[FOM] accord with mathematical practice
Nik Weaver
nweaver at math.wustl.edu
Thu Feb 23 19:22:52 EST 2006
There seems to be some confusion about my views as to the
significance of the accord of various foundational stances
with ordinary mathematical practice.
Harvey Friedman wrote:
> Weaver asserts that predicativity can do more functional analysis
> than the PRA level, but apparently concedes that the difference
> is not substantial.
The difference is substantial. I'm not sure how you got "not
substantial" out of my message
http://www.cs.nyu.edu/pipermail/fom/2006-February/010005.html
Do you think the difference is not substantial?
> > Predicativism doesn't suffer from these defects, but there are
> > still going to be occasional topics out at the margins that won't
> > be allowed. So it's a matter of degree.
>
> MATTER OF DEGREE! Thank you for now agreeing with me ...
> Sounds like you agree with me that "predicativity" doesn't
> have any special place.
The extent to which various foundational stances accord with
normal mathematical practice is a matter of degree. The second
statement is a non sequitur. It only makes sense if you think
that all that matters about a foundational stance is how well
it accords with mathematical practice (which Friedman apparently
does think).
Indeed Friedman seems to think that I share this view of his, and
hence:
"Weaver advocates predicativism because it accords well with
normal mathematical practice."
No. I claim that predicativism is special because it has a clear
philosophical basis and impredicative mathematics does not. That's
because impredicative mathematics is justified on platonic grounds
which do not hold up under scrutiny. *The fact that predicativism
accords well with normal practice is merely additional evidence
that it is special.* (Though the exactness of the fit is, to me,
quite compelling. Remember that we're talking about mathematics
which developed naturally in a climate in which Cantorian set
theory was the de facto foundational standard. The fact that
what came out matches predicativism much better than platonism
seems significant.)
Friedman seems to concede that impredicative systems can only be
justified on platonic grounds. Instead he challenges my position
by claiming that predicativism also rests on platonism. This
argument seems to me not very convincing. You justify, say, Z_2 in
terms of the existence of an abstract metaphysical world of sets
in which the axioms hold. You justify predicativism in terms of
marks on paper. Calling the latter "platonism" is a bit of a
stretch. Key difference: sets are supposed to be *unique,
canonical* abstract objects. Marks on paper may be imagined
objects, but they're not really abstract. And there is no
requirement of uniqueness; indeed, belief in the intelligibility
of *any* omega structure is enough. "Marks on paper" is just a
picturesque description.
If you're going to say that I have to be a platonist in order to
be a predicativist, you have to say that one cannot believe in
the intelligibility of the concept "omega structure" without
being a platonist. Does anyone believe this?
Nik
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