Les mathématiques grecques de Pythagore à Eudoxe

Histoire des sciences-Mathématiques-Philosophie
Les mathématiques grecques de Pythagore à Eudoxe
Institut Henri Poincaré-Amphithéâtre Hermite
28-30 novembre 2012
Avec le soutien de : IMJ-Paris 7-ASHIC-HSM-IHP
RÉSUMÉS DES CONFÉRENCES
Mercredi 28 novembre
12h30-14h (Séminaire ASHIC, JUSSIEU Couloir 15-16 salle 413)
Henry MENDELL (California State University, Los Angeles): Multitudes of Numbers in Aristotle
All known Classical and Hellenistic theories of number involve treating a number as a
concatenation of units, and this is reflected in the standard number system of the 4th cent. BCE.,
although alphabet letters were used as ordinals. There are two puzzles in Aristotle's account of
number, one is a distinction between what we count and what we count with in his account of
time. I am not sure that we can ever resolve this one adequately, although I shall give it a
try. Another lies in his definition of line and number in the Metaphysics. Here Aristotle calls a
multitude (plêthos) a countable quantity, a number a limited plêthos (multitude), as well as a line a
limited length, etc. This is odd, since Aristotle doesn't believe that there are infinite lengths or
infinite multitudes. It is always possible that he conceptually allows the possibility of multitudes
that are infinite, even if in fact they are impossible. I would like to explore a different route,
through Euclid. Euclid has three distinct concepts of number in the Elements, plêthos (commonly
translated as 'multitude'), arithmos (commonly translated as 'number'), and the homonym of a
number (not translated further, but corresponding to 'quantième'). An arithmos is a collection of
units as such. The more general plêthos is also a collection of other things, including collection of
collections of units. A plêthos is counted out by a one to one correspondence of its parts to the units
in an arithmos. Let us return to Aristotle and the Academy. All but one treatments of number in
the Academy involve an ultimately intuitive one to one relation of secondary numbers to a primary
sort of number. It is reasonable that Aristotle holds an Aristotelian version of the same view. With
this, I shall attempt an answer to both puzzles.
Mercredi 28 novembre (IHP, Amphi Hermite)
15h-16h30
Michel CRUBELLIER (Lille III) : Aristote critique de Platon : à propos de l'ontologie des objets
mathématiques
Pour Platon – et pour Aristote, qui de ce point de vue est toujours resté un Platonicien – le réel est
connaissable, et ce qui est éminemment connaissable est éminemment réel. Les mathématiques
sont, dans la Grèce du 4e siècle avant notre ère, le modèle d'une connaissance particulièrement
achevée de sorte que la question du type de réalité des objets mathématiques est un enjeu
ontologique important. A partir de quelques passages des Dialogues et de la discussion par Aristote
des thèses envisagées dans l'ancienne Académie, on cherchera à préciser la conception
platonicienne du statut ‘idéal’ de ces objets et les questions qu'elle a soulevées en ce qui concerne
(1) la structuration du domaine des mathématiques et (2) leur relation au monde des phénomènes
naturels. On reviendra, à la lumière de ces discussions, sur la nature de la connaissance
mathématique, du point de vue d'Aristote cette fois.
Jeudi 29 novembre (IHP, Amphi Hermite)
14h30-16h
Vassilis KARASMANIS (Université technologique nationale d’Athènes): The Axiomatization of
Mathematics and Plato's Conception of Knowledge in the Meno and the Republic
Euclidean geometry is a system of hierarchically ordered propositions. Each proposition of
this system is inferred deductively by others 'prior' or more elementary ones. But, not allgeometrical knowledge is demonstrative. In Euclidean geometry everything starts from some basic
principles which are considered "plain to all" (Plato, Rep. 510d1). Therefore, there is an asymmetry
in our knowledge of geometrical propositions between a) principles, which are unprovable and 'selfevident' or "plain to all", and b) derivative theorems which are justified by demonstration from the
principles or other 'more elementary' theorems. We can schematically represent the Euclidean
model (hereafter model-e) as a pyramid having the principles at its top.
It is almost common place among platonic scholars that Plato adopts model-e in his theory
of knowledge. The main evidence is found in Books VI and VII of the Republic where we find a
hierarchy among the objects of knowledge (Forms) and their relations. The metaphors of 'up' and
'down' suggest the hierarchical character of his model of knowledge which may be represented as a
pyramid with the non-hypothetical (Good) first principle at its apex. In the Republic we have also
the first evidence that Greek mathematics were organized axiomatically.
In this paper I am going to question the belief that Plato adopts model-e in the Meno. I shall
argue that it is legitimate to read the Meno as presenting a different conception of knowledge that
might remind us of modern coherentist theories. More specifically, I claim that there is no evidence
in the Meno of an asymmetry in knowability between first principles and derivative propositions. In
the Meno all knowledge seems to be inferential, requiring justification or reasoning into the causes
(aitias logismon – 98a), but there is no need of fixed and unprovable first principles from which
such reasoning proceeds. His system of knowledge consists of interrelated elements and several
accounts can explain the interrelations among the elements of the system. Such a model can be
represented as a network rather than as a pyramid (hereafter I shall call it model-n). In such a
model, circular regress may not always be vicious. I am not going to maintain that Plato consciously
proposes such a model of knowledge, but rather that he himself is exploring such directions. After
the axiomatization of mathematics (that takes place within the Academy), Plato adopts the model-e
of the mathematicians.
Jeudi 29 novembre (IHP, Amphi Hermite)
16h30-18h
Luigi BORZACCHINI (Dipartimento di Matematica, Università di Bari) : To know What is Not.
Incommensurability and Negative Knowledge
The discovery of incommensurability is probably the greatest achievement of mathematics before
Euclid and, we could say, maybe also the birth of our idea of mathematics. The genesis and the
dynamics of such discovery is difficult to make clear, it is almost a myth like the birth of Athena,
born thoroughly armed from the head of Zeus.
Common opinion is that in the IV century a Grundlagenkrisis in the ancient Pythagorean
mathematics followed that discovery, and led to the end of the Pythagorean mathematics, to the
development of an autonomous geometrical style and to the axiomatic-deductive structure we find
in Euclid’s Elements. This opinion is probably based on a surface-analogy with the foundational
problems in mathematics between the end of the XIX and the beginning of the XX century, and on
the thesis that critical results require always a greater ‘rigor’ which in turn can be achieved by a
sharper axiomatic-deductive approach.
Against this thesis a diffused criticism has been raised, but I want only to remark that the proof of
the incommensurability required the refutation by absurd, that, before the second half of the IV
century, was credibly not a recognized mathematical proof technique, so that the
incommensurability property had to be established together with its method of proof.
I shall try first to establish something more precise about the historical genesis and development
of the discovery (following the musical hypothesis of Paul Tannery). However I believe it would be
even more important to show how the crisis concerned, beyond mathematics, first and foremost
directly the very idea of knowledge in the classical Greek culture, as witnessed in the works of Plato
and Aristotle, facing the problem of negative knowledge, entailing the evolution of formal thinking
and founding the establishment of the Aristotelian logic.
The fortune and the creative role of the negative knowledge in the history of mathematics was not
limited to the technique of the proof by absurd: the infinite and the continuum were concepts whose
evolution was marked by the negative’s career. And also the history of the liar, wherein we can
recognize the simultaneous presence of being, truth and negation, is a karst river which periodically
and suddenly reappears in the history of mathematics and logic: in modern times for example
(somehow concealed in the set-theoretical and in the proof-theoretical lexicon) in Gödel’s
incompleteness theorems and in Cantor’s diagonal argument.
Vendredi 30 novembre (IHP, Amphi Hermite)
14h30-16h : Salomon OFMAN (IMJ-Paris 7) : Quelques questions autour d’une preuve
pythagoricienne d’irrationalité de √2
Une ancienne tradition fait remonter l’origine des grandeurs irrationnelles, soit directement à
Pythagore, soit aux Pythagoriciens de son époque. La démonstration d’irrationalité étant, pour
reprendre les termes d’Aristote, celle de ‘l’incommensurabilité de la diagonale du carré’, ce que
l’on traduit pour les modernes par ‘l’irrationalité de √2’.
Cela implique une série de conséquences, à la fois sur la datation (l’irrationalité est connue dès le
VIème ou tout au début du Vème siècle BCE) et sur la manière de raisonner des Grecs anciens, cette
démonstration passant par la ‘preuve par l’absurde’ (appelée encore ‘preuve par l’impossible’).
Celle-ci, en effet, paraît absente des textes mathématiques antérieurs, Égyptiens, Babyloniens ou
autres, et caractériserait ainsi les mathématiques grecques et au-delà.
Les plus anciens témoignages concernant la démonstration d’irrationalité se trouvent dans un
ouvrage d’Aristote, les Analytiques1. C’est pourquoi les tentatives de reconstructions se fondent sur
eux. Ces lignes, très brèves, ne donnent guère de détails, l’objectif du philosophe n’étant pas de
rapporter une étude mathématique, mais d’illustrer, dans un cadre syllogistique, une forme de
raisonnement, celui par l’absurde.
Dans cet exposé, nous voudrions montrer que les reconstructions standards de cette démonstration
se fondent sur des traductions de ces quelques lignes des Analytiques, par les historiens des
mathématiques, différente de celle des philosophes/philologues attachés à la littéralité textuelle.
Nous présenterons ici une démonstration2, qui à la fois s’accorde avec les témoignages textuels des
Analytiques d’Aristote, mais aussi indirectement de certains dialogues de Platon, et avec ce que
nous pouvons savoir de l’état de l’arithmétique grecque au Vème siècle. Cette démonstration est
compatible avec, et donc renforce la possibilité d’une origine pythagoricienne de l’irrationalité.
Nous terminerons en discutant une objection importante à cette démonstration.
1
Premiers Analytiques, I, 23, 41a26-32 ; 44, 50a37-38.
Cf. mon article : Une nouvelle démonstration de l’irrationalité de racine carrée de 2 d’après les Analytiques d’Aristote,
Philosophie antique, n° 10, 2010, 81-138
2
Vendredi 30 novembre (IHP, Amphi Hermite)
16h30-18h
Henry MENDELL (California State University, Los Angeles): Two definitions of the Continuous
In my paper "Two Traces of Two-Step Eudoxan Proportion Theory in Aristotle", I argued that in his
definition of 'faster', Aristotle follows a procedure from his Posterior Analytics, where one gives a
preliminary definition of a term and then proceeds to prove a fundamental theorem, where the
conclusion of the theorem is the definiens transformed into a proposition and the demonstration
transformed is the definition that provides the essence. It is remarkable that Aristotle also defines
'continuous' in this way. In Physics V 3, he famously defines the continuous as things whose limits
where they touch are the same, but later in Physics VI 2 he defines the continuous as that which is
always divisible into the divisible. Clearly, the latter is the more important property for Aristotle,
but why does he call the latter a definition? Well, he has a theorem in Physics VI 1 that what is
continuous is always divisible into divisibles. So this is an example of a fundamental theorem
transformed into a proposition being the definition of the essence. In this regard, I look carefully at
the argument of Physics VI 1 to see how it uses the definitions of V 3. An important side point is
that the definitions of V 3 should be seen as definitions within a mathematical physics and not
within geometry.