The Arché Papers on the Mathematics of Abstraction / Edition 1

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More About This Textbook


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.

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Product Details

  • ISBN-13: 9781402042645
  • Publisher: Springer Netherlands
  • Publication date: 2/3/2008
  • Series: Western Ontario Series in Philosophy of Science, #71
  • Edition description: 2007
  • Edition number: 1
  • Pages: 454
  • Product dimensions: 6.00 (w) x 9.40 (h) x 1.40 (d)

Table of Contents

Preface: by Crispin Wright

Introduction: by Roy T. Cook

Part I: The Philosophy and Mathematics of Hume's Principle

Boolos, G. [1997], "Is Hume's Principle Analytic?", In Language, Thought, and Logic, R. Heck (ed.), Oxford, Oxford University Press: 245 – 261.

Wright, C. [1999], "Is Hume's Principle Analytic?", Notre Dame Journal of Formal Logic 40: 6 - 30.

Heck, R. [1997], "Finitude and Hume's Principle", Journal of Philosophical Logic 26: 589-617.

Clark, P., ''Frege, Neo-Logicism and Applied Mathematics ''

Fraser MacBride, [2000], "On Finite Hume", Philosophia Mathematica 8:150-9.

Fraser MacBride, [2002], "Could Nothing Matter?", Analysis 62: 125-135.

Demopoulos, W. [2003], "The Philosophical Interest of Frege Arithmetic" Philosophical Books 44: 220-228

Part II: The Logic of Abstraction

Shapiro, S. & Weir, A. [2000], "Neo-logicist logic is not epistemically innocent", Philosophia Mathematica 8, 160-189.

Cook, R. [2003], "Aristotelian Logic, Axioms, and Abstraction", Philosophia Mathematica 11: 195-202.

Rayo, A. [2002], "Frege's Unofficial Arithmetic", Journal of Symbolic Logic 67: 1623-1638.

Part III: Abstraction and the Continuum

Hale, R. [2000], "Reals by Abstraction", Philosophia Mathematica 8: 100-123.

Cook, R. [2002], "The State of the Economy: Neologicism and Inflation", Philosophia Mathematica 10: 43-66.

Wright, C. [2000], "Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege's Constraint", Notre Dame Journal of Formal Logic 41: 317-334.

Shapiro, S. [2000], "Frege Meets Dedekind: A Neologicist Treatment of Real Analysis", Notre Dame Journal of Formal Logic 41: 335-364.

Part IV: Basic Law V and Set Theory

Shapiro, S. & Weir, A. [1999], "NewV, ZF and Abstraction", Philosophia Mathematica 7: 293-321.

Uzquiano, G. & I. Jané [2004], "Well- and Non-Well-Founded Extensions", Journal of Philosophical Logic 33: 437 – 465.

Hale, R. [2000], "Abstraction and Set Theory", Notre Dame Journal of Formal Logic 41: 379-398

Shapiro, S. [2003], "Prolegomenon to Any Future Neo-Logicist Set Theory: Abstraction and Indefinite Extensibility", British Journal for the Philosophy of Science 54: 59-91.

Weir, A [2004], "Neo-Fregeanism: An Embarassment of Riches", Notre Dame Journal of Formal Logic 44: 13 - 48

Cook, R. [2004], "Iteration One More Time", Notre Dame Journal of Formal Logic 44: 63 - 92

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