Epistemology versus Ontology

1. Front Matter

   Pages i-xxvii

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2. Philosophy of Logic and Mathematics

   1. Front Matter

      Pages 1-1

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3. PHILOSOPHY OF LOGIC AND MATHEMATICS

   1. Kant and Real Numbers

      • Mark van Atten

      Pages 3-23

   2. Wittgenstein’s Diagonal Argument: A Variation on Cantor and Turing

      • Juliet Floyd

      Pages 25-44

   3. Truth and Proof in Intuitionism

      • Dag Prawitz

      Pages 45-67

   4. Real and Ideal in Constructive Mathematics

      • Giovanni Sambin

      Pages 69-85

   5. In the Shadow of Incompleteness: Hilbert and Gentzen

      • Wilfried Sieg

      Pages 87-127

   6. Evolution and Logic

      • Jan M. Smith

      Pages 129-138

   7. The “Middle Wittgenstein” and Modern Mathematics

      • Sören Stenlund

      Pages 139-159

   8. Primitive Recursive Arithmetic and Its Role in the Foundations of Arithmetic: Historical and Philosophical Reflections

      • William W. Tait

      Pages 161-180

4. Foundations

   1. Front Matter

      Pages 181-181

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5. FOUNDATIONS

   1. Type Theory and Homotopy

      • Steve Awodey

      Pages 183-201

   2. A Computational Interpretation of Forcing in Type Theory

      • Thierry Coquand, Guilhem Jaber

      Pages 203-213

   3. Program Testing and the Meaning Explanations of Intuitionistic Type Theory

      • Peter Dybjer

      Pages 215-241

   4. Normativity in Logic

      • Jean-Yves Girard

      Pages 243-263

   5. Constructivist Versus Structuralist Foundations

      • Erik Palmgren

      Pages 265-279

   6. Machine Translation and Type Theory

      • Aarne Ranta

      Pages 281-311

   7. Constructive Zermelo-Fraenkel Set Theory, Power Set, and the Calculus of Constructions

      • Michael Rathjen

      Pages 313-349

   8. Coalgebras as Types Determined by Their Elimination Rules

      • Anton Setzer

      Pages 351-369

   9. Second Order Logic, Set Theory and Foundations of Mathematics

      • Jouko Väänänen

      Pages 371-380

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?

• Constructive Type Theory
• Dummett
• Gentzen
• Intuitionism
• Martin-Löf
• Martin-Löf's type theory
• Mathematical structuralism
• Meaning-as-use
• Prawitz
• Proof Theory
• Wittgenstein
• axiom of choice (ZFC)
• category theoretic foundations
• predicativistic contructivism
• set theory

• Goteborg, Sweden

  P. Dybjer

• Dept. Philosophy & Linguistics, University of Umea, Umea, Sweden

  Sten Lindström

• Dept. Mathematics, Uppsala University, Uppsala, Sweden

  Erik Palmgren

• Leiden, Netherlands

  G. Sundholm

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