[FOM] Friedman on Urquhart on Corfield

[Alasdair Urquhart]

I certainly have no objection at all to
Harvey Friedman writing about my posting!

If I might clarify my mild grumble about people
discussing Corfield's book without having
read it, I should say that my feeling about
philosophy books is that they are more essentially
discursive than most mathematics books.  
For example, I can discuss an isolated argument
from a mathematics book without having read
the whole thing, but to get a real feel for
a philosophy book, I think you have to read all,
or most of it.  Philosophers on the list, do 
you agree with me?

> >I don't always agree with it,
> > but I think I may have more points of agreement
> > than disagreement with the author.
> 
> >From reading your posting, I wasn't clear just where you disagree, except
> with regard to Lakatos.

Well, actually, I have quite a few criticisms.  The discussion
of computer proofs is rather weak, because it makes the mistake
of thinking that the "Robbins problem" was an important 
problem in logic.  That's just an example.

> > Corfield's book has two main aspects.  It
> > is a polemical argument for a re-orientation in
> > the philosophy of mathematics.
> 
> 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.

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.

> Does the book suggest an interesting thread for the FOM regarding
> computer-assisted proofs? There already has been some discussion about such
> issues as "what level of certainty should be associated with
> computer-assisted proofs".

See above remarks about the Robbins problem.

> 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.  

> First of all, mathematicians are not generally concerned with "general
> intellectual interest", "wider conceptual issues", "foundational issues",
> "philosophical understanding", etc. The vast preponderance of them do not
> see the value for their subject of such things, and develop some sense of
> "furthering mathematics" that is left unanalyzed. The occasional attempts to
> discuss such things are not impressive and not valuable. There is rarely any
> attempt to start anything from first principles, even though it is
> definitely possible to do so.
> 
> So for philosophers of mathematics to "incorporate what mathematicians
> think" into their work, is quite difficult, at least if we are talking about
> reasonably modern mathematicians.

> > Corfield advocates a "practice-oriented" philosophy of
> > mathematics, rather in the style of recent philosophy of
> > physics.
> 
> 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.  Hence, they should
be interested in knot theory, quantum groups etc. because he thinks
they are on the "cutting edge" of mathematics.  
 
Well, there is a lot more in Harvey's comments that I can't go into
here.  Philosophers of mathematics on the list, what do you think?