Finitist prejudices

[Joe Shipman]

Finitism (the view that there are no infinite sets) has some attractive
philosophical and methodological features.  However, it is important to
recognize the difference between a commitment to finitism based on these
features and a commitment to finitism based on actual evidence.  I think
there is general agreement that the epistemological principle of Occam's
razor allows us to pass from "absence of evidence" to "evidence of
absence" in this case (really, we are talking about the purest possible
case of Occam's "entities are not to be multiplied without necessity"!),
but it seems that the absence of evidence for infinite sets is being
taken too much for granted.

The standard fundamental theories of physics deal freely with classes of
operators on function spaces, and even though Feferman has argued that
most of theoretical physics can be rendered in a fragment of type theory
that is conservative over predicative arithmetic, that does not refute
the case (argued by Quine, Putnam, and Maddy) for an ontological
commitment to infinite (even uncountably infinite) sets.  Only a
reformulation of physics in which physically meaningful entities are
somehow represented by finite sets could do that (leaving aside the
issue of whether the universe itself is finite).  We are VERY far from
such a reformulation.

Freeman Dyson, who first brought fundamental quantum field theories to
their present (imperfect) level of mathematical rigor, entitled his
autobiography "Infinite in All Directions".  This was partly a reference
to the infinity of the universe in the small (divisibility of space), in
the large (considered as an open space-time manifold), and in terms of
information (he argued that in an open universe meaningful activity
could go on forever, though subjective time would slow down; Tipler made
the corresponding argument for a closed universe, in which case
subjective time would speed up).  The potential energy of gravitating
systems is theoretically infinite.  These are all consistent with
Feynman's principle that it shouldn't take an infinite amount of
information to describe what is going on in a finite amount of
spacetime, but even Feynman's principle fails in the current "best"
physical theories.

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

Another problem for finitism is that it has serious implications for
"general intellectual interest and the unity of knowledge".  I imagine
that a significant fraction of the FOM's members are theists; the most
basic truth of theology is that God exists and the most basic attribute
of God is infinity.  A seriously finitist ontological position is in
fact an atheistic position; while an atheistic "Monday-Friday" stance
may be justified for theistic scientists on methodological grounds
leaving them free to go to church on Saturday or Sunday, and a
Platonistic weekday attitude may be adopted by working mathematicians
who will retreat to finitism in their "off hours" or if pressed, the
unity of knowledge and general intellectual integrity demand that we at
least attempt to reconcile our schizophrenic attitudes.  A first step is
to recognize these connections.

It matters professionally whether you "really believe" in finitism.  For
example, a committed finitist will not be at all bothered by the
undecidability of CH and won't care about the implications of large
cardinals for determinacy of projective sets of reals, but may think the
absence of an algorithm to solve Diophantine Equations troublesome; on
the other hand, a Platonist who cares about CH and is frustrated by its
undecidability may have no problem with the result that all r.e. sets
are Diophantine because he knows that "the truth is out there" even if
no finitary machine can recognize it.  Maybe this could even serve as a
test of philosophical attitude: which is more bothersome, our inability
to solve Hilbert's First Problem (CH) or our inability to solve
Hilbert's Tenth Problem in its original terms (effective procedure for
Diophantine solvability)?

-- Joe Shipman