FOM: Re: defining ``mathematics''

[Karlis Podnieks]

-----Original Message-----
From: Stephen G Simpson <simpson at math.psu.edu>
Date: 1999. 22. dec. 2:26
Subject: FOM: defining ``mathematics''


...

The idea of identifying mathematics with rigor/objectivity per
se, or
the rigorous/objective part of our thinking, has a long pedigree
going
back to Descartes.  Nevertheless, in my opinion, this idea is
fundamentally flawed.
...

It seems to me that the right way to distinguish the various
sciences
from each other is not in terms of methodological issues, but in
terms
of subject matter.  Thus mathematics, like every other science,
is to
be defined as the study of a specific subject matter.  To
delimit that
subject matter may be difficult, but as a first attempt let's
call it
``quantity''.  In other words, I am suggesting to define
mathematics
as the science of quantity.

KP> For me as a former marxist, this should sound good. Marx and
Engels defined mathematics as "the study of quantitative
relations in the real world". In 1930s, Kolmogorov tried to
justify this definition by extending the notion of quantity to
cover graphs, groups and all the other mathematical structures.
I could prove that this totally extended notion of quantity
coincides with the notion of self-contained models (i.e. models
that could be used by robots). In other words, I am suggesting
to define mathematics as the science of self-contained models.

Since self-contained models represent the rigorous part of our
thinking, we arrive to the "wrong" (yet equivalent with the
"right") definition of mathematics in terms of methodological
issues. No problems to defend this position against arguments
from biology or history of mathematics. Still, see some other
arguments against it on the page "Digital mathematics and
non-digital mathematics. Trying to understand anti-formalists"
at  http://www.ltn.lv/~podnieks/ .

Karlis Podnieks
University of Latvia
Institute of Mathematics and Computer Science