[FOM] Re: Example of beef: nonrigorous heuristics

Harvey Friedman friedman at math.ohio-state.edu
Sun Nov 16 00:24:50 EST 2003

Reply to Chow.

On 10/29/03 8:58 PM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:

> Stephen G Simpson wrote:
>> The program sounds interesting.  However, until somebody carries it
>> out, or at least makes a start, there is no beef here.  What you are
>> calling beef is purely imaginary.

>... I think that contributions to subjects that are still in
> their infancy should be judged with more leniency than subjects that
> are more mature.  Many successful endeavors exist today only because
> a pioneer had the courage to take a promising but untested and immature
> idea, and work at it for a long time (perhaps without success at first)
> until it yielded fruit.
>... However, I think
> that some degree of tolerance should be exhibited towards new ideas
> that might eventually lead somewhere, and that they should not be too
> quickly and summarily dismissed.  I try to be tolerant even of people
> that I don't think are comparably tolerant of my own favorite subjects.

Let me say how I think that these comments fit into my own views.

1. It is extremely difficult to combine scientific depth, rigor,
philosophical relevance, and general intellectual interest, all in a single
area of research. 

2. F.o.m. is currently the supremely successful and clear example of 1.
There are some other promising emerging enterprises, but they have not yet
reached a comparable level of development.

3. The essentially unique position of f.o.m. in the intellectual landscape
will prove to be of transcendent importance - even more important than the
specific developments of f.o.m.

4. By 3, I mean that f.o.m. has built something of tremendous substance out
of interesting ideas of obvious philosophical relevance, gii, etcetera -
where it could have easily got stuck in the mud with merely interesting
descriptive commentary, historical insights, or gotten bogged down by
abandoning its roots and focusing on technicalities for their own sake.

5. F.o.m. handles well only a TINY FRACTION of philosophically relevant
issues surrounding the nature of mathematical thought, of obvious gii. In
the future, there will be a huge expansion of the scope and depth of f.o.m.

6. It is comparatively trivial to offer possible topics for f.o.m.
expansion. It is also important to offer possible topics for f.o.m.

7. In offering up possible topics for f.o.m. expansion, one should always
bear in mind the great successes of f.o.m., and look for ways to learn from
these successes. Failure to learn from these successes adds substantially to
the risk of failure.

8. When looking at the development of f.o.m. since, say, 1900, one is deeply
struck by the steady increase in the level of aspects of actual mathematical
practice that is addressed by f.o.m. This steady increase will continue for
the forseeable future.

9. However, any premature *insistence* or *requirement* that f.o.m. take
into account such and such feature of mathematical thought will be
needlessly counterproductive. The steady expansion of the scope of f.o.m.
must take place at its own pace - if the traditional f.o.m. standards for
deep insight, gii, profound surprise, etc., are to be maintained.

10. Any substantial expansion of the scope of f.o.m. is likely to seriously
borrow from the existing development of f.o.m. If some initial insight is
made that does not seem to seriously borrow from existing f.o.m., then
existing f.o.m. will be used to substantially develop that initial insight.
I.e., I have full confidence that existing f.o.m. is the proper initial
segment of f.o.m. development.

11. Consequently, attempts at downgrading the status of f.o.m., as we
witness from time to time in rather determined form here on the FOM, are not
only entirely inappropriate, wrongheaded, and easily refuted, but are
completely counterproductive in light of 10.

Harvey Friedman

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