Visual Proofs as Counterexamples to the Standard View of Informal Mathematical Proofs?

Abstract

A passage from Jody Azzouni’s article “The Algorithmic-Device View of Informal Rigorous Mathematical Proof” in which he argues against Hamami and Avigad’s standard view of informal mathematical proof with the help of a specific visual proof of 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ⋯ = 1 is critically examined. By reference to mathematicians’ judgments about visual proofs in general, it is argued that Azzouni’s critique of Hamami and Avigad’s account is not valid. Nevertheless, by identifying a necessary condition for the visual proof to be considered a proper proof in the first place, and suggesting an appropriate way to establish its correctness, it is shown how Azzouni’s assessment of the epistemic process associated with the visual proof can turn out to be essentially correct. From this, it is concluded that although visual proofs do not constitute counterexamples to the standard view in the sense suggested by Azzouni, at least the visual proof mentioned above shows that this view does not cover all the ways in which mathematical truth can be justified.