"Gonthier believes that ordinary mathematicians may start formally verifying their proofs within the decade" according to Julie Rehmeyer in How to (really) trust a mathematical proof .
I have given several false proofs myself. Here is a nice article by my guru John Stallings HOW NOT TO PROVE THE POINCARE CONJECTURE . Excerpts:
"I have committed the sin of falsely proving Poincare's Conjecture. But that was in another country; and besides, until now no one has known about it. Now, in hope of deterring others from making similar mistakes, I shall describe my mistaken proof....
There are two points about this incorrect proof worthy of note.
The first is that when we try to prove Theorem 0 in dimension two, we always run up against the problem of trying to simplify, by some geometric trick, the situation. But any little homotopy that would simplify the picture always in fact, greatly complicates it. This phenomenon has characterized every attempt that I have made or heard of to prove Poincare's Conjecture. This is the place to look for flaws in any asserted "proof".
The second point is that I was unable to find flaws in my 'proof" for quite a while, even though the error is very obvious. It was a psychological problem, a blindness, an excitement, an inhibition of reasoning by an underlying fear of being wrong. Techniques leading to the abandonment of such inhibitions should be
cultivated by every honest mathematician."
The first two sentences in the quote are a take off on lines from "The Jew of Malta" by Christopher Marlowe quoted by T.S. Eliot. Stallings originally wanted to be a poet but took some aptitude tests in school which suggested that he would do well in mathematics or physics.
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