FOM: modeling PRA in the physical world

[Stephen G Simpson]

In my posting of 1 Mar 1999 23:35:34 I suggested the following
approach to f.o.m.:

 > to argue that PRA is consistent because the physical world provides
 > a model of it, and then to justify at least a significant fragment
 > of mathematics by reducing it to PRA, a la Hilbert's program of
 > finite reductionism.

Raatikainen 3 Mar 1999 19:49:54 asked

 > isn't it the received view in it that the universe is finite ?
 > PRA, on the other hand, requires an infinite universe of discourse.

The universe is finite, but Aristotle posits a distinction between
actual and potential infinity.  The earth has rotated around the sun a
finite number of times, but this number is potentially infinite.  A
physical line segment may be finite yet potentially infinite by
division, i.e. capable of being subdivided an unlimitedly large number
of times.  One can argue that PRA is implicit in the physical fact
that many kinds of discrete acts can be repeated a potentially
infinite (or at least an unlimitedly large finite) number of times.
The act of adding 1 can be repeated indefinitely, and this gives
addition.  The act of addition can be repeated indefinitely, and this
gives multiplication.  Etc.  PRA has been identified with finitism.
Hilbert described finitism as an unproblematic part of mathematics,
which eschews the infinite and is indispensable for all scientific
reasoning.

Shipman 02 Mar 1999 09:42:49 asked

 > What arguments that the physical world provides a model of PRA do
 > not also apply to PA?

In a nutshell: Potential infinity may be enough to justify PRA, while
actual infinity is needed to justify PA.

To elaborate:

One can argue that nested quantification over a domain (i.e. locutions
such as `for all x in the domain D there exists y in the domain D such
that ...') requires an ontological commitment to the (actual)
existence of that domain.  I think Quine has argued this way.  Thus,
since PA involves nested quantification over the natural numbers, it
would appear that PA represents an ontological commitment to actual
infinity.  The physical world may not contain any actual infinities.
PRA on the other hand does not entail any such commitment, because it
can be formalized as a quantifier-free theory.

My paper on Hilbert's program
<http://www.math.psu.edu/simpson/papers/hilbert/> contains a few pages
of material elaborating on these and similar questions.

There is also my longish posting from last October

  Subject: FOM: arithmetic, geometry, natural science, formal systems, ...
  Date: Thu, 15 Oct 1998 13:06:14 -0400 (EDT)

I'm aware that much more needs to be said, but this seems like a
promising line for f.o.m.

-- Steve