Mathematical proof is foolproof, it seems, only in the absence of fools
Putting Proof in Its Place
The law seeks proof beyond a reasonable doubt, but mathematics sets a higher standard. In a tradition that goes back to Euclid, proof is taken as a guarantee of infallibility. It is the flaming sword of a sentry standing guard over the published literature of mathematics, barring all falsehoods. And the literature may need guarding. If you view mathematics as a formal system of axioms and theorems, then the structure is dangerously brittle. Admit just one false theorem and you can prove any absurdity you please.
The special status of mathematical truth, setting the discipline apart from other arts and sciences, is a notion still cherished by many mathematicians, but proof has other roles as well; it's not just a seal of approval. David Bressoud's book Proofs and Confirmations gives what I believe is the best-ever insider's account of what it's like to do mathematics. Bressoud emphasizes that the most important function of proof is not to establish that a proposition is true but to explain why it's true. "The search for proof is the first step in the search for understanding."
And of course there's more to mathematics than theorems and proofs. A genre calling itself experimental mathematics is thriving today. There are journals and conferences devoted to the theme, and a pair of books by Jonathan Borwein and David Bailey serve as a manifesto for the field. Not that practitioners of experimental math want to abandon or abolish proof, but they give greater scope to other activities: playing with examples, making conjectures, computation.
Still, there are ideas that never could have entered the human mind except through the reasoning process we call proof. Which brings me back to the trisection of angles.