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Jérôme Havenel
Peirce’s Topological Concepts,
Draft version of the book chapter published in
New Essays on Peirce’s Philosophy of Mathematics,
Matthew E. Moore ed, Open Court, 2010.
Abstract
This study deals with Peirce’s writings on topology by explaining key aspects of
their historical contexts and philosophical significances. The first part provides
an historical background. The second part offers considerations about Peirce’s
topological vocabulary and provides a comparison between Peirce’s topology and
contemporary topology. The third part deals with various relations between
Peirce’s philosophy and topology, such as the importance of topology for Peirce’s
philosophy of space, time, cosmology, continuity, and logic. Finally, a lexicon of
Peirce’s main topological concepts is provided1.
Introduction
In Peirce’s mature thought, topology is a major concern. Besides its intrinsic
interest for mathematics, Peirce thinks that the study of topology could help to
solve many philosophical questions. For example, the idea of continuity is of
prime importance for Peirce’s synechism, and topology is “what the philosopher
must study who seeks to learn anything about continuity from geometry” (NEM
3.105)2. In Peirce’s reasonings concerning the nature of time and space,
topological concepts are essentials. Moreover, in his logic of existential graphs
and in his cosmology, the influence of topological concepts is very apparent.
However, Peirce’s topological vocabulary is difficult to understand for three
reasons. First, in Peirce’s time topology was at its beginnings, therefore both
terminology and concepts have evolved a lot in the 20th century, and we find
Peirce struggling to find light in a quite unexplored environment. In particular, to
the best of my knowledge, Peirce has never studied - nor mentioned -, Poincaré’s
major contribution to algebraic topology, his “Analysis situs” (Poincaré, 1895),
followed by a series of five addenda from 1899 to 1904. Indeed, Poincaré’s papers
have provided some of the key concepts of modern topology.
Secondly, in almost all contemporary introductions since Lefschetz, topology
seems to be grounded in the notion of set, and topological spaces are defined as
sets3. But to ground topology in the theory of sets is historically misleading. If the
notion of continuous transformation can be characterized within the framework
of set theory, the main intuitions of the founders of topology were related to
general geometry. As Gauss, Listing, Klein and Poincaré, Peirce considers
I offer my grateful thanks to André De Tienne, Marc Guastavino, Matthew Moore, Marco Panza,
and Fernando Zalamea for their very helpful comments.
2 RLT, p. 246, MS 948, 1898.
3 Lefschetz, 1949, p. 3.
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topology as a general geometry. Indeed, what Peirce calls topology corresponds
roughly to algebraic topology and some aspects of differential topology4. If Peirce
is concerned with point-set topology when he characterizes the continuum as
being supermultitudinal, point-set topology is not topology for Peirce.
Thirdly, Peirce frequently changed the names he used to refer to topological
notions. It is therefore useful in introducing Peirce’s topological vocabulary to
give the main definitions, their historical context, and philosophical significance.
Hence, there is a glossary at the end of this study.
As a forerunner in topological researches, Peirce offers various definitions of
topology, but we can start with the following: “topology, or topical geometry, is
the study of the manner in which places are intrinsically connected, irrespective
of their optical and metrical relations” (NEM 2.165). One can notice Peirce’s
terminological hesitation between topology and topical geometry. In the
Cambridge Lectures of 1898, he explains that what Listing calls “topology”, he
prefers to call it “topic”, to “rhyme with metric and optic” (NEM 3.105). One
should remember that for Peirce, there is no distinction of meaning between
topology, topical geometry, geometrical topics, topics and topic. Likewise, there is
no distinction of meaning between graphics, optic and projective geometry5.
Metrics, metric and metrical geometry also mean the same thing for Peirce.
1) Historical Background
As the son of Benjamin Peirce, considered by many to have been America’s first
truly creative mathematician, and being himself an important figure in the field,
Charles Sanders Peirce was well informed of the works of many prominent
mathematicians of his day. In particular, we know that Peirce was deeply
interested by mathematicians such as Cantor, Cayley, Clifford, Dedekind, Gauss,
Helmholtz, Klein, Lobatchewski, Riemann, Story, and Sylvester6. For example,
Clifford was a friend of Benjamin Peirce and was one of the first to recognize the
importance of the paper on the founding of geometry published by Riemann in
1867. In 1873, Clifford translated Riemann’s work in English, and Charles
Sanders Peirce has read this translation7. Peirce attended several meetings at the
New York Mathematical Society, before it became the American Mathematical
Society in 1894. However, from the 1890s to the end of his life, Peirce’s contacts
with the mathematical community shrunk dramatically.
Peirce met several mathematicians when he was a member of the Johns
Hopkins University from 1879 until 1884. Among his colleagues were Sylvester
and Kempe, and it is likely under Kempe’s influence that Peirce became very
interested in the topological question of the Four Colour Problem8. “On
November 5, 1879, Dr. Story presented a communication by Mr. A. B. Kempe …
This question will be elaborated later on.
NEM 2.625, MS 145: “… Projective Geometry, which I prefer to call, with Clifford, Graphics”.
6 NEM 3.979-980; NEM 3.883.
7 Riemann, 1873.
8 The Four Color Problem is a topological question since the sizes and shapes of the map regions
do not matter; what matters is only the way they join together.
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‘On Geographical Problem of the Four Colors.’ … Mr Peirce is recorded as then
discussing…” (Eisele, 1979, p. 217).
Sylvester’s close friend Arthur Cayley was invited in 1882 to give a course of
lectures at Johns Hopkins University9. During his stay Cayley undoubtedly
influenced Peirce. Cayley also influenced both Sylvester and Clifford in their idea
of using chemical diagrams to represent algebraic invariants. In a letter to Oscar
Howard Mitchell, on December 21, 1882, Peirce explained that he was greatly
impressed by this idea10. Unsurprisingly, since Peirce was both a professional
chemist and logician, deeply interested by new mathematical ideas.
Cayley was one of the few mathematicians of his time to be interested in
Listing’s Census-theorem, published in 1861, which represents the first important
mathematical step in topology since Euler. Cayley’s interest in Listing’s work was
probably related to his work on the theory of the combinatorial machinery for
obtaining various chemical graphs11. The same year Listing published his Censustheorem, Cayley published a paper “On the Partitions of a Close”, which he
thereafter recognized to be “a first step toward the theory developed in Listing’s
memoir”12. Cayley reported on Listing’s Census-theorem to the London
Mathematical Society in 1868, and he published a summary of it in the
Messenger of Mathematics, in 1873.
It is very likely under Cayley’s influence that Peirce read Lisiting for the first
time, but as shown by his “Notes on Listing” in MS 159, Peirce was at first
disappointed and not convinced by Listing’s work. As quoted by Murphey (1993,
p. 197), in MS 159, Peirce says that Listing’s work is “Talkee, Talkee” and that he
does not “understand this so far, at all”. The dating of MS 159 is difficult; Robin
says: “1897?”; whereas Murphey seems to think that it is rather 1882-188413. In
his article “theorem” written for the Century Dictionary, so in 1884-1889, Peirce
gives a definition of Listing’s theorem, but a short one: “Listing’s theorem, an
equation between the numbers of points, lines, surfaces, and spaces, the cyclosis,
and the periphraxis of a figure in space: given in 1847 by J. B. Listing. Also called
the Census theorem” (CD, “theorem”); and Peirce does not define the Census
theorem in the CD. However, in the SV, such a definition is given which was likely
written by Peirce, as it is suggested by the reading of MS 1597. Whatever the case
may be, in MS 137, written in 1904, Peirce speaks with great respect of Listing’s
Census-theorem as being a masterpiece.
Around the same time of Listing’s first works in topology, the theory of
invariants was invented by George Boole who is best remembered as one of the
creators of mathematical logic. The theory of invariants was subsequently
developed by mathematicians such as Arthur Cayley and James Sylvester. In his
1859 “Sixth Memoir on Quantics”, which is widely quoted by Peirce, Cayley
developed the geometrical application of the theory of invariants. It was already
known that for projective geometry, in its legitimate transformations, lengths,
Parshall and Rowe, 1994, p. 79-80.
L 294.
11 Murphey, 1993, p. 196.
12 Cayley, quoted in Murphey, 1993, p. 196.
13 Murphey,1993, p. 197.
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angles, areas and volumes are not always preserved. The distance between two
points is a numerical value that is invariant in metrical geometry. However, in the
realm of projective geometry, this quantity does not remain invariant and is
therefore useless. But in projective geometry there is one numerical invariant
which is called the cross-ratio. Because it is invariant under projective
transformations, the cross-ratio, which is a ratio of ratios of distances, is a
fundamental quantity for projective geometry.
In 1859, Cayley set himself the task of establishing the metrical properties of
Euclidean geometry on the basis of projective concepts. Cayley generalized
Laguerre’s idea of defining the size of an angle in terms of the projective concept
of cross-ratio. Cayley’s idea is that metrical properties are not intrinsic properties
of a figure but are properties of the figure in relation to an absolute, which is
invariant under projective transformations. For three-dimensional geometry, this
absolute is geometrically a quadric surface, or algebraically an equation of the
second order. Cayley concluded that metrical geometry is a part of projective
geometry. However, Bertrand Russell showed in 1897 that Cayley is wrong when
he claims that metrical geometry logically follows from projective geometry.
Russell pointed out that it is not possible to obtain the usual metrical geometry by
Cayley’s method without adding a new postulate to projective geometry: “If a
logical error is to be avoided, in fact, all reference to spatial magnitude of any
kind must be avoided; for all spatial magnitude … is logically dependent on the
fundamental magnitude of distance.”14 However, Russell remarked that Klein
succeeded in avoiding this error.
In 1871, Klein took up Cayley’s idea to show that not only Euclidean geometry
but also the non-Euclidean geometries could be founded in projective geometry.
In 1872, in his Erlangen Program, Klein proposed a unified approach of geometry
that became the accepted view. Although different geometries focused on
different geometrical properties according to whether or not those properties are
invariants under the allowed transformations of the geometry in question, Klein
found a way to unify those different geometries. His idea was to define a
geometry as the study of the properties that are invariant under a given group of
transformations. Although group theory is an algebraic domain, it is a relevant
tool to represent the ideas of transformations and symmetries. For example,
Euclidean geometry is the science of invariants of the metrical group, and
projective geometry is the science of invariants of the projective group. From the
idea that each geometry is the invariant theory for a special group of
transformations, Klein could draw the classification of geometries. It is Klein’s
work which convinced almost the whole mathematical community that nonEuclidean geometries should be accepted as legitimate.
Klein’s classification of geometries had a strong influence on Peirce, and while
discussing non-Euclidean geometry, Peirce claimed that, “a new synthetic
exposition is much needed, and might well accompany a collection of the
contributions of Lobachevski, Bolyai, Riemann, Cayley, Klein, and Clifford.” (CP
8.96).
14
Russell, 1956, p. 32.
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After the previous sketch on the historical context of the development of
geometry in the 19th century, we can now deal with topology. What is topology
from the contemporary point of view? Contrary to Euclidean geometry, which
deals with metrical properties, topology is that branch of mathematics that
studies topological properties of geometrical figures. But what are these
topological properties? They do not involve measurement, and they are preserved
when a geometric figure is deformed continuously, whereas non-topological
properties change under such a deformation. For example, if you continuously
deform a circle, it will remain a circle, but it will still keep its topological property
of being a closed curve that does not intersect itself. Topology is a large and active
branch of mathematics today, one that is attracting attention from other
disciplines, like theoretical physics, molecular biology and cosmology.
Topology is par excellence the mathematical doctrine that is incompatible with
the widespread idea that mathematics is a science about nothing else than
quantities, geometrical quantities and numerical quantities. Leibniz dismissed
this idea and he coined the term analysis situs to describe a geometrical method
which expresses position (situs) in the same way algebra expresses the quantity.
However, there is a debate regarding whether or not Leibniz should be
considered to be a founder of topology15.
Peirce’s technical writings on topology are for the most part an attempt to
develop what is now called ‘Euler characteristic’16. This kind of mathematical
research was called combinatorial topology and is now called algebraic topology.
However, it would be misleading to say that Peirce’s topological considerations
are related to algebraic topology. First, Peirce works on continuous structures
useful for analysis (like lines, circles, disks, spheres, torus, etc), rather than
discrete structures. Second, Peirce’s topological considerations also deal with
topological singularities, like in differential topology.
Fig.1
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Pont, 1974, p. 2-13. See also Lakatos, 1976, p. 6
NEM 2.189.
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Until the mid-19th century, the history of topology coincides essentially with
researches on Euler’s characteristic. And these researches achieved a mature but
not a definitive form after Poincaré’s works, which developed what is now called
the Euler-Poincaré characteristic. The central idea of the Euler-Poincaré
characteristic is that continuous geometrical phenomena can be understood by
the use of discrete invariants.
The first mathematical achievement in topology was made by Euler, who wrote
in 1736 a paper in which he solved the problem of the seven bridges of
Königsberg (now Kaliningrad). In this town there is an island, called “Kneiphof”,
with two branches of the Pregel river flowing around it (fig.1).
The question is whether a person can walk in such a way that he will cross each
of these seven bridges once but not more than once. In other words, is it possible
to cross the seven bridges without retracing one’s steps? Euler’s method was to
represent the situation as a graph, that is, a collection of points (vertices) and
lines (edges) where each edge connects exactly two vertices. This means that each
bridge was represented as a line connecting a land region. Euler obtained the
following graphs:
Fig.2
Fig.3
He then reworded the problem by asking whether or not it is possible to draw
the above graph with one stroke of the pencil. To solve this problem, Euler
introduced the notion of the degree of a vertex, which represents the number of
edges connected to that vertex. A vertex is considered to be even or odd if it is the
end of an even or odd number of edges respectively. Euler proved the general
theorem that in order to be able to draw a graph with one stroke of the pencil and
without drawing the same edge more than once, it is necessary that the graph has
either no odd vertices or only two odd vertices. Since the graph representing the
seven bridges of Königsberg has four odd vertices, crossing each of the seven
bridges once but not more than once, is impossible17.
While attempting to solve this problem, Euler noticed that some geometrical
problems can be considered without using any notion of distance; a graph
17
Artemiadis, 2004, p. 348.
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contained all the relevant information. Euler thought that such a method could be
used to classify polyhedra, for such a classification is essentially a question of
position rather than quantity. Classifying polygons is very easy, since it is
sufficient to count the number of edges. And for every polygon, the number of
vertices (V) is the same as the number of edges (E), E = V18. But it is much more
complicated to classify polyhedra, because the number of faces can leave
undetermined which polyhedron is described. Euler discovered that if one counts
the number of vertices (V), edges (E) and faces (F) of any closed convex
polyhedron19, like a cube for instance, then:
(1) V – E + F = 2
For example, a cube has 8 vertices, 12 edges and 6 faces: 8 – 12 + 6 = 2.
Fig.4
F = 6, E = 12, V = 8
V–E+F=2
Fig.5
F = 0, E = 6, V = 6
V–E+F=0
Fig.6
F = 0, E = 4, V = 5
V–E+F=1
Lakatos, 1976, p. 6.
Euler was wrong in thinking that his formula holds for all polyhedra ; it is true only for all
closed convex polyhedra.
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Euler’s formula is also useful for linear figures, in which there is no face. If the
linear figure is closed, like a triangle or any polygon, then Euler’s formula equals
zero. Because for any polygon F = 0 and E = V, it follows that V – E + F = 0. If
one consider a linear figure that is not closed, then, providing that F = 0, the
number of vertices is always greater by one unit over the number of edges, and it
follows that V – E + F = 1.
Although Euler did not consider invariance under continuous transformation,
from a contemporary point of view Euler’s formula for polyhedra is considered to
be the first theorem of topology since the formula still holds when a continuous
transformation is applied to the polyhedron in question. Indeed, Euler’s new way
of treating geometry was to give birth to topology. Unlike Leibniz’s analysis situs,
Euler called it geometria situs. The similar term of “geometry of position” was
chosen by von Staudt as the name for projective geometry. Finally, Listing coined
the modern term of topology20, but the use of the term analysis situs did prevail
until around the death of Poincaré. Although it is Riemann who convinced the
mathematical community of the importance of topology by showing its usefulness
for the study of functions of a complex variable, it is Poincaré who developed the
modern science of algebraic topology, notably through his work on Euler’s
formula, renamed after him the Euler-Poincaré characteristic.
If Leibniz gave the general idea of what could be a topological science, and if
Euler demonstrated the first topological theorem, it was not until about 1850 that
interest in topology did become general. In 1847 Listing published his
“Preliminary studies on topology”. He defined this new term by the following:
By topology we mean the doctrine of the modal features of objects, or of the
laws of connection, of relative position and of succession of points, lines,
surfaces, bodies and their parts, or aggregates in space, always without regard
to matters of measure or quantity21.
However, this essay does not contain any new important topological concept.
But in 1862, Listing published his Census of spatial aggregates, or
generalization of Euler’s theorem on polyhedra22. It is surprising that the title
does not contain the word topology, but the core of the book consists in the
establishment of what Listing calls the “Census-Theorem”, which is an extension
of Euler’s theorem. Listing probably chose the word “Census” because the
‘Census-Theorem’ aims at establishing a wider taxonomy of topological spaces
than Euler’s classification of polyhedra.
Between Euler and Listing, Lhuilier had discovered that there are some
pathological polyhedra which can be classified, but for which Euler’s formula
does not hold. One way to dismiss this difficulty was to reject Lhuilier’s
pathological polyhedra as not being true polyhedra, and to restrain the definition
of polyhedra to those for which Euler’s formula holds well. But Listing took the
opposite approach, which will reveal much more fruitful. Listing’s idea was to
generalize as much as he could Euler’s formula in order to find a general formula
Breitenberger, 1999, p. 909. See also Pont, 1976, p. 43.
Quoted in Breitenberger, 1999, p. 916.
22 The translation from German is taken from Breitenberger, 1999, p. 919.
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that would hold not only for closed convex polyhedron but also for Lhuilier’s
pathological polyhedra and for many other structures.
As it has been previously seen, for any closed convex polyhedron Euler’s
formula is: V – E + F = 2. But for any closed linear figure, Euler’s formula is: V –
E + F = 0 ; and for any open linear figure, Euler’s formula is: V – E + F =
1. Listing thought that the number 2 in Euler’s formula for closed convex
polyhedron should become a variable in his “Census-Theorem”. As an extension
of Euler’s formula, we could dub this new variable S for spaces, and we would
obtain :
(2) V – E + F – S = 0.
More precisely, S would be for Listing the number of disjoint spaces. For
example, a cube in the three-dimensional space has 8 vertices, 12 edges and 6
faces, and it determines one internal space and one external space ; therefore S =
2, and we obtain: 8 – 12 + 6 - 2 = 0.
In pursuit of the highest generality, Listing considers aggregates or complexes
of spatial constituents. An aggregate or complex of spatial constituents is for
Listing any configuration of points, lines and surfaces in space, provided that
these spatial constituents must be connected together23. A constituent can be a
point, a line, a surface. But there is also a fourth category which can either be a
three-dimensional solid body or an amplex, which is Listing’s name for the
infinite space in which a complex of spatial constituents is immersed. Hence,
there are four categories of constituents.
By the letter ‘a’, Listing represents the number of points of the complex of
spatial constituents. By the letter ‘b’, Listing represents the number of lines of the
complex of spatial constituents. The number of surfaces depends on the numbers
of borders within the complex. The letter ‘c’ represents the number of surfaces of
the complex of spatial constituents. The number of spaces depends on the
numbers of borders within the solids belonging to the complex, and it also
depends on the amplex. The letter ‘d’ represents the number of spaces of the
complex of spatial constituents which corresponds to the number of disjoint
spaces.
Besides the letters: a, b, c, d, Listing’s formula also contains the letters: x, x’, x’’,
π, π’ and w. What do they mean?
The variable x denotes cyclosis. x is line-cyclosis, x’ is surface-cyclosis, x’’ is
space-cyclosis. Periphraxis is denoted by the variable π. π is surface-periphraxis
and π’ is space-periphraxis.
Because the purpose of this study is to examine Peirce’s topological vocabulary,
the exact definitions of cyclosis and periphraxis will be given later on when
examining Peirce’s terminology. However, to give a first idea, cyclosis is a
topological property related to the notion of being cyclical or not, and only
topological spaces of one or higher dimension are concerned with the topological
property of the value of their cyclosis. Likewise, periphraxis is a topological
property related to the notion of being spherical or not, and only topological
Indeed, Listing’s notion of complex is restricted to what we call today simplicial complexes. The
notion of ‘simplicial complex’ will be discussed later on.
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spaces of two or higher dimensions are concerned with the topological property
of the value of their periphraxis. In a nutshell, cyclosis and periphraxis are related
to the notion of closure for spaces of respectively one and two dimensions24.
The introduction of ‘w’ is used to correct a technical difficulty in Listing’s
formula ; if there is any constituent that reaches infinity, then w = 1 ; otherwise w
= 0. But Peirce will show that behind ‘w’ there is an important topological notion,
for which Peirce used the terms “apeiry” or “immensity”.
Finally, the general formula or Census-theorem is:
(3) a – (b – x) + (c – x’ + π) – (d – x’’ + π’ – w) = 025.
This theorem has the same form as:
(2) V – E + F – S = 0,
with
V=a
E = (b – x)
F = (c – x’ + π)
S = (d – x’’ + π’ – w)
Returning to the cube example (fig.4), it is made of 8 vertices (a = 8), 12 edges
(b = 12), 6 faces (c = 6), one internal space and one external space (d = 2). All
these constituents are continuously reducible to a point, hence, for a cube, x, x’,
x’’, π and π’ are all null. A cube does not reach infinity, therefore: w = o.
Hence, in case of a cube, x, x’, x’’, π, π’ and w are all null, and the Censustheorem correspond to : [a – b + c – d] : 8 – 12 + 6 – 2 = 0.
Listing introduced some other interesting topological concepts that influenced
Peirce. One of these is the “chorisis”, which is the number of disjoint pieces of a
spatial complex and which corresponds to the first Betti number. Contrary to
Listing’s terminology, the Betti numbers are still in use. The Betti numbers
correspond roughly to what Peirce calls the Listing numbers: the chorisis, cyclosis
and periphraxis. However, the Betti numbers do not originate with Listing, but
from a short paper of 1857 in which Riemann clearly defines the connectivity of a
surface by means of separating and non-separating cuts.
It is almost certain that Listing knew this paper by Riemann, but had he read it
more carefully, what is now called the Betti numbers would perhaps be called the
Listing numbers. Riemann’s brilliant idea had a strong influence on Betti, who
published in 1871 a memoir on topology that contained what we now call the
“Betti numbers”. The Betti numbers of a topological space are a sequence of
topological invariants, and each one is either a natural number, or infinity. The
Betti numbers were so named by Poincaré who was inspired to study topology
through Betti’s work on the subject. The first Betti number of a topological space
is, in intuitive terms, a way of counting the maximum number of cuts that can be
made without dividing the space into two pieces. For a surface that has edges,
each cut must be a “crosscut”, one that goes from a point on an edge to another
point on an edge. The first Betti number of a square is 0 because it is impossible
to crosscut a square without leaving two pieces. Poincaré uses the Betti numbers
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Breitenberger, 1999, p. 920.
Murphey, 1993, p. 202.
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to define what is now called the Euler-Poincaré characteristic of a topological
space.
What we now call the Euler-Poincaré characteristic is a wider generalization of
the Euler characteristic than the one proposed by Listing. The general idea is to
consider a subdivision of a geometric object into simplices. Because one can pave
a plane as a subdivision of triangles, such a subdivision into simplices is called
triangulation. A n-simplex is an n-dimensional analogue of a triangle. For
example, a 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a
triangle, a 3-simplex is a tetrahedron, a 4-simplex is a pentachoron, and so forth.
These simplices are used as building blocks to construct the class of topological
spaces called simplicial complexes. These spaces are built from simplices glued
together in a combinatorial manner. Simplicial complexes are used to define what
is called simplicial homology. Intuitively, homology, whose etymology means
discourse on the identical, is able to associate topological spaces that are
qualitatively equivalent by means of the use of an algebraic structure. In the case
of simplicial homology, the algebraical structures that are implied are the ndimensional simplices.
The Euler-Poincaré characteristic is a powerful topological invariant whose
value is independent of the subdivision of a geometric object into simplices, or
independent of the manner in which it is paved by simplices. Within the
theoretical framework of simplicial homology, one can demonstrate that the
Euler-Poincaré characteristic of an object mainly depends on its dimension and
on its Betti numbers26.
Fig.7
1 circle
Fig.8
1 segment and 1 point
χ=1–1=0
Fig.9
8 segments and 8 points
χ=8–8=0
The Euler-Poincaré characteristic is dubbed χ and it is an alternate summation
that is defined (for three-dimensional spaces) by the number of points, minus the
numbers of segments, plus the numbers of surfaces, minus the numbers of
volumes. For example, a circle is decomposable into 1 segment and 1 point (fig.8),
hence χ = 1 – 1 = 0. But a circle is also decomposable into 8 segments and 8
points (fig.9), hence χ = 8 – 8 = 0.
The surface of a sphere is decomposable into 4 points, 6 segments, and 4 faces
(fig.10 and fig.11), hence χ = 4 – 6 + 4 = 2.
26
Kline, 1972, p. 1173-1174.
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But the surface of a sphere is also decomposable into 8 points, 12 segments, and
6 faces, hence χ = 8 – 12 + 6 = 2.
Fig.10
4 points, 6 segments, and 4 faces
χ=4–6+4=2
Fig.11
4 points, 6 segments, and 4 faces
χ=4–6+4=2
2) Peirce’s Topological Vocabulary
For Peirce, topology is pure geometry, whereas projective geometry, like
metrical geometry, is not independent of facts about real space27. Topology “is the
most fundamental kind of geometry (all projective geometry being merely a
special department of topical geometry, as metric is of projective geometry)…”
(MS 1170, article “topical”)28. Topology “deals with only a portion of the
hypotheses accepted in other parts of geometry” (MS 94)29.
Now, what are the relations between topology, projective geometry and metrical
geometry? Peirce’s classification of geometry bears the influence of Cayley and
Klein. To classify geometries, Klein uses the possibility to define each geometry
according to the properties that are invariant under a given group of
transformations. To classify geometries, instead of group theory, Peirce uses the
broad concepts of specification and generalization. For Peirce, a geometry is more
specific if it distinguishes properties which are indistinct under a more general
geometry.
CP 4.219, 1897.
See also NEM 2.477 ; NEM 2.479, MS 137, 1904.
29 Quoted in Murphey, 1993, p. 216.
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In other words, topology cannot help the study of projective properties nor
metrical properties. A purely topological definition of a ray of light is impossible,
because the notion of a ray of light depends on our experience. When we imagine
any spatial figure, we imagine it as seen, and thus the familiar properties of light
are taken for granted by our imagination. The notion of a straight line comes
from our visual experiences and its simplicity is due to our minds having been
formed under the influence of physical phenomena30. For Peirce, “a body left to
its own inertia moves in a straight line, and a straight line appears to us the
simplest of curves. In itself, no curve is simpler than another” (CP 6.10).
Likewise, according to Peirce, for metrical geometry, it is necessary to introduce
the notion of a rigid body, but such a notion is not independent of contingent
aspects of the physical world31. The general science of topology is so broad that
the space of topology is bigger than the space of all possible experiences32.
Peirce’s topology considers as similar two objects that can be continuously
deformed into one another, provided that during such a deformation “no parts
are separated which were at first continuously connected and no parts are joined
which were at first not continuously connected” (MS 1170, article “topical”). As it
is shown in NEM 2.652, Peirce was aware of the necessity that for topology, such
a continuous transformation has to be continuous in both directions. In modern
terminology, such a continuous transformation is a homeomorphism.
The definition of topology as the pure geometry (NEM 2.652) that studies “the
modes of connection of the parts of continua” (NEM 3.105, RLT, p. 246, MS 948,
1898), leads Peirce to a research program. For Peirce, the characters which
topology studies are: “continuity and betweenness, dimensionality, topical
singularities or places within places which differ in their connections from
neighboring places within the same places, and Listing’s numbers” (MS 1170,
article “topical”).
Now if topology is a branch of geometry, the most fundamental kind of
geometry, topology itself has many branches. Indeed, within the topological
science, one must distinguish the number of dimensions of the space under
consideration.
There is a topics of a Space of one dimension, a topics of a Space of any whole
number of dimensions. Nor is that all. A Space of any number of dimensions greater
than one may have different shapes, although it be perfectly continuous in every part.
(NEM 2.480, MS 137, 1904).
But what does Peirce mean by “dimension”? Roughly speaking, the dimension
of an object is the number of coordinates needed to specify a point on the object.
For example, a line segment is one-dimensional, a square is two-dimensional,
and a cube is three-dimensional. Thus, the dimension of an object seems to be a
simple and invariant property; a line is not a square which itself is not a cube.
This property is sometimes called the “dimensionality” of an object.
CP 6.10, 1891.
CP 7.306-307.
32 NEM 2.480, MS 137, 1904.
30
31
13
14
From the contemporary point of view, the notion of dimension can be
generalized in various ways: infinite dimensions, fractional dimension, etc; and
outside of pure mathematics, different fields of study can define their spaces by
their own relevant dimensions, or independent parameters within a model.
The “dimensionality” is a qualitative property in the sense that if you drag a
point to trace out a line segment, a line segment to trace out a square, a square to
trace out a cube, the result is an object that is qualitatively “larger” than the
previous object. In other words, the purpose of the notion of dimension is to
capture this “qualitative” topological property.
But Cantor wrote to Dedekind in 1877 a proof that there was a one-to-one
correspondence between points on the interval [0, 1] and points in n-dimensional
space, n being a whole number. In other words, if considered as sets, finite
dimensional Euclidean spaces are indistinguishable. In particular, it involves that
a segment of a line, a square and a cube are indistinguishable from the point of
view of Cantor’s theorem discovered in 1877. Cantor was so surprised by his own
results that he wrote to Dedekind, “je le vois, mais je ne le crois pas”33.
Although Dedekind thought that there was nothing wrong in Cantor’s proof, he
did not think that it was the collapse of the notion of dimension as a fundamental
invariant for geometrical objects. Dedekind conjectured that the invariance of
dimension is only valid for continuous transformations. In 1907, Brouwer gave
the proof of Dedekind’s conjecture, rescuing “dimensionality” as an invariant
property of a geometrical object.
It is very likely that Peirce died without having ever heard of Brouwer’s proof,
but he was aware of the theorem Cantor discovered in 1877. In 1893, Peirce
claims that Cantor’s proof is right (CP 6.118). But around 1897, Peirce states the
opposite when he writes that: “the multiplicity of points upon a surface must be
admitted, as it seems to me, to be the square of that of the points of a line, and so
with higher dimensions. The multitude of dimensions may be of any discrete
multitude” (CP 4.226). Thus, it seems that Peirce had the same intuition
Dedekind had, and indeed, even in 1893, Peirce never explicitly said that
dimension was not an invariant property of a geometrical object.
For Peirce, it is by abstraction that an object of a lesser dimension can generate
an object of a higher dimension. For him, geometry should begin with the notions
of particle and of motion.
A particule is somewhere quite definitely. It is by abstraction that the mathematician
conceives the particule as occupying a point. The mere place is now made a subject of
thought. The particle moves ; and it is by abstraction that the geometer conceives it as
describing a line. This line which … is merely a fact turned into a substantive, is
regarded as so substantial that we talk of the line as moving, and as generating a
surface, which is a new abstraction ; and even the surface is made to move…
geometry … should translate everything about points, lines, and surfaces, into terms
of particles and their motions… (NEM 4.11, MS 691, 1901).
But Peirce wants to avoid saying something like ‘a moving point generates a
line’. In MS 137 and MS 143 Peirce explains why he is not satisfied with the
33
I see it, but I do not believe it.
14
15
common terminology of points, lines, etc. The reason is that for him “the terms
‘point’, ‘line’, ‘surface’, and ‘space’ designate absolute loci. No point, line, etc., can
move – only things occupying points, lines, etc., are capable of motion”
(Murphey, 1993, p. 206, note 42).
In the 1903 Lectures on Pragmatism, Peirce defines a family of new terms to
substitute for the classical notions of points, lines, surfaces, spaces. Peirce defines
a particule as a primary substance, “as a portion of matter which can be, and at
every instance of time is, situated in a point” (NEM 4.163). The use here of the
notion of matter can be surprising, but matter is for Peirce “that out of which
anything is made … that of which anything is or may be composed…” (CD, article
“matter”). Then Peirce defines a line as “the place which a moving particle
occupies on the whole in the course of time” (NEM 4.163).
Peirce can now introduce the notions of filament, film, and solid body. Peirce
sometimes uses the term “pellicle” in the same sense as “film”. A filament is “a
portion of matter which at any one instant is situated in a line” (NEM 4.163). The
mode of being of a filament is such that “it consists in there being a particle in
every point that a moving particle might occupy” (NEM 4.163). The particles are
conceived as being primary substances, and the filaments are abstractions.
Finally, Peirce defines in the same manner the notions of film (or pellicle) and
solid body. “A film or that portion of matter that in any one instant occupies a
surface will be still more abstract. For a film will be related to a filament just as a
filament is related to a particle. And a solid body will be a still more extreme case
of abstraction” (NEM 4.163, 1903). There is another text, written in 1904, in
which Peirce gives definitions of: point, particle, line, filament, surface, pellicle,
space, and body:
A pellicle is a movable object which at every instant occupies a whole surface. A
pellicle may occupy quite different surfaces (i.e. surfaces having no common
superficial part) at different instants. A space is a place which may be occupied by a
pellicle during a lapse of time … A solid is a movable object which at every instant
occupies a space. (NEM 2.495).
In order to clarify various topological questions, Peirce undertook the task of
enhancing Listing’s Census theorem. He kept some of Listing’s terminology and
invented new terms. Peirce kept the notions of: Census, chorisis (which he
sometimes spelled choresis), constituent of a spatial complex, cyclosis,
periphraxis; and he called ‘Listing’s numbers’ the chorisis, cyclosis and
periphraxis.
Peirce introduced, among many others, the notions of apeiry (apeiry is called by
Peirce the fourth Listing’s number), artiad, connectivity, perissid (or perissad),
retrosection (or loop-cut), shape-class, singularity, synectic singularity, synesis.
We will return to the definitions of these terms.
Before giving Peirce’s definitions of chorisis, cyclosis and periphraxis, it can be
helpful to introduce the modern way of dealing with similar questions. A
topological space is said to be “contractible” if and only if it can be reduced to a
point by a continuous deformation, provided that this deformation moves along
the space or path of the object itself.
15
16
Intuitively, a contractible space is one that can be continuously shrunk to a
point, provided that this deformation occurs within the boundaries of the space
itself. For any contractible space, the Euler-Poincaré characteristic is 1. Examples
of contractible spaces are the one-dimensional interval, the two-dimensional
disk, the three-dimensional ball.
Fig.12
one-dimensional interval
Fig.13
two-dimensional disk
Fig.14
three-dimensional ball
All non-closed lines (fig. 12) are contractible, but there is no closed line (fig. 5
and fig. 7) that is contractible. For example, in order to contract a circle (fig. 7)
into a point, the intermediate circles must go outside the space occupied by the
first circle (fig. 15), hence a circle is not contractible.
Fig.15
On the contrary, a disk (fig.13) is contractible since it can shrink to a point by a
continuous deformation that remains within the boudary of the initial disk.
Peirce’s conception of topological transformations is similar, although there are
two important technical differencies in his approach.
1) Peirce uses scarcely the terms of “contractible” (MS 1170, “apeiry”), because
he does not consider the reducibility to a point, but the reducibility to a
lower dimension. For points and lines, there is no difference, but for
surfaces or in general objects of more than one dimension, this is really
16
17
different. Hence, one should use with care the term “contractible” when
discussing Peirce’s topology. Usually, Peirce merely uses the common notion
of “to shrink”
2) For Peirce, a line segment, a disk or a ball can be shrunk, but closed lines,
considered in themselves, are cannot. But the important difference lies in
the case of a figure in a space, for example a circle (which is a closed line) on
a surface. For Peirce, on a surface, a circle can be shrunk continuously to a
point without extending outside the surface. On the contrary, from a modern
point of view, a circle on a surface is not contractible because in order to
shrink continuously to a point, it must extend outside its own space as a
circle.
Of course, these differences between Peirce’s topology and modern notion of
“contractibility” have a reason. In short, it is because Peirce uses notions that will
later on be separated in homotopy and homology. To explain that, it is now useful
to introduce very briefly and intuitively the notions of “connectedness” and
“genus”.
In modern topology, a geometrical object is simply connected if it consists of
one piece and if it does not have “holes”. Spaces that are connected but not
simply connected are called multiply connected, in which the “multiplicity” is
related to the numbers of “holes”. Intuitively, the genus of a connected surface34
can be defined as the number of “holes”, or also as the number of “handles”.
For a disk (fig.13), the genus is zero, but for a torus (fig.16 and fig.17), the genus
is 1.
Fig.16
For a closed surface, the genus can be defined in terms of the Eulercharacteristic χ, via the relationship χ = 2 - 2g, where g is the genus. Hence, there
is a deep relation between the genus and the Euler-characteristic.
However, the notion of “connectedness” is very different than that of
“contractibility”. A closed circle on a surface is not contractible, but such a circle,
for example on a sphere, can be shrunk to a point without meeting any hole,
although in such a transformation, the circle moves outside its own space (fig.25).
The surface should also be orientable, but I cannot in this article deal with this important
notion.
34
17
18
On the contrary, such a circle can be unable to shrink to a point on a torus
(fig.26).
Fig.18
genus: 1
Fig.17
genus: 1
Fig.19
genus: 2
Fig.20
genus: 2
Fig.21
genus: 3
Fig.22
genus: 3
The case of the torus is interesting and important, for some loop can be
contracted (on the surface) to a point (fig.23), whereas some others cannot
(fig.24).
18
19
Fig.23
Fig.24
Fig.25
A sphere is simply connected
because every loop can be contracted
(on the surface) to a point.
Fig.26
A torus is not simply connected
Now, is there a relation between “connectedness” and “contractibility”? To
examine quickly the question on an example, the surface of a sphere is simply
connected (fig.25), because any loop on the surface of a sphere can contract to a
point, even though it has a “hole”, as its center is hollow. But the surface of a
sphere is not contractible, as it is shown by the figure 27, and as it will be more
precisely explained later on.
In short, the notion of “contractibility” is stronger than that of “connectedness”,
since “contractibility” required that the object have no hole at all. A hollow sphere
is simply connected whereas it is not contractible.
19
20
Fig.27
1 point, 1 segment, and 2 surfaces
χ=1–1+2=2
The distinction between “connectedness” and “contractibility” can shed light on
the two important technical differencies noticed above between Peirce’s topology
and modern conception of “contractibility”. As a forerunner of topological
researches, Peirce was working at a time in which those distinctions were not as
clear as they are for contemporary mathematicians.
Now let us introduce Peirce’s notion of Listing’s numbers. At the end of this
article there is a lexicon that provides additional definitions.
Listing’s numbers … serve to define certain topical characters of any geometrical
figure, especially a figure without topical singularities. Though there are only four for
space of three dimensions there will be one more for each added dimension. These
numbers express the number of simple non singular interruptions of a continuous
place that are requisite in order to enable an object filling that space to shrink to a
nothing without breaking and without leaving the space. The four numbers are as
follows: … 1. The choresis, 2. The cyclosis, 3. The periphraxis, 4. The apeiry… (MS
1597, article “number”, subentry “Listing’s numbers”).
Now, the notion of choresis that is sometimes also called by Peirce chorisis (or
cho’risy). It is the number of separate pieces. In modern terminology, it
corresponds to the first of Betti’s numbers (called betti zero), namely the number
of connected components of the topological space.
The Cho’risy is the number of simple parts of a place of the dimensionality of the
place itself which must be filled up in order to leave no room for a single particle. (In
simpler language, it is the number of separate pieces.) (NEM 2.500).
Now, the notion of cyclosis which is also sometimes called by Peirce cyclosy.
The cyclosis of a place is the number of times it has to be cut across to make it
impossible for a self-returning filament to exist in it which can by no motion in it
shrink to a point. ( NEM 2.186).
How is the cyclosis calculated? For example, as shown in figures 28, 29 and 30,
the cyclosis of a torus is 2.
Here is this calculation in Peirce’s words:
20
21
The cyclosy of an anchor-ring surface is such that a filament passing round the ring in
the directest way has no room in the surface to shrink to nothing. To prevent the
presence on the surface of such a ring, we will fill up a line round the bar of the ring.
But still any other filament round the bar has no room to shrink to nothing. We
therefore fill up a line along the whole length of the bar, from the former interruption
round to that interruption again. This done, there is no longer room for an
unshrinkable filament; and consequently, the cyclosy of the surface was 2. (NEM 2.
502, MS 137).
It is interesting to notice that in modern terminology, the second of Betti’s
numbers (called betti one), namely the number of independent tunnels of the
topological space, is also 2 for the torus, as shown in the figure 24. A torus has
two tunnels: one that goes through the center and one that goes around the
center.
In a note written on an interleaf in his own copy of the Century Dictionary, at
the sub-entry “Listing’s numbers” of the article “number”, Peirce writes that: “A
simple self-returning line has cyclosis, 1; because if cut at one point it can without
rupture and without extending beyond the original place shrink to nothing.” (MS
1597). This means that a closed line cannot shrink to a point without extending
beyond its own space (fig.5 and fig.7). However, if one makes a cut on a closed
line (fig.7), it can shrink to a point (fig.8). In the same text, Peirce claims that a
“spherical film has a cyclosis 0; for any ring on it can shrink to nothing.” (MS
1597, article “number”, sub-entry “Listing’s numbers”).
Fig.28
A torus can be subdivided into 1 contractible surface, 2 line segments and 1 point.
A torus can also be subdivided into 2 line segments and 2 points (fig.17 and
fig.18).
21
22
Fig.29
Fig.30
In the case of the solid torus, namely a torus that has a hole in its center but that
is not hollow, like a donuts, it has a cyclosis 1.
As a single example of the cyclosy of a space, we may consider the place occupied by
the solidity of an anchor-ring. Obviously, in this space there is room for a filament
round the ring which would not have room in the annular space to shrink to nothing.
But let one surface through the bar be filled up, and there will no longer be room for
an annular filament. The cyclosy is, therefore, 1. (MS 1597, article “number”, subentry “Listing’s numbers”).
For Peirce, as shown in figure 25, “the cyclosy of the surface of a sphere is 0; for
any filament upon it has room upon it to shrink until, by a collapse at the last
instant, it becomes a particle.” (NEM 2. 501, MS 137).
Now, the notion of periphraxis which is also sometimes called by Peirce
periphrasis, or periphraxy, (or periphrax’y). In a simple but underdeveloped
definition given in the CD, Peirce defines “periphractic” as “Having, as a surface,
such a form that not every closed line within it can shrink to a point without
breaking”, and periphraxy is “The number of times a surface or region must be
cut through before it ceases to be periphractic”. The periphraxis is defined only
for spaces of dimension two or higher. Hence, for every space of dimension less
than 2, it periphraxis is 0.
The periphrasis of a place is the number of times it has to be pierced to make it
impossible for a closed sac to exist in it which can by no motion in it shrink to a line.
(NEM 2.186).
It is important to notice that Peirce says “shrink to a line” and not “shrink to
nothing”; hence it is not equivalent to the notion of “contractibility”. In another
definition of periphraxis he says that it is the:
number of simple annihilations of parts having 2 dimensions less than the object
that are requisite to enable a simple closed surface in the object to shrink to nothing
or to a line without rupture and without leaving the original place of the object. (MS
1597, article “number”, sub-entry “Listing’s numbers”).
How is the periphraxis calculated? In NEM 2. 504, Peirce claims that the
periphraxis of a disk is 0 (fig.13). Since a disk is contractible, it is easy to see that
22
23
it has periphraxis 0. Peirce also claims in NEM 2. 504 that the periphraxis of the
surface of a torus is 1 (fig.16 and fig.26). The case of the surface of the torus
belongs to a more general case, and is the same than the case of the surface of a
sphere for which its periphraxis is 1. Peirce explains that “the external surface of
any body has periphaxy one, since a film covering the surface must have a hole,
broken in it in order to shrink away from any part; and that once done, it can
shrink indefinitely toward nothing.” (MS 1170).
Let us now turn to the notion of apeiry (a’peiry). Apeiry is a property that can
fundamentally distinguish spaces of three dimensions, in the sense that if two
spaces have the same apeiry they are topologically alike. The apeiry is defined
only for spaces of dimension three or higher. Hence, for every space of dimension
less than 3, its apeiry is 0.
The A’peiry is the number of simple parts of a place of dimensionality three less than
that of the place itself which must be filled up in order to leave no room for a single
solid that shall not be geometrically capable of shrinking indefinitely toward a
surface, line or point by an ordinary motion in the inoccupied place. (NEM 2.500).
It is important to notice that Peirce says “shrinking indefinitely toward a surface”
and not “shrink to nothing”; hence it is not equivalent to the notion of
“contractibility”. In another definition of apeiry he says that it is the:
number of simple annihilations of parts having 3 dimensions less than the object that
are requisite to enable a simple non-singular solid in the object to shrink to nothing
(or to a surface) without rupture and without extending beyond the original place of
the object. (MS 1597, article “number”, sub-entry “Listing’s numbers”).
How is the apeiry calculated? Peirce states in the same text from MS 1597 that
the apeiry of every limited solid is 0. For Peirce, apeiry is zero for every limited
space, but “it is one for our unlimited space” (NEM 2.504). Since apeiry belongs
only to spaces of at least three dimensions, the main question is what is the apeiry
of our space? If our space is limited it can have various topological properties, but
its apeiry must be zero. If our space is unlimited, its apeiry is likely to be one, but
Peirce claims that “by training one can acquire the power of imagining with
facility a space whose apeiry should be more than one.” (NEM 2.504). In other
words, the main question related to the notion of apeiry is to determine whether
or not our space is unlimited.
Peirce was not satisfied with Listing’s theorem for several reasons. In MS 141
and MS 154, Peirce blames Listing for not taking into account the topological
singularities in his Census-theorem. “Listing’s investigation is insufficient in that
it takes no notice of topical singularities, or places where the ways of moving are
greater or less than the ordinary points” (MS 154, quoted by Murphey, 1993, p.
206-207). Peirce also objects that there are case of spatial complex for which the
Census-theorem is true, although such a spatial complex is impossible. “For
instance, the Census-theorem would permit a net upon a spheroidal surface
having twelve pentagonal faces, or regions, one hexagonal face, twenty-two
coigns, and thirty-three boundaries. But this is easily seen to be impossible” (MS
141, quoted by Murphey, 1993, p. 206).
Peirce’s purpose was to develop a new Census-theorem that would be true of all
kinds of shapes, and that would be false of all impossible shapes. Listing was
23
24
already aware that the chorisis, cyclosis and periphraxis were intrinsic topological
properties, or topological invariants. But Peirce noticed that for spaces of three
dimensions there is a fourth topological invariant, which he names apeiry. It was
not really discovered by Peirce, for he claims that Listing was aware of this
notion: “the word is of my invention … but Listing has the idea perfectly clear”
(MS 1597, article “number”, sub-entry “Listing’s numbers”).
As noticed by Pont, in his Census-theorem, Listing has shown that for a lot of
spatial complexes, there is a balance between the chorisis, cyclosis and
periphraxis35. But Listing failed to state clearly that the measure of this balance,
the value of the Census, gives a profound topological property and lead to a
classification of spatial complexes. One of Peirce’s deepest topological ideas was
to be one of the first to understand that one could classify spatial complexes
according to the value of its Census number. For the accomplishment of such a
classification, Peirce introduced the notion of ‘shape-class’ that corresponds
roughly to what contemporary mathematics call homotopy classes. Intuitively,
two objects are called homotopic if there is a continuous function which can
deform one object into the other. This continuous function is a homotopy, which
can be used to define a homotopy class.
Two places may be said of the same shape-class if and only if it is possible for a thing
precisely occupying the one to come by a mere movement, strict or otherwise, to
precisely occupy the other. Listing’s numbers … together with the singularities and
the dimensionality sufficing to distinguish all shape-classes. (NEM 3.1080-1081).
Now, we can remember Listing’s Census-theorem, which states that:
(3) a – (b – x) + (c – x’ + π) – (d – x’’ + π’ – w) = 0
and corresponds to:
(2) V – E + F – S = 0,
with V = a, E = (b – x), F = (c – x’ + π), S = (d – x’’ + π’ – w).
In NEM 2.505, Peirce introduced a new notation for his version of the theorem.
The letter X stands for chorisy, K for cyclosy, Π for periphraxy, and A for apeiry ;
and the signs p0, p1, p2, p3, denoting the points, lines, surfaces and solids. Thus,
Xp0, Xp1, Xp2, Xp3, are the four numbers of chorisy ; Kp1, Kp2, Kp3, are the three
numbers of cyclosy ; Πp2, Πp3, are the two numbers of periphraxy ; Ap3, is the
number of apeiry. If we compare with Listing’s formulation, we obtain the
following equivalence where the sign ‘=’ act as a mirror:
– (d – x’’
+ (c – x’
– (b – x)
a
Listing
+ π’ – w)
+ π)
Ap3
=
=
=
=
Peirce
+ Kp3 - Xp3
2
+ Πp - Kp2 + Xp2
+ Kp1 - Xp1
+ Xp0
Πp3
In “The branches of geometry”, Peirce explains in what consists his CensusTheorem:
35
Pont, 1974, p. 57.
24
25
Every place has a ‘Census-value’ which consists of the Census number of its points
minus that of its lines plus that of its surfaces minus that of its solids. The Census
number of any homogeneous space is equal to its Chorisy minus its Cyclosy plus its
Periphraxy minus its Apeiry. The ‘Census-Theorem’ is that the Census Value of any
place is unaffected by cutting it up by boundaries of lower dimensionality. (NEM
3.487).
If one notes χ the value of the Census of a topological space, χ = (Ap3
- Πp3
3
3
2
2
2
1
1
0
+ Kp - Xp ) + (+ Πp - Kp + Xp ) + (+ Kp - Xp ) + (Xp ) = - (-Ap3 +
Πp3 - Kp3 + Xp3) + (+Πp2 - Kp2 + Xp2) - (- Kp1 + Xp1) + (Xp0) = Co – Cy + Per –
Ape = X – K + Π - A
How is the Census number calculated?
In NEM 3.487 (also in NEM 2.189), Peirce gives the example of the calculation
of the Census-value of the surface of a torus. Because the surface of a torus has
one chorisis, two cyclosis, one periphraxis and zero apeiry (since it is a surface, its
apeiry is zero), therefore, its Census-value = 1 – 2 + 1 – 0 = 0. As a comparison,
one can calculate the Euler-Poincaré characteristic of a plain ring-surface, and
verify that its value is also zero, like its Census-value (figures 28, 29 and 30).
In NEM 2.187, Peirce gives the example of the calculation of the Census-value of
the surface of a sphere36. Because the surface of a sphere has one chorisis, zero
cyclosis, one periphraxis and zero apeiry, its Census-value = 1 – 0 + 1 – 0 = 2. As
a comparison, one can calculate the Euler-Poincaré characteristic of the surface
of a sphere, and verify that its value is also two, like its Census-value (figures 10,
11, and 27).
One can find several other examples on how to calculate the Census in NEM
2.186-191, MS 165, section “Topical Census”. Contrary to Listing’s work, Peirce’s
Census-theorem covers singularities. Peirce gives the example of a line segment.
It has one chorisis, zero cyclosis, zero periphraxis and zero apeiry, “But it has two
extremities which count 2. Hence the number is 1 less 2…” (NEM 2.187). In other
words, the Census-value of a line segment is (1 – 0 + 0 – 0) – (1 + 1) = 1 – 2 = - 1.
3) Topology and Philosophy
In many respects, Peirce’s work on topology had a strong influence on his
philosophy. In “The Architecture of Theories”, written in 1892, Peirce states that
recent advancements in geometry, with the development of projective geometry
and the rise of topology, should have a deep impact on metaphysics.
Metaphysics has always been the ape of mathematics. Geometry suggested the idea of
a demonstrative system of absolutely certain philosophical principles; and the ideas
of the metaphysicians have at all times been in large part drawn from mathematics.
The metaphysical axioms are imitations of the geometrical axioms; and now that the
latter have been thrown overboard, without doubt the former will be sent after them.
(CP 6.30).
See also NEM 2.187 “The Census number of the surface of a ball is 2 ; for it has 1 piece, no
Cyclosis, Periphraxis 1, and Immensity 0.”
36
25
26
For Peirce, there is a strict analogy between the relations various kinds of
spaces can have from a mathematical point of view and the relations various
kinds of realities can have from a metaphysical point of view.
Namely, as metaphysics teaches that there is a succession of realities of higher and
higher order, each a generalization of the last, and each the limit of a reality of the
next higher order, so space presents points, lines, surfaces, and solids, each generated
by the motion of a place of next lower dimensionality and the limit of a place of next
higher dimensionality. (CP 1.501, c.1897).
In what follows, a sketch will be given on the influence of Peirce’s topological
work on his conceptions of the nature of space, time, and logic.
3.1) Topology and Continuity
For Peirce, the mathematical theory of collection is not enough to investigate
the continuum. Projective geometry is important, since a projective line has no
discontinuities at its extremities, unlike the Euclidean line. According to Peirce, it
is a theorem that “every continuum without singularities returns into itself”
(NEM 2.184, MS 165, c.1895), and this likely the reason why when discussing the
linear continuum, Peirce usually does not refer to the straight line, as most
mathematicians do, but to the circle. More generally, the reason for such a choice
of the circle as a paradigmatic instance of the linear continuum, can be explained
by Peirce’s deep interest in projective geometry, for in projective geometry and
contrary to what occurs in Euclidean geometry, an infinite line is intuitively like a
circle of an infinite radius.
But for the study of continuity, topology is even more important than projective
geometry for it is the only abstract geometry which purely deals with properties
of continuity and discontinuity. Topology is for Peirce “the full account of all
forms of Continuity” (NEM 2.626, MS 145). However, the question of the
importance of topology for Peirce’s theory of continuity cannot receive in this
article a full and detailed account.
In a text written in 1904, Peirce claims that Cantor and Dedekind’s conception
of continuity is unsatisfactory and that topology can afford a better definition of
continuity. He claim that what analysts call continuity is not continuity at all, but
should be called “pseudo-continuity”:
“The topicists say that what the analysts call continuity is not continuity at all : we
call it pseudo-continuity… the analysts say no more perfect continuity than theirs is
possible; and that the descriptions of continuity given by the topicists are
nonsensical. I must confess that the attempt I made in The Monist, vol. II, pp. 542 et
seqq. to define true continuity was a failure. What I there defined as continuity was
nothing but the pseudo-continuity of the analysts…” (NEM 2.482, MS 137, 1904).
However, in the above quoted text, Peirce says that he “can here only give a
partial explanation of true continuity” (NEM 2.482), and the difference he gives
between “pseudo-continuity” and true continuity is the supermultitudinous
property37. This is not satisfactory, since in Peirce’s sense, the
supermultitudinous property is not really a topological property.
37
Havenel, 2007, section 4.
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27
I have shown elsewhere that the fifth and last period in Peirce’s evolution in his
mathematical and philosophical conceptualizations of continuity can be called
the ‘Topological Period (1908-1913)38. In this topological period, Peirce ceases to
think that the main property of a continuum is being supermultitudinous, and he
is looking for a way to explain how the parts of a continuum form a continuous
whole.
Peirce was already aware in 1897 that, if considered as mere multitude, all
continua are supermultitudinous and therefore incapable of discrimination
according to their multiplicities. Another criterion was therefore required in
order to distinguish the various kinds of continua. For Peirce, the nature of the
differences between continua “depends on the manner in which they are
connected. This connection does not spring from the nature of the individual
units, but constitutes the mode of existence of the whole.” (CP 4.219, “Multitude
and Number”).
This means that Peirce’s conception of the continuum is incompatible with
modern point-set topology which defines the continuum in a Cantorian spirit. In
particular, Johanson remarks that Peirce’s continuum “is not a Hausdorff space,
and hence does not have the nice separation properties” (Johanson, 2001, p. 10)
that is so useful in mathematical analysis. A Hausdorff space is a space in which
points can be separated by neighbourhoods, and it is true that Peirce’s continuum
is not a Hausdorff space. But there is another way to do mathematical analysis.
Within the mathematical theory of category, in Smooth Infinitesimal Analysis
(SIA), the elements of a continuum are not all distinguishable39.
Moreover, there is a third way between point-set topology and algebraic
topology that could have been used by Peirce to formalize his conceptualization of
continuity. This third way is topology without points. In his “Modern Topology
and Peirce’s Theory of the Continuum”, Arnold Johanson concludes that “though
many of Peirce’s ideas about continua are in conflict with modern point-set
topology, they are in substantial agreement with many of the conceptions of
topology without points” (Johanson, 2001, p. 10).
But what is topology without points? It is a topological theory in which points as
ultimate parts do not exist. Hence in topology without points the connectedness
must have a different definition than in point-set topology40. The main
philosophical interest of a definition of continuity within point-set topology is
that it is a satisfactory topological framework to deal with an Aristotelian
continuum. And among the two main families of theories on continuity, the
Cantorian and the Aristotelian, Peirce’s continuum clearly belongs to the
Aristotelian, although it integrates some of Cantor’s insights41.
Peirce distinguishes a perfect continuum and an imperfect continuum, this last
being a continuum “having topical singularities” (CP 4.642). He maintains that
according to his concept of continuity, “a topical singularity … is a breach of
Havenel, 2007, section 5.
Havenel, 2006, p. 290-312.
40 For a more detailed presentation of “Topology without points”, see Johanson, 1981 and
Johnstone, 1983.
41 Havenel, 2006, p. 391.
38
39
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continuity”. This idea is likely to be an evolution from his previous idea that a
continuum cannot contain actual points since in a continuum every part consists
of parts, but an actual point in a continuum is a part “which does not consist of
parts” (NEM 3.748, 1900). But if there is in a continuum no topical singularity,
then it is a perfect continuum for which hold the law that “all the parts of a
perfect continuum have the same dimensionality as the whole” (CP 4.642).
Then, Peirce states that his:
…notion of the essential character of a perfect continuum is the absolute generality
with which two rules hold good, first, that every part has parts; and second, that every
sufficiently small part has the same mode of immediate connection with others as
every other has. This manifestly vague statement will more clearly convey my idea
(though less distinctly) than the elaborate full explication of it could. (CP 4.642).
The new aspect in this definition of a perfect continuum is “that every
sufficiently small part has the same mode of immediate connection with others as
every other has”; and this idea is related to Peirce’s work on topology.
For Peirce, topology is essential for continuity because topology is “the study of
the continuous connections and defects of continuity” (CP 4.219). One can
distinguish two ways in which topology helps understanding continuity. First,
with his Census-theorem and his notion of ‘shape-class’, Peirce tries to establish a
classification of several kinds of continua according to their various dimensions
and topological singularities. This is what I propose to call external continuity, for
it deals with properties shared by objects belonging to the same class, which is
itself defined according to a homeomorphism. As we have seen, although Listing
was close to this idea, Peirce is perhaps, in the history of topology, the first to
have understood that one could classify spatial complexes according to the value
of their Census number.
Second is what I propose to call internal continuity, for it deals with the mode of
immediate connection of the parts of a continuum. As we have seen, for Peirce,
the nature of the differences between continua “depends on the manner in which
they are connected. This connection does not spring from the nature of the
individual units, but constitutes the mode of existence of the whole.” (CP 4.219,
“Multitude and Number”). The term “synesis” could be used for such a purpose.
In order to define “synesis”, Peirce says that it “cannot be defined in terms of
Riemann’s connectivity” (NEM 3.471), nor can it be defined “in terms of Listing’s
cyclosis and periphraxis, notwithstanding the value of those somewhat artificial
conceptions” (NEM 3.471). For Peirce, the synesis of the surface is:
The number of non-singular lines that can be drawn upon a single piece of surface
while still leaving it possible that a point should move continuously on that piece
from any position to any other, without crossing any of those lines. Or it may be
defined as the number of self-returning cuts that can be made in the surface without
increasing the number of pieces. (NEM 3.471).
For example, Peirce says that the synesis of a torus is 2. As we have seen, the
same result is obtained with the modern terminology of the second of Betti’s
numbers (called betti one). It is also 2 for the torus, namely the number of
independent tunnels of the topological space, as described in the figure 24.
Nonetheless, Peirce’s notion of synesis is, I think, closer in spirit with the notion
of “genus”, whereas the genus of a torus is 1, and whereas Peirce asserts that
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29
“making holes in a surface never affect the synesis” (NEM 3.471, 1900). As we
have seen, the genus of a surface corresponds intuitively to the number of pieces
to be added in order for the surface to be in one piece and to have no hole. But
neither does the notion of “synesis”, nor that of “genus”, or “Riemann’s
connectivity”, or “Listing’s cyclosis and periphraxis” can account for the mode of
immediate connection of the parts of a continuum. In a nutshell, I think that
Peirce’s notion of synesis is a failed attempt to define a topological concept that
would give such an account.
To the best of my knowledge, Peirce did not fully succeeded in his attempt to
give a satisfactory account of what are the mode of immediate connection of the
parts of several kinds of continua. For instance, Peirce states that what is
homogeneous in all the parts of a perfect continuum, is the regularity of a certain
kind of relation of each part to all the parts as a continuous whole42. Then he
claims that because continuity is unbrokenness, the relation he is looking for is a
continuous “passage” from one part to a contiguous part, but to explain what
means such a “passage”, Peirce needs to refer to time. Now, Peirce is not satisfied
for “time is a continuum; so that the prospect is that we shall rise from our
analysis with a definition of continuity in general in terms of a special continuity”.
(CP 7.535, note 6). In what follows, I will explain why time is a special continuity.
3.2) Topology and the nature of Space
One could sum up Peirce’s conception of space and time by saying that it is a
realistic interpretation of the Kantian doctrine made by a philosopher who is fully
aware of the mathematical and physical debate of the end of the 19th century, and
who supports that space and time are real and continuous.
Since Kant it has been a very wide-spread idea that it is time and space which
introduce continuity into nature. But this is an anacoluthon. Time and space are
continuous because they embody conditions of possibility, and the possible is
general, and continuity and generality are two names for the same absence of
distinction of individuals. (CP 4.172, in “Multitude and Number”, 1897).
As a mathematical theory, topology “studies purely hypothetical space, without
caring at all whether it agrees with physical space or not” (NEM 2.652). Topology
is the only “Pure Geometry” (NEM 2.652), in the sense that if topology studies the
properties of bodies, “it studies only those properties which those bodies share
with space itself” (NEM 2.652). This means that topology utterly ignores some
properties that are essential for projective geometry or for Euclidean geometry,
like the opposition between straight lines and non-straight lines.
When analyzing the nature of space, it is necessary to distinguish the
mathematical and the physical point of view. From the mathematical point of
view, Peirce states in a letter written to E.H. Moore that:
The geometer ought to consider space itself to be perfectly continuous, because he
has not the slightest reason to suppose it has anything exceptional about it, and
because it is simpler to suppose it has anything exceptional. (NEM 3.926, 1902,
March 20).
42
CP 7.535, note 6.
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30
In “Topical Geometry”, Peirce claims that if one can quite easily admit that
hypothetical space of the mathematician is “taken as continuous” (NEM 2.169,
MS 165, c.1895), “it is a question of physics whether it be true of real space, or
not, that it is continuous” (NEM 2.169).
Before examining what the properties of space are according to Peirce, it is
important to bear in mind that for him space is an objective reality and not just a
mere form of intuition. The fact that space has contingent properties is for Peirce
an indication that it does not only embody conditions of possibility, but that
space is a real thing.
That space is an objective reality and not merely a form of intuition (the Kantian
argument for which I consider illogical) seems to be shown by three characters, first,
that it is tridimensional, or perhaps quadridimensional, while there is no necessity
for its having any particular number of dimensions, second, that calculations founded
upon proper motions of stars show distinct indications that it is hyperbolic, and thus
has a constant the value of which cannot be necessary ; and third, as Newton pointed
out, velocity of rotation is absolute and not merely relative. (NEM 3.62-63).
Now to describe the specific properties of space, some topological concepts are
necessary. Although topological concepts arise within mathematics, which is a
purely hypothetical science, and although the properties of space are arbitrary
facts about space, without any logical necessity, it is with topological concepts
that the properties of space can be described. For space:
Its cyclosis and periphraxis, whether these be supposed equal to 0 or to 1 are
apparently arbitrary Facts… As to the Fourth Listing number, all must admit that its
value is 1… Here again, then, is an arbitrary existential fact about Space, which is
simply the way it insists upon being, without any logical necessity. (CP 7. 488).
In particular, the physical or cosmological question of the apeiry of our space
arises. Of course, as a purely mathematical theory, topology cannot answer this
question, but this physical and cosmological question can be answered
adequately only after the topological notion of apeiry is made clear. According to
Peirce, the apeiry of our space seems to be between 0 and 1: “Whether our actual
space is of apeiry zero or of apeiry one is not known” (MS 1170). But in another
text, Peirce claims that: “Apeiry … is one for our unlimited space and is zero for
every limited space.” (NEM 2.504, 1904). For Peirce, “in our space, all space has
its apeiry equal to 1; but all finite parts of space have apeiry 0, since even
interrupting space at a point reduces its apeiry to zero.” (MS 1597, article
“apeiry”). In a nutshell, the question as to whether our actual space is of apeiry
zero or of apeiry one roughly corresponds to the question of whether or not our
universe is infinite.
Peirce’s conception of space was also influenced by Cayley and Riemann. Peirce
read Clifford’s translation in English of Riemann’s book On the Hypotheses
Which Lie at the Foundations of Geometry, in which Riemann argues that
metrical properties of space are due to the physical entities that exist within
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31
space43. Peirce holds a Newtonian conception of space and he believes in the
reality of Cayley’s Absolute44.
Concerning the nature of our physical space, Peirce believes that “there are
some facts to support the proposition that space really returns into itself” (NEM
2.185). In “The Architecture of Theories”, Peirce examines three possibilities
concerning the nature of space:
The doctrine of the absolute applied to space comes to this, that either:
First, space is, as Euclid teaches, both unlimited and immeasurable, …, in
which case the sum of the three angles of a triangle amounts to 1800; or,
Second, space is immeasurable but limited, … in this case the sum of the three
angles of a triangle is less than 1800…; or,
Third, space is unlimited but finite (like the surface of a sphere), … , in which
case the sum of the three angles of a triangle exceeds 1800... (CP 6.28).
In CP 1.249, Peirce considers that the main alternative is between a hyperbolic
space, infinite and limited, or an elliptic space, finite and unlimited. Peirce
considers in “the Architectures of theories”, the measure of “the star’s parallax”
(CP 6.29, 1891), as a way to determine, by experimentation, the nature of space.
For him, the answer to this question should be decided by “the exactest
measurements upon the stars” (CP 1.249) but in fine, “the question cannot be
answered without recourse to philosophy” (CP 1.249). Thus, Peirce has a
philosophical reason to support that space is hyperbolic, and he argues that space
is likely to be hyperbolic (NEM 3.62-63).
In “Logic and Spiritualism”, written around 1905, Peirce explains the relations
with the metrical properties of space and philosophy.
“In regard to the principle of movement, three philosophies are possible.
1. Elliptic philosophy. Starting-point and stopping-point are not even
ideal. Movement of nature recedes from no point, advances towards no point,
has no definite tendency, but only flits from position to position.
2. Parabolic philosophy. Reason or nature develops itself according to one
universal formula; but the point toward which that development tends is the
very same nothingness from which it advances.
3. Hyperbolic philosophy. Reason marches from premisses to conclusion;
nature has ideal end different from its origin.” (W6.392, 1890)45.
I cannot here expose in much detail the reasons why Peirce defends an
hyperbolic philosophy, but for our present purposes, it is roughly enough to say
that Peirce thinks that the universe evolves like a great argument, from an
original chaos to an ideal end.
“… the course of logic as a whole … proceeds from the question to the answer, - from
the vague to the definite. And so likewise all the evolution we know of proceeds from
the vague to the definite.” (RLT, p. 258).
This idea is essential for Einstein’s general relativity.
CP 6.82
45 This text is also found in CP 6.582, but André De Tienne informed me that it is wongly dated
c.1905.
43
44
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One can remark that it is more the orientation of evolution from the vague to an
ideal end that supports Peirce’s hyperbolic philosophy, rather than
considerations on the metric of space.
3.3) Topology and the nature of Time
The relation between topology and time seems quite simple when we consider
Peirce’s claim that: “The doctrine of topics presupposes the doctrine of Time,
because it considers motions”. (NEM 2.481, MS 137, 1904). In other words, to
define topology, one needs the idea of transformation, but such an idea
presupposes time. However the relation between topology and time is more
complex than it seems at first glance.
In 1908, Peirce states that “continuity is … a … generality among all of a certain
kind of parts of one whole” (CP 7.535, note 6). He then tries to describe the
“passage” from one part to an another:
This passage seems to be an act of turning the attention from one part to another
part; in short an actual event in the mind. This seems decidedly unfortunate, since an
event can only take place in Time, and Time is a continuum; so that the prospect is
that we shall rise from our analysis with a definition of continuity in general in terms
of a special continuity. (CP 7.535).
If topology presupposes time, then it is difficult for Peirce to claim that topology
is “what the philosopher must study who seeks to learn anything about continuity
from geometry” (NEM 3.105), that topology studies “the modes of connection of
the parts of continua” (NEM 3.105, RLT, p. 246, MS 948, 1898), and that
topology “is the full account of all forms of Continuity” (NEM 2.626, MS 145).
This would lead to the idea that topology studies the various kinds of continuities,
but that to grasp the very idea of continuity, one should study the nature of time
by non-topological methods. Indeed, one can find such an idea in some of
Peirce’s writings when he claims that continuity is mostly related to mental
phenomena and to his scientific realism.
The reality of continuity appears most clearly in reference to mental phenomena; and
it is shown that every general concept is, in reference to its individuals, strictly a
continuum. … the doctrine of the reality of continuity is simply that doctrine the
scholastics called realism; and … Dr. F. E. Abbot has proved [that] … it is the doctrine
of all modern science. (CP 8, Bibliography, General, 1893).
However, Peirce also claimed that there is a topological reason to consider time
to be a belated reality, and that the topological properties of the original cosmos
where such that time did not exist:
We must suppose that as a rule the continuum has been derived from a more general
continuum, a continuum of higher generality … If this be correct, we cannot suppose
the process of derivation, a process which extends from before time and from before
logic, we cannot suppose that it began elsewhere than in the utter vagueness of
completely undetermined and dimensionless potentiality. (RLT, p. 258).
Why does this process extend before time, space and logic? Peirce thinks that
time is a belated reality in the evolution of the Universe, because from a
topological point of view, time only has one dimension, and therefore time is not
a continuum of high generality. Moreover, because according to his hyperbolic
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philosophy time is not cyclical, then time has topological singularities at its
beginning and end, since “every continuum without singularities returns into
itself” (NEM 2.184). But for Peirce, having topological singularities is an
imperfection for a continuum (CP 4.642, 1908). Hence, if evolution, at its
beginning, is a process of derivation from a continuum of higher generality to a
continuum of a lower generality, it follows that time is a belated reality in the
evolution of the Universe.
Since at least 1873, Peirce has argued in favor of the continuity of time46. In his
mature thought, Peirce argues in favor of the continuity of time, in a manner that
implies topology. In his cosmological reasoning, Peirce assumes that “the whole
universe and its laws are the result of evolution” (NEM 3.63), and as “the course
of evolution is from homogeneity to heterogeneity” (NEM 3.63). He concludes
that “it seems impossible that there should have been any action making time and
space discontinuous after they were once continuous” (NEM 3.63). This
argumentation rests implicitly on the fact that time is not a continuum of a high
generality, and that it has topological singularities. However, this argument
assumes too much to be really compelling, and I have shown elsewhere that in
order to explain the nature of time, Peirce rather uses his theory of infinitesimals
than is topology47.
3.4) Topology and Logic
In a letter to the Editor of Science, written on March 16, 1900, Peirce states
that: “The subject of topical geometry has remained in a backward state because,
as I apprehend, nobody has found a way of reasoning about it with demonstrative
rigor...” (CP 3.569). Had he read Poincaré’s works on topology, he surely would
have had a different opinion, but this shows that Peirce was looking for the
development of a logic for topology.
In a text written in 1900-1901 for the Baldwin Dictionary, Peirce expressed
more explicitly what kind of development was needed in logic in order to help the
growth of the topological science. “The logic of continua is the most important
branch of the logic of relatives, and mathematics, especially geometrical topic, or
topical geometry, has its developement retarded from the lack of a developed
logic of continua.” (CP 3. 642).
Now, does Peirce give an account of what a logic of continua would be? He
explains in his article “Mathematical Logic”, also written for the Baldwin
Dictionary, that in a continuous system,
… the principle of excluded middle, or that of contradiction, ought to be regarded as
violated … there is a limitation here to the applicability of the relation of negation
which those principles define. The definitions of otherness and of identity proper …
presuppose a universe of individuals; in a universe not consisting of individuals,
where every part consists of parts of the same kind, they are only applicable so far as
that universe admits individuals… Without this conception it is impossible to set
46
47
W 3.105-106 and also in MS 377.
Havenel, 2006, p. 191-195.
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forth the logic of topical geometry; and the philosopher will fall into insoluble
difficulties about the flow of time, and our consciousness of it. (NEM 3.747, MS 1147).
Because topology is pure geometry dealing with continua, and because there are
different logical properties for each kind of continuum, Peirce needs to
investigate these various cases. In particular, he finds that for a continuum:
Each additional dimension restricts the modes of reasoning which are applicable to
continua. General logic applies to them all, provided we take care not to assume that
objects have independent identity, after it has already been assumed that they have
not. To a continuum of an infinite multitude of dimensions it seems that no reasoning
can be applied, except that of general logic; so that we, thus, return to that very
logical system which belongs, in a peculiar way, to the simplest of all possible systems
of value, that of two values. (NEM 3.748).
In the history of mathematics, there is a permanent tension between
geometrical tendencies and arithmetical tendencies, and most of the deeper
mathematical theories are the results of new kinds of weddings between those
two tendencies. In logic, there is a similar tension between an algebraic approach
and a diagrammatic approach. In Peirce’s mature thought, which corresponds to
his works on topology, there is a huge interest for the diagrammatic approach of
his Existential Graphs. Such an approach in logic clearly bears a topological bias.
This is explicit in “Prolegomena for an apology to Pragmatism”, written around
1906, in which Peirce uses his Census-theorem to describe how it is possible to
work with his Existential Graphs48.
As pointed out by Valentine Dusek, such an approach in logic is still relevant
today:
There is a scattering of topological methods in logic today. Examples are the
compactness theorem, topological completeness proofs, the topological
interpretation of intuitionnistic logic, the theory of topoi based on category theory,
and fixed point theorems in recursive function theory… Perhaps there is a general
field of the topology of logic, of which Peirce’s existential graphs are our as yet most
comprehensive portion, but which are a precursor of developments yet to come.
(Dusek, 1993, p. 58).
The most promising field of research seems to be the application in logic of the
mathematical category theory, which fully employs diagrammatic reasoning. For
example, Zalamea has shown that Lawvere’s “Geometrizing Logic” program can
be seen as a modern development of Peirce’s Existential Graphs whose logic rests
on his theory of continuity (Zalamea, 2003, p. 157).
Conclusion
The importance of Peirce’s topological work lies mainly in its consequences for
his philosophy. It is true that Peirce cannot be considered to be one of the
founders of the mathematical science of topology, but his own achievements are
impressive when we consider the fact that Peirce was not mainly a mathematician
and that he relied on Listing’s work. One can speculate that had Peirce had the
opportunity to study Poincaré’s work on topology, it would have had a strong
48
NEM 4.323-324.
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influence on his own approach. Apart from historical interests, and despite its
exotic vocabulary, Peirce’s topological work remains of great importance for the
understanding of Peirce’s philosophy of mathematics, logic and cosmology.
LEXICON
The following lexicon is not exhaustive, but it provides several important
topological notions used by Peirce that have not been previously defined in this
article, or that worth other definitions.
* Apeiry (or a’peiry)
apeiry, n. [Gr. , the boundlessness of space] A respect in which one space of
three dimensions may differ fundamentally from another, being the number of
simple cavities that would have to be made in any plastic, contractible, and
expansible body that should completely fill that space, in order to permit that
body to shrink to a mere film without being anywhere ruptured. Whether our
actual space is of apeiry zero or of apeiry one is not known. (MS 1170).
* Artiad
“An artiad, or globular surface is a closed surface upon which every pair of selfreturning lines intersect an even number of times, zero being considered as an
even number”. (NEM 2.283, MS 94).
As Carolyn Eisele pointed out in NEM 2.283, a sphere is an example of an artiad
surface.
* Astringibility
“In the account of singularities it is necessary to distinguish different kinds of
astringibility, or capacity for shrinking to lower dimensions. It is partly because
there is no such distinction among lines that the law of their singularities is so
simple.” (NEM 2.504)
* Chorisis (choresis, cho’risy)
The choresis, or number of separate pieces of which the place consists. This may
be regarded as the number of annihilations of parts of the dimensionality of the
extended object which will annihilate the object. (MS 1597, article “number”, subentry “Listing’s numbers”).
* Connectivity
The measure of connectivity is a number belonging to any surface and fulfilling
the following conditions :
1. No gradual deformation which does not create nor destroy any contact can
alter the measure of connectivity.
2. Every puncture of the surface increases the measure of connectivity by unity.
3. Every cross-cut of the surface diminishes the measure of connectivity by
unity.
4. The measure of connectivity of a simple globular surface is zero. (NEM
2.290).
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Connectivity, n. The number q - p + 2, where p is the number of simply
connected pieces into which the a surface will be separated by q crosscuts. A
retrosection or cutting round out a piece cut out is not a cross-cut. A sphere has
connectivity 0; for q = 0, p = 2… A disk has connectivity 1; q = 0, p = 1. So has the
plane of projective geometry although a retrosection does not necessarily cut it
into two parts. The connectivity of an anchor-ring is 2, because one retrosection
reduces it to a flat ring and one cross-cut makes 1 simply-connected piece of it.
The question of connectivity must not be confounded with the question of passing
from the inside to the outside of an oval. In this, a sphere is the same as a disk.
This number, say the circuity, is Q - P + 2, where P is the number of simple
regions, produced by Q lines from boundary …back to the same. A complete oval
is reckoned. The boundary of the surface itself a boundary of a region. The
circuity of sphere and disk are 1. Projective plane 2. Disk with hole, 1. Anchorring, 3. If we reckon the number of regions that can touch one another, on a
sphere or disk 4, a ribbon returning into itself with a half twist 5, a projective
plane 6, an anchor ring 7. (Cf MS 1597 A, article “connectivity”).
* Cyclosis (Cyclo’sy)
The Cyclo’sy is the number of simple parts of a place of dimensionality one less
than that of the place itself which must be filled up in order to leave no room for a
single filament that shall not be geometrically capable of shrinking indefinitely
towards a point by an ordinary motion in the inoccupied place. (NEM 2.500).
* Fibre or filament
“A fibre, or filament, which at each instant occupies a line merely, in the course
of time occupies a surface; so that the character of being in a surface is more
general still.” (NEM 2.273).
* Lamina
“A lamina, which at each instant occupies a surface, in the course of time
occupies a solid space, which is thus another generalization.” (NEM 2.273).
* Listing’s numbers
For three-dimensional space there are four Listing numbers; the cho’risy (Gr.
…, a separation), or punctual separativeness exemplified in any point as being
distinct from any different point; the cyclo’sy (Gr. …, a surrounding), or linear
separativeness, exemplified in a self-returning line; the periphrax’y (Gr. …, a
fencing round), or superficial separativeness, exemplified in a closed surface; and
the apei’ry (Gr. …, immensity), or solid separativeness. (NEM 3.1081, c.1905?).
* Loop-cut:
“loop-cut, n. The puncturing of a surface followed by a cross-cut so as to excise a
piece. Called also a retrosection, though one would think excision would do better
if any name be wanted for an operation so composite.” (MS 1597 A, article ‘loopcut’).
36
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* Periphraxis, periphrasis, or periphraxy, (or periphrax’y)
The Periphrax’y is the number of simple parts of a place of dimensionality two
less than that of the place itself which must be filled up in order to leave no room
for a single pellicule that shall not be geometrically capable of shrinking
indefinitely toward a line or point by an ordinary motion in the inoccupied place.
(NEM 2.500).
Periphraxy belongs only to surfaces and spaces. The periphraxy of a surface is
the number of points that must be filled up in order that a pellicle filling the rest
of the surface should be geometrically capable of shrinking within its own original
place to a line or point. (NEM 2.503).
periphraxis [Gr. , a fencing round] A certain respect in which the form of
connection of the parts of one spatial extent or place of two or more dimensions
may differ from another, being the number of holes that it might be geometrically
necessary to make in a plastic, extensible, and contractile material film before
this film could shrink, by deformations and displacements within that place,
without rupture or the reverse of rupture (the bringing of separate parts into
continous connection), indefinitely toward occupying no superficial area at all.
(MS 1170).
* Perissad or perissid
A perissad surface is a closed surface on which any number of lines may be
drawn, every pair of these lines cutting one another in an odd number of points,
all such points being distinct. An unbounded plane is an example of such a
surface. (NEM 2.283, MS 94)
The surface which is divided into regions may be bounded by a line or
unbounded. If it be unbounded and separates [a] solid into two parts, I call it
artiad; if it does not, I call it perissid… The cyclosy of the simplest perissid
surface, such as an unbounded plane… (CP 5.490).
Then why should projective geometry bother its head about infinity ? It
supposes of course that the plane is perissid, or one sided. (NEM 3.926).
* Retrosection
retrosection, n. A cut which is made by first punching a hole in a surface and
then making a cross-cut beginning and ending at this puncture. It might be called
an excision. It has also been called a loop-cut; but it has nothing to do with a
loop. (MS 1597 A, article ‘retrosection’)
* Self-returning
A line is self-returning if it can be generated by a particle which ends its motion
where it began… A surface is self-returning if it can be generated by the motion of
a self-returning fibre whose motion ends where it began. A solid can only be selfreturning if it is conceived to fill the space it occupies at least twice or else fills all
Space. (NEM 2.285, MS 94)
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* Shape-class:
Two places may be said of the same shape-class if and only if it is possible for a
thing precisely occupying the one to come by a mere movement, strict or
otherwise, to precisely occupy the other. Listing’s numbers are certain quantities
measuring the degrees in which any place possesses separativeness of different
kinds, where by separativeness is meant something exemplified in the action of
any two-sided closed surface in separating all space into two parts, these numbers
together with the singularities and the dimensionality sufficing to distinguish all
shape-classes. (NEM 3.1080-1081).
* Singular place
An ordinary place of a place is a place … from which the modes of departing
movement are the same as from innumerable other places in its neighborhood in
the same place… A singular place … is a place … from which the modes of
departing movement are fewer or more than from ordinary places of the same
dimensionality. (NEM 3.1079)
Fig.31
* Surfaces
Surfaces may be described by means of adjectives showing the number of
tunnel-bridges and limiting the noun perissad or artiad, which signify what the
surface would be reduced to if its bridges were burnt, and the open tubes closed
each by the shrinking of its encircling fibre…
The following are the simplest categories of closed surfaces :
Simple perissad. Such is a plane interleaf.
Simple artiad, or globular surface, like the exterior of an ordinary stone.
Unitoroidal perissad. Imagine the surface of the earth to be plane, or perissad,
and to have a rabbit-hole with two openings.
Unitoroidal artiad. Examples : an anchor ring or baby’s chewing ring.
Bitoroidal perissad. The same with two rabbit-holes with two openings each.
Bitoroidal artiad. Example: a globular shell, with three orifices connecting the
two faces. (NEM 2.284).
* Synectic singularity
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A synectic singularity of any continuum, A, is a continuum, say B, forming a
part of A, of fewer dimensions than A, whose synectic relations to A are different
from those of any other part of A in the immediate neighborhood of B. (MS 139).
* Tunnel-bridge or fornix:
A tunnel-bridge, or fornix, is a part of a surface which can be generated by the
motion of a self-returning fibre from one self-returning line to another, without
cutting any part of the surface. A tunnel-bridge is so called, because it is a
boundary between a tunneled solid and a bridged solid. A tunnel through a
mountain is a bridge connecting two parts of the air… A surface is said to be
unitoroidal, bitoroidal, tritoroidal, etc. according to the number of tunnel-bridges
belonging to it which might be cut through round a self-returning generator
without separating the surface into two parts. (NEM 2.284).
ABBREVIATIONS
Some sources are referred to throughout this article by the following
abbreviations.
CD
CP
L
Century Dictionary, 1889
Collected Papers of Charles Sanders Peirce
Peirce’s correspondence; numbers indicate those from
Richard Robin’s:
Annotated Catalogue of the Papers of Charles S.
Peirce
Microfilm version of the Peirce manuscripts in
Houghton Library, Harvard University; numbers
indicate those from Robin, 1967; 1971.
The New Elements of Mathematics
Reasoning and the Logic of Things: The Cambridge
Conferences Lectures of 1898
Transactions of the Charles S. Peirce Society
Supplement Volume of The Century Dictionary, 1909
MS
NEM
RLT
TCSPS
SV
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Dusek, A.
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Eisele, C.
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Robin, R.
1967
1971
Annotated Catalogue of the Papers of Charles S.
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“The Peirce Papers: A Supplementary Catalogue”,
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41