The Arché Papers on the Mathematics of Abstraction

1. Front Matter

   Pages i-xxxvii

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2. The Philosophy and Mathematics of Hume’s Principle

   1. Is Hume’s Principle Analytic?

      • George Boolos

      Pages 3-15

   2. Is Hume’s Principle Analytic?

      • Crispin Wright

      Pages 17-43

   3. Frege, Neo-Logicism and Applied Mathematics

      • Peter Clark

      Pages 45-60

   4. Finitude and Hume’s Principle

      • Richard G. Heck Jr

      Pages 61-84

   5. On Finite Hume

      • Fraser MacBride

      Pages 85-93

   6. Could Nothing Matter?

      • Fraser MacBride

      Pages 95-104

   7. On the Philosophical Interest of Frege Arithmetic

      • William Demopoulos

      Pages 105-115

3. The Logic of Abstraction

   1. “Neo-Logicist” Logic is not Epistemically Innocent

      • Stewart Shapiro, Alan Weir

      Pages 119-146

   2. Aristotelian Logic, Axioms, and Abstraction

      • Roy T. Cook

      Pages 147-153

   3. Frege’s Unofficial Arithmetic

      • A. Rayo

      Pages 155-172

4. Abstraction and the Continuum

   1. Reals by Abstraction

      • Bob Hale

      Pages 175-196

   2. The State of the Economy: Neo-Logicism and Inflation

      • Roy T. Cook

      Pages 197-218

   3. Frege Meets Dedekind: A Neo-Logicist Treatment of Real Analysis

      • Stewart Shapiro

      Pages 219-252

   4. Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege’s Constraint

      • Crispin Wright

      Pages 253-272

5. Basic Law V and Set Theory

   1. New V, ZF, and Abstraction

      • Stewart Shapiro, Alan Weir

      Pages 275-302

   2. Well- and Non-Well-Founded Fregean Extensions

      • Ignacio Jané, Gabriel Uzquiano

      Pages 303-329

   3. Abstraction & Set Theory

      • Bob Hale

      Pages 331-352

   4. Prolegomenon to Any Future Neo-Logicist Set Theory: Abstraction and Indefinite Extensibility

      • Stewart Shapiro

      Pages 353-382

   5. Neo-Fregeanism: An Embarrassment of Riches

      • A. Weir

      Pages 383-420

InSeptember2000theArchéCentrelauncheda ve-yearresearchprojecten- tled the Logical and Metaphysical Foundations of Classical Mathematics. Its goal was to study the prospects, philosophical and technical, for abstractionist foundations for the classical mathematical theories of the natural, real and complex numbers and standard set theory. Funding was provided by the then Arts and Humanities Research Board (now the Arts and Humanities Research Council) for the appointment of full-time postdoctoral research fellows and PhD students to collaborate with more senior colleagues in the project, and at the same time the British Academy awarded the Centre additional resources to establish an International Network of scholars to be associated with the work. This was the beginning of the serial ‘Abstraction workshops’ of which the Centre had staged no less than eleven by December 2006. We gra- fully acknowledge the generous support of the Academy and Council, sine qua non. The project seminars and Network meetings generated—and continue to generate—a large number of leading-edge research papers on all aspects of the project agenda. The present volume is the rst of what we hope will be a number of anthologies of these researches. With two exceptions,—the contribution by the late George Boolos and the co-authored paper by Gabriel Uzquiano and Ignacio Jané,—the papers that Roy Cook has collected in the present volume are all authored by sometime members of the project team or of the British Academy Network.

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

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