Conférence dans le cadre du colloque « The Central Role of Pythagorean Music in pre-Euclidean Greek Mathematics and Philosophy » (25 mars 2022) à la Misha (Université de Strasbourg).

Résumé de la conférence :

There are three distinct periods and stages of Pythagorean Music.

--> In the early era (Sections 1-8), the success of the Pythagoreans in the arithmetization of pre-existing empirical music leads them to a primitive theory of Numbers, evolving, with the great discovery of the Euclidean algorithm, into the first half of Book VII of Euclid’s Elements. We reconstruct this process on the basis of

(a) the striking peculiarities of the Arithmetical Books in Euclid’s Elements,

(b) Aristotle’s Topics principle on the dynamic interaction between Postulates and Definitions, and

(c) the accounts by Nicomachus and Iamblichus of the Pythagorean arithmetization (including the genuine experiments by Hippasus) of earlier, probably Babylonian, empirical music.

--> In the middle period (Sections 9-12), Pythagorean music is dominated by Pythagorean Geometry. The great Pythagorean discovery of incommensurability by means of (infinite) periodic anthyphairesis, the natural transfer of Euclidean algorithm to Geometry, has its musical counterpart in the corresponding infinite, but not periodic, musical anthyphairesis of the dyad under composition, described by Philolaus in Fragment 6, but perhaps going back to Hippasus, who is probably the discoverer of the geometric incommensurability. The musical result shown by the infinity of the musical anthyphairesis, corresponding to the geometric incommensurability, is the impossibility of simultaneous equal temperament for the octave and the fifth. The construction of the octachord with two units, rather than a single one, namely the tone and the diesis, and the division of the octave in 12 semitones, but never quite equal to each other, are seen as compromises made necessary by the musical “incommensurability”. Pythagorean music influenced Plato’s philosophy, first his description of the Harmony of the Spheres in the Republic 616-617, in which the planetary system is seen as a cosmic octachord and secondly the construction of the Soul of the Cosmos in the Timaeus 34-36, where a rather artificial periodicity on 33 musical intervals is imposed. The Soul of Cosmos plays a crucial role in Plato’s final explanation, by means of the Receptacle and the four of the five canonical solids, of the way in which the sensibles participate in the intelligibles. A curious modification of Plato’s construction, by pseudo-Timaeus of Locris, increased the musical intervals to 35, in an attempt to produce a palindromic periodicity, perhaps imitating Theaetetus discovery of the palindromic periodicity of quadratic incommensurabilities.

--> In the final era (Sections 13-14) Archytas realized that stronger results on the impossibility of equal temperament and more general proofs of geometric incommensurabilities could be obtained by abandoning the anthyphairetic methods of the middle period, dominated by Plato, with progress in the establishment of the Fundamental Theorem of Arithmetic in the second half of Book VII, its application in Book VIII of the Elements less arithmetical, less deep but more general than the anthyphairetic, proofs if incommensurabilities, and the corresponding proofs on the impossibility of (rational) equal temperament of the Pythagorean musical intervals in the (pseudo-) Euclidean Sectio Canonis. Archytas’ approach made necessary the development of a more general, necessarily non-anthyphairetic, theory of ratios of magnitudes in Book V of the Elements, achieved by his student the great mathematician Eudoxus, a theory identical with the modern foundation of the real numbers in terms of Dedekind cuts that has shaped modern Analysis.

Biographie :

Stylianios Negrepontis (grec Στυλιανός Νεγρεπόντης, né le 22 février 1939 à Thessalonique) a étudié la physique à l'Université de Rochester (bachelor en 1961) et ensuite les mathématiques (master 1963). Il soutient une thèse de doctorat en 1965 avec W. Wistar Comfort (A Homology Theory of Real Compact Spaces). A partir de 1965, il fut professeur adjoint à l'Université de l'Indiana à Bloomington et à partir de 1966, professeur à l'Université McGill à Montréal, où il est resté jusqu'en 1976. A partir de 1973, il fut aussi professeur à l'Université d’Athènes. De 1983 à 1989 et de 1992 à 1994, il a été directeur de la faculté de mathématiques et de 1986 à 1988 vice-recteur de l’université. Il a pris sa retraite en 2006, et depuis, il est professeur émérite à l'Université d’Athènes. Il est spécialiste de topologie, d’espaces de Banach, et de philosophie des mathématiques, spécialement Platon et les Pythagoriciens.

Soutien : CREAA