Finitist prejudices and ontological commitment

[Joe Shipman]

Martin Davis wrote:

> I do not know whether the universe is finite or infinite, assuming this to
> be a well-posed question (and I'm not sure that it is).

"Does it take a finite amount of information to describe the universe
completely?" is well-posed if "the universe" is well-defined.  There are
several different senses in which the universe might be infinite, but this
seems the most general one.  If there is no "theory of everything" and the
universe is in some sense "lawless" then the infinitude of the universe would
not necessarily contradict *mathematical* finitism; but even the current
incomplete mathematized theories of fundamental physics are highly infinitary
and would have to be actually *wrong* (not just incomplete but correct
descriptions of an at least partially lawful universe) for mathematical
finitism to be true.

I believe Hersh's position is that there is no "T.O.E." and that all physical
theories are just mathematical ways of talking about a primordial physical
reality that cannot be fully captured by such a theory; this avoids Quine's
argument for ontological commitment because the imperfect match between theory
and reality means an ontological commitment to physical reality does not
commit one to any ontological commitment about mathematical objects.  This is
not unreasonable, but again the view that there is no theory of everything, no
fully mathematical description of physical reality, seems an a priori one that
isn't well-supported by the evidence (the trend for the last few decades has
been for greater "unification" towards a T.O.E.).

It seems to me that those who view the free use in mathematics of infinite
sets with suspicion ought to view quantum electrodynamics (and other quantum
field theories) and general relativity with even more suspicion and ought to
demand finitary versions of these theories.  Feferman has shown that much of
fundamental physics can be developed in a version of type theory that is
conservative over weak systems of arithmetic; but the type theory itself has
an infinitistic ontology.  The complex reductions involved do not leave us
with any physical theory expressible in arithmetic with a finitary ontology;
and even if such a theory could be constructed, it would probably be so
cumbersome that it would be better described as a self-imposed narrowing of
attention than a meaningful description of what actually exists.

For example, one could attempt to talk only about experimentally observable
relationships between measurable quantities, casting one's formalism in
algorithmic terms; but the algorithms would be completely opaque in the
absence of all the mathematized physics which had informed their creation.  By
analogy, one can adopt an extreme behavioristic view and talk about what is
going on in a poker game or a debate entirely in mechanical and
neurobiological terms; but such a description, which eschewed any
consideration of "mental states", would be useless because one's attention has
focused too narrowly.

-- Joe Shipman