On Theodorus' lesson in the Theaetetus 147d-e

TL;DR

This paper examines Plato's account of Theodorus' lesson on incommensurability, arguing that linguistic evidence supports an anthyphairetic method in Theodorus' proofs, contrary to previous interpretations.

Contribution

It provides a new interpretation of the evidence for Theodorus' proof method, emphasizing the significance of the plural 'hai dunameis' and its implications for understanding ancient mathematical techniques.

Findings

Linguistic analysis supports an anthyphairetic method.

Distributive rendering of 'hai dunameis' implies infinite powers.

Contradicts previous neutral or set-theoretic interpretations.

Abstract

In the celebrated Theaetetus 147d3-e1 passage Theodorus is giving a lesson to Theaetetus and his companion, proving certain quadratic incommensurabilities. The dominant view on this passage, expressed independently by H. Cherniss and M. Burnyeat, is that Plato has no interest to, and does not, inform the reader about Theodorus' method of incommensurability proofs employed in his lesson, and that, therefore, there is no way to decide from the Platonic text, our sole information on the lesson, whether the method is anthyphairetic as Zeuthen, Becker, van der Waerden suggest, or not anthyphairetic as Hardy-Wright, Knorr maintain. The main part of the paper is devoted to a discussion of the evidence for a distributive rendering of the plural 'hai dunameis' in the crucial statement, 'because the powers ('hai dunameis') were shown to be infinite in multitude' in 147d7-8 (contrary to the Burnyeat collective, set-theoretic rendering), necessarily resulting in an anthyphairetic method for Theodorus' proofs of incommensurability (contrary to the Cherniss-Burnyeat neutrality thesis).