FOM: f.o.m./TIME Magazine; alternative foundational schemes
Stephen G Simpson
simpson at math.psu.edu
Wed Feb 28 14:28:12 EST 2001
Charles Silver Tue, 27 Feb 2001:
> 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.
...
>competence in its highly specialized, more difficult reaches seems a
>prerequisite for philosophizing about mathematics.
I disagree. There are many ways for philosophers to comment
effectively on f.o.m. issues, without being experts in technical set
theory. For instance, they can comment on philosophical aspects of
constructivism, or of other alternative foundational schemes. See
also below.
I concede that contemporary philosophers of mathematics ought to be
familiar with the basis of the contemporary claim that ZFC is capable
of formalizing contemporary mathematics. But this much understanding
of set-theoretic foundations is readily acquired, e.g., from an
undergraduate textbook like Enderton's Elements of Set Theory.
> In fact, I'd go further. It seems to me that set theory has been
> set up as *the* foundation,
...
Maybe some people have tried to insist on this, but I could not agree
with such a formulation. From my perspective ant that of many other
f.o.m. researchers, ZFC is just one of many foundational schemes that
can be and are being studied.
Of course I (and presumably other f.o.m. researchers) have to admit
that ZFC-style foundations is currently first among equals, in the
sense that ZFC is the one foundational scheme that is accepted by the
majority of contemporary mathematicians. Evidence for this is the
fact that many rigorous core mathematics textbooks -- in disparate
branches such as analysis, algebra, topology, geometry, combinatorics
-- start out with a common chapter on set-theoretic "preliminaries",
i.e., foundations. These authors seem to like the fact that set
theory provides a simple, unified, flexible framework or language, in
which mathematical definitions are easily formulated, and connections
between various mathematical topics are easily described.
> 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.
I see no reason why philosophers couldn't comment on underdeveloped
possibilities, e.g., alternative foundational schemes based on
functions rather than sets, or maybe alternative schemes based on
Turing computability, etc etc. No great expertise in set theory would
be required.
-- Steve
8
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