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The familiar quote is often incorrectly attributed to Kronecker directly. Actually a colleague of his named Weber claimed after Kronecker's death that Kronecker said this. I have doubts about this because Kronecker would not have used the term "integer". He was almost as suspicious of the negative numbers as he was of transcendental numbers. Furthermore he specifically wrote that the numbers are a creation of the human mind, implying a measure of contingency not conveyed by Weber's quote at all.

Note. Thanks to Colin McLarty for providing the appropriate quote, this one literally from Kronecker.

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    $\begingroup$ Why not include the "familiar quote"? It would be gentler to lay readers. $\endgroup$
    – Rodrigo de Azevedo
    Apr 27, 2022 at 5:31

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Weber was a pretty reliable witness. Probably Kronecker did tell the 1886 Berliner Naturforscher-Versammlung something like "the whole numbers were made by dear God (der liebe Gott), the rest is the work of man."

But Kronecker also published his endorsement of Gauss saying the opposite:

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)

It is extremely unlikely that Kronecker or Weber intended the remark as anything serious about theology.

The late Walter Felscher has pointed out on the list HistoriaMatematica (26 May 1999) archived at mathforum.org that:

"Lieber Gott" is a colloquial phrase usually used only when speaking to children or illiterati. Addressing grown-ups with it contains a taste of being unserious, if not condescending. . . ; no priest, pastor, theologian or philosopher would use it when expressing himself seriously. There is the well known joke of Helmut Hasse who, having quoted Kronecker's dictum on page 1 of his yellow Vorlesungen uber Zahlentheorie (1950), added to the index of names at the book's end under the letter L the entry" Lieber Gott p. 1."

Kronecker did seriously aim to replace the algebra of irrational numbers with the "pure arithmetic" of natural number polynomials.

As to mutability and contingency, there seem to be two questions: Would numbers exist if people (or God) had not thought of them? Could some other version of numbers have been created instead of the ones we have? I know no evidence that Kronecker ever considered either question -- or that he did not!

As to the second, compare Dedekind. He was explicit that the natural and real numbers are both our thought-creations yet he never breathed a word of any idea that we could have created them differently than we did.

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  • $\begingroup$ Colin, Unlike the other quotation whose authenticity is questionable, the quote you provide directly from Kronecker shows that Kronecker viewed number as a product of the human mind. Weber's alleged quote on the contrary makes a distinction between the work of man and something else that includes number. Therefore the two quotes would seem to contradict each other, whatever the L entry may be in Hasse. My point is that Kronecker should be more reliable on Kronecker than Weber. Incidentally do you have a source for the English translation of the Uner den Zahlbegriff? $\endgroup$
    – Mikhail Katz
    May 15, 2016 at 15:45
  • $\begingroup$ @MikhailKatz If Kronecker had some serious view on the divine versus human origin of the integers, then the evidence points to human. But then either Weber made a gratuitous mistake, or he willfully falsified Kronecker's view, or Kronecker changed his mind between 1886 and 1887. I believe to the contrary Kronecker had no serious interest in theology. And notice he does not declare his faith that the numbers are created by us. He only approves of Gauss saying it. $\endgroup$
    – Colin McLarty
    May 15, 2016 at 17:25
  • $\begingroup$ Colin, this is not an issue of "theology" so much as an issue of whether Kronecker attached an immutable status to the natural numbers, or on the contrary thought that there was possibly a degree of contingency implied by his comment about them being man-made. I edited the question accordingly. $\endgroup$
    – Mikhail Katz
    May 15, 2016 at 17:30
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It seems Kronecker actually said

Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk

and did not mention "integers" or any other English word.

citation ... Speech at the Berlin meeting of the Society of German Scientists and Doctors in 1886, quoted by Weber in Jahresbericht der Deutschen Mathematiker-Vereinigung 2 (1893), p. 19.

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    $\begingroup$ "ganze" means "whole" and does not refer to natural numbers. The link you provided states clearly that this is a claim by Weber which Weber attributes to Kronecker. There is no published text by Kronecker himself making such a statement, and with good reason: Kronecker is not likely to have subscribed to it. $\endgroup$
    – Mikhail Katz
    May 12, 2016 at 14:50
  • $\begingroup$ I read the link to say that the quote is in the Jahresbericht for 1886, and the translation is by Weber for the obituary in 1893. To tell for sure we need someone whose library has Jahresbericht 1886. (I doubt it has been digitized for the Internet.) $\endgroup$
    – Gerald Edgar
    May 12, 2016 at 14:55
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    $\begingroup$ Gerald, your link is not the unique source on Weber's comment. It is well known that Kronecker never published anything of the sort. A good source is Y. Gauthier's recent book, where he says the same thing. $\endgroup$
    – Mikhail Katz
    May 12, 2016 at 15:04
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The mathematician Adolf Kneser (1862 - 1930), who in 1884 did his doctorate with L. Kronecker and E. E. Kummer in Berlin, reported on the occasion of Kronecker's 100 birthday in a speech on 19. December 1923 at the Mathematical Society of Berlin from his personal memory:

Da trat eines schönen Tages auch Kronecker auf dieses Gebiet über und tat den Ausspruch: Die ganzen Zahlen hat der liebe Gott gemacht; alles andere ist Menschenwerk.

(Adolf Kneser: Leopold Kronecker. Jahresbericht der Deutschen Mathematiker-Vereinigung 33 (1925) 210-227, p. 221.)

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    $\begingroup$ Kneser's "memory" may very well have been influenced by Weber's account quoted by @GeraldEdgar. $\endgroup$
    – Francois Ziegler
    Jun 4, 2017 at 20:28
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    $\begingroup$ Note that Weber said: "As many of you (the audience) will certainly remember, Kronecker said at the Berliner Naturforscher-Versammlung 1886 ...". If none had remembered this, certainly some questions would have come up. $\endgroup$
    – Franz Kurz
    Jun 5, 2017 at 20:58
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To me as a German and after reading all your posts and uncertainty I would explain it like this and I base this on the truth of what you have written:

  • On a meeting in a speech Weber tried to remember people about what somebody else said. It can be very well that Weber just added the word "lieber" to "Gott". Remembering somebody on something, with an ironic vain (which a lot of mathematicians have), is somehow like to talking to a child. Here I would like to mention the citation of Walter Felscher in the first entry by Colin McLarty.

If it is the case that Kronecker said the sentence without the above word, one, as a German native, cannot deduce with so much certainty which part of the sentence should be stressed. So then one could not deduce the intentions of Kronecker, but rather the intentions of Weber as to see in the next paragraph.

  • Let us now analyze with an native linguistic feeling the above sentence: "Die ganzen Zahlen hat der lieber Gott gemacht, alles andere ist Menschenswerk."

-My claim is that the stress lies on the second half "the rest is done by human work".

Proof: Suppose the contrary. The stress was on the first part. The grammatical structure of the sentence is easy and consists of object, subject, predicate; in this ordering. The object, the "whole" numbers are not new to the listener; they were common knowledge; they are not a new invention. So the stress does not lie on the "whole" numbers. They are common knowledge, so if the stress was there and he meant something else he would had to redefine the term "whole" numbers by specifying it in a subordinate clause.

Also with the aid of native language feeling if I wanted to stress the object I would put it to the end of the sentence. I would rather say: "Der [liebe] Gott hat die ganzen Zahlen [gemacht]". But I would even then use the word "erschaffen" which is like "create" rather than "made=[gemacht]".

So we see that the information to be distributed if the stress lied on the first half of Kronecker's sentence would then be the predicate and subject. But the predicate is simple. It is only "has made". It does not contain any information. I would use as said before the word "create" if the information shall be put in the predicate. Hence, what rests is the subject "[lieber] Gott" - "lovely god". But natives in German would then stress the subject by pronouncing it more so to make clear the information to be sent. But this sounds ridiculous if I say : "Die ganzen Zahlen hat der liieeeebe Gott gemacht" where this is to be translated as " the whole numbers were made by the looooveely god". It does sound ridiculous because I would never talk about god seriously with the adjective "liebe(r) - lovely". If I want to stress somebody in the sentence I would not try to harm the subject or make it childish by using the word "liebe(r)" to the subject. Here I again want to make reference to the citation of McLarty about somebody using the word "liebe(r)". Another argument, I would even change the ordering of the sentence as before so to put the subject in front and then say what he/she did.

So we see that the stress really lies on the second half of the sentence. -End of proof.

  • In the commentaries somebody said that it is known Kronecker to have approved/accepted the dictum by Gauss which contains the information that numbers are product of human mind. If this is so then it even becomes more clear that Kronecker really wanted to say that the "rest is done by human".

  • The sentence has the structure: even ..... is true, we still have .....

In German one puts the stress/ or main information normally at the end of the sentence. (The only exception I can think of is the subject which one puts in front if one wants to stress it - ** the second half of the sentence contains no subject because a general law or fact is stated. "Something is done by..." The subject here has become an object "by ...")

The end of the sentence is that 'the rest is done by human kind'.

  • As result I would say:

If Kronecker really said it like this then I see it in the following ironic manner:

My dear friends, remember shortly that we can be lucky that lovely god has given as at least some numbers to be existent, but the real thing is that the rest is done by human work! (stress on the second part)

So I don't believe the sentence to be about God. It only uses an existence property related to God as an assumption.

  • The term "ganze Zahlen" (whole numbers):

As said before the stress might not lie in this part of the sentence. The "whole numbers" are to be seen as common knowledge. But common knowledge for whole numbers back then was integer numbers I guess. In old texts people use the words "ganze Zahlen" as "integer numbers".

Still, "ganz" can also be meant in German in the sense of "vollkommen" which is like "perfect" or "not extendible".

I personally do not believe he meant only natural numbers in this sentence, but well, I actually don't know much about Kronecker. This is just a linguistic attempt to the meaning of sentences.

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    $\begingroup$ (1) Do you have any background in constructive mathematics? (2) Have you read Schappacher's text on Kronecker? $\endgroup$
    – Mikhail Katz
    Feb 10, 2018 at 19:28
  • $\begingroup$ Hey Mikhail, no, I am sorry, I dont have any background in constructive mathematics. I had some thoughts and I wanted to share them. I havent read Schappacher's text but I will soon. I also dont know much about Kronecker actually. I only tried to analyze the sentence linguistically. Maybe it helps maybe it does not. I edited the way of such certainty I wrote the text. Maybe this is disturbing for somebody else. If you have some good references about Kronecker's religous philosophy I would like to read them. In the future I want to dive a little bit into his collected works.. $\endgroup$
    – Robin
    Mar 16, 2018 at 17:25

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