I would recommend that the participants in this debate read Imre Lakatos's seminal book 'Proofs and Refutations: The Logic of Mathematical Discovery'. The debate here is not about mathematics it is about the philosophy of mathematics. Lakatos directly addresses what I see as the subject of the debate here in his book when he makes the distinction between formal and informal mathematics. Formal mathematics is contained in the proofs published in mathematical journals. Informal mathematics are the strategies that working mathemeticians use to make their work a useful exercise in mathematical discovery. I see the objection to the four color proof here. It was undoubtedly a valid formal proof. However it did nothing to advance the cause of mathematics beyond this. The reason that Lakatos equates proofs and refutation in his title is his contention that it is the refutations that are developed that show mathematicians the deficiencies in their theories. It is their attempts to deal with unwanted and unexpected refutations - to preserve a valuable theory in the face of imperfect axioms and proof methods - that teach mathemeticians the true depths of their conceptions and to point the way to new and deeper ones. Lakatos shows this by an account of the historical development of the concept of proof in mathematics and by showing in historical detail how certain valuable 'proofs' were preserved in the face of refutation. To this point Lakatos shows that the 'proofs' of the truth of Euler's number are no proofs at all. The great mathemetician Euler noticed that for any regular polyhedron the formula V-E+F=2 where V is the number of vertexes, E is the number of edges and F is the number of faces. Euler's and his successors proofs fall before any number of counterexamples. Does this prove that the theorem is 'incorrect?' Or does it mean as the mathemetician's actions show that they thought it meant was that their concept of what constituted a regular polyhedron was deficient. Lakatos shows how these conceptions were modified over a couple of hundred years as counterexample after counterexample was faced. These counterexamples all made mathematics stronger by deepening the conception of what polyhedra really are and by discovering new classes of them. In the end Euler's formula turned out not to have a proof but to be in effect a tautology. It is true for the regular polyhedra for which it is true by the definition of what constittues a polyhedron. It is true because human mathematicians in order to make progress need it to be true. The computer proof of the four color theorem was a triumph of formal mathematics. Its critics complained and. if interpreted according to what Lakatos wrote, they complained because it defeated the progress of informal mathematics. Mathematical proofs are useful tools. The tell us what we need to know. Formal mathematics is about finding them. Informal mathematics is about making them useful. Mathematics is not some Platonian ideal divorced from humanity, painting, poetry ... It is a human endeavor to meet human needs. [log in to unmask] wrote: > In a message dated 3/12/01 6:21:53 PM Eastern > Standard Time, > [log in to unmask] writes: > > > > I'm of course exaggerating ... . Theory is nothing > to gloss over and can be > > very very useful, elegant and quick. But it's a > theory, and theories have a > > history of being overturned in practice. People > are often blinded by the > > beauty of an elegant theory, but often the real > test for a theory is when > > we apply them to the world; that's usually where > things start going wrong. > > And therefore I think there is definitely > something to say for being able > > to prove something 'uitputtend' as we say in > Dutch, exhaustive. It's not > > always necessary, it's not always elegant, but > it's rock solid. You also > > often really need it when applying a theory to the > world, because when you > > use a theory in practice you also have an impure > domain to cover; practical > > > > You better get caught up on Wittgenstein or Benjamin > Whorf, because you're > saying things that don't make any sense. What's > "rock solid" about examining > every person in the world to discover, say, that > nobody in the world has an > s-shaped scar on their left shoulder? What is that > supposed to "prove" about > whether there ever was such a person in the past or > might be in the future? > > p > __________________________________________________ Do You Yahoo!? Yahoo! Auctions - Buy the things you want at great prices. http://auctions.yahoo.com/