Subject: Re: [HM] arithmetization
From: Abe Shenitzer (shenitze@pascal.math.yorku.ca)
Date: Thu Jan 06 2000 - 05:03:45 EST
Gordon Fisher asked:
What was the place of irrational numbers in this discussion? They may be
considered as infinite sequences of natural numbers, or set up using cuts
of rational numbers, but was this to be an acceptable basing of analysis
on natural numbers, as far as Kronecker was concerned?
A brief answer to his second question is "no". I think that the following
two quotes answer both of his questions.
(A) [Kronecker] categorically rejected the real-number constructions of
his day on the ground that they cannot be achieved through finite
processes only. He is said to have asked Lindemann of what use was the
proof that \pi is not algebraic inasmuch as irrational numbers are
nonexistent. Sometimes it is reported that his movement died of
inanition. We shall see later that it can be said to have reappeared later
in a new form in the work of Poincare and Brouwer. (Boyer, A History of
Math., p.570)
(B) Kronecker classified all mathematical disciplines except geometry and
mechanics as arithmetical, a category that specifically included algebra
and analysis. He never actually stated his intention of recasting analysis
without irrational numbers, however, and it is possible that he did not
take his radical notions altogether seriously himself. (From Biermann's
fine essay on Kronecker in the DSB.)
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It is not altogether out of place to point out that terms sometimes take
on a confusing life of their own. I have in mind Klein's term
"arithmetization of mathematics". In his 1895 talk (cited earlier by
Julio) Klein said among other things: "I would like to include all of
these developments under one word: the arithmetization of mathematics
... In this there lies, as you well know, both a complete understanding of
the extraordinary importance of the developments connected with this, and
a rejection of the view that the true contents of mathematics [is
presumably] completely contained in a sort of extract of
arithmetic." (P.290 of Bottazzini's "The Higher Calculus".)
Well, what term should Klein have introduced? Elementary, my dear
Watson: "The arithmetization and nonarithmetization of mathematics".
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Abe Shenitzer
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