Robert,
thanks for the additional quotations.
However, you made a slip of the fingers when you keyed in one of the passages.
To head off needless controversy, I key it in correctly below. The capitalized
word is where the slipup occured.
"Mathematics as we practice it is much MORE formally complete and precise than
other sciences, but it is much less formally complete and precise for its
content than computer programs."
Easy slip to make because of the structure of the sentence.
n
Nicholas S. Thompson
Emeritus Professor of Psychology and Ethology,
Clark University ([email protected])
http://home.earthlink.net/~nickthompson/naturaldesigns/
http://www.cusf.org [City University of Santa Fe]
----- Original Message -----
From: Marcus G. Daniels
To: [email protected]
Sent: 12/15/2009 1:09:23 PM
Subject: Re: [FRIAM] A little Proof, Dr Thurston! It aint Elementary!
On 12/15/09 12:27 PM, Robert J. Cordingley wrote:
I think this is him:
http://en.wikipedia.org/wiki/William_Thurston
The essay that Russ mentioned only mentions programming in passing.. He
doesn't say anything about it relative to `intellectual challenge', but he does
talk a lot about what is the deep value of his enterprise. The message is in
some sense that the rigor is not the end, it's the means. Some quotes:
"When one considers how hard it is to write a computer program even approaching
the intellectual scope of a good mathematical paper, and how much greater time
and effort have been put into it to make it "almost" formally correct, it is
preposterous to claim that mathematics as we practice it is anywhere near
formally correct.
Mathematics as we practice it is much less formally complete and precise than
other sciences, but it is much less formally complete and precise for its
content than computer programs. The difference has to do not just with the
amount of effort: the kind of effort is qualitatively different. In large
computer programs, a tremendous proportion of effort must be spent on myriad
compatibility issues: making sure that all of the definitions are consistent,
developing "good" data structures that have useful but not cumbersome
generality, deciding on the "right" generality for functions, etc. The
proportion of effort spent on the working part of a large program, as
distinguished from the bookkeeping part, is surprisingly small. Because of the
compatibility issues that almost inevitably escalate out of hand because the
"right" definitions changes as generality and functionality are added, computer
programs usually need to be rewritten frequently, often from scratch."
and
"The standard of correctness and completeness necessary to get a computer
program to work at all is a couple orders of magnitude higher than the
mathematical community's standard of valid proofs. Nonetheless, large computer
programs, even when they have been very carefully written and very carefully
tested, always seem to have bugs."
..and then he goes on to talk about how mathematics lacks a holistic sort of
factoring process like goes on with large [evolving] programs. And many other
interesting, reasonable remarks.
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