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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.

Rodrigo de Azevedo
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user51309
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1 Answers1

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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.

Mauro ALLEGRANZA
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  • 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. – Mikhail Katz Nov 02 '17 at 15:53
  • @MikhailKatz based on this it is probably not apocryphal. – Spencer Aug 01 '20 at 17:09
  • @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. – Mikhail Katz Dec 05 '22 at 09:17
  • @MikhailKatz So, assuming Weber is correct in claiming Kronecker made the famous quote, how do you reconcile the two? – Spencer Dec 05 '22 at 16:23
  • @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. – Mikhail Katz Dec 05 '22 at 19:47
  • Mauro, you claim that Kronecker's statement "I consider mathematics only as an abstraction of the arithmetical reality" (where is this from by the way?) expressed what you claim to be "the same idea" as the comment reported by Weber. Could you elaborate on why you think they are the same? The latter could be interpreted as a Platonist statement, but the former is not necessarily so. – Mikhail Katz Dec 05 '22 at 19:59