[FOM] PA Incompleteness

[Feng Ye]

I am always wondering what will be the answer if 'normal mathematics' is
replaced by 'mathematics with potential applications in sciences', or
'mathematics relevant to this physical universe'. The problem is that all
current independent propositions are related to fast growing functions, but
the scope of the physical universe that sciences are dealing with is so
'ridiculously small' from the mathematical point of view. 

The ratio of the linear cosmological scale to the Planck scale is less than
10^100, and the number of fundamental particles in the universe is also
quite small (compared with numbers produced by those fast growing functions
for very small arguments). Current sciences describe *only* things within
that small range, from the Planck scale up to the cosmological scale. At
least, after excluding physics theories about things at or below the Planck
scale, all mathematical applications are applications to strictly finite
things whose physics quantities are represented by numbers that are
'ridiculously small' (in terms of absolute magnitudes or precisions). 

It seems unlikely that the operation patterns in Goodstein sequences, or in
Hercules and the Hydra, will be directly related to any real physical
processes to make those independent theorems directly applicable in to real
things. This certainly does not rule out the possibility that those theorems
are applicable in some indirect ways, but I am always wondering what that
could be.

Quine and some philosophers following him called mathematics with no
potential applications in sciences 'recreational mathematics'. Realizing
that sciences deal with only such a small range of things in the universe,
we may suspect that anything *essentially* beyond PA or even a weaker system
(in the sense that it strictly, logically indispensably requires the logical
power of all the axioms in the system) has to be 'recreational' in that
Quinean sense. This is related my post last month (an initial post), that
is, there is a chance that strict finitism is already *in principle*
sufficient for scientific applications to things within such a small range
of the universe dealt with by current sciences.

Feng Ye

> -----Original Message-----
> From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of
Harvey
> Friedman
> Sent: Sunday, October 14, 2007 10:03 PM
> To: fom
> Subject: [FOM] PA Incompleteness
> 
> PA incompleteness is now a reasonably well defined and reasonably well
> developed area within mathematical logic. After the initial results (see
> below), the focus, for many years, has been on the integration of such
> exotic combinatorics into normal professional mathematical activity, with
> its normal motivation and normal culture.
>