Math Proofs
2001-11-11 by GAmoore@aol.com
>A testimony to this is the fact that _any_ half-decent mathematician >knows what's meant when talking about a "beautiful proof". The proof >of the 4-colour-map theorem (or whatever it's called in English) is >generally regarded to be an ugly proof, and people actually spend Thats about right. "Four Color Theorem" I think. It was proven about 1976 by a team of mathematicians at the University of Illinois, using computer programs involving some 1,700 cases as I recall and could only be verified by other programs. That is not eloquent or beautiful. >time to come up with a more esthetically satisfying one -- even >though the plain _fact_ of the theorem has already been proven. To a >"layman" this most often seems absurd. A proof is a proof, and it's >all just boring stuff anyway, so why bother? There are over 100 proofs of Pythorogus's theorem. But even a shorter or better proof of a known result can be considered original. The "Fundamental Theorem of ALgebra" was Gauss's PhD thesis. During his life he published several different proofs of this theorem. However, its just a few lines in complex variables (a class taken by most junior/senior college math majors). >No, this is not true. What constituted a proof 500 years ago still >is a proof today. What has changed is the scope of the field, and >hence the amount of "tools" you can use to construct a proof. The >concept of what a valid proof is though has remained unchanged. >Otherwise: why would we still study Euclid's "Elements" today (and >not just for historical reasons, but to learn math from it)? Well the Greek proofs were valid as far as I know. But those topics were clearly laid out with the 10 axioms, and it was fairly straight forward. What passed for proof in Newton's day would not be considered rigorous today, at least in some cases. Even Gauss's original proof of the FTA was later seen to not be completely rigorous - which is ironic because that was one of Gauss's contributions - bringing greater rigor to mathemaical proof. >>After all, there's a very good argument for saying that the whole >>idea of 'proof' transcends mathematics. > >Yes. Its based on logic. >>Philosophers also try to use logical argument. So do lawyers. > >Yes, "try to use" is the correct expression :-). LOL. Mathematicians make good lawyers for that reason. This reminds me of a story my brother told me recently : The vice-president was complaining to the head of the physics department about their budget. Evidently they need a lot of expensive equipment. He said "look at the math department.. they do fine with pencils, paper, and trash cans"...." and look at the philosophy department, they do fine with pencil and paper." (Apologies to any philosophers among you.) >>As you well know, many great scientists and mathematicians have been >>deeply religious, and have built impeccable proof structures built >>upon their religious assumptions whether they be Islamic, Jewish or >>whatever. I don't know about 'many' nor do I know about 'impeccable' proofs about the hereafter. I think I heard about one nut case physcist who wrote a book prooving God's existence. There is no way to prove anything about God. >>What Poincare had done was change the game yet again by changing the nature >>of proof, and by changing what was an acceptable solution. Pretty much -- >>creating a new branch of mathematics with new ways to tackle the solutions >>there was regarded as a 'correct' solution. What exactly are you talking about here? >> > GA says, rightfully, that most math-heads believe >> >in, or are able to appreciate, the beauty of mathematics "in itself" >> >-- contrary to most mathematical "laymen" who think mathematics is >> >just a boring bunch of formulas and incomprehensible stuff, without >> >any inherent "quality" or esthetics. Well I have never in my life used the term 'math heads' but yeah I said mathematicians appreciate and seek beauty. Imagine the discipline it takes to spend all day in the library into the night studying these wondrous things ... in isolation... day after day... appreciating a beauty no one in the same room can even understand at all. >>And ... my sole point is that this is a very modern approach. Earlier >>epochs did not really stress its importance in this kind of way, although >>they surely at least implicitly recognized it. Well the ancient Greeks seemed to have a healthy appreciation for pure mathematics. Archimedes, who E.T. Bell (author of "Men of Mathematics") considers to have been one of the three greatest mathematical minds to have lived (along with Gauss and Newton) invented many practical things ... the Archimedes "screw" pump, parabolic reflctors, etc. However, when he died he asked that a pure mathematical achievement be placed on his grave. >>However, since it does appear that, statistically, most of the >>mathematicians who have ever lived are probably alive today (I once >>read that, but cannot remember where so cannot validate the >>assertion with a reference), There probably more people alive today than in history too. You can get a rough comparison by considering the height of the function f(x) = C*exp(kx) = current population, to its area from 0 to present... A(x) = C*exp(kx)/k... >>I am more than happy to concede this point to the both of you and >>accept that 'most mathematicians who have lived' should take >>precedence over 'most of mathematical history'. What are we talking about? >>Laplace, Lagrange, Fourier, Euler and the like would not have >>recognized, for example, that there was a SEPARATE field of study >>called 'mathematical logic' although all of them would probably have >>recognized instantly what that particular subject was up to. The calculus we learn in high school or college now was cutting edge research 300 years ago. Every thing we know is built upon the knowledge of those who came before us. Even Newton said, that intellectually, he stood on the shoulders of giants. >>as to the issue of what numbers actually are, not even Gauss really achieved a >>viable definition. To do that, you have to go to other arenas of >>mathematics, and mathematical logicians are even now trying to find >>and plug holes in the definition of number. "And look at the mathematical logicians, they only need paper and pencils also"