FOM: Dedekind on Numbers
Julio Gonzalez Cabillon
jgc at adinet.com.uy
Sat Mar 14 19:53:56 EST 1998
As usual, Walter, thank you very much for your thoughtful comments.
| [Fri, 13 Mar 1998 14:21:23 +0100]
| I hope to find the agreement of Messrs. Tait and Pratt that
| it is only an inessential rephrasing of their words when I
| formulate the insight, which they draw from Dedekind's work, as
| E "what" the numbers "are" is explained by exhibiting the
| structure of the system of numbers, and the numbers are
| better understood in terms of their structure than by
| the nature of their elements.
| For the reals, Dedekind had no such isomorphism theorem yet;
| it was established only in 1890 (for complete Archimedian
| ordered groups) by Rodolfo Bettazzi (Teoria delle Grandezze,
| Pisa 1890) and rediscovered by Otto H"older in 1901). The
| insight E is not explicitly stated in Dedekind's articles,
| nor in his letters as far as they are known to me. There
| is, however, in Dedekind's letter to Lipschitz from June 10,
| 1876, the sentence
| dasselbe gilt von der Darstellung der Herrn Heine und
| Cantor in Halle, die nur "auszerlich von der meinen
| verschieden ist
| implying that Dedekind was aware that his axiomatization of
| the reals with help of cuts was equivalent to that of
| Heine-Cantor with help of fundamental (or Cauchy-) sequences
| [whatever else may be said (and has been said so splendidly
| by Frege) about the original Heine-Cantor approach].
| It seems to be a matter of debate whether Dedekind's
| awareness here can be counted as witnessing implicitly his
| support of the insight E ; based on Dedekind's general
| methodological attitudes, I am inclined to do so. Still, a
| reference to Dedekind for insight E about the reals cannot,
| it seems, be supported by an explicit quotation, but would
| require a closer report on the tangled web within which the
| 'foundations of analysis' developed one hundred years ago.
It's a long time now since I don't put a finger on Dedekind's memoirs.
Anyhow, if I am remembering things correctly, Mohamed A. Sinaceur,
in a useful paper [*], provides his insights from which to evaluate
Dedekind's general methodological attitudes. According to Sinaceur,
Dedekind should be regarded as a pioneer in posing philosophical
problems to maths, finishing up the "imperialism of the calculus".
In my opinion, this is far from clear -- but not the main point
here, I presume. To my memory, the essay [*] contains translations
into the French of correspondence exchanged between Lipschitz and
[*] Sinaceur, Mohamed A.: "La methode mathematique de Dedekind",
_Revue d'histoire des sciences et de leurs applications_, v. 32,
no. 2, pp. 107-142, 1979.
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