FOM: infinity of the universe...

[Allen Hazen]

A propos Holmes's post on the infinity of the universe.
   Suppose the universe IS infinite in extent (for every natural number
there is a region of space of nore than that number of cubic light-years in
volume).  It doesn't automatically follow that interesting mathematical
assertions, such as Goldbach's conjecture, have physical interpretations:
for that we need not only lots of physical entities (to take the place of
the numbers) but also physical RELATIONS to interpret the mathematical
predicates (& function-expressions), and it isn't obvious to me that these
will be available.
   Simple example.  Suppose that the infinite physical universe is like the
American Mid-west: a vast and boring plain stretching out farther than the
eye can see.  And suppose the objects chosen to represent the numbers are
the (indistinguishable except by their position) fence-posts in a fence
that stretches off to infinity.  Now, one of the mathematical predicates we
will want to interpret is DIVIDES.  This is a relation relating any number
to others arbitrarily much larger, and (if the first number is >1) not
relating it to their immediate successors.  So we want a physical relation
that relates a given fencepost DIFFERENTIALLY (YES to this one, NO to that)
to posts arbitrarily far away.   Without cheating and defining the relation
in MATHEMATICAL terms, and thinking of PHYSICAL relations as those whose
holding or failing to hold between two objects can, at least in principle,
be ascertained by observational and experimental techniques, are there
likely to be such relations?  I have an ugly feeling that, if the physics
of my imagined hyper-Iowa is at all like that of the real world, that for
any specified PHYSICAL relation there will be a distance such that it will
relate a given post to either all or to none of the posts more than this
distance past it. And similarly for physical relations in the perhaps
spatially (& temporally) infinite real world.  But this is just a suspicion
on my part.
    (Mathematical-- model-theoretic, to be precise-- point strengthening my
suspicion: there are infinite structures in whose First-Order theories
First Order arithmetic cannot be interpreted. I see no A PRIORI reason for
confidence that the "structure" having the set of physical "objects" as its
domain and "physical" relations as its relations isn't one of them.)
    Moral.  I don't feel attracted to the sort of ultra-finitism and
anti-platonism Kanovei and Sazonov have been arguing for, but I don't think
it is easy to use physical cosmology to establish the meaningfulness
(independent of proof) of infinitistic mathematical assertions.