FOM: defining "mathematics"

[Sam Buss]

   (This is in reply to the message of Simpson included below.)

   I phrased my definition of mathematics fairly delicately, and
one should read it very carefully to see what I intended.

   The first sentence was intended to be "the definition" of mathematics.
It read: "Mathematics is the study of objects and constructions, or of
aspects of objects and constructions, which are capable of being
fully and completely defined."

Please note the inclusion of the modifier "capable of being" before "fully
and completely defined."  This was to permit the study of mathematics before
modern day rigor to still qualify as mathematics.  Likewise, new areas
of mathematics, such as chaos, may well include the study of concepts which
have not yet reached full and complete definition.  But these areas
are still mathematics, because they study phenomena which are *capable of
being* fully and completely defined.

I should stress that according to this definition, one can be doing 
mathematics even if one is not meeting the robust and objective standard
of rigor mentioned in the second sentence.  It is the subject matter
that determines whether one is doing mathematics, not the method of
reasoning.

The second sentence was not part of the official "definition", but was
intended to clarify the nature of mathematics.  This sentence read:
"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."

Steve Simpson had a complaint about this point also, namely, that
biologists could reasonably claim to be using rigorous logic.  To 
this I have two responses:  (a) I never claimed that mathematics is
everything that is studied using logic and rigor, that is to
say, other endeavors such as biology can certainly use
rigorous reasoning,   and (b) I disagree with even the possibility
that a concept like "arachnid" is capable of being fully
and completely specified.  I think that biologists would agree that the
notion of "species" or "class" is fluid, and subject to change and to
interpretation.  Further, one could always imagine a continuum of possible 
species ranging from some arachnid species to, say, some crustecean species,
with no clear dividing lines between distinct species, much less clear dividing
lines between distinct classes.  Given suitable evolutionary conditions, these
species could actually occur.  (By the way, I hear there is currently debate
among biologists about a complete revamping of the taxonomy of species,
based on genetic histories.  Also the old rule of thumb that a species is
a set of animals which could mate and have fertile offspring is not universally
true as the relationship is not transitive.  All this is just to say that the
physical world of animal and plant life, is a complicated place, with
few mathematically rigorous definitions.)

In the end, Simpson and I are at least partially in agreement (although
he seems to have misinterpreted my definition).   In fact Simpson says 
in his email:   "Thus mathematics, like every other science, is to
be defined as the study of a specific subject matter."  I agree.

  --- Sam Buss

-------------------------------------
-----In reply to:

> From owner-fom at math.psu.edu Tue Dec 21 16:09 PST 1999
> From: Stephen G Simpson <simpson at math.psu.edu>
> Date: Tue, 21 Dec 1999 19:07:24 -0500 (EST)
> To: fom at math.psu.edu
> Subject: FOM: defining ``mathematics''
> 
> 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/
> 
>