Mathematics springs from creative acts of the human imagination; yet at the same time the creativity of the mathematician is constrained by *the fact of the matter.* It is not up to the mathematician whether there are infinitely many prime numbers – either there are or there aren’t, and thanks to one of Euclid’s theorems, we know that there are; in fact, it is even provable.

About provability, much is known. For example, many mathematical proofs can be mechanized, that is, checked by a computer. In fact, one could imagine a completely automated practice, where one could theoretically build, say, a Turing Machine, into which one could input any mathematical conjecture and the machine would output a definite answer, yes or no, or true or false, in a finite amount of time.

One way of describing the Incompleteness Theorems (1931) of the Austrian logician Kurt Gödel is to say that he proved, in the form of a mathematical theorem, that the possibility of a fully automated mathematics can never be realized.

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