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 BarHillel and Azriel Levy in their mathematicophilosophical
classic Foundations of Set Theory [1973]. Cantor and Dedekind of course believed
they had bridged the gap with the creation of their arithmeticoset
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 mathematicophilosophical thesis, the transference of reals'
purported continuity to the continuity of the Euclidean straight
line has come to be called the CantorDedekind 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
CantorDedekind 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 nonCantorian theories of the infinite (and infinitesimal)
pioneered by Giuseppe Veronese [1891], Tullio LeviCivita [1892;
1898], David Hilbert [1899] and Hans Hahn [1907] in connection with
their work on nonArchimedean ordered algebraic and geometric systems,
and by Paul du BoisReymond [187071; 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 NeumannBernaysGödel 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 wellsuited 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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the last donut? 