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::: center home >> events >> lunchtime >> 2005-06 >> abstracts

Tuesday, 11 April 2006

Lunchtime Colloquium

The Absolute Arithmetic Continuum
and the Unification of all Numbers Great and Small

Philip Ehrlich, Ohio University, Philosophy

12:05 pm, 817R Cathedral of Learning


Abstract:  “Bridging the gap between the domains of discreteness and of continuity, or between arithmetic and geometry, is a central, presumably even the central problem of the foundations of mathematics”.  So wrote Abraham Fraenkel, Yehoshua Bar-Hillel and Azriel Levy in their mathematico-philosophical classic Foundations of Set Theory [1973].  Cantor and Dedekind of course believed they had bridged the gap with the creation of their arithmetico-set theoretic continuum of real numbers, and for roughly a century now it has been one of the central tenets of standard mathematical philosophy that indeed they had.  In accordance with this view the geometric linear continuum is assumed to be isomorphic with the arithmetic continuum, the axioms of geometry being so selected to ensure this would be the case. In honor of Cantor and Dedekind, who first proposed this mathematico-philosophical thesis, the transference of reals' purported continuity to the continuity of the Euclidean straight line has come to be called the Cantor-Dedekind axiom.  Given the Archimedean nature of the real number system, once this axiom is adopted we have the classic result of standard mathematical philosophy that infinitesimals are superfluous to the analysis of the structure of a continuous straight line.


More than twenty years ago, however, we began to suspect that while the Cantor-Dedekind theory succeeds in bridging the gap between the domains of arithmetic and of classical Euclidean geometry, it only reveals a glimpse of a far richer theory of continua that not only allows for infinitesimals, but leads to a vast generalization of portions of Cantor's theory of the infinite, a generalization that also provides a setting for Abraham Robinson's nonstandard approach to analysis [1961] as well as for the profound and all too often overlooked non-Cantorian theories of the infinite (and infinitesimal) pioneered by Giuseppe Veronese [1891], Tullio Levi-Civita [1892; 1898], David Hilbert [1899] and Hans Hahn [1907] in connection with their work on non-Archimedean ordered algebraic and geometric systems, and by Paul du Bois-Reymond [1870-71; 1875; 1877; 1882], Otto Stolz [1883], Felix Hausdorff [1907; 1909] and G. H. Hardy [1910] in connection with their work on the rate of growth of real functions. Central to the theory is J. H. Conway's theory of surreal numbers [1976] and the present author's amplifications and generalizations thereof.  In a number of earlier works we suggested that whereas the real number system should merely be regarded as constituting an arithmetic continuum modulo the Archimedean axiom, the system of surreal numbers may be regarded as a sort of absolute arithmetic continuum (modulo von Neumann-Bernays-Gödel set theory with Global Choice).  In the present discussion we will draw attention to the unifying framework the absolute arithmetic continuum provides not only for the reals and the ordinals but for the various other systems of numbers great and small alluded to above. Finally, we will suggest that by limiting the absolute arithmetic continuum to the substructure consisting of its finite and infinitesimal members, one obtains a remarkably well-suited formal replacement for the intriguing Peircean continuum sketched by Charles Sanders Peirce [1898; 1900] in the years bracketing the turn of the twentieth century.

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Revised 3/6/08 - Copyright 2006