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From: steve@Brl-Bmd.ARPA
Newsgroups: net.ai
Subject: Mathematical Methods
Message-ID: <730@sri-arpa.UUCP>
Date: Sun, 10-Jun-84 06:47:53 EDT
Article-I.D.: sri-arpa.730
Posted: Sun Jun 10 06:47:53 1984
Date-Received: Tue, 12-Jun-84 00:37:53 EDT
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From:  Stephen Wolff 

Not at all deep; maybe others will find our gropings briefly amusing .....

    Date:     Fri, 8 Jun 84 11:19:30 EDT
    From:     Brint 

"The usual attitude of mathematicians is reflected in their published
research papers and in mathematics textbooks.  Proofs are revamped and
polished until all trace of how they were discovered is completely
hidden.  The reader is left to assume that the proof came to the originator
in a blinding flash, since it contains steps which no one could possibly
have guessed would succeed.  The painstaking process of trial and error,
revision and adjustment are all invisible."

Alan Bundy


    From:     Stephen Wolff 

I have the greatest respect for Alan Bundy, and I agree with his words.  I
shall however adamantly disagree with his (or anyone's) implication that

"The painstaking process of trial and error, revision and adjustment....."

should NOT be invisible -- in a MATHEMATICS paper.  The purpose of such a
paper MUST be FIRST to advance knowledge; proofs MUST be as spare, concise
and lucid as it is within the author's talent to make them -- for sloppy or
wordy proofs are just that much harder to verify.  And, indeed, the paper is
diminished to PRECISELY the extent that the author's trials and fumbles are
displayed -- for they may prejudice the world-view of a reader and lead him
to the same (POSSIBLY erroneous) result.

If you say that there are too few (maybe no) places to publish mathematicians'
thought processes, methods of hypothesis, &c., then I shall agree.  And,
further, state my belief that UNTIL we are able to read how both successful
and unsuccessful mathematicians derive the objects of their study, then all
successful efforts at automated reasoning will be just blind beginners' luck.


    From:     Paul Broome 

Bundy was not implying that the dead end paths in the search for a proof
should be in the paper that publishes the proof.  Just before the portion
that Brint quoted, he discussed Polya's books, "How to Solve It" and
"Mathematical Discovery" and introduced the paragraph containing the
aforementioned quote with, "Polya's attitude in trying to understand the
'mysterious' aspects of problem solving is all too rare."  His next
paragraph begins with "The only attempt, of which I am aware, to explain
the process by which a proof was constructed, is B.L. van der Waerden's
paper, 'How the proof of Baudet's conjecture was found', .."

He's giving motivation for a book on the modeling of mathematical reasoning.


    From:     Brint 

Perhaps, as in so many endeavors, several bright people actually
agree:

        1. Mathematics papers are not the place for discussing
trial_and_error, inspirational flashes, false starts, and other
means for "discovering" truth and error.

        2. Forums are needed for the discussion of such ideas in
order to advance our understanding of the process at least toward
the end of improving mathematical reasoning by computer.

        3. In some limited way, such forums exist.  We need to
encourage and motivate our mathematicians to contribute to them.

Brint