FOM: infinity of the universe...

[Jeffrey Ketland]

Allen Hazen (5 November 2000, FOM: Infinity of the universe):

>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.

[snip]

>    (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.

I agree -- whether PA is interpretable within a physical theory T containing
geometry depends upon which relations to count as "physical".

If the physical geometry T is too weak, it might even be complete, via
Tarski's famous result for first-order geometry. Tarski 1951: A Decision
Method for Elementary Algebra and Geometry. Also, Tarski 1959, "What is
Elementary Geometry?", in Hintikka 1969, "The Philosophy of Mathematics".

BUT, in general, I think Holmes is right: e.g., Hartry Field (1980: "Science
Without Numbers") HAS discussed a way of developing physical theories (with
"quasi-second-order" quantifiers ranging over space-time regions) such that
PA *is* interpretable within them.

One such physical geometric theory discussed by Professor Field is called N.
This "nominalistic" theory N is meant to be a purely physical or
nominalistic theory of space-time, containing physical primitives "x is
between y and z" and "xy is congruent to wz", and primitive predicates for
fields on space-time.
Field mentions in his book that you can model PA in N. In particular, you
can apply Goedel's Incompleteness Theorem to N. There is a physical
statement Con(N) which is not provable in N. (I think these ideas that
Goedelian incompleteness will apply to formal axiomatizations of
nominalistic physics are due to Moschovakis and Burgess).
Stewart Shapiro wrote an important paper in 1983 ("Conservativeness and
Incompleteness", Journal of Philosophy 80) discussing this.
So, this is an important topic in recent philosophy of mathematics (which is
also connected to the sort of physical nominalism which Kanovei and Sazonov
seem to think is obvious).

Anyway, to cut a long story short, there may be interesting pure
mathematical statements (such as Goldbach's Conjecture or even the Continuum
Hypothesis) which may have "physical equivalents" (as statements about
physically definable sets of space-time points). E.g., let GC have a
physical equivalent GC*. Then GC is true in the numbers iff GC* is
physically true about some set of space-time points. So, even if GC is
independent of ZFC, we might (in principle) we able to figure out whether GC
is true by *physical experiments/measurements*.
Joe Shipman and I have discussed this idea before here on FOM.

Regards - Jeff

~~~~~~~~~~~ Jeffrey Ketland ~~~~~~~~~
Dept of Philosophy, University of Nottingham
Nottingham NG7 2RD United Kingdom
Tel: 0115 951 5843
Home: 0115 922 3978
E-mail: jeffrey.ketland at nottingham.ac.uk
Home: ketland at ketland.fsnet.co.uk
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~