steve@Brl-Bmd.ARPA (06/10/84)
From: Stephen Wolff <steve@Brl-Bmd.ARPA>
Not at all deep; maybe others will find our gropings briefly amusing .....
Date: Fri, 8 Jun 84 11:19:30 EDT
From: Brint <abc@BRL-TGR.ARPA>
"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 <steve@BRL-BMD.ARPA>
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 <broome@BRL-TGR.ARPA>
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 <abc@BRL-TGR.ARPA>
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.
Brintsher@rochester.UUCP (David Sher) (06/13/84)
Personally, I have done mathematics upto the beginning graduate level for various courses. When I do any difficult piece of mathematics I find that after the fact I can never remember how I came upon the proof. I can reconstruct my steps but the reconstruction has no real relationship to what I really did. The sensation of finishing a proof is highly analogous to waking up from a dream. This is possibly the most important reason why I am doing artificial intelligence rather than mathematics today. If other real mathematicians also operate in this manner then it is not surprizing that they are reluctant to write up their resoning processes. They literally can not remember them. -David
Bundy%edxa@ucl-cs.arpa (06/14/84)
From: BUNDY HPS (on ERCC DEC-10) <Bundy%edxa@ucl-cs.arpa>
I support Broome's and Brint's interpretations of what I was
trying to say in my book. I was not trying to criticise mathematics
papers per se, but to point out that they do not contain some of the
information that AI researchers need for computational modelling and to
make a plea for a forum for such information.
But let me add a caveat to that. The proofs in a paper are at
least as important a contribution to mathematics as the theorems they
prove. Future mathematicians may want to use these proofs as models for
proofs in analogous areas of mathematics (think of diagonalization
arguments, for instance). So it will improve the MATHEMATICAL content
of the papers if the author points out the structure of the proof and
draws attention to what s/he regards as the key ideas behind the proof.
Alan Bundyhbs%BUGS%Nosc@sri-unix.UUCP (06/15/84)
From: Harlan Sexton <hbs%BUGS@Nosc> It is true that most mathematics papers contain little of the sort of informal, sloppy, and confused thinking that always accompanies any of the mathematical discovery that I have been a party to, but these papers are written for and by professional mathematicians in journals that are quite backlogged. Also, although I have always been intrigued by the differences beween modes of discovery among various mathematicians of my acquaintance, I never found knowing how others thought about problems of much use to me, and I think that most practicing mathematicians are even less inclined to wonder about such things than I was when I was a "real" mathematician. However, in response to the comment by David ???, I can only say that I, and most of my fellow graduate students to whom I talked about such things, had no trouble recalling the processes whereby we arrived at the ideas behind proofs (and the process of proving something given an "idea" was just tedious provided the idea was solid). The process used to arrive at the idea, however, was as idiosyncratic as the process one uses to choose a spouse, and it was generally as portable. I found it very useful to know WHAT people thought about various things, and I learned a great deal from my advisor about valuable attitudes toward PDE's, for example (sort of expert knowledge about what to expect from a PDE), but HOW he thought about them was not useful. (With the exception of the infamous Paul J. Cohen, I felt that I appreciated HOW these other people thought; it was just that it felt like wearing someone else's shoes to think that way. In Cohen's case we just figured that Paul was so smart that he didn't have to think, at least like normal people.) In the last year or so of my graduate career, someone came to the mathematics department and interviewed a number of graduate students, including me, about something which had to do with how we thought about mathematical constructs (of very simple types which they specified). Presumably this information, and related things, would be of some interest to Bundy. I'm sorry that I can't be more specific, but if he would contact the School of Education at Stanford (or maybe the Psychology Dept., but I think this had to do with some project on mathematics education), they might be able to help him. There is also a short book by J. Hadamard, published by Dover, and some writings by H. Poincare', but as I recall these weren't very detailed (and he probably knows of them already anyway). Finally, I know that for a while Paul Cohen was interested in mathematical theorem proving, and so he might have some useful information and ideas, as well. (I believe that he is still in the Math. Dept. at Stanford. The AMS MAA SIAM Combined Membership List should have his address.) --Harlan Sexton
jcz@ncsu.UUCP (John Carl Zeigler) (06/19/84)
It is not surprising that mathemeticians cannot remember what they do when they first contsrtuct proofs, especially 'difficult' proofs. Difficult proofs probably take quite a bit of processing power, with none left over for observing and recording what was done. In order to get a record of what exactly occurs ( a 'protocol' ) when a proof is being constructed, we would have to interrupt the subject and get him to tell us what he is doing - interferring with the precise things we want to measure! There is much the same problem with studying how programmers write programs. We can approach a recording by saving every scrap of paper and recording every keystroke, but that is not such a great clue to mental processes. It would be nice if some mathemetician would save EVERY single scrap of paper ( timestamped, please! ) involved in a proof, from start to finish. Maybe we would find some insight in that. . . John Carl Zeigler North Carolina State University