothesis of the Idealist is laid bare. – I now turn my
attention to the hypotheses of the idealist theory.
It lies on a supposed representation, near to that which we ordinarily pass over
without paying it attention and which could be considered more than any other as the true
origin of every obscurity and difficulty of conception, in the principles not only of
mathematics, but also of exact physical sciences. This representation is precisely as
unreal as that of the infinitesimal in which it results. The initial hypothesis on which the
idealist barely insists is the existence of exact measurement. It serves as foundation to all
these considerations and conclusions. From the representation of exact length he deduces
the infinitesimal, which he serves to prove the existence of limit, then his infinitesimal
ladder of different types of quantities. He has recourse to this representation to render
intelligible the formation of a decimal fraction which continues without law up to infinity.
Finally, he relies upon the exact temperature of space, to render probable the effective
existence of parallel "empirical numbers." Let us ignore of temperature, which is a very
complex representation; the Idealist attaches his most important consequences to the
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concept of exact length, and to the identity of the nature of length and of its divisions as
small as we wish, from which he deduces the brusque sectioning of a line, the division to
infinity, and all the rest.
It is precisely here, concerning exact measurement, where our paths separate. I
hold this as imaginary, in a word, unknowable. To avoid a false conception of my
manner of understanding, I will immediately put forth evidence of the point which is in
question. I do not deny the representation of the exact, but we do not, and cannot have a
scientific concept of exact geometric quantities.
Our geometrical perceptions and representations are not inexact, when our mind
does its duty. With young eyes we see the distant field of snow which is separate
from [-88-] the sky with the cleanest contours, and when later we become myopic, and it
happens that after some time we became familiar with spectacles, the first glance thrown
on these forms precisely limited brings us joy. We are plainly satisfied by this certitude
because perceptions do not reveal the larger thins, and consequently we cannot imagine a
greater certitude than the sensible impression that produces for us the lines of separation.
It is essentially the perceptions themselves and entirely new representations, installed
over time in memory, which suffices to engender the image of exact delimitations like the
other images in the memory. If it is not renewed, over time, it will lose its precision.
When without the help of sensible objects we wish to represent to ourselves exact
geometrical images with a newness at least approaching a pristine newness, this costs us18
18
In observing myself, I have had the impression that I work, so to speak, with the help of two forms of
representations, the one coarser, the other finer, as others have made themselves analogue observations in
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some effort. But that which remains and cannot be destroyed like an image, is the
concept of the exact in sensible perception. It is attached to this sign: the word.
We continue following this concept through the unconscious development to the
creation of the concept of the exact ideal. Here we are hardly closer to what happens.
Changing experience teaches that straight lines which appear exact, like the corners of
houses seen from afar, examined more closely, reveal an irregular form, and other straight
lines preserve this appearance, even observed with a microscope. We possess enough
examples of geometric images cited as perfect, lines, corners, surfaces, construed by
mechanical processes, [-89-] which display an absolute exactitude, some to the naked eye,
others to the eye armed with a microscope. We see thus that: we possess the
representation of the perceivably exact, as well as the suspected exact, which we sense
with extraordinary delicateness. From this is born a series of ideas the conclusions of
which dominate our system of concepts in every direction; wherever an endless series of
representations lacks a representation for the limit, the analogy of innumerable series of
other scientific domains. If a certain effort is necessary to represent geometrical images with every
precision of primitive perceptions, trying to replicate the precision of our young years in view of purifying
our concepts of space, in particular, our mechanical concepts, and develop them under a more and more
abstract form, this effort, I say, will reinforce our aptitude into a seeming work of thought, but not our taste
for this work. It is thus that later we believe we can specify certain mechanical abstractions, because one
feels these efforts are quite unfruitful. In the ordinary work of my thought, these imperfect images of
points, lines, surfaces, arise in my mind when, for example, in the study of a geometrical problem, I
interrupt my lecture to submit to analysis the representations at which I have arrived. I ordinarily represent
figures in a picture or stereoscopic images to myself.
314
representations which ends with a limit makes us assume in spite of ourselves that there
either is or is not a limit. But as it does not respond to a real perception, an image of the
object cannot exist as a representation; this is simply a representation of words, and it
comes to be added to our of system concepts like a concept of something that is real, that
is representable. This is, briefly the origin of what I could call the Idealism common to
all men, and to every metaphysics. It is an essentially unconscious phenomenon that is
passed not only between individuals but even that it throughout the species.19 Thus we
believe in the possibility of perfect exactitude without having ever seen it; or rather, we
do not even reflect upon it. It becomes like many other analogue concepts, and without
giving rise to any discussion, joins the system of concepts of the ordinary man.
31. – The idea of exact measurement is subjected to the critique of the
Empiricist. – Thus, we are idealists without realizing it. But from the scientific point of
view we must be able to give an account of our hypotheses and of our fundamental
concepts, and not unconsciously accept any point of interruption. Our scientific problem
thus consists in returning to the idea of the exact and comparing it to reality; we must
research whether the limit for which we have a word in our system of concepts responds
to the limit which exists. Thus the question is posed: Are we well founded in
recognizing the objective reality of geometric ideas? This is related to the question on the
19
This is not the place to analyze more closely this sketch of a border in our thought, or to clarify with
numerous examples; it is this problem which is addressed by a work already announced which will appear
immediately after this one. However, until this passage has appeared in German, the extreme importance of
315
concept of exact measurement, to examine whether it obviously suffices to imagine the
exact measurement of length, which is to say of rectilinear extension.
[-90-] In the sensible image of the limited straight line, we distinguish three
impressions: 1st, this is a line; 2nd, this is a straight line; 3rd, it begins brusquely, and is
terminated brusquely. We will examine the first two characteristics at the same time.
We cannot tell with certitude that any form in nature is exactly a straight line –
with a brusque beginning and end. We cannot speak of an arrow or a thread, which both
have a density, nor even of a ray of light, which as Newton said, has caprices, nor of the
sharp ridge, which always necessarily presents divisions in the magnitude of grains of
powder, which constitute the point; but physical considerations oblige us still to regard as
digression the continual differences, and small as they are, from the exact idealized
geometrical image, with all the corners and planes prepared in as perfect a manner as
possible, crystal corners freshly formed and in the best condition, the surface of liquid
capillaries, the edge of the floating drop, etc. For we could not assume any substance to
be susceptible to changing volume, being combined, being exhaled, being electrified, or
being composed of chemical atoms, unless it fills the space in a rigorously uniform
manner. Every portion of space, as small as it is, is composed of similar matter confined
to differences in composition, so that ultimately, in the world which is offered to our
senses, every line as regular as it appears will appear as sufficiently dense, like an object
the method that it describes to our understanding of the origin of that which is called metaphysics, has
already been remarked upon.
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extended in the sense of length, with numerous twists and thickenings. On identical
grounds cannot be founded the hypothesis of exact extremities of a length.
But what is of the highest importance, it is that even if there existed in nature
exactly straight lines, we could never know it. For the character of the straight line is for
us an indirect character. – It requires that of certain perceptions of the image disappear.
For example, if one makes a line between two fixed points turn back on itself, the
appearance of this cannot changing, and we cannot imagine it possible to travel around
the line in such a fashion that two of its points always fall at the same points of the retina.
If one represents the unlimited straight line, or else one makes an abstraction from its
extremities, one can only otherwise express the criteria of the line in finding in the
condition that, contrary to other geometrical images, it is determined by two points,
whereas three points are necessary to fix [-91-] other geometrical figures in space. For
the corner this physically signifies that if it runs between two settings, its microscopic
image must not change, even while it changes direction.
Now, being shown that what every instrument emerging from the hand of man
permits us to observe the deviations the non-existence of which characterizes the straight
line, no matter what definition we use, no one will believe that they never permit
measurement of these deviations as small as we wish. But even if we had received our
machinery from divine hands, and if they were endowed with a marvelous power, they
would never permit us, in the presence of the perfect line, to recognize it as such; for to
measure it would require the establishment of certain deviations or, more generally,
certain quantities of whose existence is possibly non-existent, which is rigorously
worthless. But this proof requires that the optical procedures, mechanical or of any other
317
nature, provide for infinite in the sense of the idealist enlargement; the enlargement must
be without limit, which, in any fashion that we attempt, is unimaginable for man. But
finally, it is not a question of the possibility of perfect processes of measurement, but of
the possibility for us of convincing ourselves of their exactitude; we must be able to
control these mechanisms. For, to have the right of trusting our verification, it is
necessary that these mechanisms of verification are made of a well-known substance,
fabricated with the help of technical processes which are familiar to us, so that we could
appreciate their surety. A similar appreciation could only ever furnish us with a certain
quantity of exactitude; due to the inexactitude of our senses and our processes, we always
are at the point of departure. We understand that: human knowledge escapes the exact
absolute in every circumstance, if we admit nothing for certain in respect to the
combinations at which our systems of representation end up.
The ideal rectilinear extension is thus only the final arbitrary admission of a series
of always more precise representations of measurements, but a series for which we cannot
prove the existence of an end. In the world such as it is offered to our perceptions,
nothing speaks in favor of the existence of the geometrical ideal, and further everything
speaking against it.
What remains after these reflections of the Idealist’s system?
The unit of length, if for example we dream only of these extremities, only
scientifically has an arbitrary exactness; it is only a representation oscillating between
differences as small as one would wish, like a scale [-92-], i.e. a fundamental
measurement construed with every scientific precaution an epoch could arrange. The
parts of the unit length are evidently subjected to the same inexactitude that we can
318
suppose as restricted as we wish. We could represent equal parts in innumerable quantity
on each extension, as small as one wishes, precisely because the inexactitude diminishes
as we are permitted to push the division of an extension as far as we wish. But the
pursuit of this division, when we do not halt at any part, finishes by being lost in vague
hand waving. Thought can push this division as far as we want, this act does not prove
the infinitely small, but rather fatigue discourages the mind which cannot see the end of
its march, hidden in a uniformly cloudy region.
In other aspects, mathematical sciences evidently do not have a need of perfect
exactitude of geometric representations. For the science which can require
representations at the highest level, geometry, in general the precision of straight lines of
nature suffices; if we dream that we can represent lines as long as we wish, it is to render
the inevitable inexactitudes as small as we wish, by relationship to the length. Stars
appear as points, even with the strongest magnification -- an example which satisfies our
need of rigor, and the bundle of rays that sends to our eye one fixed star is certainly a
straight enough line to suffice for every requirement of geometry.
The empiricist conception, by leading me to see in exact measurement only a
verbal product of thought, outside of all reality, to declare even that, were it a reality, we
could not recognize it, makes me reject this concept as fundamental to the system of
mathematical concepts. I must still separate the conclusions of the Idealist, of which the
foundational uniqueness was the idea of exact length, from his theory of the infinitesimal,
and, with it, the proof that he deduces for the existence of a limit of irrational numbers, in
the same way as the limit itself. Meanwhile, if I deny this limit of decimal fractions
319
without irrational ends, I must add that to my eyes it can no longer be a question of the
limit of the rational fraction without end, as an exact ideal value.
If we wish to put ourselves in the shelter of the products of thought such as the
infinitesimal of the Idealist with its strange properties, we cannot leave the domain of [93-] the real representations of man, for our laws of thought, that is the natural filiations
of representations, are abstracted from the world of phenomena in the course of
development of the human species, and it is not truly possible that it occur it anew, which
would contradict our most familiar forms of thought, as for example the concept of
equality enlarged upon by the Idealist which is in general admitted. In preserving in
thought the concepts which correspond to non-reality, to intuition, we give way to
multiple paradoxes which befall Idealism in all domains of thought.
THE IDEALIST
32. – The existence of exact measurement is founded by the theory of the
Idealist. – The principle objection that the Empiricist makes against my system is the
non-reality of exact measurement taken as a basis of our logical operations.
For, that an exactly straight line could not truly be composed with the material
which forms the body of our knowledge, I agree with him without restriction. I believe
too that even if we take in our hands an absolutely exact prism, we could not know by any
means whether it had defects.
Only, all our scientific representations are riddled with geometrical ideas, exact
measurement is found equally in the thought of all, that it is only critical intelligence
which sees it as an ideal, which is to say as something which does not belong to the
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domain of perception, and the justification of its admissibility, on the contrary, will
astonish the impartial man. He will wonder when the distance between two points has had
need of being clarified as an exact concept, this concept so familiar to every schoolboy.
A mode of intuition as profound and as generally rooted, which is not conveniently
destroyed, and we could perhaps better succeeding by establishing scientifically what we
wish to remove. For the thought that exact measurement does not exist in the world
would be as troubling as those which reflect on the stability of things, and they would end
up by returning to belief in geometrical ideals. In every combination of quantities, from
which we wish to deduce something, we have regrets. The geometric ideals could follow
from the march of thought, and draw form it scientific reason for being.
Of the multiple representations from the world of phenomenon, the concept of
space is the last abstraction. Space is [-94-] a quality common to representations of every
sense, particularly of vision, of touch, and of muscular sensation.
Objects placed aside each other, one before the other, or one behind the other at
comparable intervals, the relative movement of the perceived object and of that which is
perceived, all this remains residual to that which we call the concept of space. By thought
we suppose ourselves and objects moving through space, as well as others which are put
in our place, so that the contents of finite space become accessories in our representation.
This is the origin of the abstraction of empty space which is of a completely scientific
nature. For what we have in view, it suffices to report on the representation of
homogenous space, which has been abstracted from perceptions of liquid and transparent
bodies. Meanwhile the scientific mind doubts the homogeneity of these natural bodies,
321
so that we are obliged anew to return to empty space for the foundation of our
considerations.
The representation of empty space is without doubt a limit of real representations,
but we cannot say if it is itself real, i.e. if there is or has been a space in which bodies are
moved exactly according to the law of inertia, which does not let any ray of light pass,
into which no force can penetrate, briefly, if thanks to acquisitions that the holds for
physics, besides pneumatic machines, there will be pumps destined to abolish the
unthinkable. For those who consider the material world as limited or as formed of
isolated sets, empty space is still a necessary consequence of their conception. In any
case, I could give empty space as fundamental to my considerations, for every chamber is
the image, the real representation of a space and as I can draw out in large part that which
it contains, supposing that I arrange of sufficient means, I have the representation of this
space emptied of all the material it contained, limited completely naturally by our real
representations, and science would have to renounce much, if it refuses to admit this
limit. I thus part from empty space or more generally from homogenous space.
Evidently in all empty space not only can we introduce by thought shapes without
number, but still they are real. In this block of marble is carved our portrait, then the
sculptor can with his chisel and according to his talent arrive at a greater or worse
resemblance. Similarly I affirm that in this space that I imagine as homogenous as the air
is for our senses, or rather empty, every shape of [-95-] precise form can not only be
assumed but is actually contained. If I imagine a sphere having a certain point for its
center and the surface has been passed through a second point in a certain fashion, the
system of these two points exists in space in a parallel sphere, absolutely like the block of
322
marble where one could make something escape with the chisel. This is a representation
so natural that it is even clear to the child.20
But one does not need to appeal to the imagination to be assured of the existence
of the geometrical exact. The study of the last elements of bodies leads to the same
result. Some representation that could be made of matter which fills space in such a way
that it becomes a system of moving points at a distance from one another, or else becomes
a homogenous and continuous matter (logically one does not know to make a third
hypothesis for the idea of a heterogeneous matter is repugnant to the mind until the point
at which, having decomposed, it has been restored to representations of homogenous
space or of space without content), the thought which in analysis must always result in
these exact images. Let us suppose the matter homogeneous; it is necessary that it always
have limits of some type, which form with every desirable precision the end of a
homogeneous body and the commencement of another. But the mind does not find
satisfaction in homogenous and continuous material, it decomposes it and it restores it to
its formal elements empty of all content, increasing the distance between elements; then
these elements, to attain the limit of thought, must be known as simply as exact images,
as surfaces, as lines, as ideal points. If we wish to restrict ourselves to points, their
distance is a straight line which has at the very least exact extremities, and this is the
fundamental hypothesis of the idealist system.
It is to my eyes a mistake that we wish to admit only non-exact forms. What
advantage has a certain inexactitude over exactitude? Isn't this certain inexactitude
20
These reflections have been communicated to the author when a child, by his father.
323
necessarily exact in its contours, so that we are forced to find the exact in the inexact?
We must accord to the geometrical ideal the same right of existence that any character of
the ideal has!
Thus men are certainly idealists. Why do they give such pains to be convinced to
doubt the evidence of geometrical ideals, if they only thereby attain [-96-] a changing of a
clear and precise intuition against a mode of uncertain representations?
The infinitesimal, inseparable from the ideal, so that the Empiricist represents it as
a danger against which we must guard, it's certainly not yet given an incontestable place
in the system of concepts common to all men, and this is sufficiently explained, as I have
shown. But the infinitesimal corresponds in an incontestable fashion to the infinitely
large, which immediately finds admittance, and even must be considered as belonging to
the system of concepts common to all men. It is notably the concept of generalized
equality, following which α + an infinitesimal = α, if α is finite, which seems repugnant to the Empiricist and which is called the strange property of the infinitesimal. For in
mathematics for a long time the equality ∞ + a finite quantity = ∞ has been granted
citizenship. This property of the infinitely small is not formally distinct from that of the
infinite. The former equality is so to speak less elevated by a degree than the latter.
But we can still understand this in other ways. We can say that the sign of
equality is only applied to the finite parts of the two members of the equation just as we
decide in certain calculations that this sign only concerns the real part or the imaginary
part of each of two members. In this fashion we do not introduce into the calculation this
property of the infinitely small, on the grounds that it does not change the finite by
324
addition; at least we do not introduce it in a formal fashion, for at the bottom the sense of
operations supposes it.
33. – On the double concept of geometrical ideals. – The definition of
geometrical ideals is not a clear definition; on the contrary, we find ourselves faced with
these two essentially different manners of conceiving the point, the line, and the surface,
one of which has the infinitesimal in an important role, and this is the place to insist some
on this question. There are two ways of conceiving of the formation of geometrical
ideals, as limits of ordinary representations. One presents to thought as its point of
departure a closed space, limited by surfaces, and the fact thus departs from the ideal
surface which only forms an exact limit of a single side, where it separates the given
space from that which it surrounds, and which the other side neglects as it is, in the
representation of the angle, the space comprised between these sides. Two surfaces are
cut following a corner, that which is limited by two other surfaces which gives rise with
these two extreme points to the extension limited in length.
The other mode of formation of these geometrical ideals lies [-97-] in the
representation of the infinitesimal limit, and develops fundamental geometrical
representations in the inverse order. Here above all the point is given as equivalent to a
space which is contracted to the infinite in each of its directions, i.e. to a space which can
never be measured in any direction with the help of a finite unit. Several authors create
lines by the continuous movement of this infinitesimal point through space, and the
surface by the movement of a line. Abstraction made from this mode of representation
requires a certain relationship of measurement between the three fundamental geometrical
ideas, to which they are not themselves submitted, this mode of representation is uniquely
325
created from a special image almost like the creation of hollow spaces in unleavened
bread, when we place it among the force of solid bodies; it returns in sum to movement
and does not possess in general the degree of abstraction to which we must aspire in the
determination of these very important concepts.
Without considering the mode of formation of the point, we are forced to conceive
of the line as the limit of a slender thread. Following this principle of definition the
surface must be considered as an infinitely thin body along a single dimension, according
to the representation of Gauss (Disquisitionnes circa superficies curvas, art. XIII). It is
incontestable that the second mode of formation is better connected to the mathematic
representation of fundamental geometrical figures. Only, it assumes the concept of
infinitesimal, at which we only arrive in leaving behind exact representation of the first
type.
The most logical series of ideas, which leads to the geometrical ideals of
mathematics, could be this: We at first would follow the primary mode of representation,
according to which we conceive of the unlimited line with a corner. Then the
representation of the exact intersection proceeds, as I have shown in the system of the
Idealist, from the concept of the infinitely small, which definitively permits us to
conceive the geometrical ideals following the second mode.
THE EMPIRICIST
34. – Conclusion of his critique of Idealism. – That which precedes is sorted by
our author into for and against Idealism, and suffices to clearly show the opposition and
326
incompatibility of our two modes of intuition, which must reach not only to the
fundamental concepts of mathematics but to [-98-] the collection of our conception of the
world. A series of representations, which excites in us a desire to pursue it to its end, can
by its nature not possess an equal end in our system of representations. I am content to
pursue the series of representations as long as it implies representations, but Idealism
gives its conclusion as a fact, even when it cannot prove its existence, or, more strongly,
form a representation of it. The Idealist is pulled by his concern with axioms.
This axiom is and remains the existence of exact measurement, for I cannot see
any conclusive character in the proof attempted (of this subject). It rests first of all on a
non-representable limit, empty space, or homogeneous space. However, to admit this
limit, one needs to agree that empty space in reality includes every shape. In fact, it
includes nothing. We take from our memory images of shapes passably exact and we
represent to ourselves in place of empty space a space which includes these shapes. This
is incontestably all that happens. The method which leads the Idealist to conclude this
fact of the existence of the ideals seems to me false. For his argumentation, following
which the geometrical ideals are present in space, under the pretext that every block of
marble contains thousands of statues which wait for the sculptor to deliver them, and
what furthermore the ideal must be as real as a certain deviation, already rests absolutely
on the hypothesis of the ideal – a determined deviation, from which the geometrical
definition is of the same nature as these ideals, having to obey the same laws that it does.
When I speak of inexact geometrical shapes, that is to say oscillating between limits
which are established in some manner, I have in mind not a special shape, but all possible
shapes included in these limits.
327
Finally, the Idealist invokes these last elements from the corporeal world of
bodies, where elements must have exact limits, for, in the contrary case, we could not
arrive at the first degree of abstraction. However we do not know how to simply
conclude the existence of exact shapes, for the reason that these last elements are
precisely the limits of representation as difficult to represent as the geometrical ideals; I
judge them equally inadmissible. I have not objected to corpuscular theories when I
reject atoms.
The difference in our logical methods is shown most clearly in the determination
of concepts to which the Idealist attaches a completely [-99-] specialized importance; I
wish to speak of his distinction between "unlimited" and "infinite" which I wish to attack
in finishing up. For my part, with the best will in the world, I can only can only see
things of arbitrary largeness or smallness; to my eyes, it is an unhappy chance that
"infinite" is used rather than “non-finite".
The primitive sense of the word is assuredly "without end," synonymous with
"unlimited."
Who doubts that our sensory perceptions only allow us to penetrate absolutely
imperfectly into the essence of things?
In this penchant to leap over natural limits of our powers of representation with
limits of unknowable concepts, I see our mistaken instinct to know and I take as wise the
restraint of this instinct as well as others, and I am resigned to it when profound
knowledge is refused to me.
328
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Curriculum Vitae
Lisa Keele
[email protected]
1525 Edna Avenue
Natchitoches, LA 71457
Education
Indiana University, Bloomington
• Ph.D. in Philosophy, 2008.
Coursework concentration: Logic and foundations of mathematics. Ph.D. Minor:
Gender Studies.
• M.A. in Philosophy, 2000.
University of Utah
• B.A. in Philosophy, Minor in Anthropology, 1995.
Areas of Specialization
•
Logic, Philosophy of Mathematics.
Areas of Competence
•
Metaphysics and Epistemology, Gender Studies, Feminist Philosophy.
Dissertation
Title: Theories of Continuity and Infinitesimals: Four Philosophers of the Nineteenth
Century
Dissertation Committee: David McCarty (chair), Larry Moss, Timothy O'Connor,
Fred Schmitt
Dissertation Summary: My dissertation concerned the problem of numerical
continuity, as debated in the late nineteenth century, focusing on the philosophy of
Richard Dedekind, Georg Cantor, Paul du Bois-Reymond, and Charles S. Peirce.
Honors and Awards
•
Outstanding Associate Instructor, Department of Philosophy, Indiana University, May
1999.
Publications
•
"Let Them Eat (Barley) Cake: An Epicurean Recipe to Share with your Students,"
with Rondo Keele, APA Newsletter on Teaching Philosophy, vol. 02, no. 1 (Fall
2002): 208-210.
•
"Hidden Worship: The Religious Worship of Orthodox Jewish Women," Women in
Judaism: Contemporary Writings, 1998, URL =
<"http://www.utoronto.ca/wjudaism/contemporary/articles/a_keele1.html">.
Guest Lectures
•
"Barley Cakes and Friendship," discussing the relationship of Epicurean ethics to
Epicurean science. Presented in the Louisiana Scholars' College Texts and Traditions
course, 2007 and 2008.
•
Will the Real Socrates Please Stand Up," discussing Socrates as the rebel, in the
Apology, who would rather die than obey the state, and Socrates the loyal citizen, in
the Crito, who would rather die than disobey the state. Presented in the Louisiana
Scholar's College Texts and Traditions course, 2008.
Teaching Appointments
•
Adjunct Faculty, School of Social Sciences, Northwestern State University,
Natchitoches, Louisiana, 2007-2008.
Classes Taught: Ethics (theory and applied, 4 classes), Introduction to Philosophy (3
classes), Logic (formal and informal, 1 class).
•
Teaching Assistant, Philosophy Department, Indiana University Bloomington, 19952000.
•
Grader, Philosophy Department, University of Utah, 1994-1995.
Non-teaching Appointments
•
Special Assistant to the Ambassador; Embassy of the Republic of Korea in Cairo,
2004-2005.
•
Research Assistant; Forced Migration and Refugee Center, American University in
Cairo, 2003-2004.
•
Program Coordinator, Indiana University Logic Group, Indiana University,
Bloomington, 1999-2000.
Service
•
Second reader on a Senior Thesis at the Louisiana Scholars' College. Title: "Ethics
and Multimedia Journalism." 2007-2008.
•
Fraser Snowden Award in Philosophy committee member; 2008.
•
Mentor, McNair Scholar's Program, Indiana University Bloomington, 1997 - 1998.
Reading Languages
•
French