Subject: Re: [HM] the term 'arithmetization'
From: Michael Detlefsen (detlefsen.1@nd.edu)
Date: Mon Jan 03 2000 - 13:55:33 EST
Julio locates a use of the term 'arithmetization' occurred earlier than
Kronecker's 1887 essay. My intention (in keeping with Prof. Shenitzer's
emphasis on Klein's use of the term) was to locate uses of the term having
the same basic sense as Klein's. Kronecker's clearly does; the earlier use
cited by Julio clearly does not. I'm sorry if anyone was mislead by my
remark. My only aim was to clarify the origin of the term as Klein used it.
I believe it is correct to say, as I did, that use of the term, in Klein's
sense, originated with Kronecker. (I might add that I did so because I have
so often heard it said that it was Klein who coined the term
'arithmetization' (as applied to mathematics) in his 1895 lecture.
My view is that Kronecker and Klein used the term in the same BASIC sense,
and that when one inquires after the origins of this BASIC usage, the true
source is Kronecker's 1887 lecture and not Klein's 1895 lecture. They did
not, however, use it in exactly the same sense. There are, in particular,
two differences that seem worth pointing out.
(i) The first concerns a possible difference in extension. In particular,
it concerns the disposition of geometry and indicates a possible important
difference for distinguishing between Kronecker's 'arithmetization' and
Klein's 'arithmetization of mathematics'. Kronecker, following Gauss,
accepted the basic distinction between the arithmetic and geometric sides
of mathematics and did not therefore maintain the 'arithmetizability' of
geometry but only of those disciplines belonging to the broadly
'arithmetic' side of mathematics (most notably, what he referred to as
'algebra' and 'analysis'). (Note the similarity here with Brouwer and
various of the intuitionists.) Klein's views on this division are not
altogether clear. He did, however, advocate what he referred to as the
'fusion' of arithmetic and geometry and said that arithmetic formulae
facilitated the most precise formulations of geometric facts. Because of
this, Klein, unlike Kronecker, may have self-consciously understood his
'arithmetization' as extending not only to algebra and analysis but to
geometry as well--a 'total' arithmetization of 'mathematics' and one which
therefore differed in its intended scope from that envisioned by Kronecker.
(ii) The second concerns a possible difference in intension. In particular,
it concerns a divergence in the epistemological meaning of the term
'arithmetization'. Klein did not seem to share Kronecker's epistemological
attitude regarding the allegedly 'special' character of our knowledge of
the arithmetic of the positive whole numbers. He would therefore not have
seen his arithmetization of mathematics as serving the same 'reductionist'
epistemological ends as Kronecker saw his arithmetization of the
arithmetical side of mathematics as serving. Hence, even within the
arithmetical side of mathematics, arithmetization may not have had the same
epistemological meaning for Klein as it did for Kronecker.
Perhaps this adds a little useful detail (and orientation) to the current
discussion of the term 'arithmetization' and its origins.
Finally, let me say that I am curious and unclear concerning the intended
significance of the distinction Julio makes between the would-be term
"Arithmetisirung" and the would-be term "Arithmetisierung". Would you
please elaborate, Julio?
Happy new year to all,
Mic Detlefsen
P.S. If one is interested in ALL uses of the term 'arithmetisiren' (and its
cognates and parallel words in other languages in which German authors may
have written), regardless of their basic meaning, I would be surprised if
the first use isn't considerably earlier than the latter half of the 19th
century. Having never made an effort to locate such uses, however, I do not
have specific knowledge of this. Lambert and Leibniz (who would have
employed it in a sense roughly like that of the use Julio cites from the
review of McColl) are likely places to look, though.
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