Reals by Abstraction. An Inquiry about Epistemic Economy in Mathematics: Volume I

This first of a two-volume book discusses the process of conceptual recasting of mathematical contents under a new and more convenient form, complying with the ideal of epistemic economy (expressing the same content by appealing to fewer or minimal conceptual resources). It accounts for this ideal and discusses its philosophical significance, also in opposition to the most frequently considered ideal of ontological parsimony, in the context of an original approach to mathematical knowledge. If it is admitted that mathematical objects have no external existence, the former ideal is much more sensible and more plausible to be pursued, despite lacking any clear and specifically dedicated account. To illustrate it, this first volume considers many examples of second-order definitions of natural numbers, by comparing Peano Arithmetic with Frege one, and showing the greater epistemic economy of the latter over the former. This ensures the book is of interest to scholars and students in the philosophy of mathematics and logic.

Discusses the conceptual recasting of mathematical contents under a new and more convenient form Accounts for this ideal in the context of an original approach to mathematical knowledge Considers many examples drawn from second-order definitions of natural numbers, by comparing Peano Arithmetic with Frege

Autorentext

Marco Panza is The Kennedy Professor of Philosophy at Chapman University and a Research Director of First Class at the CNRS. Historian and philosopher of Logica and mathematics, he has studied in Milano and Paris and taught at the University of Geneva, the University of Nantes, the UNAM (Mexico City), the Pompeu Fabra Univ. in Barcelona, the universities of Paris 7 and Paris 1, and Chapman University. Author of about 150 papers and books in different Languages (including English, French, Italian, Spanish, and Portuguese) that appeared in the most recognized journals in his domain, and for some prestigious publishers, like Flammarion, Vrin, and Springer. His scientific expertise includes the history of Greek and Early modern mathematics (especially, Euclid, Viète, Descartes, Newton, Euler, and Lagrange), Frege's foundational program and his contemporary legacy, the philosophy of mathematical practice (he is a founding member of the Association of Philosophy of Mathematical Practice).

Inhalt

Introduction.- Part I: Epistemic Economy.- 1 Mathematical Content(s).- 2 Epistemic Economy.- 3 Different Sorts of Formal Definitions.- 4 And Their Respective Epistemic Costs.- 5 The Philosophical Significance of Epistemic Economy.- 6 Semantics and Recasting.- Part II: Second-Order Theories of Natural Numbers.- 7 Z2 and its Subsystems.- 8 Frege Arithmetic and Alike.- Appendices.- Bibliographical References.