Intuitionistic Proof Versus Classical Truth: The Role of Brouwer’s Creative Subject in Intuitionistic Mathematics

Description

This book examines the role of acts of choice in classical and intuitionistic mathematics. Featuring fifteen papers – both new and previously published – it offers a fresh analysis of concepts developed by the mathematician and philosopher L.E.J. Brouwer, the founder of intuitionism. The author explores Brouwer’s idealization of the creative subject as the basis for intuitionistic truth, and in the process he also discusses an important, related question: to what extent does the intuitionistic perspective succeed in avoiding the classical realistic notion of truth? The papers detail realistic aspects in the idealization of the creative subject and investigate the hidden role of choice even in classical logic and mathematics, covering such topics as bar theorem, type theory, inductive evidence, Beth models, fallible models, and more. In addition, the author offers a critical analysis of the response of key mathematicians and philosophers to Brouwer’s work. These figures include Michael Dummett, Saul Kripke, Per Martin-Löf, and Arend Heyting. This book appeals to researchers and graduate students with an interest in philosophy of mathematics, linguistics, and mathematics. Front Matter ....Pages i-xiii Brouwer, Dummett and the Bar Theorem (Enrico Martino)....Pages 1-14 Creative Subject and Bar Theorem (Enrico Martino)....Pages 15-22 Natural Intuitionistic Semantics and Generalized Beth Semantics (Enrico Martino)....Pages 23-26 Connection Between the Principle of Inductive Evidence and the Bar Theorem (Enrico Martino)....Pages 27-30 On the Brouwerian Concept of Negative Continuity (Enrico Martino)....Pages 31-45 Classical and Intuitionistic Semantical Groundedness (Enrico Martino)....Pages 47-51 Brouwer’s Equivalence Between Virtual and Inextensible Order (Enrico Martino)....Pages 53-62 An Intuitionistic Notion of Hypothetical Truth for Which Strong Completeness Intuitionistically Holds (Enrico Martino)....Pages 63-73 Propositions and Judgements in Martin-Löf (Enrico Martino)....Pages 75-84 Negationless Intuitionism (Enrico Martino)....Pages 85-95 Temporal and Atemporal Truth in Intuitionistic Mathematics (Enrico Martino)....Pages 97-111 Arbitrary Reference in Mathematical Reasoning (Enrico Martino)....Pages 113-131 The Priority of Arithmetical Truth over Arithmetical Provability (Enrico Martino)....Pages 133-146 The Impredicativity of the Intuitionistic Meaning of Logical Constants (Enrico Martino)....Pages 147-156 The Intuitionistic Meaning of Logical Constants and Fallible Models (Enrico Martino)....Pages 157-163 Back Matter ....Pages 165-170