Proceedings of the International Congress of Mathematicians
August 16-24, 1983, Warszawa
HANS EKEUDENTHAL
The Implicit Philosophy of Mathematics
History and Education
Here you are : both of the subjects of our section of this congress united
in one title, and as though this were not yet enough, philosophy, once the
third in this illustrious company, resurrected — though, of course, philosophy not meant the way it is understood in section II, that is, as a mature
offspring of mathematics itself and equally ranking with other offsprings.
I mean it rather the way it is meant when you ask someone "tell me,
what is your philosophy?" It is implicit philosophy which does not need,
nor ask for, formalized language to be made explicit.
I have omitted one thing in the title, for brevity's sake. I should have
added "in their mutual relation". Should I really have? I introduced
"philosophy" in the singular rather than in the plural, that is, one philosophy behind both history and education, or if they are two, that one
that is common to both.
Philosophy of history often means what we can learn from the past to
cope with the future. In our particular case it could mean what we can
learn from the history of old mathematics for the sake of teaching young
people. Strangely enough nobody has ever looked at the converse idea,
that is, what can we learn from educating the youth for understanding
the past of mankind? This idea looks odd, but I will show you it is not as
far-fetched as you might believe.
À century ago biologists were the first to assert the so-called biogenetic
law: that ontogenesis is an abridged recapitulation of phylogenesis, that is,
that the individual in its development briefly repeats the development of
its kind. We know for sure that this law is not true in this trivial way. But
neither is it true that the new generation starts just at the point where its
predecessors finished. Our biological, social, and mental life starts somewhere in the past of our race, at stages where man was not yet what he is now.
[1695]
1696
Section 19: H. Freudenthal
The young learner recapitulates the learning process of mankind, though
in a modified way. He repeats history not as it actually happened but as it
would have happened if people in the past had known something like what
we do know now. It is a revised and improved version of the historical
learning process that young learners recapitulate.
"Ought to recapitulate" — we should say. In fact we have not yet
understood the past well enough to really give them this chance to recapitulate it. Let me show this by examples, which are more convincing than
abstract statements can ever be.
Negative numbers
Negative numbers were a conquest of the 16th century. Why weren't they
welcomed earlier? Well, there is little need to calculate 3—5 or to solve
equations like 2a?+ 7 = 3 . Even quadratic equations, known as early
as Babylonian antiquity, like x% + x — 2 =^=0, did not provide a strong
enough incentive to extend the number domain to negative numbers. In
contrast, fractions and even irrational numbers are almost as ancient as
natural numbers, thanks to the necessity of dividing and measuring. It
was the destiny of the marvellous formula used in solving the cubic equation to open up the clogged channel of history; Three solutions forced
themselves upon the enchanted solver. Who would dare to despise this
wealth and throw away part of it? So negative numbers knocked at the
door and they were welcomed as were imaginary numbers, which knocked
as forcefully. Welcomed? Yes, and no. It was only as late as the 19th
century that the last resistance to negative and imaginary numbers was
conquerred. Eor a long time people seriously doubted whether man was
allowed to create new numbers beyond the realm created, as they believed*
by nature.
Meanwhile experience and history have taught us revolutionary
'lessons. If some beautiful formula, some theorem, some theory refuses to
apply as generally as we would like it, we now put the blame not on the
formula, the theorem, the theory, but on the problem to be solved. Problems are often being adjusted to the solution rather than the other way
round.
The cubic equation was one of the first examples of this behaviour.
Pressing the solution at any price led to negative and imaginary numbers.
The first extension of natural numbers, towards fractions, had been much
less controversial. Prom the first mathematical documents onwards we
meet with fractions. Soul-searching in this domain was of a much later
The Implicit Philosophy of Mathematics History and Education
1697
date, in Greek mathematics, when philosophers forbade breaking the unit.
Greek mathematicians replaced fractions with ratios while calculators in
commerce and science persisted in using fractions. Indeed, fractions are
the natural tool if magnitudes are measured and divided.
* Mathematics in the Greek sense is about numbers, and as far as
geometry is concerned, about magnitudes — a view that mathematicians
in more recent times have tried to share, at least in theory. The negative
numbers originated from the formal algebraic need for general validity
of solving formulae, but not until the algebraization of geometry (the
so-called analytic geometry of former times) did they become effective —
I mean effective in terms of real contents. The idea to use algebra to
describe geometric figures and solve geometric problems is older than
Descartes. We owe to Descartes the tendency to use one coordinate system
(to express it in modern terms), independently of the figure and the problem. Descartes still had some trouble with negative numbers. Indeed,
numbers were introduced as magnitudes; letters indicated magnitudes,
thus positive numbers. But those who applied Descartes' method could no
longer avoid having letters mean negative numbers also. If straight lines
are to be described algebraically in their totality, if curves are to be described algebraically in any situation, one cannot but admit negative values for
the variables. The need for
general validity of algebraic solution methods,
to which the negative numbers owed their existence, is from the 17th
century onward reinforced by the need for
general validity of descriptions of geometric relations.
The second need, more directed towards contents than the formal algebraic
one, is the most natural and compelling. It is actually responsible for the
success story of negative (and also of complex) numbers.
If negative numbers are introduced, it does not suffice to claim their
mere existence — this is often didactically overlooked, as also happens
with rational numbers. Negative numbers become operational by their use
in calculations, obeying certain laws which are uniquely determined as
extensions of certain laws governing the positive numbers. This is
the algebraic permanence principle,
which includes what I just called the
general validity of algebraic solution methods,
and virtually it is the same idea, albeit formulated in a broader view.
1698
Section 19: H. Freudenthal
I recall a few examples of the algebraic permanence principle.
(_3) + (-4) - - ( 3 + 4)
is proved by starting with the definition equations for — a9
(~3)+3 ^ o ,
(-4) + 4 - 0 ,
adding them formally, then using commutativity and associativity, in
order to arrive at the definition equation
((-3) + (-4)) + ( 3 + 4 ) = 0
for - ( 3 + 4).
Or: Starting with the same definition equations, one proves
( - 3 ) . ( - 4 ) = 3.4,
by multiplying distributively the first by 4 and the second by —3,
4-(-3) + 4-(3) = 0 ,
( - 3 ) - ( - 4 ) + (~3).4 = 0
and subtracting them from each other.
Or: With Va defined as the x making x2 = a9 one gets
j/fl }/b
= ]/àb
by multiplying the definition equations
œ2 — a9
y2
=b
to get
(wyY =>x2y2 = ab.
Similarly, if operations are to be extended,
a1*" = n\/ä
because both terms have the same w-th power. **
Por a century the algebraic permanence principle has been ridiculed as
a sham. The axiomatic method should have taught us sounder lessons. It is
the way we always proceed when extending mathematical definitions. It is
the way we find out how to extend mathematical objects in a reasonable
and unique way, to prove the uniqueness of the extension, and to prepare
the construction that proves the intended extension. It is the way negative
numbers have been taught until quite recently, when new didactic ideas
emerged. I will focus on one of them only, the number line, on which
Tho Implicit Philosophy of Mathematics History and Education
1699
negative numbers are viewed as movable vectors which are being operated
on as such. It is so splendid an idea that one marvels why it has not
worked didactically. P. M. van Hiele has been the first to indicate the
reason, which is so simple that one marvels even more why nobody before
him hit upon it: dimension one is the least appropriate to give vectors
the chance they deserve. If you do not believe it, look up in all those
textbooks the desperate attempts visibly to separate, add and subtract
vectors, which unfortunately in one dimension cover and eclipse each
other.
In Van Hide's newest approach negative numbers arise in a two-dimensional frame. A number pair
[~~3,
4 -]
P—3, 4 ~~|
r"3, —4 H
[""—3, —4 H
means
means
means
means
3
3
3
3
steps
steps
steps
steps
to
to
to
to
the
the
the
the
right, 4 steps upwards,
left, 4 steps upwards,
right, 4 steps downwards,
left, 4 steps downwards.
The left-right and up-down are those of the drawing plane, with horizontal
and vertical axes on which the numbers of steps can be read in units. By
performing such operations in succession one describes or prescribes
rectilinearly constructed drawings in the plane. Adding the vectors is
nothing but performing these operations in succession. Thus
T3, - 4 n + r - ß , 2 n
arises in a natural way and defines as naturally what it is
3 + (-6) and (~4) + 2.
In the two-dimensional model the laws governing the operations of addition
and multiplication are visually obvious by virtue of the model's geometric
-meaning.
In van Hide's many-sided approach this extension of the number
domain is applied to extend the definitions of functions introduced ea-rlier
by means of tables such as
X-+X — 3 ,
x->x + 3,
X-+Z—X,
x->2x.
1700
Section 19: H. Freudenthal
Let us now have instruction starting with these functions. But before
doing so let us turn once moïe back to history and remember that negative
numbers were first invented to * extend broader validity to certain algebraic solving methods while soon their indispensability in geometry became
overpowering — I mean their indispensability in the algebraized geometry
such as developed after Descartes. Negative numbers would have remained
a nice plaything, and the operations, motivated by algebraic permanence,
mere rules of a game, which could have been fixed in another way, were it
not that geometry had seized upon them. Negative numbers are indispensable if the whole plane is to be described by coordinates and planar
figures are to be grasped in their whole extension by equations. The
simplest figures in the plane, lines, are then translated by the simplest
equations, those of the first degree, called linear because of their relation
to straight lines; circles and other conies are fitted by second degree
equations. I think that both in phenomenological analysis and didactics
too little/emphasis is laid upon this fact:
the justification of the numerical operations and their laws by the simplicity
of the algebraic description of geometrical figures and relations.
Briefly stated:
Algebra is valid because it functions in geometry.
It is strange that so far this insight has not, if at all, strongly enough been
pronounced. In history it has never been used against people who argued
against negative numbers. In teaching algebra it should be our duty to
convince the learner of the validity of the operations and their properties
so forcefully that he cannot but accept them. The most convincing argument is to show him the operationality of algebra in geometry. This,
I believe, should be our policy in teaching negative numbers.
Here "geometry" does not mean an axiomatic structure but what is
visually obvious or conceptually follows from what is visually obvious —
a visuality that neither requires involved explanations in the vernacular'
nor sophisticated elaboration. The one-dimensional medium, the straight
line, has not enough visual structure; two dimensions is the minimum
that is required, and with a view to the graphic possibilities the most
appropriate medium. **
Let us repeat history in a modified way : turning
the algebraic permanence principle
The Implicit Philosophy of Mathematics History and Education
1701
into
the geometrical algebraic permanence principle,
now applied not to extend solving formulae but to extend such functions as
X-+X — 3 ,
x->x + 3,
x->3—x,
x->2x9
which only imperfectly reflect geometric figures.
-i—•
1
2
3
A
5
I need not explain to you in detail how negative numbers, their operations
and the laws governing them arise didactically in this context.
And perhaps you will also grasp what I meant when at the start I
claimed that teaching the young can teach us historical lessons.
Let us now turn to another subject.
Variables
* Por centuries "variable" meant — in mathematics and elsewhere —
something that really varies, something in the
physical, social, mental
as well as in the
mathematical
world that is
perceived, imagined, supposed
1702
Section 19: H. Freudenthal
as varying, that is in addition to
the
the
the
the
the
the
the
the
the
time that passes,
path that is covered,
aim that changes,
water that is rising,
temperature that oscillates,
wind that is changing,
days that lengthen,
mortality rate that decreases,
progressive rate of income tax,
also the
variable mathematical objects
by which these phenomena are described. From the
variable physical, social, mental
phenomena one is led to
variable numbers, magnitudes, points, sets,
in general
variable mathematical objects.
Locutions like
the
the
the
the
number
point P
element
number
e approaches (converges to) 0,
runs on the surface S9
x runs through the set S,
e is approached by the sequence ( 1 + 1 \n)n
if n goes to inf
witness this kinematic aspect of the "variable". I t is true that in the
course of, say, the past half century such locutions have been outlawed by
purists. Indeed one can dispense with them,
xn converges to 0
can be written as
iimnxn = 0
and be defined, with no kinematics involved, by
for every e > 0 there is an n0 such that \xn\ < e for
x runs through the set S
n^n0*9
The Implicit Philosophy of Mathematics History and Education
1703
can very simply be written as
x G S.
Well, one can dispense with that kind of kinematics provided one has once
possessed it, learned to use it and then to eliminate it — this is a general
didactical feature. #*
As soon as names were needed for mathematical variables they were
indicated by letters. But at that time letters had already been in use in
mathematics for about two millennia, in geometry to indicate arbitrary
points, in geometrical algebra for arbitrary magnitudes, and in number
theory for arbitrary whole numbers, as witnessed by Euclid's Elements.
I did not say variable points, magnitudes, numbers — I did say arbitrary
ones, and this is a fundamental difference. One letter meant one individual point, one magnitude, one number, though it did not matter, or
was taken as unknown, which one. Letters were used in mathematics as
polyvalent names.
Polyvalent names are a well-known feature in the vernacular, too.
Proper names such as "Warsaw" for a particular city, and "Poland" for
a particular country, are rare — we cannot afford too many. We cannot
afford a proper name for each particular mouse, or table, or stone, so we
use the same for each of them, and in each particular case by a specific
way of binding indicate which one is meant : this mouse, or that table or
the first stone of the Academy building. "I" is such a polyvalent name,
bound by the mere fact of pronouncing it to the person that says it. "Here"
and "there" are polyvalent names which can be bound to places, "thing" is
one that may apply to anything whatsoever.
Mathematics has proper names, even an infinity of them, — I mean the
vocabulary of natural numbers, constructed in an algorithmic way. Compared with the rich variety of polyvalent names in the vernacular, the
stock available in mathematics looks poor: the letters of the alphabet or of
a few alphabets, sometimes enriched by subscripts, accents, dots and
dashes. Unlike the polyvalent names in the vernacular they are not restricted to particular species of things such as mice, tables, stones. They are
general purpose polyvalent names, which can mean everything mathematical. Accordingly they were used to formulate general laws like
(a + b)* =a2 + 2ab + b*
or to ask for a solution as in
x2 + x — 2 = 0 ;
55 — Proceedings..., t. II
1704
Section 19: H. Freudenthal
such polyvalent names were called indeterminates or unknowns. Por a long
time it was a much discussed topic whether literal algebra instruction
should start with the one rather than the other use of the letters.
What is your reaction to this story? Shrugging, incredulity, astonishment, or a half-smile? Or are you sorry about the lost paradise of the good
old time? Today, it is all variable, indeed, as you know, and nothing else.
But as a historian you may ask who brought this change about, and when it
happened. You may even ask when people became accustomed to it and
when they started teaching it this way. I confess I have not investigated
this. I do not even know when I myself got acquainted with the extended
use of the word "variable" and when I myself started using it.
Anyway it is clear that it started in formal logic. When logicians looked
into the status of the "letters", the fact that for such a letter you may
substitute whatever you like might have suggested the term "variable",
but with the reservation that it was a mere metaphor, because there was
nothing in it that really varies. I t was a highly suggestive metaphor; yet
while r usurping the term "Variable", logicians even went one step further
in hollowing it out : a variable was not even a name, let alone a polyvalent
name, but a mere placeholder, that is, a hole to be filled by names, and
only for opportunity's sake are different kinds of holes being distinguished
by different symbols.
Even more serious things happened as logicians and purists usurped
the term covering the genuine mathematical variable. Its original meaning
shifted, got lost. Mathematical terminology was stripped of its kinematic
undertones. Terminology like that mentioned earlier, as time that passes,
numbers approaching a limit, points running on a surface, was exorcised.
* Lumping concepts of various origin together, using one name for
things that stripped of their frills boil down to the same, is one of the
important characteristics of our mathematical activity. Here we have met
with such an historical occurrence:
polyvalent name
and
variable object
related with each other and confounded. #*
You may like it or not but it is a fact in mathematics that more unity
is a precondition of more profound understanding and continuing progress,
and that reshaping mathematical language is a highly valuable activity.
But as historians we are not asked what we like. We have to be conscious
of the reasons why things were viewed differently in the past — in fact
The Implicit Philosophy of Mathematics History and Education
1705
they were good reasons — and of the reasons why things changed — these
were equally good reasons. Moreover, as educators we have to find out at
which point learners should start to recapitulate history. Finally, if we are
both historians and educators, the one ought to learn from the other to
understand his task more profoundly.
* Fortunately : expellas naturam furca, tarnen usque recurrit — nature
though driven out with the fork, nevertheless returns. The mathematical
purism — of high value within mathematics — is a forced and less satisfactory language as soon as one steps out of mathematics. The abundance of
variable objects in the half-way mathematized vernacular can be eliminated by linguistic sophistication but by this linguistic measure they
are not disposed of. And — even more important — in order to be eliminated by linguistic tricks, they must once have been experienced by the
learner. Indeed, there is no other way to guarantee that he will be able to
restore them as he needs the vernacular to recognize and apply mathematics in the real world. The world is a realm of change, describing the
world is describing change, and to do this one creates variable objects —
physical, social, mental, and finally mathematical ones. There exist many
languages of description, or rather many levels of describing. On a high
level of formalization the variable mathematical objects may be forsaken,
but on less formalized levels they are a genetically and didactically indispensable link with the physical, social, and mental variables, which on their
part are indispensable tools. **
It is shocking that textbooks in the wake of New Math try to teach
mathematics as though it were nothing but an impeccable language, which
as it happens most of the students are unable to grasp and speak, and
that researchers conduct subtle investigations to find out whether students
understand variables as polyvalent names or as mere place-holders while it
never crossed their mind that variables should and could be understood
as variable objects. It is no less shocking that historians are not alarmed
enough by the obvious failure of this kind of instruction to look more profoundly into history in order to find out what history can teach education.
Let me put this in an even broader context by my third and last
example.
Inversion and conversion
Viewed historically, mathematics has grown not only in substance and
subject matter. It is as much, and perhaps even more, a process of reshaping and remodelling, of turning things upside down and inside out.
1706
Section 19: H. Freudenthal
In Greek mathematics conies originated from the problem of solving
quadratic equations, or, as the Greeks formulated it, the problem of
applying an area F to a, line segment a * either exactly, that is ax = F,
or so that a square falls short: (a—x)x = F,
x2
F
J
[
^
>
ü
or so that a square exceeds: (a+x)x
= F.
1
X
*2
^
a
„
These three cases are distinguished with the Greek words for agreement,
falling short, and excess as
parabolic, elliptic, hyperbolic
application. This, then, is the origin of our terms for the conies.
If F is given as a square with side y, then the above equations become
ax
= y*,
(a—x)x = y%,
(a+x)x
=y2,
respectively, which are indeed the equations of parabola, ellipse, hyperbola. ** This was the way these curves were first encountered, by solving
quadratic equations, and only afterwards was it discovered that they
represent the planar sections of a cone. I t was Apollonius who inverted
the course of history and started with conic sections to derive their "symptoms", that is equations, from the geometrical data.
After Descartes had developed what was called analytic geometry,
the view was again turned round: quadratic equations, now in a more
The Implicit Philosophy of Mathematics History and Education
1707
general setting, became the source of conies, and their relation to cones,
though easily proved by Daoidelin's method, was played down, as a minor
subject, at present probably unknown to many users of mathematics.
Or look to a side track, projective geometry: Pascal's theorem on the
inscribed hexagon used as a defining property in Steiner's approach to
conies by means of projectively related pencils. Or to projective invariance
of the harmonic quadruple as a means of defining projectivity of mappings
in von Staudt's approach.
The foregoing are examples of straight inversion of view. A more
sophisticated example of refashioning, again of Greek mathematics — let
us call it conversion — is the elimination of proportion and similarity
arguments from elementary geometry by means of area transformation.
There can be little doubt that the so-called Pythagorean theorem, known
as early as 2000 B.O. in Babylonia, was first discovered and proved by
similarity arguments, by which it is almost trivial. Though it can also
easily be proved by congruence arguments, neither Euclid's proof nor the
one that very likely preceded it in history, was that easy. They show that
Euclid's predecessors had exerted themselves to avoid similarity arguments.
The tool they invented was the transformation of areas, say of rectangles,
that is, the replacement of the proportion
a : b = c: d
by the equality of >areas
ad = be.
In the case of the Pythagorean theorem, which deals with areas, this is
not a far-fetched idea. But they extended similarity avoidance as a principle, which is pursued in the most terrifying way in the construction of
the regular pentagon — important for the regular solids. The construction,
if carried out by simila-rity and the golden section, which is a proportionality
concept, is almost trivial. Euclid's way, by congruence arguments, is
a masterpiece of contorted thought, and an appalling example of blocking
understanding by dogmatism.
When did this proportions and similarity avoiding dogmatism come
about? Was it some time before a satisfactory theory of proportion was
developed ? But why should it have been preserved afterwards ? By mere
tradition or because it was such a marvellous piece of mental gymnastics ?
Or was it craving for purity of method, "do not do with similarity what
you can achieve by congruence"?
1708
Section 19: H. Freudenthal
Unfortunately here Euclid's Elements stopped the pre-Euclidean
process of reshaping and remodelling. This situation lasted two millennia.
It needed a new undogmatic view to rescue the mathematics that was at
a dead end, driven there by the rigorous Greek mental discipline. History
always repeats itself. However, today traditions and dogmatisms enjoy
a much shorter life than those that once blocked the development of
mathematics for almost two millennia. Today no structure of and within
mathematics is safe from inversion and conversion.
But the phenomena of inversion and conversion as a mathematical
virtue are not restricted to what can be called the macro level. Individual
inventors and inventions are permeated by their influence. Eb mathematical idea is published in the way it was discovered. Techniques have been
developed and are used, if a problem has been solved, to turn the solution
procedure upside down, or if it is a larger complex of statements and
theorems, to turn definitions into propositions, and propositions into
definitions, the hot invention into icy beauty. This is what happens on
what I would call the meso level of the history of mathematics. But it
extends to even smaller constituents of our mathematical activity, the
micro level. As an example let us look at the definition of continuity.
Intuitively: a small change of one variable causes a small change of another.
By formalization this is inverted so that the e precedes the <5: "for every
e > 0 there is a <5 > 0 such that...". This inversion is required by the
difference in flavour between two usep of the word "small": small enough,
and as small as you like, where the arbitrariness of the second conditions
the sufficiency of the first. This is a paradigm of the micro inversion which
takes place whenever we switch from one view to the other : in order to
effect some B we have to adjust D to arrive at B.
Let us now turn to education. Years ago I coined the term "antididactical inversion" (see for instance Mathematics as an Fducational TasJc,
p. 122) and illustrated it by a number of examples. One of them was
Peano's axioms, deriving complete induction from them, in order to
apply this principle. The historical course was the inverse, and so should it
be in didactics. ïTobody can become conscious about complete induction
before having unconsciously applied it, and nobody can formulate complete
induction unless he has noticed it. Nobody can grasp Peano's axioms
unless he can formulate complete induction. This is the didactical order and
the historical order. Applying complete induction unconsciously, becoming
conscious of it, formulating it, and building it into Peano's axioms. People
who teach mathematics as a ready-made system prefer antididactical
inversion.
The Implicit Philosophy of Mathematics History and Education
1709
Let us be satisfied with this one example. If mathematics teaching
proves to be a failure, the reason is often, if not always, that we do not
realize that young people have to start somewhere in the past of mankind
and somehow repeat the learning process of mankind. This is the lesson
historians and educators can learn from each other.
Acknowledgement
The present lecture contains pieces from my forthcoming book "Didactical
Phenomenology of Mathematical Structures", where these ideas were
developed in a broader context. This has been made possible by the courtesy of the Publisher D. Bei del, Dordrecht-Boston. To the pieces between an
asterisk and the next double asterisk applies : Copyright © 1983 by D. Beidel
Publishing Company, Dordrecht, Holland.