What is a proof?
Yes- there is a big leap when we prove something from everyday axioms. But do they not assume the title of axioms once proved? If they were axioms, what is the need to prove it? Does it not reflect the inability of a person to actually see the ‘obvious’ when he is searching for proofs? Let me be clear that I enjoy a well written and elegant proof and am immensely discontent when I am unable to prove a seemingly obvious statement. But is not the fact that my intuition suggests that the statement has to be true reason enough? You may argue that a proof leaves nothing to chance and gives you a guarantee that the statement has to be true. But when you dissect a proof right down to its base axiom, there you will have to rely on an assumption- that our intuition deems valid.
I am not here to argue on the basis of the correctness in our underlying assumption of the base axiom. Proofs are merely a way to deduce results from a given premise. And the premise here is the truth of the axiom. I am uncomfortable about the fact that people are willing to accept the intuition behind the base axiom but not the intuition behind the results that follow. True, in most cases it is easier to be aware of the former- and the latter may be hard to see as obvious.
A case in point would be Fermat’s last theorem. Andrew Wiles did come up with an absolutely marvellous proof of the theorem- something that puzzled the greatest minds for three and a half centuries. But in those three and a half centuries, what if some person saw it as obvious. Does he/she have to prove it in Wiles’ way or any other way to actually believe in it? Indeed Fermat himself could have been one such person. ‘The truly marvellous proof that this margin is too small to contain’ may have very well referred to a product of intuition that words find hard to explain.
Let a world in which everybody speaks