What Are Numbers and What Should They Be

Hardcover. All profit from the sale of this monograph will be used to support research at the RESEARCH INSTITUTE FOR MATHEMATICS & to cover costs for further publication of the RIM Series of monographs. PREFACE BY EDITORS. The unique creativity of Dedekind blazes across this essay on arithmetic like lightening across a serene sky. Indeed, it is a highly educational experience for college students to see & learn from this mathematical giant as he stumbles, forgets, worries, slips, trips, sweats & unearths his precious mathematical mines single-handedly before our eyes in his essay. Dedekind's essay unearths the following branches of mathematics: AXIOM SET THEORY, hinting implicitly at most of the Zermelo axioms, which surely led Zermelo to his formalization of them. AXIOMATIC ARITHMETIC, hinting at Peano axioms, which Peano must have noted. RECURSIVE FUNCTION THEORY, hinting at Computability, which Skolem, Goedel & Kleene must have noted for their work. FOUNDATIONS OF MATHEMATICS, which surely guided Zermelo & Bertrand Russell in their seminal foundational enterprises. It should be noted that the true founder of the set-theoretic foundations of mathematics is Dedekind & not Cantor as many imagine. The footnotes in this translations are by the editors & serve mainly to make a more explicit connection between Dedekind's essay & the above mentioned branches of mathematics that Dedekind helped to pioneer. We have also taken the liberty of modernizing Dedekind's original notation, terminology & even his German text. Also, Dedekind's prefaces & footnotes, some of which have been incorporated by the editors into the text are omitted. Finally, we trust that this effort at modernizing Dedekind's essay does not obscure his intent, as often happens when attempts are made at improving upon the Masters, but rather that it should serve to unfold to students the beauty, romance & style of true mathematical creativity. To order call Research Institute for Mathematics at (207) 866-7712 or write to 383 College Ave., Orono, ME 04473.Dedekind, Richard is the author of 'What Are Numbers and What Should They Be', published 1995 under ISBN 9780964302310 and ISBN 0964302314.