Subject: Re: [HM] the term 'arithmetization'
From: Julio Gonzalez Cabillon (jgc@adinet.com.uy)
Date: Sun Jan 02 2000 - 11:14:26 EST
[Abe Shenitzer]
> ...
> Incidentally, the term "Arithmetization of Analysis" is due to Klein.
[Mic Detlefsen]
>
> Contrary to what Prof. Shenitzer writes, the term 'arithmetization'
> ('arithmetisierung', auf Deutsch) did not appear first in Klein's
> writings, but in Kronecker's 1887 essay.
[Abe Shenitzer]
>
> There are two terms. One is "arithmetization" and the other is
> "arithmetization of mathematics". I fully rely on Mic Detlefsen
> when he says that the first of these terms appeared in an essay by
> Kronecker. As for the second term, I ascribed it to Klein on the
> basis of a statement in Pierre Dugacs' essay "On the foundations
> of analysis". ...
[Bill Everdell]
>
> Am I right in assuming that Mic Detlefsen means (by "Kronecker's 1887
> essay") Kronecker's "Ueber den Zahlbegriff" in Crelle's 101(1887)...?
> ...
[Julio Gonzalez Cabillon]
>
> Yes, indeed. The key-text is:
>
> ... ich glaube auch, dass es dereinst gelingen wird, den
> gesammten Inhalt aller dieser mathematischen Disciplinen
> zu "arithmetisiren", d.h. einzig und allein auf den im
> engsten Sinne genommenen Zahlbegriff zu gruenden, also
> die Modificationen und Erweiterungen dieses Begriffs
> wieder abzustreifen ...
(snip)
>
> And, certainly, the expression "Arithmetisirung der Mathematik" (as
> such) was introduced by Felix Klein in 1895 during his address before
> the _Koeniglichen Gesellschaft der Wissenschaften zu Goettingen_ ...
(snip)
>
> "I would like to define all these developments by just one word:
> _Arithmetization of Mathematics_." (Klein)
> ...
And contrary to what Prof. Detlefsen writes, the term 'arithmetization'
('arithmetisirung', auf Deutsch) did not appear first in Kronecker's
1887 essay, but in earlier writings. I shall agree with Mic that the
context is different, but I find the term 'arithmetisirung' as early as
1879 in a review of McColl's "Calculus of Equivalent Statements" (Proc.
LMS, vol X, pp 16-28):
In diesem dritten Artikel (die frueheren s.F.d.M. X. 34, JFM10.
0034.02) setzt Herr McColl seine zuerst unter dem Namen "Symbolical
Language" veroeffentlichte und zunaechst zur Verwendung auf
mathematische Probleme bestimmte Arithmetisirung der Logik fort
und bringt sie zu einem gewissen Abschluss. (JFM11.0049.01)
PS: Mind that the term "Arithmetisirung" (rather than "Arithmetisierung")
was the one used by Klein et al.
PPS: I haven't checked McCall's "Calculus".
Regards to all,
Julio Gonzalez Cabillon
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