ISBN:
9789400744356
Language:
English
Pages:
Online-Ressource (XXVII, 385 p. 18 illus, digital)
Series Statement:
Logic, Epistemology, and the Unity of Science 27
Series Statement:
SpringerLink
Series Statement:
Bücher
Parallel Title:
Buchausg. u.d.T.
Keywords:
Philosophy (General)
;
Genetic epistemology
;
Logic
;
Ontology
;
Logic, Symbolic and mathematical
;
Philosophy
;
Philosophy (General)
;
Genetic epistemology
;
Logic
;
Ontology
;
Logic, Symbolic and mathematical
Abstract:
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?
Abstract:
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?
Description / Table of Contents:
Epistemology versus Ontology; Contents; Introduction; 1 Background; 2 Martin-Löf: Pioneer and Land Clearer; 3 Contributions to This Volume; 3.1 Part I: Philosophy of Logic and Mathematics; 3.2 Part II: Foundations; Acknowledgments; On the Philosophical Work of Per Martin-Löf; Notes on the Contributors; Part I Philosophy of Logic and Mathematics; Chapter 1: Kant and Real Numbers; 1.1 Introduction; 1.2 Mathematics Within Subjective Limits; 1.3 Kant's Discussion with Rehberg; 1.4 Infinite Sequences as Concepts and as Objects; 1.5 Concluding Remark; References
Description / Table of Contents:
Chapter 2: Wittgenstein's Diagonal Argument: A Variationon Cantor and Turing2.1 Introduction; 2.2 Three Diagonal Arguments; 2.2.1 The Halting Problem; 2.2.2 Turing's First Argument; 2.2.3 The Argument from the Pointerless Machine; 2.3 Wittgenstein's Diagonal Argument; 2.4 The Positive Russell Paradox; 2.5 Interpreting Wittgenstein; References; Chapter 3: Truth and Proof in Intuitionism; 3.1 Early Intuitionistic Accounts of Propositions, Assertions, and Proof; 3.1.1 Heyting on Propositions and Assertions; 3.1.2 Heyting on Proofs; 3.1.3 The BHK-Interpretation; 3.2 Dummett's Verificationism
Description / Table of Contents:
3.2.1 A Correction of the Intuitionistic Meaning-Theory3.2.2 Truth in Verificationism and the Knowability Principle; 3.3 Martin-Löf's Type Theory; 3.4 Martin-Löf's Siena Lectures and a Subsequent Paper; 3.5 The Epistemic Approach to Meaning and Truth Being Abandoned; 3.6 Reasons for the Shift; 3.6.1 Is the Ontological Standpoint Compatible with Intuitionism?; 3.6.2 Reasons for Rejecting the Knowability Principle; 3.6.3 Are the Proofs of the BHK-Interpretation Representations of Proof Acts?; 3.6.4 An Alternative Argument for the Epistemic Nature of Proof-Objects; References
Description / Table of Contents:
Chapter 4: Real and Ideal in Constructive Mathematics4.1 Explanations from Above and Explanations from Below; 4.2 The Dynamic Process in Logic and in Foundations; 4.3 Real and Ideal Notions in Constructive Topology; References; Chapter 5: In the Shadow of Incompleteness: Hilbert and Gentzen; 5.1 A Puzzle; 5.2 Results, Methods, and Problems; 5.3 Unprovability in General, First; 5.4 Unprovability of Consistency, Second; 5.5 Hilbert's Response; 5.6 The New Student; 5.7 An Impasse; 5.8 Toward a Solution; 5.9 New Perspectives; References; Chapter 6: Evolution and Logic
Description / Table of Contents:
6.1 Hume's Analysis of Causality6.2 Evolutionistic Understanding of Causality; 6.3 Evolutionistic Understanding of Logic; 6.4 Foundations of Mathematics; 6.5 Martin-Löf Type Theory and the Synthetic A Priori; 6.6 Ontology; 6.7 Concluding Remarks; References; Chapter 7: The ``Middle Wittgenstein'' and Modern Mathematics; 7.1 Introduction; 7.2 Grammar and Geometry; 7.3 Grammar and the Axiomatic Method; 7.4 Grammar and the Theory of Relativity; 7.5 Mental Verbs and the Method of Ideal Elements; 7.6 Wittgenstein and Hertz; References
Description / Table of Contents:
Chapter 8: Primitive Recursive Arithmetic and Its Role in the Foundations of Arithmetic: Historical and Philosophical Reflections: In Honor of Per Martin-Löf on the Occasion of His Retirement
Note:
Description based upon print version of record
DOI:
10.1007/978-94-007-4435-6
URL:
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