### The Origin of Geometry

#### **Michel
Serres**

From ed. Harari, Josue V. And David
F. Bell, Hermes; Literature, Science, Philosophy. the Johns Hopkins
University Press, Baltimore, 1982

Renan had the best reasons in
the world for calling the advent of mathematics in Greece a miracle. The
construction of geometric idealities or the establishment of the first
proofs were, after all, very improbable events. If we could form some
idea of what took place around Thales and Pythagoras, we would advance a
bit in philosophy. The beginnings of modern science in the Renaissance
are much less difficult to understand; this was, all things considered,
only a reprise. Bearing witness to this Greek miracle, we have at our
disposal two groups of texts. First, the mathematical corpus itself, as
it exists in the *Elements *of Euclid, or elsewhere, treatises made
up of fragments. On the other hand, doxography, the scattered histories
in the manner of Diogenes Laertius, Plutarch, or Athenaeus, several
remarks of Aristotle, or the notes of commentators such as Proclus or
Simplicius. It is an understatement to say that we are dealing here with
two groups of texts; we are in fact dealing with *two languages*.
Now, to ask the question of the Greek beginning of geometry is precisely
to ask how one passed from one language to another, from one type of
writing to another, from the language reputed to be natural and its
alphabetic notation to the rigorous and systematic language of numbers,
measures, axioms, and formal arguments. What we have left of all this
history presents nothing but two languages as such, narratives or
legends and proofs or figures, words and formulas. Thus it is as if we
were confronted by two parallel lines which, as is well known, never
meet. The origin constantly recedes, inaccessible, irretrievable. The
problem is open.

I have tried to resolve this question three times.
First, by immersing it in the technology of communications. When two
speakers have a dialogue or a dispute, the channel that connects them
must be drawn by a diagram with four poles, a complete square equipped
with its two diagonals. However loud or irreconcilable their quarrel,
however calm or tranquil their agreement, they are linked, in fact,
twice: they need, first of all, a certain intersection of their
repertoires, without which they would remain strangers; they then band
together against the noise which blocks the communication channel. These
two conditions are necessary to the diaIogue, though not sufficient.
Consequently, the two speakers have a common interest in excluding a
third man and including a fourth, both of whom are prosopopoeias of
the,powers of noise or of the instance of intersection.(1)Now
this schema functions in exactly this manner in Plato's *Dialogues*,
as can easily be shown, through the play of people and their naming, *
their resemblances and differences*, their mimetic preoccupations and
the dynamics of their violence. Now then, and above all, the
mathematical sites, from the *Meno *through the *Timaeus*, by
way of the *Statesman *and others, are all reducible geometrically
to this diagram. Whence the origin appears, we pass from one language to
another, the language said to be natural presupposes a dialectical
schema, and this schema, drawn or written in the sand, as such, is the
first of the geometric idealities. Mathematics presents itself as a
successful dialogue or a communication which rigorously dominates its
repertoire and is maximally purged of noise. Of course, it is not that
simple. The irrational and the unspeakable lie in the details; listening
always requires collating; there is always a leftover or a residue,
indefinitely. But then, the schema remains open, and history possible.
The philosophy of Plato, in its presentation and its models, is
therefore inaugural, or better yet, it seizes the inaugural moment.

To be retained from this first
attempt at an explanation are the expulsions and the purge. Why the
parricide of old father Parmenides, who had to formulate, for the first
time, the principle of contradiction. To be noted here again is how two
speakers, irreconcilable adversaries, find themselves forced to turn
together against the same third man for the dialogue to remain possible,
for the elementary link of human relationships to be possible, for
geometry to become possible. Be quiet, don't make any noise, put your
head back in the sand, go away or die. Strange diagonal which was
thought to be so pure, and which is agonal and which remains an agony.

The second attempt contemplates
Thales at the foot of the Pyramids, in the light of the sun. It involves
several geneses, one of which is ritual.(2) But I had
not taken into account the fact that the Pyramids are also tombs, that
beneath the theorem of Thales, a corpse was buried, hidden. The space in
which the geometer intervenes is the space of similarities: he is there,
evident, next to three tombs of the *same *form and of *another
*dimension -the tombs are *imitating *one another. And it is the
pure space of geometry, that of the group of *similarities *which
appeared with Thales. The result is that the theorem and its immersion
in Egyptian legend says, without saying it, that there lies beneath the
*mimetic operator*, constructed concretely and represented
theoretically, a hidden royal corpse. I had seen the sacred above, in
the sun of Ra and in the Platonic epiphany, where the sun that had come
in the ideality of stereometric volume finally assured its diaphaneity;
I had not seen it below, hidden beneath the tombstone, in the incestuous
cadaver. But let us stay in Egypt for a while.

The third attempt consists in noting the double
writing of geometry.(3) Using figures, schemas, and
diagrams. Using letters, words, and sentences of the system, organized
by their own semantics and syntax. Leibniz had already observed this
double system of writing, consecrated by Descartes and by the
Pythagoreans, a double system which represents itself and expresses
itself one by the other. He sometimes liked, as did many others, to
privilege the intuition, clairvoyant or blind, required by the first [diagrams]
over the deductions produced by the second [words]. There are, as is
well known, or as usual, two schools of thought on the subject. It
happens that they trade their power throughout the course of history. It
also happens that the schema contains more information than several
lines of writing, that these lines of writing lay out indefinitely what
we draw from the schema, as from a well or a cornucopia. Ancient algebra
writes, drawing out line by line what the figure of ancient geometry
dictates to it, what that figure contains in one stroke. The process
never stopped; we are still talking about the square or about the
diagonal. We cannot even be certain that history is not precisely that.

Now, many histories report that the Greeks crossed the
sea to educate themselves in Egypt. Democritus says it; it is said of
Thales; Plato writes it in the *Timaeus*. There were even, as
usual, two schools at odds over the question. One held the Greeks to be
the teachers of geometry; the other, the Egyptian priests. This dispute
caused them to lose sight of the essential: that the Egyptians wrote in
ideograms and the Greeks used an alphabet. Communication between the two
cultures can be thought of in terms of the relation between these two
scriptive systems (*signaletiques*). Now, this relation is
precisely the same as the one in geometry which separates and unites
figures and diagrams on the one hand, algebraic writing on the other.
Are the square, the triangle, the circle, and the other figures all that
remains of hieroglyphics in Greece? As far as I know, they are ideograms.
Whence the solution: the historical relation of Greece to Egypt is
thinkable in terms of the relation of an alphabet to a set of ideograms,
and since geometry could not exist without writing, mathematics being
written rather than spoken, this relation is brought back into geometry
as an operation using a double system of writing. There we have an easy
passage between the natural language and the new language, a passage
which can be carried out on the multiple condition that we take into
consideration two different languages, two different writing systems and
their common ties. And this resolves in tum the historical question: the
brutal stoppage of geometry in Egypt, its freezing, its crystallization
into fixed ideograms, and the irrepressible development, in Greece as
well as in our culture, of the new language, that inexhaustible
discourse of mathematics and rigor which is the very history of that
culture. The inaugural relation of the geometric ideogram to the
alphabet, words, and sentences opens onto a limitless path.

This third solution blots out a portion of the texts.
The old Egyptian priest, in the *Timaeus*, compares the knowledge
of the Greeks when they were children to the time-wom science of his own
culture.(4) He evokes, in order to compare them, floods,
fires, celestial fire, catastrophes. Absent from the solution are the
priest, history, either mythical or real, in space and time, the
violence of the elements which hides the origin and which, as the *
Timaeus *clearly says, always hides that origin. Except, precisely,
from the priest, who knows the secret of this violence. The sun of Ra is
replaced by Phaethon, and mystical contemplation by the catastrophe of
deviation.

We must start over -go back to those parallel lines
that never meet. On the one hand, histories, legends, and doxographies,
composed in natural language. On the other, a whole corpus, written in
mathematical signs and symbols by geometers, by arithmeticians. We are
therefore not concerned with merely linking two sets of texts; we must
try to glue, two languages back together again. The question always
arose in the space of the relation between experience and the abstract,
the senses and purity. Try to figure out the status of the pure, which
is impure when history changes. No. Can you imagine (that there exists)
a Rosetta Stone with some legends written on one side, with a theorem
written on the other side? Here no language is unknown or undecipherable,
no side of the stone causes problems; what is in question is the edge
common to the two sides, their common border; what is in question is the
stone itself.

Legends. Somebody or other who conceived some new
solution sacrificed an ox, a bull. The famous problem of the duplication
of the cube arises regarding the stone of an altar at Delos. Thales, at
the Pyramids, is on the threshold of the sacred. We are not yet, perhaps,
at the origins. But, surely, what separates the Greeks from their
possible predecessors, Egyptians or Babylonians, is the establishment of
a proof. Now, the first proof we know of is the apagogic proof on the
irrationality of
.
(5)

And so, legends, once again. Euclid's *Elements*,
Book X, first scholium. It was a Pythagorean who proved, for the first
time, the so-called irrationality [of numbers]. Perhaps his name was
Hippasus of Metapontum. Perhaps the sect had sworn an oath to divulge
nothing. Well, Hippasus of Metapontum spoke. Perhaps he was expelled. In
any case, it seems certain that he died in a shipwreck. The anonymous
scholiast continues: "The authors of this legend wanted to speak through
allegory. Everything that is irrational and deprived of form must remain
hidden, that is what they were trying to say. That if any soul wishes to
penetrate this secret region and leave it open, then it will be engulfed
in the sea of becoming, it will drown in its restless currents."

Legends and allegories and, now, history. For we read
a significant event on three levels. We read it in the scholia,
commentaries, narratives. We read it in philosophical texts. We read it
in the theorems of geometry. The event is the *crisis*, the famous
crisis of irrational numbers. Owing to this crisis, mathematics, at a
point exceedingly close to its origin, came very close to dying. In the
aftermath of this crisis, Platonism had to be recast. The crisis touched
the logos. If logos means proportion, measured relation, the irrational
or alogon is the impossibility of measuring. If logos means discourse,
the alogon prohibits speaking. Thus exactitude crumbles, reason is mute.

Hippasus of Metapontum, or another, dies of this
crisis, that is the legend and its allegorical cover in the scholium of
the *Elements*. Parmenides, the father, dies of this crisis-this is
the philosophical sacrifice perpetrated by Plato. But, once again,
history: Plato portrays Theaetetus dying upon returning from the the
battle of Corinth (369), Theaetetus, the founder, precisely, of the
theory of irrational numbers as it is recapitulated in Book X of Euclid.
The crisis read three times renders the reading of a triple death: the
legendary death of Hippasus, the philosophical parricide of Parmenides,
the historical death of Theaetetus. One crisis, three texts, one victim,
three narratives. Now, on the other side of the stone, on the other face
and in another language, we have the crisis and the possible death of
mathematics in itself.

Given then a proof to explicate as one would a text.
And, first of all, the proof, doubtless the oldest in history, the one
which Aristotle will call *reduction to the absurd*. Given a square
whose side *AB = b*, whose diagonal *AC = a*:

We wish to measure *AC *in terms of *AB*. If
this is possible, it is because the two lengths are mutually
commensurable. We can then write *AC/AB = a/b*. It is assumed that
*a/b* is reduced to its simplest form, so that the integers *a
*and *b *are mutually prime. Now, by the Pythagorean theorem: *
a*² = 2*b*². Therefore a² is even, therefore a is even. And if
*a *and *b *are mutually prime, *b is an odd number*. If
a is even, we may posit: *a *= 2*c*. Consequently, *a*² =
4*c*². Consequently 2*b*² = 4*c*², that is, *b*² = 2*c*².
Thus, *b is an even number*.

The situation is intolerable, the number *b is at
the same time even and odd*, which, of course, is impossible.
Therefore it is impossible to measure the diagonal in terms of the side.
They are mutually incommensurable. I repeat, if logos is the
proportional, here *a/b *or 1/,
the alogon is the incommensurable. If logos is discourse or speech, you
can no longer say anything about the diagonal and
is
irrational. It is impossible to decide whether b is even or odd. Let us
draw up the list of the notions used here. 1) What does it mean for two
lengths to be mutually commensurable? It means that they have common
aliquot parts. There exists, or one could make, a ruler, divided into
units, in relation to which these two lengths may, in turn, be divided
into parts. In other words, they are *other *when they are alone
together, face to face, but they are *same*, or just about, in
relation to a third term, the unit of measurement taken as reference.
The situation is interesting, and it is well known: *two irreducibly
different entities are reduced to similarity through an exterior point
of view*. It is fortunate (or necessary) here that the term *
measure *has, traditionally, at least two meanings, the geometric or
metrological one and the meaning of non-disproportion, of serenity, of
nonviolence, of peace. These two meanings derive from a similar
situation, an identical operation. Socrates objects to the violent
crisis of Callicles with the famous remark: you are ignorant of geometry.
The Royal Weaver of the *Statesman *is the bearer of a supreme
science: superior metrology, of which we will have occasion to speak
again. 2) What does it mean for two numbers to be mutually prime? It
means that they are radically different, that they have no common factor
besides one. We thereby ascertain the first situation, their total
otherness, unless we take the unit of measurement into account. 3) What
is the Pythagorean theorem? It is the fundamental theorem of measurement
in the space of *similarities*. For it is invariant by variation of
the coefficients of the squares, by variation of the forms constructed
on the hypotenuse and the two sides of the triangle. And the space of
similarities is that space where things can be of the *same *form
and of *another *size. It is the space of models and of imitations.
The theorem of Pythagoras founds measurement on the representative space
of imitation. Pythagoras sacrifices an ox there, repeats once again the
legendary text. 4) What, now, is evenness? And what is oddness? The
English terms reduce to a word the long Greek discourses: *even *
means equal, united, flat, *same*; *odd *means bizarre,
unmatched, extra, left over, unequal, in short, *other*. To
characterize a number by the absurdity that it is at the same time even
and odd is to say that it is at the same time *same *and *other*.

Conceptually, the apagogic theorem or proof does *
nothing but* play variations on the notion of same and other, using
measurement and commensurability, using the fact of two numbers being-
mutually prime, using the Pythagorean theorem, using evenness and
oddness. It is a rigorous proof, and the first in history, based on *
mimesis*. It says something very simple: *supposing mimesis, it is
reducible to the absurd*. Thus the crisis of irrational numbers
overturns Pythagorean arithmetic and early Platonism.

Hippasus revealed this, he dies of it -end of the
first act.

It must be said today that this was said more than two
millennia ago. Why go on playing a game that has been decided? For it is
as plain as a thousand suns that if the diagonal or
are
incommensurable or irrational, they can still be constructed on the
square, that the mode of their geometric existence is not different from
that of the side. Even the young slave of the *Meno*, who is
ignorant, will know how, will be able, to construct it. In the same way,
children know how to spin tops which the *Republic *analyzes as
being stable and mobile at the same time. How is it then that reason can
take facts that the most ignorant children know how to establish and
construct, and can demonstate them to be irrational? There must be a
reason for this irrationality itself.

In other words, we are demonstrating the absurdity of
the irrational. We reduce it to the contradictory or to the undecidable.
Yet, it exists; we cannot do anything about it. The top spins, even if
we demonstrate that, for impregnable reasons, it is, undecidably, both
mobile and fixed. That's the way it is. *Therefore*, all of the
theory which precedes and founds the proof must be reviewed, transformed.
It is not reason that governs, it is the obstacle. What becomes absurd
is not what we have proven to be absurd, it is the theory on which the
proof depends. Here we have the very ordinary movement of science: once
it reaches a dead-end of this kind, it immediately transforms its
presuppositions.

Translation: *mimesis *is reducible to
contradiction or to the undecidable. Yet it exists; we cannot do
anything about it. It spins. It works, as they say. That's the way it is.
It can always be shown that we can neither speak nor walk, or that
Achilles will never catch up with the tortoise. Yet, we do speak, we do
walk, the fleet-footed Achilles does pass the tortoise. That's the way
it is. *Therefore*, all of the theory which precedes must be
transformed. What becomes absurd is not what we have proven to be absurd,
it is the theory as a whole on which the proof depends.

Whence the (hi)story which follows. Theodorus
continues along the legendary path of Hippasus. He multiplies the proofs
of irrationality. He goes up to
.
There are a lot of these absurdities, there are as many of them as you
want. We even know that there are many more of them than there are of
rational relations. Whereupon Theaetetus takes up the archaic
Pythagoreanism again and gives a general theory which grounds, in a new
reason, the facts of irrationality. Book X of the *Elements *can
now be written. The crisis ends, mathematics recovers an order,
Theaetetus dies, here ends this story, a technical one in the language
of the system, a historical one in the everyday language that relates
the battle of Corinth. Plato recasts his philosophy, father Parmenides
is sacrificed during the parricide on the altar of the principle of
contradiction; for surely the *Same *must be *Other*, after a
fashion. Thus, Royalty is founded. The Royal Weaver combines in an
ordered web rational proportions and the irrationals; gone is the crisis
of the reversal, gone is the technology of the dichotomy, founded on the
square, on the iteration of the diagonal. Society, finally, is in order.
This dialogue is fatally entitled, not *Geometry*, but the *
Statesman*.

The Rosetta Stone is constructed. Suppose it is to be
read on all of its sides. In the language of legend, in that of history,
that of mathematics, that of philosophy. The message that it delivers
passes from language to language. The crisis is at stake. This crisis is
sacrificial. A series of deaths accompanies its translations into the
languages considered. Following these sacrifices, order reappears: in
mathematics, in philosophy, in history, in political society. The schema
of Rene Girard allows us not only to show the isomorphism of these
languages, but also, and especially, their link, how they fit together.(6)
For it is not enough to narrate, the operators of this movement must
be made to appear. Now these operators, all constructed on the pair Same-Other,
are seen, deployed in their rigor, throughout the very first geometric
proof. just as the square equipped with its diagonal appeared, in my
first solution, as the thematized object of the complete intersubjective
relation, formation of the ideality as such, so the rigorous proof
appears as such, manipulating all the operators of *mimesis*,
namely, the internal dynamics of the schema proposed by Girard. The
origin of geometry is immersed in sacrifical history and the two
parallel lines are henceforth in connection. Legend, myth, history,
philosophy, and pure science have common borders over which a unitary
schema builds bridges.

Metapontum and geometer, he was the Pontifex, the
Royal Weaver. His violent death in the storm, the death of Theaetetus in
the violence of combat, the death of father Parmenides, all these deaths
are murders. The irrational is mimetic. The stone which we have read was
the stone of the altar at Delos. And geometry begins in violence and in
the sacred.(7)

**Michel Serres**

http://pratt.edu/~arch543p
#### Notes:

(1) The line
from Speaker 1 to Speaker 2 represents the channel of communication
thatjoins the two speakers together. The line from Noise to the Code or
Repertoire represents the indissoluble link between noise and the code.
Noise always threatens to overwhelm the code and to disrupt
communication. Successful communication, then, requires the exclusion of
a third term (noise) and the inclusion of a fourth (code). See "Platonic
Dialogue," chapter 6 of' the present volume. See also Michel Serres. Le
Parasite (Paris: Grasset, 1980). -Ed.

(2) See "Mathematics
and Philosophy: What Thales Saw...... chapter 8 of the present volume. -Ed.

(3)This third
explaiiatioii appears as "Origine de la geometrie, 4" in Michel Serres,
Hermes V.- Le Passage du Nord-Ouest (Paris: Minuit, 1980), pp. 175-84. -Ed.

(4) Plato,
Timaeus, 22b ff.

(5) An apagogic
proof is one that proceeds by disproving the proposition which
contradicts the one to be established, in other words, that proceeds by
reductio ad absurduni. - Ed.

(6) The
reference is to Rene Girard's theory of the emissary victim. See chapter
9, note 9 in the present volume. - Ed.

(7) It is just
as remarkable that the physics of Epicurus, as Lucretius develops it in
De Rerum Natura, is framed by the sacrifice of Iphigenia and the plague
of Athens. These two events, legendary or historical, can be read using
the grid of phvsics. But, inversely, all this physics can be read using
the same schema, since the term inane means "purge" and "expulsion." I
have shown this in detail in La Naissance de la physique dans le texte
de Luctice: Fleuves el turbulences (Paris: Minuit, 1977). (See also "Lucretius:
Science and Religion," chapter 9 of the present volume. -Ed.)