Fw: FOM: Re: f.o.m./TIME Magazine

[charles silver]

>Reply to Silver Tue, 27 Feb 2001 17:54.
>
>Friedman wrote:
>
>>>The problem with the first is that the Mathematics and Philosophy
>>cultures
>>>have grown very far apart, with virtually no common language.

>Silver wrote:
>
>>    One problem I see is that so-called foundations seems to depend
>>too much on technical stuff about set theory....Therefore,
>>philosophers just can 't do it.


Friedman:
>I would like to distinguish two levels of involvement in f.o.m.
>
>1. One is having a clear and sophisticated understanding of the main
>findings and main open issues in f.o.m. This does NOT entail any
>understanding of any proofs.


      I believe anyone discussing the nature of proof (a fundamental
f.o.m. topic) will have to have seen, followed, digested, and composed
a fair number of them.


>2. The second is making serious contributions to f.o.m. Here I agree
with
>the thrust of what Silver says in the next paragraph. Sharply put,
genuine
>f.o.m. research requires a combination of substantial mathematical
and
>philosophical power, and this combination is rarely seen in
sufficient
>quantities.

    I agree.   Those in math departments doing logic tend
to be unschooled and naive philosophically, and those in philosophy
departments tend to be unschooled and naive mathematically.

>However, there is another important aspect of this. We must have an
>environment where appropriately gifted people find f.o.m. research an
>attractive choice.

    It would be welcome to see such environments emerge.

>Firstly, a mathematics student with interest in f.o.m. will normally
be
>sent to mathematical logicians who may have no interest and little
>understanding of f.o.m., or worse still, may identify f.o.m. with
>mathematical logic.

    Yes.

>To put it frankly, despite my many good friends who are
>mathematical logicians, mathematical logic cannot be expected to
currently
>look attractive to people with philosophical inclinations, since
>philosophical discussion and philosophical motivation has been nearly
>expunged from the current mathematical logic literature and culture.

    Yes.  Then there are the journals, which hold up as exemplars
technical but often meaningless results (therefore ratifying the
elimination of "philosophical motivation"--and any other motivation
except technical excellence for its own sake).

>Secondly, a philosophy student with interest in f.o.m. will normally
be
>sent to philosophers of mathematics. Here the problem is less severe,
but
>still very significant: the philosopher of mathematics may not have
enough
>understanding and appreciation of the mathematical aspects of f.o.m.
to
>orient the student properly towards f.o.m. There might be the
attitude that
>the more mathematical aspects of f.o.m. have less philosophical
importance,
>which is false. (The more mathematical aspects of mathematical logic
DO
>have less philosophical importance - often none at all). Thus f.o.m.
may
>appear to the student to be a disconnected subject with mathematical
stuff
>and philosophical stuff, unintegrated, and one must wear two
unrelated
>hats. This appearance cannot be expected to be attractive.


    Agreed.

>Thirdly, the reports that I have from attempts at joint programs in
>mathematics and philosophy have not been positive.

    Don't some of these joint programs really become just weak math +
a
tiny bit of philosophy (just enough to make it look philosophical)?


>But the bottom line is that
>students tend to be forced to go with one side or the other.


>But at an intellectual level, what can be done? A viable intellectual
>movement would go a long way towards facilitating 1 or 2 above.
>
>Perhaps the single clearest intellectual suggestion is:
>
>**the development of philosophically critical expositions of
mathematical
>subjects, and specifically a philosophically critical exposition of
f.o.m.**


    I think there needs to be a support system, such as journals,
jobs,
prestige, etc.   Without such support, it's hard to envision great
development in a joint area.

>Personally, I am not satisfied with any philosophically critical
>treatment
>I have seen of any mathematical subject, including f.o.m. Because of
>the
>lack of such philosophically critical expositions, it seems natural
that
>students will normally develop at most one side of their
>mathematical/philosophical thinking.



>A philosophically critical treatment of f.o.m. would definitely be
readily
>understandable to at least the analytic philosophy community, and to
many
>other communities. Such a treatment need not contain any detailed
proofs of anything.


    Well, as I said above, *some* ability to do and understand proofs
seems necessary.   I think the question is: how much?


>Such a treatment must contain a philosophically critical treatment of
set
>theory, as set theory has a special role in f.o.m. One could call
this
>macroscopic global foundations of mathematics. Macroscopic because of
the
>high strength of the axioms. Global because one is founding all of
>mathematical thought.

    Are there good books in this area?   Have you ever considered
writing a philosophically motivated set theory book?   Such a book
would be very welcome. In general, more philosophically motivated
 f.o.m. articles and books would greatly help the cause.

Charlie Silver