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“As Philolaos the Pythagorean Said”
Philosophy, Geometry, Freedom
Published online by Cambridge University Press: 28 February 2024
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In his collection of anecdotes, Lives, Opinions, and Remarkable Sayings of the Most Famous Ancient Philosophers, Diogenes Laertius devotes a chapter to the life of Zeno of Elea. Zeno's reputation is based on his celebrated paradoxes, amply discussed by Aristotle: a moving body will never reach its (pre-defined) telos, since it first has to cover half (or more than half) the remaining distance; the faster will never catch up with the slower, since it first has to get to the point from which the slower has just left. Zeno's style is laconic, like that of an Aesop fable. Maxima e minimis: there is no superfluous word. Everything needed to arrive at the conclusion is explicitly stated.
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- Copyright © 1998 Fédération Internationale des Sociétés de Philosophie / International Federation of Philosophical Societies (FISP)
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* The author has translated and paraphrased (in italics) quotations from the Greek himself (N.d.I.R.)
1. Certainly a provocative personality, we must recognize (if we can trust Plato's portrait of him in the first part of Parmenides) an intellectual dandyish quality in the handsome and elegant Zeno, who, at forty years old, is the lover of Par menides, with whom he travels with to Athens. Parmenides, a handsome man himself, imposing in his noble presence, white beard and hair, is just over sixty-five.
2. In his Academicorum historia, Phiolodemus speaks of Plato as the architect of the metrologia - a very successful terminological choice thanks to its semantic proximity to the current technical expression theory of measure.
3. Alfred North Whitehead, the great mathematician and philosopher, speaks of mathematics as “divine madness” - certainly a metastasis of Epinomis and Phaedrus.
4. At least on this point, Aristotle agrees with Plato. In his Nicomedean Ethics he comes back to the idea that mathematical pleasures are not mixed with pain and in his Topics he refers to the act of contemplation of the incommensurability of the diagonal of the square, as example of a pleasure that does not know the opposition of pain, exactly like the act of carnal love, whereas spiritual love is linked to its opposite, hate. And, in his Metaphysics, he expresses himself even more categorically, when he writes that those who state that mathematical sciences have nothing to say about Beauty and Goodness are certainly wrong.
5. Such a narrow link between geometry and political thought may very well give the impression of being far-fetched and naive, but it has experienced a true renaissance centuries later. Under the somewhat strange title: “Géomètre,” d'Alembert wrote an article in l'Encyclopédie, which he, along with Diderot, published. Completely unexpectedly, we read in it the following text, which has the tone of a political manifesto: “Geometry is perhaps the only way to stir up, little by little, in certain countries in Europe, the yoke of oppression and ignorance under which they suffer. Give birth, if possible, to geometricians among these peoples. Soon the study of geometry will lead to true Phi losophy, which by the generalized and immediate light it will shed will soon be stronger than all forces of superstition.”
In a geometry textbook published in 1817 in Erlangen by the German math ematician, Georg Simon Ohm, we find at the beginning of the phalanx of severely structured theorems - the following text, certainly quite unusual in the work of a specialist: “Geometry, only geometry is capable of instilling men with the spirit of independence, it alone can preserve the biases of a spiritual despotism.” Actually, the masterly and unique work that is Greek geometry is the supreme product of the first society founded on the principles of demo cracy. Without Greek democracy - no Elements by Euclid. “There is no royal road in geometry,” Euclid of Alexandria is purported to have answered Ptolemy, King of Egypt. Yes, in the world of geometry, the young slave of Menon and the King of Egypt must obey the same universal laws of reason.
6. One of the greatest mathematicians of the nineteenth century, Leopold Kro necker, never accepted the existence of irrational numbers, and still today there are eminent mathematicians who are openly reserved and even averse to the idea, only accepting a limited class of irrational numbers whose exis tence may be founded on certain constructions.
7. Philosophy knows of and admits of no direct application, in mathematics or elsewhere. No theorem may be demonstrated by means of the philosophemes. Behind no theory or mathematical conception can one identify the teaching of a precise philosopher.
The channels through which philosophical thought nonetheless has a deci sive influence on the body of mathematical thought are topologically com plex; the network of capillaries through which the currents of philosophical thought irrigate the universe of mathematical knowledge is delicately struc tured, and next to nothing is known about the fine details.
The sole existing visible indications are the aphoristic declarations of a large number of mathematicians, as well as of some authors, not mathemati cians, who understood exactly this interpenetration and interdependence: first Plotin, Nicolas de Cuse, Marsilio Ficino and two important poets-Novalis, and above all, the amazing Edgar Allan Poe.
8. I will cite but two. In the Posterior Analytics there is the theorem called “ellipti cal geometry” - valid only in one of the two large branches of non-Euclidean geometry: if the sum of the angles of a triangle is greater than two right angles, then the parallels will meet, that is: in this non-Euclidean surface there are no straight lines which are not incident, all the straight lines meet. In On The Heavens, we find the non-Euclidean theorem, valid in both types of non-Euclidean geometry, elliptical and hyperbolic: if it is impossible for the sum of the angles of a triangle to be equal to two right angles - that is if it is impossible for a triangle to be Euclidean - then the diagonal is commensurable to the side of the square. This theorem is notable not only for its fundamental character, the richness and sophistication of its specific geometric content, but perhaps above all, for the way its hypothesis is formulated: impossible - for a triangle- to be Euclidean! The amazing character of this expression is truly astonishing because the modal predicate impossible is never attributed by Aristotle to a non-Euclidean triangle.
9. For readers interested in details about these non-Euclidean fragments which Aristotle preserved for us, allow me to recommend my book, Aristotele e I fon damenti assiomatici della geometria. Prolegomeni alla comprensione dei frammenti non-euclidei del “Corpus Aristotelicum,” Milan, Vita e Pensiero 1998.
10. In an erudite work, La comunità matematica Italiana di fronte alle leggi razziali, published in Cosenza in 1991, Pietro Nastasi provided a summary both detailed and brilliant in its historical and political sensitivity toward the events that preceded and followed the decision of the Academy.
11. “Diese ungetheilte Substanz der absoluten Freiheit erhebt sich auf den Thron der Welt, ohne dass irgend eine Macht ihr Widerstand zu leisten vermögte”- Berlin, Ed. 1842, p. 428.
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