Subject: Re: [HM] arithmetization
From: Gordon (gfisher@shentel.net)
Date: Wed Jan 05 2000 - 11:13:17 EST
Abe Shenitzer wrote:
> 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.
>
[deletion]
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?
Gordon Fisher gfisher@shentel.net
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