>
> And at 9:14 pm -0700 21/12/98, Gordon Fisher wrote:
[snip]
> > Well, would you admit that Book V gives a foundation for what have come
> > to be called real numbers (as Dedekind observed in *Was sind und was
> > sollen die Zahlen?"), and that this in turn is useful in connection with
> > dealing with what we call limit processes, and these are involved in
> > considering incommensurables?
>
> Dedekind didn't think that, and explained why he didn't in his two
> introductions. (I think, but I've lent my copy to someone and can't now
> check.) Once past the appealing similarity of the mechanisms of the two
> definitions, there are problems in your point of view, of which I highlight
> one that particularly interests me: The thing that makes numbers numbers is
> that we can do arithmetic with them, and Dedekind saw his definition of
> arithmetic with his cuts as being just as important as their completeness.
> There is almost nothing corresponding to this kind of arithmetic with the
> ratios of Greek maths (and what little there is strangely complicated
> and circumspect).
>
In Dedekind's letters to Lipschitz of June 10 and July 27,
1876 (reprinted in vol. 3 of his collected works, vol. 2 of the
Chelsea edition), he addresses this question in some detail.
Lipschitz had argued that Dedekind's "cuts" gave nothing new, since
everything was worked out already in Euclid, Book V.
Dedekind first replies (and here I will quote from the letter
of June 10, p. 471):
"At the same time, I maintain that for the most part (actually
almost all) these theorems of Arithmetic have not been proved up to
now, and in order to push this denial to the utmost, as it were, I say
that the theorem Root2.Root3 = Root6 has previously never been proved.
If anyone wishes to refute me in this, if he wishes, in other words,
to maintain that the theorem has already been proved, then the burden
of proof now falls on him, and he must show me an actually published
proof of this theorem, or of one that encompasses it. Do you really
believe that such a proof actually exists in any book? Naturally, I
have searched through a whole stack of works from various countries on
this point, and what does one find? Nothing but the most blatant
circular reasoning; basically: Root a . Root b = Root ab, because
(Root a . Root b)^2 = (Root a)^2.(Root b)^2 = ab; this is not preceded
by the least explication of the product of two irrational numbers, and
without any hesitation the theorem (mn)^2 = m^2 n^2, which has been
proved for rational numbers m, n, is applied also to irrationals. Now
is it not truly scandalous that mathematical instruction in the
schools passes for a particularly excellent means of developing the
understanding, while in fact such offenses against logic would not be
tolerated for an instant in any other discipline (e.g., Grammar)? If
one does not wish to proceed scientifically, or is unable to because
of time constraints, one should at least be honest and admit this
openly to the pupil, who in any case is only too willing to *believe*
a theorem simply on the instructor's word; that is better than to
stifle the pristine capacity for correct reasoning by presenting
sophisms as valid proofs."
[Zugleich behaupte ich, dass diese Saetze der Arithmetik zum
gro"ssten Theile (eigentlich fast alle) bisher nicht bewiesen seien, und
um wo mo"glich den Widerspruch aufs A"usserste zu reizen, sage ich, der
Satz: Root2.Root3 = Root6 sei verher noch nie bewiesen. Will Jemand
mich hierin widerlegen, will man also behaupten, der Satz sei schon
bewiesen, so liegt jetzt die Beweislast dem Anderen ob, und er muss mir
einen wirklich publicierten Beweis dieses oder eines ihn umfassenden
Satzes namhaft machen. Glauben Sie nun wirklich, dass ein solcher
Beweis sich in irgend einem Buche findet? Natu"rlich habe ich eine
ganze Menge von Werken der vershiedenen Nationen auf diesen Punct hin
gepru"ft, und was findet man da? Nichts als die rohesten
Cirkelschlu"sse, etwa so: Root a . Root b ist = Root ab, weil (Root a .
Root b)^2 = (Root a)^2.(Root b)^2 = ab ist; nicht die geringste
Erkla"rung des Productes von zwei irrationalen Zahlen geht voraus, und
ohne irgend ein Bedenken wird der fu"r rationale Zahlen m, n bewiesene
Satz (mn)^2 = m^2 n^2 auch fu"r irrationale Zahlen in Anspruch
genommen. Ist es nun nicht eigentlich empo"rend, dass der Untericht in
der Mathematick auf Schulen als ein besonders ausgezeichnetes
Bildungsmittel des Verstandes gilt, wa"hrend doch in keiner anderen
Disciplin (wie z.B. Grammatik) solche Versto"sse gegen die Logik nur
einen Augenblick geduldet wu"rden? Man sei, wenn man einmal nicht
wissenschaftlich verfahren will oder auch der Zeit wegen nicht kann,
wenigstens ehrlich und gestehe dies auch dem Schu"ler offen ein, der
ohnehin sehr geneigt ist, dem Lehrer auf dessen Wort hin an einen Satz
zu g l a u b e n; das ist besser, als durch Scheinbeweise den reinen,
edelen Sinn fu"r wahre Beweise zu erto"dten.]
I could wish that all teachers of mathematics might take to
heart this admonition of Dedekind's.
Dedekind then goes on to argue that what distinguishes his own
treatment from Euclid, Book V, is that nothing in Euclid expresses the
property that the system of magnitudes is *complete*. (Dedekind uses
both the words *stetig*, "continuous", and *vollstaendig*, "complete",
for this property.)
In the preface to "Was sind und was sollen die Zahlen"
(p. 339), he clarifies this observation by noting that (I use modern
language here for brevity and clarity) if the space R^3 (R = real
numbers) were replaced by F^3, where F = field of algebraic numbers,
then all the constructions of Euclid would remain valid.
Dedekind does not really bring out the point that David Fowler
has stressed, that ancient Greek mathematics was not arithmeticized,
although he does emphasize that his intention is to base the theory of
irrationals purely on arithmetic, not on the concept of geometric
magnitude. I think that one might perhaps say, in fact, that it was
Dedekind's work which completed the process of arithmeticization, of
which we are the heirs today.
Stacy Langton
University of San Diego
langton@acusd.edu