The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. In this period, there were also lively exchanges between the various schools culminating in the famous Hilbert-Brouwer controversy in the 1920s.
The purpose of this anthology is to review the programmes in the foundations of mathematics from the classical period and to assess their possible relevance for contemporary philosophy of mathematics. What can we say, in retrospect, about the various foundational programmes of the classical period and the disputes that took place between them? To what extent do the classical programmes of logicism, intuitionism and formalism represent options that are still alive today? These questions are addressed in this volume by leading mathematical logicians and philosophers of mathematics.
The volume will be of interest primarily to researchers and graduate students of philosophy, logic, mathematics and theoretical computer science. The material will be accessible to specialists in these areas and to advanced graduate students in the respective fields.
Author(s): Sten Lindström, Erik Palmgren (auth.), Sten Lindström, Erik Palmgren, Krister Segerberg, Viggo Stoltenberg-Hansen (eds.)
Series: Synthese Library 341
Edition: 1
Publisher: Springer Netherlands
Year: 2009
Language: English
Pages: 512
Tags: Mathematical Logic and Foundations; Logic; Philosophy of Language; Epistemology; Ontology; History of Mathematics
Front Matter....Pages I-XII
Front Matter....Pages 1-1
Introduction: The Three Foundational Programmes....Pages 1-23
Front Matter....Pages 25-25
Protocol Sentences for Lite Logicism....Pages 27-46
Frege’s Context Principle and Reference to Natural Numbers....Pages 47-68
The Measure of Scottish Neo-Logicism....Pages 69-90
Natural Logicism via the Logic of Orderly Pairing....Pages 91-125
Front Matter....Pages 127-127
A Constructive Version of the Lusin Separation Theorem....Pages 129-151
Dini’s Theorem in the Light of Reverse Mathematics....Pages 153-166
Journey into Apartness Space....Pages 167-187
Relativization of Real Numbers to a Universe....Pages 189-207
100 Years of Zermelo’s Axiom of Choice: What was the Problem with It?....Pages 209-219
Intuitionism and the Anti-Justification of Bivalence....Pages 221-236
From Intuitionistic to Point-Free Topology: On the Foundation of Homotopy Theory....Pages 237-253
Program Extraction in Constructive Analysis....Pages 255-275
Brouwer’s Approximate Fixed-Point Theorem is Equivalent to Brouwer’s Fan Theorem....Pages 277-299
Front Matter....Pages 301-301
“Gödel’s Modernism: On Set-Theoretic Incompleteness,” Revisited....Pages 303-355
Tarski’s Practice and Philosophy: Between Formalism and Pragmatism....Pages 357-396
The Constructive Hilbert Program and the Limits of Martin-Löf Type Theory....Pages 397-433
Categories, Structures, and the Frege-Hilbert Controversy: The Status of Meta-mathematics....Pages 435-448
Beyond Hilbert’s Reach?....Pages 449-483
Hilbert and the Problem of Clarifying the Infinite....Pages 485-503
Back Matter....Pages 505-512
