FOM: L.Kronecker and G.Cantor

[Michael Detlefsen]

Alexander Zenkin asks the following questions re. Cantor and his
relationship with Kronecker.


1) It is known that L.Kronecker had a very negative attitude toward
G.Cantor's set theory. What is known about mathematical argumentation of
L.Kronecker's objections?
2) There is a version (unfortunately, I can't recall a reference) that
just such the L.Kronecker attitude became the main reason of G.Cantor's
illness. But L.Kronecker passed away in 1891. So, it is quite difficult
to trust in the reliability of such the reason. What versions are there
as to the true reason of G.Cantor's illness?
3) Is anything known as to G.Cantor told or wrote apropos of his own set
theory after 1899?

Question 1 is very broad, but I would refer you to the correspondence with
Hermite for one interesting reference (cf. Meschkowski's well-known work on
the life and work of Cantor for the texts). One of the trademarks of K's
position was the special place granted to the whole numbers. Hermite took a
position on this that was both in agreement and disagreement with what K
held. Hermite wrote:

"The numbers seem to me to be constituted as a world of realities which
exists external to us and which has the same character of absolute
necessity as the realities of nature whose understanding is given to us by
our senses."

In one respect, this is in direct opposition to what K (expressly following
Gauss, cf. K's 'Ueber den Zahlbegriff', 1887 and Gauss' 1817 letter to
Olbers and 1829 letter to Bessel) had stated--namely, that it was space
that has an external reality and that numbers were "purely a product of our
intellect"--thus making the epistemic character of geometry and arithmetic
fundamentally different. In another respect, however, it is in agreement
with K's views since it gives the arithmetic of the whole numbers a
methodologically priveleged place in mathematics.

Cantor went farther--than both Gauss and K--saying that the reality of the
natural numbers was greater even than the existence of natural objects.
They exist, he said in response to Hermite, "at the highest level of
reality as eternal ideas in the Intellectu Divino". He had taken a somewhat
more cautious stance in his 1869 Habilitationsschrift saying that "Integer
numbers constitute a unity composed of laws and relations in a manner
similar to those of celestial bodies." I say 'more cautious' because I'm
not sure how Cantor understood the celestial bodies and their being. He
sometimes said things, however, that suggest a view giving them special
status in God's creative intellect. (For more on this, see Dauben's
interesting 'G. Cantor and Pope Leo XIII'.) The view is interesting because
it represents an attempt not only to accommodate K's (and Gauss') view, but
indeed to extend and strengthen it ... though not in ways that would
necessarily have met with K's approval.

Re. 2, the work you refer to but can't remember is probably Fraenkel's
bigoraphy of Cantor published in the 1930 volume of the Jahresbericht der
Deutschen Mathematiker Vereinigung. Schoenflies also wrote a paper ('Die
Krisis in Cantors mathematischen Schaffen', Acta Mathematica 1927) which
stressed the effects of K's criticisms on Cantor. A more balanced, though
not in all respects more convincing, view is given in Dauben's 'George
Cantor's Creation of Transfinite Set Theory: Personality and Psychology in
the History of Mathematics', Annals of the New York Academy of Sciences,
1979.

Re. 3, I'm not sure what you have in mind, but there is certainly
correspondence with Graf, Klein, Jourdain, G.C. Young and others from 1899
and after ... and in places it touched on his ideas in set theory.

Hope this is of some help ...

>From momentarily sunny (but colder than Bill Tait's Chicago) South Bend ...

Mic Detlefsen



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Michael Detlefsen
Department of Philosophy
University of Notre Dame
Notre Dame, Indiana  46556
U.S.A.
e-mail:  Detlefsen.1 at nd.edu
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