::: about
   ::: news
   ::: links
   ::: giving
   ::: contact

   ::: calendar
   ::: lunchtime
   ::: annual lecture series
   ::: conferences

   ::: visiting fellows
   ::: resident fellows
   ::: associates

   ::: visiting fellowships
   ::: resident fellowships
   ::: associateships

being here
   ::: visiting
   ::: the last donut
   ::: photo album

::: center home >> events >> lunchtime >> 2003-04 >> abstracts

Friday, 2 April 2004
The Rise of non-Archimedean Mathematics and the Roots of a Misconception I:
the Emergence of non-Archimedean Grössensysteme

Philip Ehrlich
Ohio University
12:05 pm, 817R Cathedral of Learning

Abstract: In his paper Recent Work On The Principles of Mathematics, which appeared in 1901, Bertrand Russell reported that the three central problems of traditional mathematical philosophy--the nature of the infinite, the nature of the infinitesimal, and the nature of the continuum--had all been “completely solved” [1901, p. 89]. Indeed, as Russell went on to add: “The solutions, for those acquainted with mathematics, are so clear as to leave no longer the slightest doubt or difficulty” [1901, p. 89]. According to Russell, the structure of the infinite and the continuum were completely revealed by Cantor and Dedekind, and the concept of an infinitesimal had been found to be incoherent and was “banish[ed] from mathematics” through the work of Weierstrass and others [1901, pp. 88, 90]. These themes were reiterated in Russell’s often reprinted Mathematics and the Metaphysician [1918], and further developed in both editions of Russell’s The Principles of Mathematics [1903; 1937], the works which perhaps more than any other helped to promulgate these ideas among historians and philosophers, of mathematics. In the two editions of the latter work, however, the banishment of infinitesimals that Russell spoke of in 1901 was given an apparent theoretical urgency. No longer was it simply that “nobody could discover what the infinitely little might be,” [1901, p. 90] but rather, according to Russell, the kinds of infinitesimals that had been of principal interest to mathematicians were shown to be either “mathematical fictions” whose existence would imply a contradiction [1903, p. 336; 1937, p. 336] or, outright “self-contradictory,” as in the case of an infinitesimal line segment [1903, p. 368; 1937, p. 368].

In a fledgling work in progress I presented at the Center a number of years ago attention was drawn to just how misleading the just-cited views of Russell are vis-à-vis late nineteenth-century mathematics and to just how mischievous those views of Russell have been. Having now completed a significant portion of that work, my talk will fill in some of the gaps contained in my earlier talk.

Revised 3/11/08 - Copyright 2006