© 2009

Logicism, Intuitionism, and Formalism

What has Become of Them?

  • Sten Lindström
  • Erik Palmgren
  • Krister Segerberg
  • Viggo Stoltenberg-Hansen
  • Contains essays by world-leading experts in the philosophy and foundations of mathematics, describing current developments in the foundations of mathematics in a historical perspective

  • Analyses the classical philosophical and foundational views of Frege, Brouwer, Hilbert, Gödel and Tarski and examines their relevance for current developments

  • Provides an in-depth analysis of various kinds of neologicist philosophies of mathematics

  • Contains a comprehensive section on mathematical intuitionism and constructive mathematics

  • Offers extensive discussions, by several authors, of the proof-theoretic programme of Hilbert and Bernays


Part of the Synthese Library book series (SYLI, volume 341)

Table of contents

  1. Front Matter
    Pages I-XII
  2. Introduction: The Three Foundational Programmes

    1. Front Matter
      Pages 1-1
    2. Sten Lindström, Erik Palmgren
      Pages 1-23
  3. Logicism and Neo-Logicism

    1. Front Matter
      Pages 25-25
    2. John P. Burgess
      Pages 27-46
    3. Stewart Shapiro
      Pages 69-90
  4. Intuitionism and Constructive Mathematics

  5. Formalism

    1. Front Matter
      Pages 301-301
    2. Mark van Atten, Juliette Kennedy
      Pages 303-355

About this book


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.


Bertrand Russell Formalism Foundations of mathematics Gottlob Frege Intuitionism Logicism Philosophy of mathematics Topology Variable calculus mathematics proof theorem type theory

Editors and affiliations

  • Sten Lindström
    • 1
  • Erik Palmgren
    • 2
  • Krister Segerberg
    • 3
  • Viggo Stoltenberg-Hansen
    • 2
  1. 1.Dept. Historical, Philosophical and Religious StudiesUmeå UniversitySweden
  2. 2.Department of MathematicsUppsala University751 06 UppsalaSweden
  3. 3.Department of PhilosophyUppsala University751 26 UppsalaSweden

About the editors

Sten Lindström is Professor of Philosophy at Umeå University and has been a Research Fellow at the Swedish Collegium for Advanced Study (SCAS). He has published papers on intensional logic, belief revision and philosophy of language, and co-edited the books Logic, Action and Cognition: Essays in Philosophical Logic (Kluwer, 1997) and Collected Papers of Stig Kanger with Essays on his Life and Work, I-II (Kluwer, 2001).

Erik Palmgren is Professor of Mathematics at Uppsala University. His research interests are mainly mathematical logic and the foundations of mathematics. He is presently working on the foundational programme of replacing impredicative constructions by inductive constructions in mathematics, with special emphasis on point-free topology and topos theory.

Krister Segerberg is Emeritus Professor of Philosophy at Uppsala University and the University of Auckland. He is the author of papers in modal logic, the logic of action, belief revision and deontic logic, as well as the books An Essay in Classical Modal Logic (1971) and Classical Propositional Operators: An Exercise in the Foundations of Logic (1982).

Viggo Stoltenberg-Hansen is professor of Mathematical Logic at Uppsala University. His main interests include computability and constructivity in mathematics.

Bibliographic information

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