This second of a two-volume book prolongs the inquiry of volume 1 by considering other examples apt to better illustrate the ideal of epistemic economy in mathematical recasting. The new examples are drawn from second-order definitions of real numbers and allow the comparison of three different approaches to the definition of these numbers: by domain extension, as ratios of magnitudes, and as bicimal developments. These definitions are presented in detail (by often going further than any available presentation) and compared with respect to their epistemic economy. In doing so, original results and insights are offered, such as a faithful and still consistent rephrasing of Frege's definitions of reals, and an original and epistemically highly economical definition of real numbers as bicimal developments. This ensures the book is of interest to scholars and students in the philosophy of mathematics and logic.

Considers other examples drawn from second-order definitions of real numbers by comparing three different approaches Rephrases Frege's definition of reals in a consistent and faithful manner Suggest an original and epistemically highly economical definition of real numbers as bicimal developments

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

Part III: Theories of Real Numbers.- 9 Defining Real Numbers through Relatives and Rationals.- 10 Defining Real Numbers as Ratios of Magnitudes.- 11 Defining Real Numbers as Bicimal Pairs.- Part IV: Conclusions.- Appendices.- Bibliographical References.