The Arché Papers on the Mathematics of Abstraction [electronic resource] / edited by Roy T. Cook.

The Philosophy and Mathematics of Hume's Principle -- Is Hume's Principle Analytic? -- Is Hume's Principle Analytic? -- Frege, Neo-Logicism and Applied Mathematics -- Finitude and Hume's Principle -- On Finite Hume -- Could Nothing Matter? -- On the Philosophical Interest of Frege Arithmetic -- The Logic of Abstraction -- "Neo-Logicist" Logic is not Epistemically Innocent -- Aristotelian Logic, Axioms, and Abstraction -- Frege's Unofficial Arithmetic -- Abstraction and the Continuum -- Reals by Abstraction -- The State of the Economy: Neo-Logicism and Inflation -- Frege Meets Dedekind: A Neo-Logicist Treatment of Real Analysis -- Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege's Constraint -- Basic Law V and Set Theory -- New V, ZF, and Abstraction -- Well- and Non-Well-Founded Fregean Extensions -- Abstraction & Set Theory -- Prolegomenon to Any Future Neo-Logicist Set Theory: Abstraction and Indefinite Extensibility -- Neo-Fregeanism: An Embarrassment of Riches -- Iteration one More Time.

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.