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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.

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"God made the natural numbers; all else is the work of man." -- Leopold Kronecker – InterestedGuest Jun 25 '11 at 21:49
As a Math student, I think it is best if you don't do a project on Math. – jspecter Jun 25 '11 at 22:04
This exact question is addressed at… – outsider Jun 25 '11 at 22:28
@Bruno: No, I mean research in the sense of "library research" not "mathematical research" or "original research". The fact that the course is required and that you don't seem to be thrilled about taking it is not really an adequate excuse for asking the internet mathematical community to help you write it. I am voting to close. – Pete L. Clark Jun 26 '11 at 0:37
I really don't understand the votes to close this question as off-topic - this is definitely a question relating to the history and development of mathematics. I therefore vote against closing following this suggestion here. The next user who wants to cast a vote to close should leave a comment cancelling my vote instead of voting. (please vote this comment up so that it appears above the "fold") – t.b. Jun 26 '11 at 9:55
up vote 14 down vote accepted

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:

Excerpt from Weber's obituary

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."

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I don't understand the down-vote. It answered what the OP asked, and it's a great answer in its own right, so I'm up-voting this answer. – Mike Jones Jun 25 '11 at 21:54
By "Die ganzen Zahlen" one indeed means $\mathbb{Z}$, while $\mathbb{N}$ is most often called "die Menge der natürlichen Zahlen". The word "ganz" is synonym for "komplett", "unbeschädigt" which you could translate as "integer". (You never say "die unbeschädigten Zahlen", though.) – leftaroundabout Jun 25 '11 at 23:19
@leftaroundabout Thanks. Feel free to add that by editing the answer if you'd like. – trutheality Jun 26 '11 at 0:54
@leftaroundabout: What you're saying is true in nowadays usage, but I'm not sure that this applies to the present discussion (lacking context). – t.b. Jun 26 '11 at 10:41
@Bruno: I added a loose translation of the passage I displayed. I attempted to retain the colorful language, I hope you can follow the idea. – t.b. Jun 26 '11 at 20:46

I would like to recommend 'The Two Cultures of Mathematics' by W. T. Gowers

In the setting of this article, personally, I prefer to say, Theory is created, while a solution to a math problem is discovered.

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I fixed your link. I know of this article and have read it before (and liked it a lot). It is of course related. Thank you. – Bruno Stonek Jun 25 '11 at 22:06

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).

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I can't claim to have read your entire post in detail, but I would like to respond to the claim "Anything we will ever be able to say about Nature will be mathematical in the end." This obviously cannot be known - while it has been our experience so far that natural phenomena can be described mathematically, there is no reason why the universe should be able to be described by math, or indeed by anything. As Einstein said, "The most incomprehensible thing about the world is that it is at all comprehensible", and we can never rule out that the universe is not completely comprehensible. – Zev Chonoles Jun 26 '11 at 5:13
Whereas there are lots of reasons to "believe" (scientifically by induction) that Nature is indeed mathematical (name every single structure and mathematical law I mentioned and all the rest), there is NO single reason to believe the contrary except the logical possibility of it. All the evidence supports the claim so far. We cannot be sure, of course, but THAT IS SCIENCE, as we cannot be sure that a relativistic black hole passes through the solar system and there is no rising sun tomorrow, all the evidence says it is quite improbable... – Javier Álvarez Jun 26 '11 at 5:20
Besides, there is a school of mathematical thought which says that natural languages can be reduced to mathematics in the end. Any information-theoretic construction that we made, be it in mathematics or in any other language, is in the end some kind of structure within the correlations of the information in our brains and the degrees of freedom perceived from the outside world. Since everything that can be said about Nature must be said in some language, everything that an inside observer will ever say will be an information-structure, and thus reducible to formal systems and mathematics. – Javier Álvarez Jun 26 '11 at 5:23
@Zev Chonoles: as I said in the post, it is my opinion based on my experience and knowledge and I hoped not to open a debate. To rebut some of your claims: most mathematicians blind themselves to the scientific method because they want complete truth, but the example of physics gives you the answer that INDEED Nature is described by Math to the level of precision we have today. The amount of progress made by that knowledge PROVES it is described at some level by math. – Javier Álvarez Jun 26 '11 at 5:33
@Zev Chonoles (cont'): We can never rule out anything? are you sure? I though science made advances because of falsifiable models of Nature. As Feynman says in the lectures I linked to, "we can only be sure about what is false". I challenge anybody to defy gravity and jump a cliff... since you could not be sure if it starts repelling that precise moment. Some things are true and those are here to stay. One of them is the usefulness of mathematics in the natural sciences... of that you can be sure. – Javier Álvarez Jun 26 '11 at 5:38

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.

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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.

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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"?)

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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.

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Thomas Kuhn's book "The History of Scientific Revolutions" is basically a treatise on precisely this question. He specifically takes up this conundrum of discovery versus invention in terms of the discovery/invention of oxygen.

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