Epistemology versus ontology : essays on the philosophy and foundations of mathematics in honour of Per Martin-Löf
著者
書誌事項
Epistemology versus ontology : essays on the philosophy and foundations of mathematics in honour of Per Martin-Löf
（Logic, epistemology, and the unity of science / editors, Shahid Rahman, John Symons, v. 27）
Springer, c2012
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注記
Others editors: Sten Lindström, Erik Palmgren, Göran Sundholm
"The present anthology has its origin in an international conference that was arranged at the Swedish Collegium for Advanced Studies (SCAS) in Uppsala, May 5-8, 2009, 'Philosophy and Foundations of Mathematics: Epistemological and Ontological aspects'. The conference was dedicated to Per Martin-Löf on the occasion of his retirement"--p. vii
Includes bibliographical references and index
内容説明・目次
内容説明
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-Loef type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Loef. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Loef 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?
目次
- Introduction.- On the Philosophical Work of Per Martin-Loef: Goeran Sundholm.-Notes on the contributors.- Part 1. Philosophy of Logic and Mathematics.- Chapter 1. Kant and Real Numbers: Mark van Atten.- Chapter 2. Wittgenstein's Diagonal Argument: A Variation on Cantor and Turing: Juliet Floyd.- Chapter 3. Truth and Proof in Intuitionism: Dag Prawitz.- Chapter 4. Real and Ideal in Constructive Mathematics: Giovanni Sambin.- Chapter 5. In the Shadow of Incompleteness: Hilbert and Gentzen: Wilfried Sieg.- Chapter 6. Evolution and Logic: Jan Smith.- Chapter 7. The "Middle Wittgenstein" and Modern Mathematics: Soeren Stenlund.- Chapter 8. Primitive Recursive Arithmetic and Its Role in the Foundations of Arithmetic: historical and Philosophical Reflections: William Tait.- Part 2. Foundations.- Chapter 9. Type Theory and Homotopy: Steve Awodey.- Chapter 10. A Computational Interpretation of Forcing in Type Theory: Thierry Coquand
- Guilhem Jaber.- Chapter 11. Program Testing and the Meaning Explanations of Intuitionistic Type Theory: Peter Dybjer.- Chapter 12. Normativity in Logic: Jean-Yves Girard.- Chapter 13. Constructivist versus Structuralist Foundations: Erik Palmgren.- Chapter 14. Machine Translation and Type Theory: Aarne Ranta.- Chapter 15. Constructive Zermelo-Fraenkel Set Theory, Powerset, and the Calculus of Constructions: Michael Rathjen.- Chapter 16. Coalgebras as Types determined by their Elimination Rules: Anton Setzer.- Chapter 17. Second Order Logic, Set Theory and Foundations of Mathematics: Jouko Vaananen.
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