This chapter investigates deductive practices in what is arguably their main current instantiation, namely practices of mathematical proofs. The dialogical hypothesis delivers a compelling account of a number of features of these practices; indeed, the fictive characters Prover and Skeptic can be viewed as embodied by real-life mathematicians. The chapter includes a discussion of the ontological status of proofs, the functions of proofs, practices of mathematicians such as peer review and collaboration, and a brief discussion of probabilistic and computational proofs. It also discusses three case studies: the reception of Gödel’s incompleteness results, a failed proof of the inconsistency of Peano Arithmetic, and a purported proof of the ABC conjecture.

This chapter critically discusses the prominent dialogical accounts of logic and deduction proposed by Lorenzen, Hintikka, and Lakatos. It is argued that, while they contain valuable insights, Lorenzen’s dialogical logic and Hintikka’s game-theoretical semantics ultimately both fail to provide a satisfactory philosophical account of logic and deduction in dialogical terms. This critical evaluation then leads to a precise formulation of the dialogical model defended in the book, the Prover–Skeptic model, which is by and large inspired by Lakatos’ ‘proofs and refutations’ model, but with some important modifications.

This chapter presents a dialogical rationale based on the Prover–Skeptic model for the three main features of deduction identified in Chapter 1: necessary truth-preservation, perspicuity, and belief-bracketing. Moreover, it addresses four important ongoing debates in the philosophy of logic: the normativity of logic, logical pluralism, logical paradoxes, and logical consequence. It is shown that the Prover–Skeptic model provides a promising vantage point to address the questions raised in these debates.