Path: utzoo!utgpu!jarvis.csri.toronto.edu!mailrus!wasatch!cs.utexas.edu!rutgers!att!ihlpb!arm From: arm@ihlpb.ATT.COM (Macalalad) Newsgroups: comp.ai Subject: Re: Natural Paradox Message-ID: <9551@ihlpb.ATT.COM> Date: 9 Feb 89 16:16:52 GMT References: <1706@tank.uchicago.edu> <9526@ihlpb.ATT.COM> <2053@buengc.BU.EDU> <0XvA3xy00Xoj82iV53@andrew.cmu.edu> <2073@buengc.BU.EDU> Reply-To: arm@ihlpb.UUCP (55528-Macalalad,A.R.) Organization: AT&T Bell Laboratories - Naperville, Illinois Lines: 49 In article <2073@buengc.BU.EDU> bph@buengc.bu.edu (Blair P. Houghton) writes: > >I don't know about how you learned Calculus, but when I went through it >the first time the book mentions Rolle's theorem (having to do with the >idea that a continuous function that is positive at one point and >negative at another must be zero at some point in-between) More precisely, (or as precisely as I can be off the top of my head :-) given a function f that is continuous and differentiable on (a,b) such that f(a) = f(b) = 0, then there exists at least one c in (a,b) such that f'(c) = 0. >and says it's >gotta be true, but that they wouldn't dare prove it. Every other theorem >in a first-year calc book gets proven. Rolle's doesn't. I've never >seen a basic Calc book that proves it. That also means I've never seen >it proven. Come on, _every_ basic Calc book _I've_ seen proves it. The proof is very basic, and I could probably prove it right now, were I sufficiently motivated. I leave it as an exercise for the reader. (:-) >I seem to remember someone along the lines of Einstein as having said >that he couldn't prove it; but it's gotta be provable; it's the >cornerstone of the proofs of almost every theorem that follows it in >the book I first used. > >Okay, so they're provable, but can we prove that they're provable? Not to belabor the point, but the question of whether or not a theorem is provable or provable that it's provable, and so on, is entirely different from the question of whether _we_ can prove the theorem is provable, and the two should not be confused. To show that it is possible for a theorem to be provable yet not provable that it is provable, it is sufficient to show that there exists such a theorem, and it is not necessary to come up with a specific example. Of course, it is probably easy to show that a theorem which is provable is provable that it is provable, and were I sufficiently motivated.... (:-) The second question, which seems to me to be the more interesting question, involves how we come up with, or fail to come up with, proofs for theorems. This is a more psychological question, and one which is certainly appropriate for this group. Comments? -Alex