FOM: Thomae and non-archimedian domains

[Julio Gonzalez Cabillon]

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 Cholera-Bacillus
|    of infinitesimals''.
| 
| While I do not know what Cantor actually may have referred
| to, Thomae (1840-1921 , since 1872 Professor in Halle, since
| 1879 in Jena) does have a documentable connection, not with
| infinitemals, but with non-archimedian 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 Bois-Reymond 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. 25-44, 1987].
In this paper Simons first gives an overview of Frege's place in
math-history, 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 Bois-Reymond and their
reception" [_Archive for History of Exact Sciences_, vol. 24, no. 2,
pp. 101-163, 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