© 2007

The Arché Papers on the Mathematics of Abstraction

  • Editors
  • Roy T. Cook


  • Comprises a survey of the philosophical and technical work being done at Arché: The AHRB Centre for the Philosophy of Logic, Language, Mathematics, and Mind at the University of St Andrews

  • Provides an overview of the Neo-Fregean position in philosophy of mathematics

  • Collects together hard to find material on Neo-Fregean set theory and real analysis

  • Contains papers by distinguished philosophers of mathematics, including George Boolos, William Demopoulos, Bob Hale, Richard Heck, Crispin Wright, Stewart Shapiro, and rising stars such as Roy T. Cook, Agustin Rayo, and Gabriel Uzquiano

  • Contains a preface by Crispin Wright


Part of the The Western Ontario Series in Philosophy of Science book series (WONS, volume 71)

Table of contents

  1. Front Matter
    Pages i-xxxvii
  2. The Philosophy and Mathematics of Hume’s Principle

    1. George Boolos
      Pages 3-15
    2. Crispin Wright
      Pages 17-43
    3. Richard G. Heck Jr
      Pages 61-84
    4. Fraser MacBride
      Pages 85-93
    5. Fraser MacBride
      Pages 95-104
    6. William Demopoulos
      Pages 105-115
  3. The Logic of Abstraction

    1. Stewart Shapiro, Alan Weir
      Pages 119-146
    2. A. Rayo
      Pages 155-172
  4. Abstraction and the Continuum

  5. Basic Law V and Set Theory

    1. Stewart Shapiro, Alan Weir
      Pages 275-302
    2. Ignacio Jané, Gabriel Uzquiano
      Pages 303-329
    3. Bob Hale
      Pages 331-352

About this book


This volume collects together a number of important papers concerning both the method of abstraction generally and the use of particular abstraction principles to reconstruct central areas of mathematics along logicist lines.

Gottlob Frege's original logicist project was, in effect, refuted by Russell's paradox. Crispin Wright has recently revived Frege’s enterprise, however, providing a philosophical and technical framework within which a reconstruction of arithmetic is possible. While the Neo-Fregean project has recieved extensive attention and discussion, the present volume is unique in presenting a thorough going examination of the mathematical aspects of the neo-logicist project (and the particular philosophical issues arising from these technical concerns). Attention is focused on extending the Neo-Fregean treatment to all of mathematics, with the reconstruction of real analysis from various cut - or cauchy-sequence-related abstraction principles and the reconstruction of set theory from various restricted versions of Basic Law V as case studies. As a result, the volume provides a test of the scope and limits of the neo-logicist project, detailing what has been accomplished and outlining the desiderata still outstanding.

All papers in the anthology have their origins in presentations at Arché events, thus providing a volume that is both a survey of the cutting edge in research on the technical aspects of abstraction and a catalogue of the work in this area that has been supported in various ways by Arché.


Abstraction Arithmetic Gottlob Frege Philosophy of mathematics Real Analysis formal logic logic philosophy of science set theory symbolic logic

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