FOM: Postscript to 'Dedekind on Numbers'

Walter Felscher walter.felscher at
Fri Mar 27 13:36:11 EST 1998

When I mailed "Dedekind on Numbers" on March 13 , I could not
mention the book

   Pierre Dugac :
   Richard Dedekind et les fondements des mathe'matiques
   Pre'face de Jean Dieudonne'
   Paris, Librairie Philosophique J.Vrin, 1976

which I had mislaid and, at that time, could not find.
Having recovered it now, I wish to recommend it to the
interested reader and to add some observations derived from it.

Dugac's book contains 142 pages of the author's text,
giving a valuable analysis of the historical context in
which Dedekind's

   72  Stetigkeit und irrationale Zahlen
   88  Was sind und was sollen die Zahlen
   93  idem, 2nd edition

arose. This is followed by pages 143-315 with excerpts from
Dedekind's correspondence with mathematicians, not published
before, and excerpts from unpublished manuscripts of Dedekind's,
most of them deposited in the university library at G"ottingen.

One of these manuscripts has a bearing upon a quotation made
in "Dedekind on Numbers" from a letter to Weber of January
24, 1888 :
               Wir sind g"ottlichen Geschlechtes ...
               [ we are of divine descent ... ]      .

Appendix LI of Dugac's book is an undated manuscript by
Dedekind, titled

  Bemerkungen zu Weber's Manuscript :
                  Elementare Mengenlehre, Nat"urliche Zahlen

published as "Elementare Mengenlehre" in Jahresber. DMV 15
(1906) 173-184 . It begins with Dedekind's comment

     Ich habe nicht gesagt: "Wir sind g"ottlicher _Natur_"
     sondern "Wir sind g"ottlichen _Geschlechts_"
     (Apostelgeschichte Cap. 17 Abs. 28-29 ).

     [I did not say "we are of divine _nature_"
      but "we are of divine _descent_"
      (Acts, Chap. 17 , para 18-29 ) ] .

So  "g"ottlichen Geschlechts" here is the formulation chosen
by Luther at the beginning of para 29  [the King James
version choses to speak of "offspring" ].

The above shows, apart from the linguistic/literary origin
of Dedekind's phrase, that he chose his words with care and,
therefore, with intent. What the intent was, that is not
evident: maybe Dedekind just wanted to make a literary
allusion to a text familiar to his contemporaries [after
all, the German title of 88 appears as an allusion to the
title of Schiller's inaugural lecture at Jena "Was heisst,
und zu welchem Ende, studiert man Universalgeschichte" ]. Or
maybe he wanted to emphasize an important conceptual
distinction between to be "of div. nature" and to be "of
div. descent" - in which case we would need to know more
about the theological doctrines with which Dedekind may
have been familiar.

What should be clear, however, is that Dedekind's
formulations were chosen explicitly to convey the author's
intentions, though at present these are not completely clear
to us.  Hence it will do no good to rephrase them imply in
the terms of today's mathematical or philosophical fads,
fashions or foundations.


PS  The case that we lack the understanding of a mathematician's
    philosophical excursions is, of course, much more
    drastic in the case of Cantor's "U"eber unendliche,
    lineare Punktmannichfaltigkeiten." No. 5 . Math.Ann. 21
    (1883) 545-586 . While mathematicians have written one
    hagiography about Cantor after the other, and while the
    philosophers produce the (n+1)-st article about Frege's
    "Sinn und Bedeutung", nobody seems to have taken it upon
    himself to pave at least an exploratory path through the
    morass of Cantor's theological-philosophical explanations.

More information about the FOM mailing list