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InSeptember2000theArcheCentrelauncheda 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 Jane,-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.