Parmenides and Zeno of Elea Criticise the Pythagorean Definition of Point, Part II

This post follows on from these five posts on Archimedes (287 BCE-212 BCE):

1. The Rediscovery of The Archimedes Palimpsest and His Use of the Infinite

2. André Koch Torres Assis And Ceno Pietro Magnaghi Write About Archimedes's Method

3. Archimedes Uses a Lever to Prove Geometry Theorems

4. Archimedes Is Reticent to Publish Proofs Using Indivisibles

5. Parmenides and Zeno of Elea Criticise the Pythagorean Definition of Point, Part I

Because of length restrictions on Substack, I have been forced to split this post into two; this is the second part. The first part is post (5) above.

In this post, I will be quoting from Paul Tannery’s (1843-1904) work Pour l’Histoire de la science hellène: De Thalès à Empédocle¹ [For the history of Hellenic science: From Thales to Empedocles]. Tannery, a prolific French mathematician and historian of science, among his numerous achievements, edited (with Charles Henry (1859-1926)) the works of Pierre de Fermat (1607-1665)², and (with Charles Adam (1857-1940)) the works of René Descartes (1596-1650)³. The quotations below are my adaptations of DeepL translations; the original French quotations can be found in the footnotes.

I think it is important to read what Tannery wrote about Zeno (c.490 BCE-c.430 BCE), who plays an important rôle in Plato’s (c.427 BCE-348 BCE) dialogue Parmenides, in which Socrates (c.470 BCE-399 BCE) is a young man, stumped by Zeno’s reasoning.

The aim of the Discourses, which he [Zeno] had written was very clearly defined by Plato, to whom we must obviously adhere: Zeno fought against the belief in plurality as a hypothesis and by showing that, if this hypothesis is admitted, we necessarily arrive at contradictions, since we are equally led to affirm for things both infinite smallness and infinite greatness, both rest and motion. So it must be clearly understood (and this is often overlooked) that, whatever his famous arguments, Zeno in no way denied motion (he was not a sceptic), he merely asserted its incompatibility with the belief in plurality.⁴ [my emphasis]

Tannery writes that Zeno’s argumentation is not focusing on the common man, who would believe quite correctly that “two sheep are not one and the same thing”. Rather, Zeno is focusing on a specific set of intellectuals.

Such, in fact, must have been Zeno's appearance from antiquity in the eyes of many people, especially in Athens, if they came to read his writings there; from the point of view of the impression he made on the common man, he can rightly be compared with the modern idealists; but no more than they, still less perhaps, did he write for this common man incapable of understanding him; it was to a restricted and learned public that he addressed himself; it was a particular theory that he fought: before this public, against this theory, he had all the success he could wish for.⁵ [my emphasis]

According to Tannery, Parmenides, in his poem On Nature, was specifically focusing on the Pythagoreans.

Parmenides had written his poem in a milieu where, as thinkers, only the Pythagoreans were in honour; he had reproduced more or less exactly their exoteric teaching on cosmology and physics, but, in any case, he had denied the truth of their dualistic thesis; on the other hand, in his ontological theory, presented as being of mathematical rigour and certainty, he had not, to tell the truth, attacked them directly. His fundamental principle, on being and non-being, was basically the same as the postulate “nothing is made out of nothing” already accepted, at least implicitly, by all the thinkers who had preceded him; to establish it, he therefore only had to refute the vulgar opinion on genesis and destruction; but, once this principle had been established, he drew entirely new consequences, and in particular those on the unity, continuity and immobility of the universe contradicted the Pythagorean doctrines.⁶ [my emphasis]

Zeno, the disciple of Parmenides, focused specifically on the Pythagorean doctrine of a point being a “unit having a position”.

What, then, was the weak point recognised by Zeno in the Pythagorean doctrines of his time? How does he present it as an affirmation of the plurality of things? The key is given to us by a famous definition of the mathematical point, a definition that was still standard at the time of Aristotle, but which historians have not considered carefully enough.

For the Pythagoreans, the point is the unit having a position, or in other words the unit considered in space. It follows immediately from this definition that the geometrical body is a plurality, the sum of points, just as a number is a plurality, the sum of units.

A body, a surface or a line are in no way a sum, a totality of juxtaposed points; the point, mathematically speaking, is in no way a unit, it is a pure zero, a nothing of quantity.⁷ [my emphasis]

The Pythagoreans, notwithstanding the critique made by Zeno, would, like in any good cult, continue on as if nothing had happened.

We should not be surprised that the Pythagoreans made this mistake, despite the development of their geometrical knowledge; they had in fact started from the vulgar prejudice, still shared by most people who are strangers to mathematics, and the only discovery that could have made them suspect the falsity of this prejudice, namely the discovery of the existence of incommensurable quantities, remained in the School, as the history of mathematics shows, a real logical scandal, a formidable stumbling block. Nevertheless, they continued their arithmetical speculations on the numbers of triangles, polygons, pyramids, etc., speculations based in fact on the idea that it is possible to form geometric figures with arrangements of points in specific numbers.

Moreover, at that time, no distinction could yet be made between a geometric body and a physical body; the Pythagoreans therefore imagined the bodies of nature as formed by the assembly of physical points; it is of little importance to discuss here whether or not they conceived of these points as being of one or two different natures (dualistic hypothesis); nor is it necessary to investigate whether or not they had preserved without alteration the doctrine of the Master, whether or not they had correctly understood his teachings.⁸ [my emphasis]

According to Tannery, Zeno’s attacks on the Pythagorean understanding of number were devastating.

I have already indicated … two different meanings that the famous expression “Things are numbers” may have had, before Philolaos. Zeno's polemic teaches us that, in his time, the first stage had been passed and the proposition was understood in the sense that bodies were considered to be sums of points, and their properties to be linked to the properties of the numbers representing these sums.

It is in fact this formula, taken in this sense, that Zeno combats by expressing it in very much the same terms, at any rate clearer to the public: Beings are a plurality (πολλά ἐστι τὰ ὄντα). Explained in this sense, his arguments appear sharp, pressing, irrefutable, even those where we ordinarily see mere paralogisms.

Zeno's success was complete; his opponents could not reply.⁹ [my emphasis]

With Zeno’s dismantling of the idea of a geometrical object as a sum of points, and his demonstration that this idea naturally leads to contradictions, it becomes clear why Archimedes refrained from considering, at least in public, a theorem in geometry to be proven if the only available proof were based on indivisibles.

The subtleties of what happens when we pull out our “infinite microscope” to examine what is happening at the infinitesimal scale are the subject of crucial debates in the histories of both physics and mathematics through the centuries. We will be returning to this debate over and over.

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1. Paul Tannery. Pour l’Histoire de la science hellène: De Thalès à Empédocle. Deuxième édition par A. Diès. Avec une Préface de Federigo Enriques. Paris: Gauthier-Villars et cie, 1930. Chapitre X: Zénon D'Èlée, pp.255-270.

2. Paul Tannery et Charles Henry, éditeurs. Œuvres de Fermat. 4 volumes. Paris: Gauthier-Villars et fils, 1891-1912.

3. Charles Adam & Paul Tannery, éditeurs. Œuvres de Descartes. 12 volumes. Paris: Léopold Cerf, 1897-1910.

4. Le but des Discours qu'il avait écrits a été trés clairement défini par Platon, auquel il faut évidemment s’en tenir: Zénon a combattu la croyance à la pluralité comme hypothèse et en démontrant que, si cette hypothèse est admise, on arrive nécessairement à des contradictions, puisqu’on est également conduit à affirmer pour les choses l’infinie petitesse et infinie grandeur, le repos et le mouvement. Ainsi il doit être bien entendu (ce qu’on oublie souvent de mentionner) que, quels que soient ses célèbres arguments, Zénon n’a nullement nié le mouvement (ce nest pas un sceptique), il a seulement affirmé son incompatibilité avec la croyance à la pluralité. [Tannery, pp.256-257]

5. Tel, en fait, dut apparaître Zénon dès l'antiquité aux yeux de bien des gens, surtout à Athènes, s'il vint y lire ses écrits; au point de vue de l’impression qu'il produisit sur le vulgaire, on peut avec raison le comparer aux idéalistes modernes; mais pas plus qu’eux, encore moins peut-être, il n’écrivait pour ce vulgaire incapable de le comprendre; c’était à un public restreint et savant qu'il s’adressait; c’était une théorie particulière qu'il combattait: devant ce public, contre cette théorie, il eut tout le succès qu'il pouvait désirer. [Tannery, p.257]

6. Parménide avait écrit son poéme dans un milieu où, comme penseurs, les pythagoriens seuls étaient en honneur, il avait reproduit plus ou moins exactement leur enseignement exotérique relatif à la cosmologie et à la physique, mais, en tout cas, il avait nié la vérité de leur thèse dualiste; d’un autre côté, dans sa théorie ontologique, présentée comme étant d’une rigueur et d'une certitude mathématiques, il ne les avait pas, à vrai dire, attaqués directement. Son principe fondamental, sur l’être et le non-être, revenait au fond au postulat «rien ne se fait de rien» déjà admis, au moins implicitement, par tous les penseurs qui l'avaient précédé; pour l’établir, il n’avait donc à réfuter que l'opinion vulgaire sur la genèse et la destruction; mais, de ce principe une fois posé, il tirait des conséquences toutes nouvelles, et notamment celles sur l'unité, la continuité, l'immobilité de l'univers contredisaient les doctrines pythagoriennes. [Tannery, p.258]

7. Quel était donc le point faible reconnu par Zénon dans les doctrines pythagoriennes de son temps? de quelle façon le présente-t-il comme étant une affirmation de la pluralité des choses? La clef nous est donnée par une célébre définition du point mathématique, définition encore classique au temps d’Aristote, mais que les historiens n’ont pas considérée assez attentivement.

Pour les pythagoriens, le point est l'unité ayant une position, ou autrement l’unité considérée dans l’espace. Il suit immédiatement de cette définition que le corps géométrique est une pluralité, somme de points, de même que le nombre est une pluralité, somme d’unités.

Or, une telle proposition est absolument fausse; un corps, une surface ou une ligne, ne sont nullement une somme, une totalité de points juxtaposés; le point, mathématiquement parlant, n’est nullement une unité, c’est un pur zéro, un rien de quantité. [Tannery, pp.258-259]

8. D’ailleurs, à cette époque, aucune distinction ne pouvait encore exister entre un corps géométrique et un corps physique; les pythagoriens se représentaient donc les corps de la nature comme formés par l’assemblage de points physiques; il importe peu de discuter ici s’ils concevaient ou non ces points comme étant d’une ou de deux natures différentes (hypothèse dualistique); il n’y a pas davantage à rechercher s’ils avaient ou non conservé sans altération la doctrine du Maître, s’ils avaient bien compris ses enseignements. [Tannery, p.259]

9. J’ai déja indiqué deux sens différents qu’a pu recevoir, avant Philolaos, la célèbre expression: «Les choses sont nombres.» La polémique de Zénon nous apprend que, de son temps, le premier stade était franchi et la proposition entendue dans ce sens que les corps étaient considérés comme sommes de points, et leurs propriétés comme liées aux propriétés des nombres représentant ces sommes.

C’est en effet cette formule, prise en ce sens, que combat Zénon en l’exprimant en termes à très peu près identiques, en tout cas plus clairs pour le public: Les êtres sont une pluralité (πολλά ἐστι τὰ ὄντα). Expliqués dans ce sens, ses arguments apparaissent comme nets, pressants, irréfutables, même ceux où l'on ne voit d’ordinaire que de simples paralogismes.

Le succès de Zénon fut complet; ses adversaires ne pouvaient lui répondre. [Tannery, pp.259-260]