Reference request: is mathematics discovered or created?
Original answer by trutheality:
Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.
-Leopold Kronecker
Translated to English:
God made the integers; all else is the work of man.
It also often appears as "natural numbers".
A quick search online suggests that "ganzen Zahlen" means integers in German. But I don't speak German, so any input from someone who does is appreciated.
Added: (Theo Buehler)
Kronecker's quote is from a talk he gave at the "Berliner Naturforscher-Versammlung" in 1886. I'm not aware of a transcript of this talk. The quote is most often cited in the form in which it appears in the very interesting obituary by H. Weber:
The obituary can be found in the Jahresbericht der Deutschen Mathematiker-Vereinigung Vol. 2, (1891/92), the quote is on page 19.
Here's an attempt at a translation (rather loose):
Concerning the rigor of notions [Kronecker] imposes highest requirements and tries to squeeze everything that should have a right of citizenship in Mathematics into the crystal clear and edgy form of number theory. Many among you will remember the dictum he made during a talk at the 1886 reunion of natural scientists in Berlin ("Berliner Naturforscher-Versammlung"): "God made the integers; all else is the work of man."
I would like to recommend 'The Two Cultures of Mathematics' by W. T. Gowers http://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf
In the setting of this article, personally, I prefer to say, Theory is created, while a solution to a math problem is discovered.
As a physicist who has recently switched to a Mathematics career, I can give you only my opinion based on my experience and knowledge of the Laws of Nature. I do believe mathematics is completely real and is discovered not invented. A similar opinion was held by physicist Richard Feynman, in particular I recommend you watch his old lectures on the Character of the Physical Law, concretely lecture no. 2 about "The Relation of Mathematics and Physics" to appreciate that mathematics seems to be the proper setting to talk about the structures we find in Nature.
If you want to deepen about the mathematical universe hypothesis concerning the (for many crazy) idea that everything is mathematical, see the preprint by Max Tegmark and his other articles in his website.
(This answer contained an excessively long digression about those ideas but I have removed it in order not to contribute to endless debates; only the previous references remain as useful).