[FOM] Two level foundations?

[Karlis Podnieks]

(Inspired by reading David Corfield's "Towards a Philosophy of Real
Mathematics")

Please, excuse my returning to old controversies, but, it seems, the problem
of the "nature" of natural numbers is no yet settled. Formalists declare
that "simply" there is PA, there are several versions of the second order
arithmetic and many versions of arithmetic in various set theories (ZFC, its
extensions by large cardinal axioms etc.). On the other hand, non-formalists
have a feeling that "behind" all these formal arithmetical systems there
exists some stable and unique structure, which is independent of these
systems, and which none of them can capture completely. By the way,
formalists also have a similar strong feeling, but they reject it
deliberately, mainly - without any attempt to explain, where could this
feeling come from.

Couldn't separating of the two following levels in the foundations of
mathematics lead to a solution?

Low level foundations - explain what makes mathematics different from other
branches of science, what makes its position unique (the difference between
mathematics and physics seems to be completely other that the difference
between physics and biology), why are "snook theories" possible in
mathematics only, how are applications of mathematics possible, etc.

High level foundations - take the low level explanation as a basis, and
explain, how real mathematics works: what is the role of each
particular mathematical structure, how are these structures emerging, how
could "snook theories" be avoided, "where mathematics is going", etc.

For me, the "nature" of natural numbers (this strange strong feeling...)
needs a high level explanation. At the bottom level, the only "safe" things
I can see, are "many formal theories"...

Couldn't some of the recent FOM threads initiated by Prof.Friedman be
regarded as steps towards a high level explanation of the "nature" of
natural numbers?

Best wishes,
Karlis.Podnieks at mii.lu.lv
www.ltn.lv/~podnieks
Institute of Mathematics and Computer Science
University of Latvia