Epistemology Versus Ontology : Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf

This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice (ZFC). This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics? Front Matter....Pages i-xxvii Front Matter....Pages 1-1 Kant and Real Numbers....Pages 3-23 Wittgenstein’s Diagonal Argument: A Variation on Cantor and Turing....Pages 25-44 Truth and Proof in Intuitionism....Pages 45-67 Real and Ideal in Constructive Mathematics....Pages 69-85 In the Shadow of Incompleteness: Hilbert and Gentzen....Pages 87-127 Evolution and Logic....Pages 129-138 The “Middle Wittgenstein” and Modern Mathematics....Pages 139-159 Primitive Recursive Arithmetic and Its Role in the Foundations of Arithmetic: Historical and Philosophical Reflections....Pages 161-180 Front Matter....Pages 181-181 Type Theory and Homotopy....Pages 183-201 A Computational Interpretation of Forcing in Type Theory....Pages 203-213 Program Testing and the Meaning Explanations of Intuitionistic Type Theory....Pages 215-241 Normativity in Logic....Pages 243-263 Constructivist Versus Structuralist Foundations....Pages 265-279 Machine Translation and Type Theory....Pages 281-311 Constructive Zermelo-Fraenkel Set Theory, Power Set, and the Calculus of Constructions....Pages 313-349 Coalgebras as Types Determined by Their Elimination Rules....Pages 351-369 Second Order Logic, Set Theory and Foundations of Mathematics....Pages 371-380 Back Matter....Pages 381-385 This book brings together philosophers, mathematicians and logicians to explore important problems in the philosophy and foundations of mathematics. Contrasts classical logic and set theory with predicativistic constructivism based on Martin-Lof type theory."