FOM: Re: [FOM] L.Kronecker and G.Cantor

William Tait wtait at ix.netcom.com
Mon Mar 15 13:55:58 EST 1999

Alaxander Zenkin wrote:
>
> 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?

Your question echoes a challenge tp Kronecker in a footnote on p.2 of
Dedekind's *Was sind und was sollen die Zahlen?* (1887) He writes

\begin{quote}In what manner this determination is brought about [viz. of
whether or not an object is in a given set], and whether we know a way of
deciding upon it, is a matter of indifference for all that follows; .... I
mention this expressly because Kronecker not long ago (Crelle's Journal,
Vol. 99, pp. 334-336) has endeavered to impose certain limitations upon the
free formation of concepts in mathematics which i do not believe to be
justified; but there seems to be no call tp enter upon this matter with
more detail until the distinguished mathematician shall have published his
reasons for the necessity or merely the expediency of these limitations.
\end{quote}

The reference to Kronecker is to his (1886) Ueber einigeAnwendungen der
Modulesysteme auf elementare algebraische Fragen''. See inparticular the
footnote in the section cited by Dedekind. (I have only Kronecker's
Collected Works and the footnote there is on p. 156 of Vol. III.)

The first sentence of the Dedekind quote reflects a passage in Cantor's
Ueber unendliche  lineare Punktmannigfaltigkeiten'' Nr 3 (p. 150 in the
collected works)

\begin{quote}
I call a manifold (an aggregate [Inbegriff], a set) of elements,
which belong to any conceptual sphere, well-defined, if on the basis
of its definition and in consequence of the logical principle of
excluded middle, it must be recognized that it is internally
determined whether an arbitrary object of this conceptual sphere
belongs to the manifold or not, and also, whether two objects in the
set , in spite of formal differences in the manner in which they are
given, are equal or not. In general the relevant distinctions cannot
in practice be made with certainty and exactness by the capabilities
or methods presently availabe. But that is not of any concern. The
only concern is the internal determination from which in concrete
cases, where it is required, an actual (external) determination is to
be developed by means of a perfection of resources. \citeyear[p.
150]{cantor:32}
\end{quote}

This was published in 1882 and so precedes Kronecker's statement. Yet, I
think that it has to be reacting to some criticism. The example he
discusses is the concept of an algebraic number. In 1874 he had shown that
every interval contains transcendental numbers, with a proof which was not
constructive. Very likely he was at least in part defending this proof.
Incidently, part of Cantor's argument is entirely constructive: He asserts
that, given any *one-to-one* enumeration F of reals, there is a real in any
given interval not in the enumeration. His construction depends only on
being able to decide F(n) < F(m) for all m and n. Since it is given that
F(n) not-= F(m) for distinct n and m, this is entirely constructive. The
non-constructive part of his argument is to go from the existence of an
enumeration of the algebraic numbers to a one-to-one enumeration. (On the
other hand, it seems that there is a constructive one-to-one enumeration of
the algebraic numbers, which he could not have known about.)

Note that Cantor's definition of a set here is very different from his
latter explanations of that concept, beginning in 1883 in Ueber
unendliche  lineare Punktmannigfaltigkeiten'' Nr 5, where he intyroduces
the transfinite numbers. The reason (of which he was completely aware) is
of course that the transfinite numbers constitute a conceptual sphere' in
which it is demonstrably not the case that every well-defined property
determines a set. For example, the property of being a transfinite number
does not determine a set, since he states as a defining condition for these
numbers that every set of numbers has a least upper bound.

> 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?

There is a good discussion of Cantor's illness in Purkert and Iigauds
biography *Georg Cantor: 1845-1918*, pp. 79-92. They say that the first
signs of illness were in 1884.

Regards from (at last) sunny Chicago,

Bill Tait

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