FOM: Thomae and nonarchimedian domains
Julio Gonzalez Cabillon
jgc at adinet.com.uy
Thu Nov 27 19:49:08 EST 1997
Dear Walter,
On Wed, 26 Nov 1997, you wrote:
 Bill Tait, on Nov.12th, wrote in connection with Goedel and
 infinitesimals that

 Cantor is quoted by Dauben as saying that Johannes Thomae
 (who had an office down the hall from Frege) was the
 first to ``infect mathematics with the CholeraBacillus
 of infinitesimals''.

 While I do not know what Cantor actually may have referred
 to, Thomae (18401921 , since 1872 Professor in Halle, since
 1879 in Jena) does have a documentable connection, not with
 infinitemals, but with nonarchimedian extensions of the
 reals: in his

 Abrisz einer Theorie der complexen Funktionen, Halle 1870

 and his

 Elementare Theorie der analytischen Functionen einer
 complexen Ver„nderlichen., 2te Aufl., Halle 1898

 he represented the orders of growth of real functions by
 lexicographically ordered semigroups of sequences of
 integers. It then was Paul du BoisReymond 1882 who
 expressed the idea that the totality of orders of growth
 (which he called the infinitaere Pantachie) should be viewed
 as an expansion of the continuum; for a more modern
 presentation of his work cf. G.H.Hardy, Orders of Infinity,
 Cambridge 1924.

 It may not be superfluous to point out that this achievement
 of Thomae's is NOT connected with his opinions on the
 foundation of numbers, analyzed so masterfully in Frege's
 "Ueber die Zahlen des Herr Thomae".

 W.F.

Peter M. Simons [Institut fuer Philosophie, Universitaet Salzburg] has
written an interesting essay entitled "Frege's theory of real numbers"
[_History and Philosophy of Logic_, vol. 8, no. 1, pp. 2544, 1987].
In this paper Simons first gives an overview of Frege's place in
mathhistory, and then goes on to introduce Frege's harsh criticisms
about theories of real numbers, such as those due to Cantor, Dedekind,
Thomae and Weierstrass.
Four years later, Gordon Fisher addresses a wonderful study on "The
infinite and infinitesimal quantities of du BoisReymond and their
reception" [_Archive for History of Exact Sciences_, vol. 24, no. 2,
pp. 101163, 1981]. In this lengthy paper, the author provides, among
many other things, a brief survey of related attempts to develop
theories of the infinite and infinitesimals by Johannes Thomae,
Giuseppe Veronese and Otto Stolz.
Sincerely, JGC
More information about the FOM
mailing list