Les mathématiques grecques de Pythagore à Eudoxe
Transcription
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.