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From wikipekia:

The calculus controversy was an argument between 17th-century mathematicians Isaac Newton and Gottfried Leibniz (begun or fomented in part by their disciples and associates – see Development of the quarrel below) over who had first invented calculus. It is a question that had been the cause of a major intellectual controversy over who first discovered calculus, one that began simmering in 1699 and broke out in full force in 1711.

I'm just curious if in the field of mathematics it means one thing to invent and another to discover or if they go totally hand in hand.

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closed as not constructive by Asaf Karagila, Qiaochu Yuan Nov 13 '11 at 21:34

As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, visit the help center for guidance.If this question can be reworded to fit the rules in the help center, please edit the question.

I don't think there is a strict distinction in everyday language, and there isn't one in mathematics either. I'd say an invention is just a certain form of discovery, namely if you discover that doing this-or-that will have such-and-such desirable consequences, then your discovery is an invention. In math, the desirable consequences could be that you can now solve or understand something or other more easily. (Patent law has its own definition of "invention", incomprehensible to man). – Henning Makholm Nov 13 '11 at 2:53
@simplicity I believe this to be a relevant question. Understanding the nature of mathematics is critical. That being said, this question could fit in both philosophy and mathematics, but I believe it fits best here – analysisj Nov 13 '11 at 3:20
@simplicity This is not debate. It is a conversation wherein someone is trying to understand an idea pertaining to mathematics. I would appreciate constructive comments, not insults, especially when they are directed at the original asker. – analysisj Nov 13 '11 at 3:35
It's a philosophical question rather than a mathematical one. But I'm not convinced that we should regard it (as "simplicity" suggests) as "stupid". – Michael Hardy Nov 13 '11 at 4:27
@simplicity: Poppin's? Did you mean Popper? – Asaf Karagila Nov 13 '11 at 5:55

Connes, the Fields medalist, and Changeux, a celebrated neurophysiologist, have had an interesting discussion on that subject.
It is this book.
And here is paper commenting on the book.

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I just want to point to the fact that this is indicative of a somewhat bigger question. Is mathematics simply descriptive of reality or does it exist on its own in a Platonic existence? For instance, was Fermat's Last Theorem true before it was proved by Wiles? Mario Livio wrote an interesting book exploring this question. It is called Is God a Mathematician. He concludes that certain concepts may be invented, such as calculus, but then the results are discovered as inexorable deductions from the invention.

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This isn't Mathematics. This isn't Meta-mathematics. – simplicity Nov 13 '11 at 3:28
Yes, and we have a meta-math tag, @simplicity. Your point being? – J. M. Nov 13 '11 at 4:12

It's not inconceivable that it is possible to rigorously define the concepts discover and invent without entirely loosing what is tried to capture with the intuitive idea. Any platonist would agree that the structure of the integers, say locations of the prime numbers are discovered. However the integers have many isomorphic representations, say set-theoretic and peano axiomatic. One can argue that these representations are invented by man, but the background structure that governs these representations are discovered.

Groups are a good example, they have many isomorphic representations, but the same structure.

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