Meaning of Kronecker's comment to Lindemann

Question

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?

Answer 1

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.

[03/2023] In Lindemann's autobiography "Lebenserinnerungen" (p. 88), published in Munich 1971, Lindemann recounts a visit (in spring 1883) to Weierstrass and Kronecker in Berlin. Weierstrass told him that Kronecker wanted to abolish irrational numbers. Then Lindemann writes :

"Accordingly, Kronecker told me that it would be interesting to see how the proof about the number $\pi$ would look like if everything was done with rational numbers alone."

Answer 2

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.

Answer 3

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.

Answer 4

As written, Kronecker's statement [the one quoted by Gerry] is very similar to the idea of doing analysis in "weak" formal systems, such as first order Peano arithmetic, primitive recursive arithmetic, hereditarily finite ZF, or based on informal principles that roughly correspond to such systems. That is, "completed infinite" constructs (such as the totality of Cauchy sequences, or a single unspecified Cauchy sequence, or manipulations of infinite sets) may have a nebulous status but are OK as a heuristic or a formalism as long as in each case, such as formulas involving $e$ and $\pi$, the resulting manipulations are seen to ultimately reduce to calculations that can be performed and proved in PA, or whatever the acceptable theory of finitary constructions.

This is interpreting the posted question, "what might have been the intended meaning of Kronecker", as what is a sensible reading of Kronecker's statement, and not as the historical question of what he truly, originally, demonstrably did intend.

(Added: to put this another way, Kronecker might say that $\pi$ is fine as a formalism for organizing finite computations about explicit finitely presented objects such as algebraic numbers or matrices with integer entries, but that $\pi$ by itself is more ontologically dubious. Transcendence proofs, from this perspective, are a sequence of estimates about finite integer calculations, similar to irrationality estimates of $\sqrt{d}$ using continued fractions, which are interpretable as Diophantine inequalities between positive integers. There is some sense to this insofar as, even with a modern theory of algorithmic objects --- finite computer programs such as those computing $\pi$ approximations --- $\pi$ is inevitably a higher-type object than integers; verifying any given integer formula such as 2+2=4 is a finite calculation but $e^{i \pi} = -1$ requires some form of induction, no matter how explicitly one constructs $e$ and $\pi$.)

Answer 5

This story was referenced by Lindemann itself in notes to German edition of Poincare's "Wissenschaft und Hypothese" of 1904. It was mentioned in the context of explaining Kronecker's point of view that irrational numbers are only auxiliary tool for studying geometry and mechanics, so algebra and analysis can be constructed without any reference to irrational numbers. Quote you mentioned is just short an radical description of this idea.

You can read the original Lindemann note (number 3, p. 246) itself and correct me.

Answer 6

Doron Zeilberger makes similar statements and has written about them in some detail. Zeilberger seems to take a stronger position, denying the existence of some integers too.

Answer 7

For the Kronecker's views, Brouwer's intuitionism and much more real motivations for Hilbert's program than the usual tales about paradoxes as main reasons of crisis (they are only a pretext and the reasons go back to 19th century reincarnation of mathematics) see for example

Jeremy Avigad and Erich Reck "Clarifying the nature of the infinite: the development of metamathematics and proof theory".

You may also look at many other important historical works referenced there.

Answer 8

There is some sense in socalled ultrafinitism (the believe or ideology that not only there is no infinite set of all integers, but that there are just finitely many integers). After all, there is an integer that is larger than any (concrete) real number that will ever be considered by a human being.

My guess would be that Kronecker did actually deny the existence of $\pi$ as a real real number (the two "real"'s are intended). He might have conceived $\pi$ as the definition of something like a number that we can approximate as precisely as we wish. This view corresponds to the view that the set of integers does not exist but every individual integer does (the actual infinite versus the potential infinite).