[FOM] Resources on the empirical foundations of mathematics

[Richard Haney]

I am interested in studying the *empirical* foundations of mathematics.
 In particular, I am interested in exploring the *empirical*
foundations of logic, number systems, and geometry.  I am interested in
exploring the possibility that conventional logic(s) may fail to yield
empirically true conclusions when applied to *real-world* phenomena
even though the premises may seem to be perfectly true.  I am also
interested in exploring the possibility, for example, that the rules of
arithmetic may regularly (or irregularly) fail for, say, sufficiently
large numbers when applied to *real-world* phenomena.  I am interested,
for example, in considering the possibility that assumptions of
constructivists may be too strong.  And, for example, I am interested
in exploring how much interesting, useful mathematics can be done by
allowing oneself, when talking about natural numbers, to talk only
about natural numbers less than or equal to some unspecified, rather
large natural number.

I have also read that some theoretical physicists have theorized that
space-time may be discrete (i.e., not continuous) at a sufficiently
small, sub-atomic scale.  This sort of thing, too, would be of interest
as to the sort of mathematics this might give rise to, say, in place of
the real numbers, especially considering that physical space on a
straight line was the origin of the ancient Greek concept of numbers. 
Insisting that every side of every idealized triangle had to have
exactly one length (number) may have led the world down one historical
path in mathematics, whereas alternatives might have been quite
different.  I don't pretend to be an expert on these things, but I
would like to learn more about them.  As far as I can tell, these
issues, with a focus on empirical relevance, seem to have gone
unexplored at the level of serious research.

I imagine that there may be a great many other interesting, relevant
issues and questions in the empirical foundations of mathematics but
that I am simply unable to imagine or formulate them at the moment.

So what I would like to do is to find some really good resources for
research in this area.

Can anyone help me out with this?

Richard Haney


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