markh@csd4.milw.wisc.edu (Mark William Hopkins) (02/10/89)
In article <9551@ihlpb.ATT.COM> arm@ihlpb.UUCP (55528-Macalalad,A.R.) writes: >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. If it's not explicitly proven, then that is because Rolle's theorem is really a theorem about compact sets -- which is a topological concept that lies a bit beyond most Calculus courses. Of course, you can avoid the word "compact", but that's exactly like avoiding the idea of "derivatives" in an Economics course by calling them "marginal rates". But changing the name "compact" does not alter the fact that you are actually doing topology when you are proving Rolle's Theorem. Rolle's Theorem goes like this: Given a function f:[a, b] --> REAL * with f DIFFERENTIABLE on the INTERIOR of [a, b]: (a, b) * and CONTINUOUS on the BOUNDARY: {a, b}, there is a point, c, in the interior (a, b) * at which the derivative of f vanishes. To prove this, you have to do the following: (1) Prove that f(x) has a maximum and minimum on the set [a, b]. This assumes that f(x) is CONTINUOUS on the set [a, b] and that the set is CLOSED (it includes its boundary points) and BOUNDED (it is bounded in size). What you are showing is that continuous functions map closed and bounded sets onto closed and bounded sets. (2) Prove that wherever f(x) has a maximum or minimum in the set (a, b), its derivative vanishes. Of course, if both the maximum and minimum occur at the endpoints a, b then the function is a constant equal to 0 (which is where the assumptions that f(a) = 0 = f(b) come in), so that its derivative is everywhere 0. (2) is used and derived separately when the Calculus course discusses maxima and minima, but when proving (1) you end up repeating a classical proof that has to do with the topological property, compactness. And the same proof appears over and over in many other disguises. (1) goes like this: * if f(x) has no maximum over the set [a, b], define a sequence of points x1, x2, ... with f(x1) > 1, f(x2) > 2, etc. * Find a convergent subsequence y1, y2, ..., --> y. (which assumes that [a, b] is bounded) * Then its limit y is a point in the set [a, b] -- possibly an endpoint (this assumes that [a, b] is closed.) * Then as yi --> y, f(yi) --> f(y) (since f is continuous). * Since f(yi) > i, then f(y) is infinite -- which violates the assumption that f(x) is continuous. >>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. Just prove it and all the difficulties will vanish. >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. By using the same ideas over and over and over ... Mathematicians are much more limited in their ability to prove theorems than AI researchers realise. What the proof outline above shows is that people (in reality) will carry out a proof in a modular fashion, exactly as a programmer is supposed to make their programs modular. The modules are much like software tools that get placed in every Mathematicians' toolbox and get used when deriving higher level results. So the net result is that the same ideas are used over and over and over ...
bph@buengc.BU.EDU (Blair P. Houghton) (02/11/89)
In article <849@csd4.milw.wisc.edu> markh@csd4.milw.wisc.edu (Mark William Hopkins) writes: >In article <9551@ihlpb.ATT.COM> arm@ihlpb.UUCP (55528-Macalalad,A.R.) writes: >>In article <2073@buengc.BU.EDU> bph@buengc.bu.edu (Blair P. Houghton) writes: >>> >>> [..something blithering about calculus...] I just Looove being shown to be dead wrong. It makes my lower cervical vertibrae squeeze together with sheer anxious tension, and those little twitches come back to the corner of my eyes... 'scuse me whilst I go soak my head... --Blair "Yeah, but it was a _good_ calc book that refused to prove Rolle's..."
bwk@mbunix.mitre.org (Barry W. Kort) (02/11/89)
In article <849@csd4.milw.wisc.edu> markh@csd4.milw.wisc.edu (Mark William Hopkins) writes: >Rolle's Theorem goes like this: > > Given a function f:[a, b] --> REAL > * with f DIFFERENTIABLE on the INTERIOR of [a, b]: (a, b) > * and CONTINUOUS on the BOUNDARY: {a, b}, > > there is a point, c, in the interior (a, b) > * at which the derivative of f vanishes. Mark, just so nobody falls off the boat, let me add the missing antecedent: * and f(a) = f(b), which should be inserted in the blank space in your statement of the theorem. --Barry Kort