5
$\begingroup$

It is said here that

In 1874 Cantor published an article in Crelle's Journal which marks the birth of set theory. A follow-up paper was submitted by Cantor to Crelle's Journal in 1878 but already set theory was becoming the centre of controversy. Kronecker, who was on the editorial staff of Crelle's Journal, was unhappy about the revolutionary new ideas contained in Cantor's paper.

However the authors of the entry don't elaborate on reasons why Kronecker was so unhappy, which prompted my curiosity.

Improve this question
$\endgroup$
4
  • 2
    $\begingroup$ Kronecker wasn't willing to accept a .... delta .... to his own theory. // rim shot $\endgroup$
    – Carl Witthoft
    Nov 1, 2017 at 12:17
  • 1
    $\begingroup$ You can see the post did-kronecker-say-that-set-theory-is-not-mathematics for links to Kronecker's attitude with set theory. $\endgroup$
    – Mauro ALLEGRANZA
    Nov 1, 2017 at 18:13
  • $\begingroup$ @MauroALLEGRANZA I can't access the book supplied in the link in mention, unfortunately. So still waiting for an answer, a concise one perhaps. $\endgroup$
    – user51309
    Nov 2, 2017 at 2:49
  • 1
    $\begingroup$ Another consideration for the answer may be the dispute among the editors of Crelle's Journal, leading up to the expulsion of Kronecker from the board. $\endgroup$
    – Gerald Edgar
    Nov 2, 2017 at 14:31

1 Answer 1

Reset to default
7
$\begingroup$

About Kronecker's views on "foundational" issues, you can see:

There is no "explicit" philosophy of mathematics in Kronecker's works.

He may be regarded as a "constructivist", or perhaps as a precursor of the finitist approach.

In his essay "Uber den Zahlbegriff" (1887) he outlined the project of "arithmetizing" Algebra and Analysis; that is, to found these disciplines on the fundamental notion of number, avoiding thus geometrical intuition.

In his arithmetization project he considered a mathematical definition acceptable only if it could be checked in a finite number of steps, criticizing the "pure" existence proofs. He stated that an existence proof for a number could be considered correct only if it contained a method to find the number whose existence was proven.

Some of his remarks belong to mathematical folklore, like his widely reported statement that:

"the Lord made the natural numbers (ganze Zahlen), everything else is the work of men";

the same idea is reiterated in the following statement:

"I consider mathematics only as an abstraction of the arithmetical reality".

Note: for some references to Kronecker's opposition to Cantorian set theory, we can see José Ferreirós, Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics, Birkhäuser (2007), sub voce.

In conclusion, it seems that Kronecker never published his critique of Cantor's theory, but we can conjecture from the sources above that he disliked Cantor's new theory of the "arithmetics of the infinite" as well as Cantor's wide use of "pure" existence proofs.

Improve this answer
$\endgroup$
6
  • $\begingroup$ You may want to clarify that the "widely reported statement" may be apocriphal as we have no document by Kronecker containing such a statement. Rather, it was reported by Weber after Kronecker was safely underground. Kronecker scholars have raised doubts about the accuracy of Weber's report. $\endgroup$
    – Mikhail Katz
    Nov 2, 2017 at 15:53
  • $\begingroup$ @MikhailKatz based on this it is probably not apocryphal. $\endgroup$
    – Spencer
    Aug 1, 2020 at 17:09
  • $\begingroup$ @Spencer, the page you linked provides the following quote from Kronecker: "The principal difference between geometry and mechanics on one hand, and the other mathematical disciplines we comprehend under the name of \arithmetic, consists according to Gauss in this: the object of the latter, number, is a pure product of our mind, while space as well as time has reality also outside of our mind which we cannot fully prescribe a priori. (Kronecker, Uber den Zahlbegriff, 1887, p. 339)" This quote is not apocryphal, and goes in the opposite direction of Weber's phrase. $\endgroup$
    – Mikhail Katz
    Dec 5, 2022 at 9:17
  • $\begingroup$ @MikhailKatz So, assuming Weber is correct in claiming Kronecker made the famous quote, how do you reconcile the two? $\endgroup$
    – Spencer
    Dec 5, 2022 at 16:23
  • $\begingroup$ @Spencer, we have on the one hand a direct quotation from Kronecker expressing anti-Platonist views, and on the other a report after Kronecker's death by Weber, phrased in reportedly "condescending" terminology (see the comment by Felscher at the post you linked). The choice is yours. $\endgroup$
    – Mikhail Katz
    Dec 5, 2022 at 19:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.