Lecture Titles and Abstracts

by Catarina Dutilh Novaes

The Dialogical Roots of Deduction

This lecture introduces the main ideas developed in my monograph The Dialogical Roots of Deduction (CUP, 2020) henceforth DRD. The main idea is that a deductive argument such as a mathematical proof can be understood as a dialogue between two (fictive) participants, Prover and Skeptic. Prover aims to produce explanatory persuasion, while Skeptic aims to ensure that the proof is valid and perspicuous. Thus seen, the ideal of explanatoriness for mathematical proofs is naturally understood as a dialogical desideratum. In the book, I show that the dialogical perspective allows for a unified account of deductive theories and practices, explaining with one stroke various puzzling and apparently disconnected features of the relevant phenomena.

Dialogical pragmatism

There is a sense in which DRD does not address head-on a fundamental philosophical question, namely that of the (transcendental) justification of deduction. What, if anything, can justify deduction? What the book offers instead is an analysis of deduction insofar as it is instantiated and embedded in actual human practices. This approach is motivated by pragmatist commitments, and yet, in DRD, the pragmatist grounding remains only implicit. This lecture rectifies this omission by articulating more explicitly the dialogical pragmatism at the core of my account. Dialogical pragmatism takes a clear stance on the justification question: there is nothing above and beyond the practices of deduction that is relevant for the question of the justification of deduction. To support this claim, I draw a comparison with pragmatist theories of truth, and discuss the recent literature on anti-exceptionalism about logic and what counts as evidence for logical theories.

Definitions and conceptual creativity in mathematics

Another topic that DRD does not address at length is the issue of definitions and conceptual change in mathematical practice, that is, the idea that mathematical proof may involve conceptual creativity: “the proof creates a new concept” (Wittgenstein, RFM III-41). Prover and Skeptic may in fact construe new concepts together in these dialogues, especially as various inferential relations are spelled out. The creative aspects of mathematical proof can be easily accommodated by the Prover-Skeptic model as long as the two characters continue to be sufficiently aligned in how they understand the key concepts at each stage. The fact that proofs create new concepts lends further support to the idea that there is nothing above and beyond the practices of deduction themselves that justifies deduction, as it is through and in these very practices that the conditions of applicability of (logical, mathematical) concepts are established. These observations in turn lead to a conception of definitions in mathematics as an essentially creative rather than descriptive endeavor.