Diagrams as joint epistemic actions: A dialogical account of diagrams in mathematical proofs

Abstract

The book The Dialogical Roots of Deduction presented a detailed dialogical account of deduction, in particular of mathematical proof. The key idea is that a mathematical proof corresponds to a dialogue between two (fictive) participants, Prover and Skeptic, where Prover attempts to establish that some conclusion follows from certain premises by producing explanatory persuasion in Skeptic. While covering many aspects of mathematical proof, the book did not discuss diagrams, despite their ubiquity in mathematical practice. In this paper, I remedy this important lacuna in the original presentation of the dialogical account. I argue that diagrams play a fundamental epistemic role in eliciting active engagement from Skeptic to understand the argument put forward by Prover. To this end, Prover relies on imperatives to invite Skeptic to construct diagrams. Thus understood, the role of diagrams in mathematical proofs is primarily operative rather than semantic/representational, eliciting ‘hands-on’ engagement. In particular, I argue that diagrams in mathematical proofs are best understood as joint epistemic actions, thus highlighting their role in the production and transmission of (mathematical) knowledge and understanding. I close with some observations on the implications of this account of diagrams for mathematics education.