Subject: Re: [HM] the term 'arithmetization'
From: Julio Gonzalez Cabillon (jgc@adinet.com.uy)
Date: Mon Jan 03 2000 - 15:56:11 EST
[Bernd Buldt]
>
> julio cabillon noticed:
Bernd, I suspect you're assuming that "Gonzalez" is my middle name. Please
notice that my (full) surname is "Gonzalez Cabillon".
Hispanic 'Familiennamen' have father's surname (first) and mother's surname
(in second place). This is the custom in Spain and Latin America (except
Brazil). Thus, the "Gonzalez" part of my name comes from my father, whereas
"Cabillon" is my mother's contribution. In Brazil, for instance, the other
way round is the case -- if I were Brazilian, my full name would be "Julio
Cabillon Gonzalez" (accents omitted in all cases).
When the father's surname is uncommon (rare), it is regarded as unnecessary,
and unusual to write our mother's surname here, there, and everywhere (time
and again). Conversely, when the father's surname is very common (eg Garcia,
Perez, Gonzalez, Rodriguez ...), it is convenient and standard to write both
surnames (eg Gabriel Garcia Marquez, Federico Garcia Lorca, Adolfo Perez
Esquivel ...).
In any case, no offense has been taken!
[Julio Gonzalez Cabillon]
>
> PS: Mind that the term "Arithmetisirung" (rather than "Arithmetisierung")
> was the one used by Klein et al.
[Bernd Buldt]
>
> just for those non-native speakers: it doesn't make a difference in
> meaning, its simply that the german orthography has changed since then.
>
Of course, it doesn't make a difference in meaning. German orthography has
changed, but the term "Arithmetisierung" rooted almost at the same time.
My intention (in pointing out Klein's use of the term) was to precise the
spelling of the title of Klein's paper. I am sorry if anyone was mislead
by my PS.
Klein's 1895 address was translated into French by Vassilief (professor at
Kazan University) & L. Laugel, and published in 1897:
"Je parle de l'_Arithmetization des Mathematiques_ (_Arithmetizirung_)"
Here we find another spelling of the term - most probably not intended.
[Mic Detlefsen]
>
> 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.)
As I said, the expression "Arithmetisirung der Mathematik" (as such) was
introduced by Klein in 1895 during his address before the _Koeniglichen
Gesellschaft der Wissenschaften zu Goettingen_. Therefore, I agree with
Abe when he remarks: "There are two terms. One is 'arithmetization' and
the other is 'arithmetization of mathematics'".
[Mic Detlefsen]
>
> 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.
My view, however, is that Kronecker and Klein used the term "arithmetization"
with a different scope and meaning. You put it very nicely when you remark:
"...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').
...
"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."
[Mic Detlefsen]
>
> They did not, however, use it in exactly the same sense. There are, in
> particular, two differences that seem worth pointing out.
And these (at least two) distinctions alone make a great difference to me.
These distinctions seemed to have been quite clear at the time -- "Die
vollkommene Arithmetisirung im Sinne Kronecker's" [i.e. the perfect
'Arithmetisirung' in the Kronecker's sense]. Felix Klein's holistic and
organizational idea of the mathematical sciences, masterly summarised in
his closing paragraph,
"I compare mathematical Science with a tree of which its roots
are, each day, rooted more deeply in the ground, whereas, above,
the branches extend freely and cast their shadows on us. As the
most essential part, do we ought to regard the roots or the
branches? The botanist teaches us that the question is wrongly
posed, and that the life of the organism depends rather on _the
mutual exchange between its various_ parts."
does not seem to have waited for too long in his mind, for Klein's 1893
address was in other direction. Perhaps, Minkowski's first announcements
of his "Geometrie der Zahlen" (during 1893) played an important role within
Klein epistemological position.
Regards to all,
Julio Gonzalez Cabillon
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