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From: bwk@mbunix.mitre.org (Barry W. Kort)
Newsgroups: comp.ai
Subject: Re: Natural Paradox
Summary: Seeley asserts a proof of Rolle's Theorem.
Keywords: True but unproven.
Message-ID: <44663@linus.UUCP>
Date: 9 Feb 89 12:44:37 GMT
References: <1706@tank.uchicago.edu> <9526@ihlpb.ATT.COM> <kXuo9-y00V4=88uEVm@andrew.cmu.edu> <2053@buengc.BU.EDU> <0XvA3xy00Xoj82iV53@andrew.cmu.edu> <2073@buengc.BU.EDU>
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In article <2073@buengc.BU.EDU> bph@buengc.bu.edu (Blair P. Houghton) writes
about unproven theorems in calculus:

 > It means we have to prove Rolle's Theorem.

I looked this up in Robert T. Seeley's book.  Although the chain
of lemma's, theorems, and proofs are lengthy, he does prove the
Mean Value Theorem and Intermediate Value Theorem (including Rolle's
Theorem) using the Maximum Value Theorem, which he proves in an
appendix.  (The Maximum Value Theorem states that a continuous
function on a closed interval necessarily achieves a maximum
(and a minimum) in the interval)  

I am not prepared to defend the rigor of Seeley's proofs, but his
running commentary acnkowledged the difficulty of exhibiting the
unbroken chain of proofs, which he evidently succeeds in doing.

--Barry Kort