LeoYard said:
Do you have a linked article to mathematical proof? I'm looking for an article that explains the importance of proofs in mathematics.
Mathematics: A Very Short Introduction by Timothy Gowers Chapter 3 is very good on this. You might be able to link into p.36 on the Amazon look ahead, Google books, or Questia. It was so good I took notes. Here's the gist:
Mathematicians demand proof, an argument that puts a statement beyond all possible doubt.
[My aside: This is obviously important. Knowing something "for certain" is obviously better than having some doubt about it!]
But although one can often establish a statement as true beyond all reasonable doubt, how can one claim that an argument leaves no room for doubt?
The proof that √2 is irrational is as follows.. [Gowers gives the steps of the standard "quick" proof]
Each step seems obviously valid. But is there really no room for doubt? [p38-39 Gowers adds some more steps to anything that seems shaky] But is there
still really no room for doubt?
The steps of a mathematical argument can be broken down into smaller and more clearly valid substeps;
“this process eventually comes to an end… you will end up with a very long argument that starts with axioms that are universally accepted and proceeds to the desired conclusion by means of only the most elementary logical rules (such as ‘if A is true and A implies B then B is true’).” [p.39]
Call this "formal proof".
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[This is me, not Gowers!: When I take notes I add any controversial brainstorms I might have, here's one on how physicists might deal with the mountain of mathematical proof that they might think they should master, and for those who feel guilty about not having read Spivak (a very informal text anyway by the standards of Bourbaki):
Even the simplest proofs, like proving √2 is irrational, can proceed to hundreds of steps if they are formalised in this fashion. It makes the learning of mathematics very difficult if you attempt to follow proofs of everything. The solution for the physicist is: don’t bother with proofs, just accept the conclusions. Instead of following hundreds of lines of dense formalism you can read one line of natural language that says “√2 is an irrational number”. You can trust this statement, if you trust that mathematicians are doing their job. And why shouldn’t you? You trust the engineers that built your television. Unless you are an electronics wizard, you simply use it and do not enquire into the finer details of its construction before you trust yourself to use it.]
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Anyway, enough of my ramblings, and back to notes on Gower. He goes on to say that formal proof, "was one of the great discoveries of the early 20th century, due to Frege, Russell, and Whitehead... This means that any dispute about the validity of a mathematical proof can always be resolved. "
"In the 19th century there were genuine disagreements about matters of mathematical substance… if there is disagreement about whether a proof is correct, it is either because the proof has not been written in sufficient detail, or because not enough effort has been spent on understanding it and checking it carefully.”
“That disputes can in principle be resolved does make mathematics unique. There is no mathematical equivalent of astronomers who still believe in the steady-state theory of the universe...”
No mathematician would bother to write out a proof in complete detail—as a deduction from basic axioms using only the most utterly obvious steps. Mathematical papers are written for highly trained readers who do not need everything spelled out. But if somebody makes an important claim, and other mathematicians find it hard to follow the proof, they will ask for clarification, and the process will then begin of dividing steps of the proof into smaller, more easily understood substeps.
[Notice. This means most mathematicians never follow the very highest standard of proof!]
Why should one accept the axioms proposed by mathematicians? Why should one accept the principle of mathematical induction? “First, the principle seems obviously valid to virtually everybody who understands it. Second, what matters about an axiom system is less the truth of the axioms than their consistency and their usefulness.”
Gower warns those seeking enlightenment from proof: "A common experience is to come to the end of a proof and think, ‘I understood how each line followed from the previous one, but somehow I am none the wiser about why the theorem is true, or how anybody thought of this argument’." [p.41]
Great stuff. The rest of the book is just as good (with lots more to say about proof) and dirt cheap, so forget the link and tell people to buy it and eat sandwiches for two days to pay for it.