I believe Thurston is referring to the formal syntax of the computer
languages he is using: there is no ambiguity .. even to the point of
having syntax rules, generally in BNF (Backus–Naur Form). And at the
time of his writing the article, he was likely using Algol, a rather
advanced and sophisticated language, and one with the first
interesting representations of data structures .. called Records in
Algol as I recall.
Thus in some ways computer programs can be validated, in that their
syntax is rigorous. And naturally the compiler enforces these rules.
Theorems do not have quite that concreteness, although being exposed
to a wide community is sorta a similar scheme of rigor.
-- Owen
On Dec 14, 2009, at 10:43 PM, Nicholas Thompson wrote:
Dear Friammers,
We have decided to carry on from our seminar on Emergence to one on
Mathematical Thinking. Although we don't meet for a month, I found
myself reading the first assignment, Thurston's On Proof and
Progress in Mathematics. Now Thurston loves mathematics and is
apparently good at it, but he is firm in arguing that the process of
proof is not as the normative account would have it. Given our
local debates about the ideal of formalism and given my suspicion
that many computer programmers suffer from math envy (the way
experimental psychologists suffer from physics envy), I was
astonished by the following paragraphs.
The standared of correctness and completeness necessary to get a
computer program to work at all is a couple of orders of magnitude
higher than the mathematical community's standard of valid proof.
Astonished, and yet, instantly convinced that it was true. Note
that Thurston is proud of how mathematicians do their work; no
criticism here.
I think that mathematics is one of the most intellectually
gratifying of huan activities. Because we have a high standard for
clear and convincing thinking and because we place a high value on
listening to and trying to understand each other, we don't engage in
interminable arguments and endless redoing of our mathematics. We
are prepared to be convinced by others. Intellctually, mathematics
moves very quickly. Entire mathmatical landscapes change and change
again in amazing ways during a single career.
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 to be put into it to make it
'almost'formally correct, it is preposterous to claim that
mathematics as we practice is any where near formally corrrect.
You would almost think that computer programming was the Queen of
the Sciences.
Nick
I wonder what you all think about it.
Nick
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