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.

Zusammenfassung
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 generatedand continue to generatea 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.

Inhalt

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.