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

[charles silver]

Harvey wrote:
>Yet, f.o.m. is not being properly supported in the Universities. It
lies
>properly between Mathematics and Philosophy, with basic ideas
permeating
>Computer Science. However, due to the inappropriately disciplinary
mode of
>operation of the Universities, it survives only in relatively
ineffective
>forms, modified in order to be palatable to the host Department.


    This seems right to me.

>THe remnants of f.o.m. in Mathematics Departments - loosely called
>mathematical logic - are not philosophical enough, whereas the
remnants of
>f.o.m. in Philosophy Departments - loosely called philosophy of
mathematics
>- are not mathematical enough.


    This also seems right.

>Certainly, mathematical logic is of some interest (not too much) to
>mathematicians, particularly when it impinges directly on standard
>mathematical topics, and philosophy of mathematics is of some
interest (not
>too much) to philosophers, particularly when it impinges directly on
>standard philosophical topics.


    Ditto.


>The problem with the first is that the Mathematics and Philosophy
cultures
>have grown very far apart, with virtually no common language. The
irony is
>that f.o.m. could have served as the common ground, preventing the
cultures
>from having grown this far apart. Also, University Administrations
have not
>recognized the serious flaws in a Disciplinary approach to University
life.


    One problem I see is that so-called foundations seems to depend
too much on technical stuff about set theory for philosophers to be
interested in and/or competent at.   In fact, I'd go further.   It
seems to me that set theory has been set up as *the* foundation, and
competence in its highly specialized, more difficult reaches seems a
prerequisite for philosophizing about mathematics.   Therefore,
philosophers just can 't do it.    So, there are a bunch of
specialists--you are one--who do set theory, and a very few of these
specialists--you are one again--happen to have strong philosophical in
terests.   But, for example, stuff about large cardinals is
inaccessible to virtually all philosophers (i.e., those philosophers
who inhabit philosophy departments).

    Awhile back, Steve mentioned (I think maybe as many as three
times) some fundamental concepts, such as number, function, etc.   If
these topics could be effectively and interestingly discussed without
someone being reasonably expert in set theory, then these fundamental
concepts might interest philosophers.   I'm not so sure of this,
however.  Maybe.

    About your (and Steve's) concern that math. logic is not
necessarily foundations: part of math logic is set theory, and set
theory features prominently in foundational talk.   Therefore, it's
natural for people to think math logic is foundations.   That is,
foundations seems to be set theory, which is also part of math. logic.
(I'm not saying this is a terrific argument, just a natural conclusion
to draw.)

    I personally think reverse mathematics seems foundational, but
several mathematicians I've spoken to seem not to hold it in high
regard.   I don't know what exactly this implies, but I'm speculating
that they don't consider it either to be "interesting" math logic or
they think it has no foundational interest.

    I would be very interested in hearing what ideas you (Harvey) or
anyone else has that could fill in the intersection of foundational
topics that could engage both mathematicians and philosophers.

Charlie Silver