A proof is some explanation to convincing others that a statement is true (or false in case of a counterexample). As Yuri Manin once wrote: "A proof becomes a proof only after the social act of accepting it as a proof".

Since Mathematics is not comprehensively 'connected' to the nature, I am trying to compare it with another situation: there were many beliefs in history that every body was convinced their truth, it could even be as far as a superstition. But as time passed their validity changed.

My question is: Do we need other type(s) of intelligent creation (like intelligent aliens) for final check whether our proofs are true or not? Or, human mind is enough for checking their truth?

Note - My question is especially focus on non-empirical Mathematics.

EDIT - Also, there have been many mathematical proofs that had very subtle inconsistency detected by smart people; checking Fermat Last Theorem's proof by a colleague of Wiles came up with a subtle mistake is an example; Is it possible that a 'proof' has that type of mistake but not easy enough to 'see' even by nowadays' people for a long long time? any historical story?

PS - migrated from math.SE


1 Answer 1


Neither human nor any other finite mind is enough for true finality, we or them may be delusional, or make subtle errors in logic no matter how many times we verify it, a computer may have glitches when checking the steps, etc. This seems far fetched for simple propositions like the Pythagorean theorem, but not so much for elaborate proofs, Wiles's proof originally had a mistake, proof of the four color theorem could only be done by a computer, and it is suspected that there are still minor flaws and gaps in the multivolume classification of finite simple groups.

Of course, we are fairly certain of our mathematical proofs, the more so the simpler they are. But even that certainty is ultimately empirical, it is based on our confidence in understanding our own workings, just as our certainty that the Sun will rise tomorrow is based on our confidence in understanding the workings of the world. The more sophisticated a mind the more complex proofs it can be confident about. But none of it is the final check, only God can provide that, if he exists.

  • Thank you. Esp. I liked "The more sophisticated a mind the more complex proofs it can be confident about."
    – MKR
    Jun 17, 2015 at 2:10

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