finitist prejudices

[Stephen G Simpson]

Joe Shipman 05 Mar 1999 18:27:46

 > The standard fundamental theories of physics deal freely with
 > classes of operators on function spaces

What if we could develop the requisite functional analysis in a
subsystem of second order arithmetic that is conservative over PRA?
PRA seems crucial here, because PRA is finitistic: it may commit us to
potential infinity, but it does not commit us to actual infinity.  See
also my paper on Hilbert's program
<http://www.math.psu.edu/simpson/papers/hilbert/> and my book on
subsystems of second order arithmetic
<http://www.math.psu.edu/simpson/sosoa/>.

[ By the way, these web addresses as well as the FOM web address may
be out of order for the next few days, because of a computer system
upgrade. ]

 > the case (argued by Quine, Putnam, and Maddy) for an ontological
 > commitment to infinite (even uncountably infinite) sets.

Could you please state Quine's case briefly?  A summary of Putnam's
and Maddy's case would also be welcome, though I think they have
changed their minds on this issue.  (Maddy's first book was `Realism
in Mathematics' and her second was `Naturalism in Mathematics'.)

 > Steve and Martin, are the grounds on which you think the universe is
 > finite empirical or a priori?

I don't have a final, well-thought-out answer to this, and even my
tentative answer may appear somewhat strange to you.  First, if I say
the universe is finite, that doesn't imply that the universe is
mathematically describable by a finite formula -- I have no evidence
for such a statement.  What I mean by `finite' in this context is
something like `definite' or `limited' or `following definite laws'.
Now, with that proviso, my grounds for believing that the universe is
finite is, loosely speaking, empirical' because the universe appears
to be orderly, to follow definite laws.  My grounds are also, loosely
speaking, a priori, because the idea of an orderly universe is a
prerequisite for all thought about anything.

 > A seriously finitist ontological position is in fact an atheistic
 > position;

I'm an atheist, so that's not a problem for me.  By the way, when
theists say that God is infinite, I think the main implication of that
is that, according to the theists, God is not limited by the laws of
nature.

 > unity of knowledge and general intellectual integrity demand that
 > we at least attempt to reconcile our schizophrenic attitudes.
 ....

 > It matters professionally whether you "really believe" in finitism.

I thoroughly agree.

-- Steve