FOM: Dedekind on Numbers

[Walter Felscher]

Mr. Tait, on Wed, 11 Mar 98 12:26:54 -0500, wrote among
other things

    Dedekind long ago answered the `tired old objection' in the
    case of the real numbers when Weber asked him why he did not
    simply identify the real numbers with the cuts of rationals.
    His answer, in essence, is that it is ungrammatical to do
    so --- to ask, e.g. of two numbers whether one is a subset of
    the other or not.

    The `tired old objection' assumes that if the natural
    numbers are some one thing, then they must be sets or
    something else other than just the numbers; and then
    continues by pointing out that there is no distinguished
    such thing that they can be. Dedekind's answer is that they
    are simply the numbers. The proper answer to *what* they are
    is answered --- as he answered it --- by exhibiting the
    structure of the system of numbers.

and from that Mr. Pratt, on  Wed, 11 Mar 1998 12:44:13 -0800,
drew the lesson

    Oh, that's a very nice bit of history I didn't know.  So
    when arguing that the real line is better understood in
    terms of its structure than its elements (the standard
    category theory line, and the only view of the reals that
    I've been able to make any sense out of), one can cite
    Dedekind as an early source of that insight (and you as
    endorsing that viewpoint). This is very satisfactory.

While nothing in these remarks is wrong, closer inspection
will show that Dedekind's "answers" are not as univocal as
suggested by the interpretation read from them by Mr. Tait,
such that the consequence drawn by Mr. Pratt may also be
phrased with some more circumspection.

  *

Dedekind's answers apear in his letter to Weber of January
24, 1888, as reprinted in Gesammelte Mathematische Werke,
Band 3 (ed. R.Fricke, E.Noether, ™.Ore), Braunschweig 1932,
p. 488-490 . There first is a discussion of natural numbers
in which Dedekind comments on Weber's proposal to introduce
them as cardinal numbers (i.e. equivalence classes of finite
sets), in contrast to his own introduction of them as
ordinal numbers in Was-sind-und-was-sollen-die-Zahlen [and
Weber's proposal was carried out in vol.1 of Weber-Wellstein,
Encyclopaedie der Elementar-Mathematik, Leipzig 1903-1907 ];
it ends with the remark

     Will man aber Deinen Weg einschlagen ..., so m"ochte
     ich doch rathen, unter der Zahl (Anzahl, Cardinalzahl)
     lieber nicht die Classe (das System aller einander
     "ahnlichen endlichen Systeme) selbst zu verstehen,
     sondern etwas Neues (dieser Classe Entsprechendes), was
     der Geist erschafft. Wir sind g"ottlichen Geschlechtes
     und besitzen ohne jeden Zweifel sch"opferische Kraft
     nicht blos in materiellen Dingen (Eisenbahnen,
     Telegraphen), sondern ganz besonders in geistigen
     Dingen.

     [If, however, one wants to pursue the way you
     propose ..., then I would suggest to understand as
     number (cardinal number) rather not the class (the
     system of all similar finite systems) itself, but
     something New (which corresponds to this class) that is
     created by the mind. We are of divine descent and we no
     doubt posses creative power not only in things material
     (railways, telegraphs), but particularly in things
     mental. ]

Only then Dedekind comes to irrational numbers, writing

     Es ist dies ganz dieselbe Frage, von der Du am Schlusse
     Deines Briefes bez"uglich meiner Irrational-Theorie
     sprichst, wo Du sagt, die Irrationalzahl sei "uberhaupt
     Nichts anderes als der Schnitt selbst, w"ahrend ich es
     vorziehe etwas Neues (vom Schnitte Verschiedenes) zu
     erschaffen, was dem Schnitte entspricht, und wovon ich
     sage, dasz es den Schnitt hervorbringe, erzeuge. Wir
     haben das Recht, und eine solche Sch"opfungskraft
     zuzusprechen, und auszerdem ist es der Gleichartigkeit
     aller Zahlen wegen viel zweckm"asziger, so zu
     verfahren. Die rationalen Zahlen erzeugen doch auch
     Schnitte, aber ich werde die rationale Zahl gewisz
     nicht f"ur identisch ausgeben mit dem von ihr erzeugten
     Schnitte; und auch nach Einf"uhrung der irrationalen
     Zahlen wird man von Schnitt-Erscheinungen oft mit
     solchen Ausdr"ucken sprechen, ihnen solche Attribute
     zuerkennen, die auf die entsprechenden Zahlen selbst
     angewendet gar seltsam klingen w"urden. Etwas ganz
     "Ahnliches gilt auch von der Definition der
     Cardinalzahl (Anzahl) als Classe; man wird Vieles von
     der Classe sagen (z.B. dasz sie ein System von
     unendlich vielen Elementen, n"amlich allen "ahnlichen
     Systemen ist), was man der Zahl selbst doch gewisz
     h"ochst ungern (als Schwergewicht) anh"angen w"urde;
     denkt irgend Jemand daran, oder wird er es nicht gern
     bald vergessen, dasz die Zahl vier ein System von
     unendlich vielen Elementen ist ?  ...

     [ It is this the same question of which you speak at
     the end of your letter about my theory of irrationals,
     where you say that the irrational number is nothing
     else but the cut itself, while I prefer to create
     something New (distinct from the cut) which corresponds
     to the cut, and of which I say that it produces,
     generates the cut. We do have the right to ascribe such
     creative power to ourselves, and besides it also is
     much more practical to proceed in this manner because
     of the equal character of all numbers. Clearly, also
     the rational numbers generate cuts, but I certainly
     will not present a rational number as identical with
     the cut it generates; and also after the introduction
     of irrational numbers we often will speak about the
     cut-phenomena with expressions which ascribe attributes
     to them that, applied to the corresponding numbers
     themselves, would sound fairly strange indeed.
     Something quite similar holds for the definition of the
     cardinal number as a class; there are many things that
     may be said about the class (e.g. that it is a system
     of infinitely many elements, namely all similar
     systems) which we certainly would utterly dislike to
     hang (as a counterweight) around the number's neck
     itself; does anyone think of it, or will he at least
     not love to forget it as fast as possible, that the
     number four is a system of infinitely many elements ? ]


  **

Summing up Dedekind's answers, I notice

D1. Dedekind proposes to view numbers as creations of the
    human mind. That they are manifestations of its creative
    power is the principal reason which Dedekind puts
    forward against Weber's reductionist propositions.

D2. Only with a "auszerdem = besides" does Dedekind mention
    his mathematical, internal reasons, which Mr. Tait so
    aptly referred to as the prevention of ungrammatical
    usages brought about by a reductionist terminology.

It is obvious that D1 is not a mathematical argumention, and
it may go against many a contemporary reader's grain to even
listen to it. Yet for Dedekind it is a foundational, epistemic
argumentation, and historical justice requires us to
acknowledge this fact. Also, it is evident from Dedekind's
own commentaries - e.g. in his letter to Lipschitz from June
10, 1876 , reprinted l.c., p.470 - that he does not consider
his view in D1 as particularly novel or original; rather, he
sees it as a description of the ways mathematicians commonly
invent new numbers. [Of course, I am aware that authors such
as Mr. Hersh will immediately conclude that Dedekind's voice
here is that of the liberal German bourgeois of his time.
Which, of course, explains nothing.]


  ***

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.

Of course, these are not the words of Dedekind, and so the
question arises whether the insight E is either explicitly
formulated in different words in Dedekind's work, or is at
least implicit in his commentaries.

For the natural numbers, E is explicitly contained in
Bemerkung 134 of Was-sind-und-was-sollen-die-Zahlen: a
statement about natural numbers (referring only to the
concepts occurring in the axioms) which holds in one system
satisfying Dedekind's axioms also holds in any other such
system. This is based on the isomorphism theorem of Satz
132, stating that any two systems satisfying Dedekind's
axioms are isomorphic.

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 is, of course, most uplifting a situation when we can
quote the great men of the past to support the insights of
the present. But history is not just a quarry from which we
may choose isolated blocks to embellish our present designs.

W.F.