Friday,
2 April 2004
The Rise of nonArchimedean Mathematics and the Roots of a Misconception
I:
the Emergence of nonArchimedean 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 philosophythe nature of the
infinite, the nature of the infinitesimal, and the nature of the
continuumhad 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
“selfcontradictory,” 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 justcited
views of Russell are visàvis late nineteenthcentury 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.
