[FOM] Friedman on Urquhart on Corfield

[Harvey Friedman]

Reply to Urquhart.

On 10/16/03 7:45 PM, "Alasdair Urquhart" <urquhart at cs.toronto.edu> wrote:

Friedman:

>> In a sense, I have been suggesting an expansion of activity in philosophy in
>> mathematics in the direction of closer ties with the foundations of
>> mathematics. However, I have never suggested that philosophers of
>> mathematics make any serious effort to do foundations of mathematics - but
>> rather to incorporate findings in f.o.m., make comments about f.o.m., make
>> philosophical interpretations of f.o.m., ask f.o.m. questions that may
>> affect f.o.m. research, etc.

Urquhart:
> 
> That's fine, but part of Corfield's argument is precisely that
> the whole "f.o.m." orientation has distorted philosophy of mathematics.
> You may disagree, but that's his argument.

That's fine, but that is not an argument. That is a statement.

First of all, f.o.m. doesn't "have philosophy of mathematics". It is
neutral. 

Second of all, if somebody can defend a position to the effect that the
whole concept of f.o.m. as currently practiced has some fundamental
unrecognized-to-practitioner problems/biases, let them defend such a
position here in the open forum of the FOM.

If you find this a serious possibility, then I would like you to outline
what you think is the elements of a case for it.

One should bear in mind that f.o.m. is production oriented, with the
sometimes realized aim of obtaining undisputable and dramatic findings of
unusual general intellectual interest regarding the nature of mathematical
thought. Although historically, f.o.m. has been spectacularly successful, it
clearly has only scratched the surface with regard to even the issues with
which it has had the most success.

We look forward to major expansion of the scope of f.o.m., but in an orderly
fashion, where the very high standards of f.o.m. are still met.

E.g., it is not clear to me that it is time for f.o.m. to seriously aim at
explaining what a good mathematical conjecture is.

On the other hand, a possibly ripe topic about which f.o.m. has had some
limited success, but not impressive success by historical f.o.m. standards,
is the question: what is a mathematical classification? (This has been
nibbled at in descriptive set theory and model theory). This is probably on
the edge of something to aim at for f.o.m.

>> Does the book suggest an interesting thread for the FOM regarding
>> Bayesianism in mathematics?
> 
> Yes, I think that the whole question of plausibility of mathematical
> conjectures is a very interesting one.  I find Corfield's discussion
> of this problem quite stimulating.

OK, so let's start a new FOM thread concerning Bayesianism in mathematics.
Let's see what the prospects are for serious f.o.m. work along these lines.
> 
>> Let me comment on this from the viewpoint of foundations of mathematics, to
>> be distinguished from philosophy of mathematics.
>> 
>> One should not forget that foundations of physics is perhaps roughly at the
>> level of the foundations of mathematics in 1800. I am not in favor of any
>> hard push towards 1800 level foundations of modern mathematics. This would
>> be at odds with the deliberate and steady expansion I discussed above.
> 
> Interesting comment!  Perhaps it's true.   I don't know.  But bear
> in mind that Corfield's model is precisely philosophy of physics.
> In other words, he wants philosophers of mathematics to emulate
> their contemporaries in philosophy of physics.

Gxx help us if current philosophy of physics is our guide. No, let me say it
differently. Even Gxx can't help us if current philosophy of physics is our
guide. 

>Hence, they should
> be interested in knot theory, quantum groups etc. because he thinks
> they are on the "cutting edge" of mathematics.

The foundations of physics is in absurd condition even for trivial physics.
The serious issues are foundational/philosophical and are not going to be
clarified by mathematical complexities.

The foundations of probability/statistics is in better shape - not 1800,
more like late 1800's - but certainly before Frege.

These nonsubjects will get off the ground in a serious way through pondering
- and only through pondering - just what makes f.o.m. work so well. Of
course, major additional ideas not present in f.o.m. are needed.

Harvey Friedman