Reference request: is mathematics discovered or created?

Question

I have to write a short monograph as an assignment for a course on the philosophy of science. Being a math student, of course I want to opt for something math-related. After some initial ideas which would have needed way too much research, I imagined I could narrow it down to a question which I have always wondered about: is mathematics discovered or created?

I'm thus asking for references to books/papers/quotes/anything which adresses this question. I hope it is not too soft for a math.SE question; I apologize if it is.

In particular, I remember a quote saying something like "Natural numbers were created by God. All else is the work of men", I'd like to know its exact statement and author.

Anything, even if tangentially related, may come in handy. Thank you.

Answer 1

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.

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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."

Answer 2

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.

Answer 3

The operative word is research. It is a search of the truth about things that already exist.

About theories - they require proofs that must be acceptable by fellow mathematicians. The sufficiency of a proof is subjective and varies with the person and time. This itself makes theories more of a hypothesis.

I came across a quote (by someone - I do not remember) - We call the theories we believe axioms and the facts we disbelieve theories.

Answer 4

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.

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(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).

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Answer 5

In his autobiography Un mathématicien aux prises avec le siècle L. Schwartz discusses the question and says that it somewhat complicated. I haven’t the book, so can't cite properly, but the reasoning was something like this. Consider, for example, complex numbers. They can be regarded as human invention. But all their properties then are discoveries.

Answer 6

An excellent discussion of these issues is given by Reuben Hersch in his book What is mathematics, really?. The general message is that mathematics is philosophically "humanist" - it has a socially created reality. This doesn't give much of an idea of what the book is about, but it's about the best account of these sorts of issues that I've seen.

Answer 7

Doug Hofstadter's book Fluid Concepts and Creative Analogies responds to this question. He adopts the metaphor of mathematician as a person feeling around in a dark cave. He feels that mathematicians use their creativity to discover natural truths.

(So, I guess his answer might be "Both"?)