Is there any difference between a math invention and a math discovery? 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.
 A: 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.
A: 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.  
A: 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.
