- Preface.- Notes on the Contributors.- Introduction
- Sten Lindstrom, Erik Palmgren.- I. LOGICISM AND NEO-LOGICISM.- Protocol Sentences for Lite Logicism
- John Burgess.- Frege's Context Principle and Reference to Natural Numbers
- Oystein Linnebo.- The Measure of Scottish Neo-Logicism
- Stewart Shapiro.- Natural Logicism via the Logic of Orderly Pairing
- Neil Tennant.- II. INTUITIONISM AND CONSTRUCTIVE MATHEMATICS.- A Constructive Version of the Lusin Separation Theorem
- Peter Aczel.- Dini's Theorem in the Light of Reverse Mathematics
- Josef Berger, Peter Schuster.- Journey in Apartness Space
- Douglas Bridges, Luminita Vita.- Relativisation of Real Numbers to a Universe
- Hajime Ishihara.- 100 years of Zermelo's Axiom of Choice: What Was the Problem With It?
- Per Martin-Lof.- Intuitionism and the Anti-Justification of Bivalence
- Peter Pagin.- From Intuitionistic to Point-Free Topology
- Erik Palmgren.- Program Extraction in Constructive Mathematics
- Helmut Schwichtenberg.- Brouwer's Approximate Fixed-Point Theorem is Equivalent to Brouwer's Fan Theorem
- Wim Veldman.- III. FORMALISM.- "Godel's Modernism: On Set-Theoretic Incompleteness, " Revisited
- Mark van Atten, Juliette Kennedy.- Tarski's Practice and Philosophy: Between Formalism and Pragmatism
- Hourya Benis Sinaceur.- The Constructive Hilbert-Program and the Limits of Martin-Lof Type Theory
- Michael Rathjen.- Categories, Structures, and the Frege-Hilbert Controversy: the Status of Meta-Mathematics
- Stewart Shapiro.- Beyond Hilbert's Reach?
- Wilfried Sieg.- Hilbert and the Problem of Clarifying the Infinite
- Soren Stenlund.- Index.
- (source: Nielsen Book Data)
The period in the foundations of mathematics that started in 1879 with the publication of Frege's "Begriffsschrift" and ended in 1931 with Godel's "A ber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme I" can reasonably be called the classical period. It saw the development of three major foundational programs: 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 programs 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 programs of the classical period and the disputes that took place between them? To what extent do the classical programs 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. A special section is concerned with constructive mathematics and its foundations. This active branch of mathematics is a direct legacy of Brouwer's intuitionism. Today one often views it more abstractly as mathematics based on intuitionistic logic. It can then be regarded as a generalisation of classical mathematics in that it may be given, firstly, the standard set-theoretic interpretation, secondly, algorithmic meaning, and thirdly, nonstandard interpretations in terms of variable sets (sheaves over topological spaces). 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.
(source: Nielsen Book Data)