Subject: [HM] arithmetization
From: Abe Shenitzer (shenitze@pascal.math.yorku.ca)
Date: Wed Jan 05 2000 - 05:03:29 EST
Apropos of the arithmetization (regardless of spelling) of analysis.
The discussion seemed to center on Kronecker's views concerning the
foundations of mathematics. In this connection I would like to cite
criticisms of Kronecker's position by Dedekind and Hilbert.
Let me repeat. In 1887, as well as earlier, Kronecker thought of the
positive integers as the foundation of analysis. He says as much in
the short quote cited by Julio. But it is one thing to base everything
on the natural numbers and quite another thing to "cast off the
modifications and extensions of this concept".
Kronecker's views of what will, or should, happen one day are made
perfectly clear by Dedekind in the following excerpt from the
introduction to his "Was sind und was sollen die Zahlen". Dedekind
tells us not only what he is against in Kronecker's program but what
to put in its place. (My translation of this excerpt differs in a few
places from Bottazzini's on p.260 of his "The Higher Calculus".)
--------------------------------------------------------
Given my view [of the primary role of numbers], it is obvious, and
anything but new, that it is possible to express every theorem of
algebra and of higher analysis, however advanced, as a theorem about
the natural numbers, a claim I heard from Dirichlet many times. But I
see absolutely no merit - nor did Dirichlet - in actually undertaking
this tedious transcription and in declining to use and admit numbers
other than the natural numbers. To the contrary. For the most part,
the greatest and most fruitful advances in mathematics and in the other
sciences have been achieved through the creation and introduction of
new concepts. What forced these developments was the frequent recurrence
of complex phenomena, whose mastery by means of the old concepts could
only be achieved with great difficulty.
([Gerade bei dieser Auffassung] erscheint es als etwas
Selbstverstaendliches und durchaus nicht Neues, dass jeder auch noch so
fern liegende Satz der Algebra und hoeheren Analysis sich als ein Satz
ueber die natuerlichen Zahlen aussprechen laesst, eine Behauptung, die ich
auch wiederholt aus dem Munde von Dirichlet gehoert habe. Aber ich
erblicke keineswegs etwas Verdienstliches darin - und das lag auch
Dirichlet ganz fern -, diese muehselige Umschreibung wirklich vornehmen
und keine anderen als die natuerlichen Zahlen benutzen und anerkennen zu
wollen. Im Gegenteil, die groessten und fruchtbarsten Fortschritte in der
Mathematik und anderen Wissenschaften sind vorzugsweise durch die
Schoepfung und Einfuehrung neuer Begriffe gemacht, nachdem die haeufige
Wiederkehr zusammengesetzter Erscheinungen, welche von der alten Begriffen
nur muehselig beherrscht werden, dazu gedraengt hat.)
-----------------------------------------------------------
Hilbert criticized Kronecker for failing to realize that the
whole numbers could and should be justified. He described him as a
dogmatist, for he (i.e. Kronecker) "took the whole numbers with their
essential properties as dogma and failed to look further back" (see p.361
of O. Becker's "Grundlagen der Mathematik"). (Of course the "looking
further back" was done by Dedekind in his "Was sind und was sollen die
Zahlen" (itself a kind of addendum to his "Stetigkeit u. Irrationale
Zahlen" of 1872) and by Peano in his axiomatization of arithmetic (1889)).
Elsewhere (op. cit., p.375) Hilbert criticized Kronecker for
extreme inconsistency. Indeed, one is surprised to learn that an "integer
fundamentalist" like Kronecker applauded the use of ideal elements (by
Kummer). (But one must not forget that Kummer was his Gymnasium teacher!)
(Needless to say, Kronecker's views on the foundations and practice of
mathematics have nothing to do with his stature as one of the great
algebraists of the 19th century.)
P.S. For a truly extensive discussion of the topic of arithmetization of
analysis see Ch.7 of Bottazzini's "The Higher Calculus".
Abe Shenitzer
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