The Reconstruction of Theaetetus' Theory of Ratios of Magnitudes

TL;DR

This paper reconstructs Theaetetus' theory of ratios of magnitudes using anthyphairesis, revealing a restricted application to certain pairs of magnitudes and avoiding Eudoxus' definition, based on mathematical and philosophical analysis.

Contribution

It offers a novel anthyphairetic interpretation of Theaetetus' ratios, focusing on finite or periodic cases and bypassing Eudoxus' definition, differing from previous reconstructions.

Findings

Reconstruction applies only to pairs with finite or periodic anthyphairesis.

Avoids use of Eudoxus' definition in the theory.

Links mathematical ratios to Plato's philosophical concepts.

Abstract

In the present chapter, we obtain the reconstruction of Theaetetus' theory of ratios of magnitudes based, according to Aristotle's Topics 158b, on the definition of proportion in terms of equal anthyphairesis. Our reconstruction is built on the anthyphairetic interpretation of the notoriously difficult Theaetetus 147d6-e1 passage on Theaetetus' mathematical discovery of quadratic incommensurabilities, itself based on the traces it has left on Plato's philosophical definition of Knowledge in his dialogues Theaetetus, Sophist and Meno. Contrary to earlier reconstructions by Becker, van der Waerden and Knorr, our reconstruction reveals a theory that (a) applies only to the restricted class of pairs of magnitudes whose anthyphairesis is finite or eventually periodic, and (b) avoids the problematic use of Eudoxus' definition 4 of Book V of Euclid's Elements. The final version of this paper will appear as a chapter in the book Essays on Topology: Dedicated to Valentin Po\'enaru, ed. L. Funar and A. Papadopoulos, Springer, 2025.