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

This post follows on from these four 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

Because of length restrictions on Substack, I have been forced to split this post into two; this is the first part.

In my last post, I wrote that the Dutch historian of science, Eduard Jan Dijksterhuis (1892-1965)¹, had noted that Archimedes used the indivisibles in the mechanical proofs which he reserved for himself and his close correspondents, but not in the geometrical proofs which he published. Here is how I concluded:

But where did this reticence to public use of the indivisibles come from? For this, we will need to look at Enrico Rufini’s (1890-1924) Il “Metodo” di Archimede e le origini del calcolo infinitesimale nell’antichitಠ[Archimedes’s “Method” and the Origins of the Infinitesimal Calculus in Antiquity], which is referred to by Dijksterhuis. Therein, Rufini explains the contradictions that Parmenides and Zeno of Elea point out in the Pythagorean understanding of matter and number, and the implications that this will have on the perception of indivisibles.

In that book, Rufini provides an Italian-language translation of Archimedes’s Method. He also provides a historical overview of the debates within the ancient Greek world leading up to the work of Archimedes. His intent, as related by Italian mathematician and historian of science Federigo Enriques (1871-1946), was the following:

Perciò aveva accolto con entusiasmo il mio invito a tentare una vasta ricostruzione storica dell’analisi infinitesimale, che, incominciando dall’antichità, doveva proseguirsi nel Rinascimento, fino a Leibniz e Newton. [p.14]

Here is the Deepl translation:

He had therefore enthusiastically accepted my invitation to attempt a vast historical reconstruction of infinitesimal analysis, which, beginning in antiquity, was to continue through the Renaissance, up to Leibniz and Newton.

Unfortunately, Rufini died of typhus soon after finishing his translation of the Method, and so was unable to bring to fruition this ambitious enterprise. There is a eulogy written in his honor by Aldo Mieli (1879-1950)³.

In this historical overview, Rufini refers to Paul Tannery’s (1843-1904) work Pour l’Histoire de la science hellène: De Thalès à Empédocle⁴ [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)⁵, and (with Charles Adam (1857-1940)) the works of René Descartes (1596-1650)⁶.

In this post, I will focus on the critiques that Parmenides (c.late 6th century BCE-c.5th century BCE) and Zeno (c.490 BCE-c.430 BCE) made of the Pythagorean concepts of number and point, which led naturally to the avoidance of the use of indivisibles.

It must be borne in mind that reading and writing about the ancient world is very different from the same for the post-Renaissance period. For much of the latter period, there is a traceability of documents that allows us to read physical documents that come directly from the authors, either in the form of manuscripts or published works, which are still available in libraries and archives in different parts of the world.

For the ancient world, on the other hand, information is truly scanty. For most thinkers, there are only fragments, and in many situations even these are only passed on by third parties. There is no clear traceability of provenance, to the point where the very existence of certain historical figures is not accepted by all.

As a result, I will not be reading original source material, nor even translations thereof, but, rather, I will be quoting from both Rufini’s book and Tannery’s. In the text, I will be using Deepl-produced translations which I have adapted as necessary. I will put the original Italian and French text in the footnotes.

According to Rufini, following from his understanding of the work of Tannery and Enriques, in the Pythagorean view, “all things are numbers”, which ultimately means that both matter and geometric figures, which are conflated, are built up from monads:

Now it is well known how such a theory is summed up in the formula “things are numbers”; which formula, in the most ancient period, meant that matter was made up of material points or monads (μονάδες, units having position) and that qualitative differences depended only on the number and position of those points. This interpretation (suggested by Tannery) could be further specified if one were to reconnect (as Enriques does) the Pythagorean views to the ideas of Anaximander. Anaximander advanced the hypothesis that the natural substratum of things was an indeterminate substance, infinitely diffusible (τὸ ἅπειρον); from this hypothesis one can derive the monadic structure of sensible matter proposed by Pythagoras, supposing a condensation of the primitive substance around certain monadic points or centres, which consequently had to remain delimited by a surrounding void, or, perhaps better, by a rarefied ethereal medium, of the nature of fire.⁷

On the theory of matter was modelled that of geometry. The substratum of geometric figures was an unlimitedly extended and infinitely divisible matter, not unlike Anaximander's cosmic substance, with which the concept of space was identified. Geometric figures were limitations or determinations of this space, and thus forms of a universal matter, which is found identical in qualitatively different sensible things. In this way, geometry, although not suddenly transported into the realm of the intelligible, rose above the vulgar empirical conception and gave geometric figures a certain degree of abstraction.⁸

But this was only a first degree of abstraction, since the empirical assumption still remained at the basis of the Pythagorean conception. For it was held that the primitive element of geometrical figures was precisely the monad; and so the geometrical point still appeared as a material point, at once extended and indivisible. And just as all things were numbers, that is, a sum of units, so every geometric figure was a sum of points.⁹

But this structure came crashing down with the discovery, when attempting to measure the diagonal of a square, of numbers that were incommensurable with respect to each other. Two numbers a and b are deemed to be mutually incommensurable if neither is a rational multiple of the other. For example, 4/3 and 5/4 are mutually commensurable, because 4/3 * 15/16 = 5/4; on the other hand, 2 and √2 are not mutually commensurable, as neither is a rational multiple of the other. The term commensurable should not be understood as irrational number; rather, it is a relationship between two numbers; for example, √2 and √8, both irrational, are mutually commensurable, since √8 = 2√2.

But the fundamental point-monad hypothesis was bound to collide with the discovery of incommensurable quantities that occurred within the Pythagorean school itself, a discovery that threatened to ruin the scientific edifice so carefully constructed.¹⁰

So what did the Pythagoreans decide to do? They made it a “state secret” that there were numbers that were mutually incommensurable, as their entire universe was created based on the idea that everything in the universe was based on relations between whole numbers. Of course, as with many state secrets, ultimately the information leaked out to the wider world.

Parmenides of Elea and Zeno of Elea are known as monists, as they rejected the many, as I will outline below. Zeno is best known for his paradoxes of motion, the three most famous being recounted in Aristotle’s (384 BCE-322 BCE) Physics:

• The dichotomy paradox: That which is in locomotion must arrive at the half-way stage before it arrives at the goal.

  If one needs to get somewhere, one must first reach half-way there, and to get to that half-way point, one must first reach half-way there, i.e., to the quarter-way point, and so on. It follows that an infinite number of tasks must be undertaken, and so one can never even start moving.

• Achilles and the tortoise: In a race, the quickest runner can never over­take the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

  For Achilles to catch the tortoise, he must first reach the point where the tortoise currently stands, but by that time the tortoise will have moved on. This process repeated will also lead to an infinite number of tasks to be undertaken, and so Achilles will never catch the tortoise.

• The arrow paradox: If everything when it occupies an equal space is at rest at that instant of time, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless at that instant of time and at the next instant of time but if both instants of time are taken as the same instant or continuous instant of time then it is in motion.

  For an arrow to leave a bow and reach its target, it must pass through a series of instants. In a given instant, the arrow will not move. It follows that the arrow cannot move through time.

We will return to Zeno’s paradoxes in the second part by examining Tannery’s discussion thereof. But let us first return to Rufini and Parmenides:

To resolve the crisis that troubled Pythagorean geometry was directed (according to Tannery) the work of the school of Elea, work that was effectively initiated by Parmenides (around 500 B.C., as Enriques has shown). In fact, Parmenides’s poem On Nature (περὶ φύσεως) acquires a particularly clear and evocative meaning, if it is considered not as a purely metaphysical treatise, but also in part (according to the interpretation put forward by Enriques) as being directed towards criticising the monadic theory of the Pythagoreans, and therefore also their empirical conception of the fundamental entities of geometry: point, line and surface.¹¹

That Parmenides was concerned with questions concerning the principles of geometry is deduced from two quotations from Proclus: one of which reports a classification of figures proposed by the Eleate; the other states that the Euclidean definition “the point is that which has no parts” conforms to Parmenides’s criterion that negative definitions are in agreement with principles. Parmenides would therefore have sought to define the first concepts of geometry, and so it will come as no surprise if we find hints of the criticism he instituted in his writings.¹²

Proclus (412-485), one of the most important Neoplatonist philosophers, was claiming that it is with Parmenides that begins the formal definitions of point, line and surface that we associate with Euclid. Rufini continues by examining two fragments from On Nature.

Two passages from the Parmenidean poem, already illustrated by Enriques, deserve special mention here:

fragment 2, which contains a hint at the concept of the surface without thickness, as that “which does not separate space from the connection of space”;

fragment 6, which is a strong indictment of the various solutions proposed by the Pythagoreans regarding the concept of the point. He reproaches them for being “both deaf and blind, stupefied, without discernment,” because they do not see the contradiction that consists in believing “that being and non-being are the same thing and not the same thing.” In other words, Parmenides condemns the concept of the point-monad as something contradictory, whose existence (or extension) is affirmed and denied at the same time, and which must have appeared to him as an infinitesimal present bastard.¹³

Parmenides’s words are hard to follow, but the critique by Zeno, described below, will clarify these issues. Rufini concludes with respect to Parmenides.

In Parmenides’s work, therefore, the rational concept of point, line and surface is affirmed for the first time; his critique tends in essence to establish that geometric entities can only be defined by abstraction, by an indefinite process of idealisation, as limits of the sensible. Now this affirmation constitutes the first recognition of the infinitesimal character of the fundamental concepts of geometry, and can therefore be regarded as the first acquisition of infinitesimal analysis.¹⁴

Zeno, in order to support the theses of his master Parmenides, explained that the monad-point leads to a contradictory situation in which everything is simultaneously infinitely small and infinitely large. Should the monad-point be with extent, i.e., of finite size, an infinite set of these would be infinitely large. Similarly, should the monad-point be with no extent, i.e., of zero size, then even any set of these would still be infinitely small. It follows that the Pythagorean monad-point is self-contradictory.

It should be understood that Zeno is not offering an explanation of what is, but clearly explaining what is not. What the work of Parmenides and Zeno leads to is an understanding that there is a difference between matter and geometry, and that in the latter we are working with idealisations, with abstractions. Geometrical objects, at least the ones in our heads, are not material, but, rather, ideal.

I think it is important to read what Tannery wrote about Zeno, 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.

This will be the subject of the second part of this post.

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1. E.J. Dijksterhuis. Archimedes. Translated by C. Dikshoorn. With a new bibliographic essay by Wilbur R. Knorr. Princeton University Press, 1987. Chapters I-V of this work were formerly published in Dutch (Archimedes, I, P. Noordhoff, Groningen, 1938). The subsequent chapters appeared in the Dutch periodical Euclides (XV-XVII, XX; 1938-1944).

2. Enrico Rufini. Il “Metodo” di Archimede e le origini del calcolo infinitesimale nell’antichità. Milano: Giangiacomo Feltrinelli, 1961. First published in 1926 by Nicola Zanichelli Editore, Bologna.

3. Aldo Mieli. Enrico Rufini (1890-1924), con Elenco degli scritti di E. R. (con 1 ill.). Archeion 5:322-324, 1924.

4. 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.

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

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

7. Ora è noto come una tale teoria sia riassunta nella formula “le cose sono numeri”; la quale formula nel periodo piú antico, voleva significare che la materia fosse costituita di punti materiali o monadi (μονάδες, unità aventi posizione) e che le differenze qualitative dipendessero soltanto dal numero e dalla posizione di quei punti. Questa interpretazione (suggerita dal Tannery) si potrebbe ancora meglio precisare, ove si vogliano ricollegare (come fa l'Enriques) le vedute pitagoriche alle idee di Anassimandro. Questi avanzò l’ipotesi che il sostrato naturale delle cose fosse una sostanza indeterminata, infinitamente diffusibile (τὸ ἅπειρον); da questa ipotesi può derivare la struttura monadica della materia sensibile proposta da Pitagora, supponendo una condensazione della sostanza primitiva intorno a certi punti o centri monadici, i quali per conseguenza dovevano rimanere delimitati da un circostante vuoto, o, forse meglio, da un mezzo etereo rarefatto, della natura del fuoco. [Rufini, p.20]

8. Sulla teoria della materia fu modellata quella della geometria. Sostrato delle figure geometriche era una materia illimitatamente estesa e infinitamente divisibile, non diversa dalla sostanza cosmica di Anassimandro, con la quale si identificava il concetto di spazio. Le figure geometriche erano limitazioni o determinazioni di tale spazio, e quindi forme di una materia universale, che si ritrova identica nelle cose sensibili qualitativamente diverse. In tal modo la geometria, se pur non veniva trasportata d'un tratto nel regno dell’intelligibile, si sollevava però sopra la volgare concezione empirica e conferiva alle figure geometriche un certo grado di astrazione. [Rufini, pp.20-21]

9. Ma era questo soltanto un primo grado di astrazione, poiché il presupposto empirico rimaneva ancora alla base della concezione pitagorica. Si riteneva infatti che l’elemento primitivo delle figure geometriche fosse precisamente la monade; e cosí il punto geometrico appariva ancora come punto materiale, ad un tempo esteso e indivisibile. E come tutte le cose erano numeri, cioè somma di unità, cosí ogni figura geometrica era una somma di punti. [Rufini, p.21]

10. Ma l’ipotesi fondamentale del punto-monade doveva fatalmente urtare con la scoperta delle grandezze incommensurabili avvenuta in seno alla stessa scuola pitagorica, scoperta che minacciava di rovinare l'edificio scientifico con tanta cura costruito. [Rufini, p.21]

11. A risolvere la crisi che travagliava la geometria pitagorica fu diretta (secondo Tannery) l'opera della scuola d’Elea, opera che fu efficacemente iniziata da Parmenide (verso il 500 a.C., come ha mostrato l'Enriques). Infatti il poema parmenideo Sulla natura (περὶ φύσεως) acquista un significato particolarmente chiaro e suggestivo, ove esso si consideri non come un trattato puramente metafisico, ma in parte anche (secondo l’interpretazione rimessa appunto in valore dall'Enriques) come diretto a criticare la teoria monadica dei pitagorici, e quindi anche la loro concezione empirica degli enti fondamentali della geometria: punto, linea, superficie. [Rufini, p.22]

12. Che Parmenide si fosse occupato delle questioni inerenti ai principî della geometria lo si ricava da due citazioni di Proclo: una delle quali riporta una classificazione delle figure proposta dall’Eleate; l'altra attesta che la definizione euclidea “il punto è ciò che non ha parti” è conforme al criterio di Parmenide, secondo cui le definizioni negative convengono ai principi. Parmenide dunque avrebbe cercato di definire i primi concetti della geometria, e perciò non farà meraviglia se nei suoi scritti ritroviamo accenni alla critica da lui istituita. [Rufini, p.22]

13. A tal proposito meritano di essere qui specialmente ricordati due passi del poema parmenideo, già illustrati dall'Enriques:

il frammento 2, che contiene un accenno al concetto della superficie senza spessore, come quella “che non separa lo spazio dalla connessione dello spazio”;

il frammento 6, che è una forte requisitoria delle varie soluzioni proposte dai pitagorici circa il concetto di punto. Egli li rimprovera di essere “sordi insieme e ciechi, istupiditi, senza discernimento,” perché non vedono la contraddizione che consiste nel ritenere “che essere e non essere sia la stessa cosa e non la stessa cosa.” In altre parole, Parmenide condanna il concetto del punto-monade come qualche cosa di contraddittorio, di cui si afferma e si nega ad un tempo l'esistenza (o l'estensione), e che doveva a lui apparire come un bastardo infinitesimo attuale. [p.23]

14. Nell’opera di Parmenide si afferma dunque per la prima volta il concetto razionale del punto, della linea e della superficie; la sua critica tende in sostanza a stabilire che gli enti geometrici non possono definirsi che per astrazione, con un procedimento indefinito di idealizzazione, come limiti del sensibile. Ora questa affermazione costituisce il primo riconoscimento del carattere infinitesimale dei concetti fondamentali della geometria, e quindi può riguardarsi come il primo acquisto dell'analisi infinitesimale. [Rufini, p.23]