[FOM] Example of beef: nonrigorous heuristics
Timothy Y. Chow
tchow at alum.mit.edu
Mon Oct 27 09:37:07 EST 2003
Simpson asks, "Where's the beef?" Let me flesh out my previous article on
the "unreasonable effectiveness of nonrigorous heuristics" to provide an
illustration of the kind of beef that I think Corfield is talking about.
One can imagine carrying out the following program.
1. Give a precise mathematical definition of "nonrigorous heuristic" and
of its "effectiveness." "Effectiveness" would be something like the rate
at which the heuristic gets concretized into rigorous theorems.
2. Develop a model, based on observations of mathematicians at work, of
the process dual to effectiveness, i.e., the rate at which heuristics
are abstracted from existing rigorous theorems through a process of
contemplation and organization.
3. Prove rigorously that under certain plausible assumptions, the
effectiveness of heuristics that exist in the real world is actually more
or less what one would expect probabilistically (given the amount of
existing rigorous mathematics), and that their apparent unreasonableness
is an artifact of human psychology (compare with the fact that a long run
of heads in a series of coin flips seems strange to us even when its
frequency is exactly what one would expect from probability theory).
Since I just came up with this "research program" on a bus ride this
morning, it has little chance of being stunningly successful. However,
I do not see any immediate reason why it must *necessarily* fail, and
so I think it succeeds as an illustration of a *possible* line of work
that
a. is of philosophical, and even general intellectual interest, inasmuch
as it might yield insights into the structure and growth of mathematical
knowledge;
b. has relationships with f.o.m. as classically understood, while at the
same time heads off in a new direction;
c. could generate what Harvey Friedman dubs "permanent value" in the sense
that new mathematical theorems could emerge;
d. is still ill-defined enough that it would benefit from some further
philosophical thinking in order to clarify the ideas in question.
Tim
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