[FOM] predicativity

[Nik Weaver]

Jeremy Avigad wrote:

> Most philosophers and mathematicians seem happy to take set
> theory as a foundation for mathematics ...

Probably so, but this could be because the alternatives of which
they (mathematicians at least) are aware violate their intuition
even more strongly than Cantorian set theory does.  I think it is
also fair to say that most mathematicians are far more comfortable
with number-theoretic platonism than set-theoretic platonism ---
which accords exactly with the standpoint I am advocating.

> One can point out that such [predicative] theories are
> strong enough to carry out a good deal of mathematics --- modulo some
> reworking --- but are not as strong, and hence not as dubious, as full
> set theory. But this is a somewhat weak defense, and has not won many
> converts.

Yes, but is this really the best defense predicative theories
have?  I gave a more principled defense in my last message and
this seems to have been ignored.  The point is that impredicative
(circular) definitions only make sense against a background of a
well-defined preexisting universe of sets; if one does not believe
there is such a thing, it becomes very hard to see why impredicative
definitions should be allowed.  On the other hand, one can argue
that constructions of length omega are meaningful because it is
possible to form a concrete mental image of them.  Combining these
two ideas --- there is no preexisting universe of sets, and valid
constructions must be concretely imaginable --- gives rise to the
conceptualist view that I am defending.  Surely this is a better
defense than simply saying "not as strong, hence not as dubious".

> One can also claim that reconstruing parts of classical mathematics
> in predicative terms has mathematical benefits as well.

Indeed, there are several grounds for making this argument.
One has a much more concrete grasp of, say, J_2 than of standard
models of ZFC whose existence we must take on faith.  This gives
someone working in J_2 an advantage roughly analogous to the
advantage enjoyed by someone who does calculus with epsilon-delta
arguments as opposed to naively using infinitesimals.  Another
advantage of the conceptualist view is that it excises vast regions
of (what appears to the working mathematician to be) set-theoretic
pathology.

A point that I have not seen emphasized is the remarkably exact
fit of mathematical conceptualism --- or predicativism given the
natural numbers, if you prefer --- with ordinary mathematical
practice.  For instance, this is very striking in duality theory
in functional analysis.  One uses second duals of Banach spaces
all the time, but third duals appear with extreme rarity.  As I
explain in my paper "Mathematical conceptualism" this exactly fits
with the conceptualist stance.  (In fact, the fit is even better
than I just indicated.)

At the risk of being repetitious, my papers on conceptualism can be
found at

http://www.math.wustl.edu/~nweaver/conceptualism.html


Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu