Q:

Meaning of Kronecker's comment to Lindemann

At the Mactutor history page, it is said that Kronecker remarked to Lindemann:
"What good your beautiful proof on [the transcendence of] π? Why investigate such problems, given that irrational numbers do not even exist?"
It is also folklore that he remarked "God made the integers, all else is the work of man" and the above quotation becomes intriguing in this light. What might have been the intended meaning of Kronecker? Surely, he does not literally mean that π does not exist? Surely, he doesn't believe that convergence and real numbers do not make sense?

A:

This story is apocryphal. In my review of Dauben's article The battle for Cantorian set theory, I wrote that Dauben repeats the well known claim that Kronecker, in a lecture at the Berliner Naturforscher-Versammlung in 1886, called Lindemann's proof of the transcendence of $\pi$ worthless since irrational numbers do not exist. The earliest references he offers for supporting this claim are Weber's article in [Math. Ann. 43, 1--25 (1893; JFM 25.0033.03)] and Kneser's [Jahresber. DMV 33, 210--228 (1925; JFM 51.0025.03)]; both articles, however, only give Kronecker's equally famous statement that God made natural numbers; all else is man's work. For a more historical evaluation of Kronecker's ``rejection'' of irrational numbers, see H. M.~Edwards [Essays in constructive mathematics (2005; Zbl 1090.11001)], as well as J.~Boniface and N.~Schappacher [Rev. Hist. Math. 7, 207--275 (2001; Zbl 1014.01003)]. 
Let me also remark that in his lectures on number theory published by Hensel (1901; JFM 32.0184.06), Kronecker gives the Leibniz series for $\pi/4$ and writes that this definition of the transcendental number $\pi$ is actually of a number theoretic character.

A:

I don't have a good answer to this question, in fact I don't think there is a good answer, because Kronecker did not always oppose the use of irrational numbers. For example, in his 1863 paper "Über die Auflösung der Pell'schen Gleichung mittels 
elliptischer Functionen" (Werke, vol. IV, pp. 221--225), Kronecker
explicitly uses the numbers
$\frac{2}{9}e^{\frac{5}{18}\pi\sqrt{17}}$,
$\frac{2}{49}e^{\frac{17}{42}\pi\sqrt{97}}$, and
$e^{\frac{1}{20}\pi\sqrt{85}}$.
Addendum. Kronecker did not stop having fun with irrational numbers in 
his old age, either. His 1889 paper "Zur Theorie der elliptischen Functionen"
(Werke, vol V, pp. 1--132) is full of fancy expressions involving $e$, $\pi$,
and square roots like those above.

A:

The biography of Kronecker at http://www-history.mcs.st-andrews.ac.uk/Biographies/Kronecker.html gives some more information about Kronecker's beliefs. Here's one paragraph: 
Although Kronecker's view of mathematics was well known to his colleagues throughout the 1870s and 1880s, it was not until 1886 that he made these views public. In that year he argued against the theory of irrational numbers used by Dedekind, Cantor and Heine giving the arguments by which he opposed:-
... the introduction of various concepts by the help of which it has frequently been attempted in recent times (but first by Heine) to conceive and establish the "irrationals" in general. Even the concept of an infinite series, for example one which increases according to definite powers of variables, is in my opinion only permissible with the reservation that in every special case, on the basis of the arithmetic laws of constructing terms (or coefficients), ... certain assumptions must be shown to hold which are applicable to the series like finite expressions, and which thus make the extension beyond the concept of a finite series really unnecessary.
EDIT: J. Pierpont, Mathematical rigor, past and present, Bull Amer Math Soc 34 (1928) 23-53, freely available on the AMS website, says Kronecker first made his views public in 1882. See especially pages 37 to 39. 

