[FOM] Defining 'Foundations of Mathematics'

[Donald]

In watching this forum and reading the latest posts on teaching
infinitesimal calculus, I have to wonder what is being included as 'The
Foundation of Mathematics'?

The discussions around ZFC seem to consist entirely of pure mathematics.
Not a bad thing, however does it constitute a Foundation that can cover all
of mathematics - including applied mathematics?  Reading Stephan Korner's
'The Philosophy of Mathematics', he analyzes 'Mathematics as Logic',
'Mathematics as the Science of Formal Systems', and 'Mathematics as the
Activity of Intuitive Constructions' and provides a criticism of each based
upon their inability to address applied mathematics.  He then goes on to
discuss 'The Nature of Pure and Applied Mathematics'.  I wonder if his
insight might also apply to this forum?

Several discussions around Constructivism appear partial attempts to bridge
into applied math. I can see this perspective of mathematics being an
attempt to include the limitations of actual mathematical processes - yet it
still appears to reside in pure mathematics (and Dr. Korner provides some
evidence in his criticism of it).  The application (or discussion) of
computer systems to mathematical issues should bring in applied aspects,
however discussions in this forum also seem to limit these to the pure side.

A potent set of mathematical tools are the numeric systems used by both pure
and applied mathematics.  In many respects numeric systems might be said to
lie on the boundary between pure and applied mathematics - being used by
both and hence be a foundational tool of both aspects of mathematics.  It
would, therefore, seem that any Foundation of Mathematics should account for
such a tool set.

I see the discussions on the Foundations of Mathematics utilizing numeric
systems, however I do not see them accounting for these systems.  In my
(albeit light) background in mathematics, 'boundary' situations were usually
where new ideas (and the more interesting work) was.  Maybe the pure and
applied boundary, including numeric systems, might prove fruitful.


Take Care,
-Don

"Only theory decides what it is that we manage to observe."  A. Einstein
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