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
 HeineCantor with help of fundamental (or Cauchy) sequences
 [whatever else may be said (and has been said so splendidly
 by Frege) about the original HeineCantor 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
Dedekind (1876).
[*] Sinaceur, Mohamed A.: "La methode mathematique de Dedekind",
_Revue d'histoire des sciences et de leurs applications_, v. 32,
no. 2, pp. 107142, 1979.
Regards,
Julio GC
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