FOM: defining ``mathematics''

[Stephen G Simpson]

Having concurred with Sam Buss on the importance of the predicate
calculus, let me take issue with Buss on another point, concerning the
definition of mathematics.

Buss Tue Dec 07 00:51:16 1999 defines ``mathematics'' as follows:

 > "Mathematics is the study of objects and constructions, or of
 > aspects of objects and constructions, which are capable of being
 > fully and completely defined.  A defining characteristic of
 > mathematics is that once mathematical objects are sufficiently
 > well-specified then mathematical reasoning can be carried out with
 > a robust and objective standard of rigor."

Now, I think this definition of mathematics is wrong on two counts.

First, Buss hangs his definition of mathematics on a methodological
issue: rigor.  But this doesn't seem to fit with the history of the
subject.  Our current standard of mathematical rigor evolved only in
the 19th and 20th centuries.  Would Buss claim that there was no
serious mathematics in the 17th and 18th centuries, prior to that
evolution?

Second, Buss's definition of mathematics in terms of rigor/objectivity
would seem to deny the possibility of rigor/objectivity in all
sciences other than mathematics.  For example, if biologists were to
come up with a rigorous definition of ``arachnid'' which permits
reasoning about arachnids to ``be carried out with a robust and
objective standard of rigor'', would that remove arachnids from the
realm of biology and make them part of mathematics?  If I were a
biologist, shouldn't I take offense at Buss's implicit suggestion that
the standards of rigor/objectivity in biology are somehow necessarily
lower than in 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.  An interesting study of this topic is David
Lachterman's book ``The Ethics of Geometry'', Routledge, 1989, 255
pages.

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.  Some people may consider this definition
old-fashioned and outmoded, but I think it has a lot of merit.

See also my short essay on the foundations of mathematics at
<http://www.math.psu.edu/simpson/hierarchy.html>.

-- Steve

Name: Stephen G. Simpson
Position: Professor of Mathematics
Institution: Penn State University
Research interest: foundations of mathematics
More information: http://www.math.psu.edu/simpson/