The mathematics of periodic anthyphairesis as a basis for the full understanding of Plato's philosophy

Abstract

Even though Plato's philosophy in ancient times was always closely associated with mathematics, modern Platonic scholarship, during the last five centuries, has moved steadily toward de-mathematization. The present work aims to outline a radical re-interpretation of Plato's philosophy, according to which the Platonic Idea, that is, the intelligible Being, has the structure of the philosophical analogue of a geometric dyad in a philosophic anthyphaeresis -- the precursor of modern continued fractions -- which was studied by the Pythagoreans, Theodorus and Theaetetus in relation with the discoveries of quadratic incommensurabilities. This mathematical structure is clearly visible in the Platonic method of Division and Collection, equivalently Name and Logos, equivalently True Opinion plus Logos, in the dialogues Theaetetus, Sophist, Statesman, Meno, and Parmenides. Equipped with this structure of an intelligible Being, we provide definitive answers to fundamental questions, that were not be resolved by Platonists, concerning the following topics: the dialectic numbers, which are based on the anthyphairetic periodicity and the plus one rule, stating that the dialectic number of terms of a sequence is the (number of) ratios of successive terms plus one (stated in the Parmenides 148d-149d); the description of the intelligible being as an Indivisible Line, a statement bordering on the contradictory; the also seemingly contradictory Sophist 's statement that ``the not-Being is a Being'', based on the equalization of the two elements of the dyad defining an intelligible Being; the more general self-similar Oneness of an intelligible Being, based on the equalization of all parts generated by the anthyphairetic division of an intelligible Being; and finally the Third Man Argument in the Introduction to the Parmenides, appearing as a threat for Plato's theory, but essentially innocuous because of the self-similar Oneness. The third part of our study aims to prove that, contrary to the presently dominant interpretation of Zeno's arguments and paradoxes as being devoid of mathematical content, the analysis of Zeno's presence in the Parmenides, Sophist (via the Eleatic Stranger), and Zeno's verbatim Fragments preserved by Simplicius, show that Plato's intelligible Beings essentially coincide with Zeno's true Beings, and hence that Zeno's philosophical thought was already anthyphairetic, and hence heavily influenced by the Pythagorean's Mathematics. These findings run against Burkert's claim that ``ontology is prior to mathematics''. Modern Platonists have never obtained a clear description of the structure of an intelligible Idea in terms of the mathematics of periodic anthyphairesis, and thus were not able to answer fundamental questions, nor to realize the close connection of Zeno's intelligible beings with Zeno's true Beings.