Logicism, Intuitionism, and Formalism: What Has Become of Them? (Synthese Library Book 341)

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. 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. Introduction: The Three Foundational Programmes -- Introduction: The Three Foundational Programmes -- Logicism And Neo-logicism -- Protocol Sentences For Lite Logicism -- Frege’s Context Principle And Reference To Natural Numbers -- The Measure Of Scottish Neo-logicism -- Natural Logicism Via The Logic Of Orderly Pairing -- Intuitionism And Constructive Mathematics -- A Constructive Version Of The Lusin Separation Theorem -- Dini’s Theorem In The Light Of Reverse Mathematics -- Journey Into Apartness Space -- Relativization Of Real Numbers To A Universe -- 100 Years Of Zermelo’s Axiom Of Choice: What Was The Problem With It? -- Intuitionism And The Anti-justification Of Bivalence -- From Intuitionistic To Point-free Topology: On The Foundation Of Homotopy Theory -- Program Extraction In Constructive Analysis -- Brouwer’s Approximate Fixed-point Theorem Is Equivalent To Brouwer’s Fan Theorem -- Formalism -- “gödel’s Modernism: On Set-theoretic Incompleteness,” Revisited -- Tarski’s Practice And Philosophy: Between Formalism And Pragmatism -- The Constructive Hilbert Program And The Limits Of Martin-löf Type Theory -- Categories, Structures, And The Frege-hilbert Controversy: The Status Of Meta-mathematics -- Beyond Hilbert’s Reach? -- Hilbert And The Problem Of Clarifying The Infinite. Edited By Sten Lindstrom ... [et Al.]. Includes Bibliographical References And Index. The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gdel's ber formal unentscheidbare Stze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational 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. A special section is concerned with constructive mathematics and its foundations. This active branch of mathematics is a direct legacy of Brouwer's intutionism. 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. 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 "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." -- Book Jacket Aims to review the programmes in the foundations of mathematics from the classical period and to assess their possible relevance for contemporary philosophy of mathematics. This work is suitable for researchers and graduate students of philosophy, logic, mathematics and theoretical computer science This anthology reviews the programmes in the foundations of mathematics from the classical period and assesses their possible relevance for contemporary philosophy of mathematics. A special section is concerned with constructive mathematics.