FOM: What is mathematics?

[Gordon Fisher]

Peter Schuster wrote:

> >From owner-fom at math.psu.edu Wed Feb 13 00:51 MET 2002
> >Date: Tue, 12 Feb 2002 15:03:14 -0500
> >From: Ayan Mahalanobis <amah8857 at fau.edu>
>
> > I have a
> >more basic and elementary question; What is mathematics? or rather What
> >should be Mathematics?
>
> ------
>
> Following the principle that some thing is by what it is distinguished
> from other things, couldn't one just say that mathematics is proving?
>
> ------

My view is that "proving" is a process of convincing oneself,
and appropriate others, that _something_ is "true", or "right"
or "correct".  And I ask, what is a _something_ which one
wants to prove is "true" (etc.)?  It may be a "proposition"
or "theorem" or "lemma" or "statement" (etc.) composed
formally using some axiomatic system, or informally
using some sort of jargon within an ordinary language.
But where does such a "proposition" (etc.) come from
in the first place?  And, even more, where did the
axiomatic system come from in the first place?

I suggest that, for example, Euclid did not set out
to prove in his _Elements_ some propositions which
he derived from axioms and postulates he found
engraved on stone somewhere (or on papyrus, etc.).
He had to formulate some axioms and postulates,
guided by untold centuries, maybe millennia, of
geometrical "results" (to use a rather neutral term)
attained by hundreds, maybe millions of individuals,
and passed on through generations using languages,
oral or written, and originally not formal languages,
but ordinary ones.  To formulate his assumptions,
he had to have some idea in advance of the sort of
things he wanted to prove, and, more impressive
still, that the "things" he wanted to prove could be
related to one another in certain ways, by way of
what we have come to call logic (or logics).

One hears sometimes that geometry began with the
ancient Babylonians, or maybe Sumerians, or that the sort of
axiomatic geometry put forth by Euclid began with Thales,
and so on.  However, I suggest that geometry, in a
comprehensive sense, began much earlier in the course of
evolution, in life forms which preceded humankind, and
that the true origin of geometry lies somewhere in the brains,
neurological structures (especially the visual systems), of
some if not all living creatures.  Humans, as a community,
somehow attained a wonderful way of communicating about
geometry (or geometries), and lots of other "things", using
languages, and eventually formal systems.

Along the way, Euclid (to go back to him), standing on the
shoulders of innumerable others, put into language (classical
Greek) what I will call _intuitions_ of these others, no doubt
along with some of his own.

So I propose that a proposition that _mathematics_ as a
whole be defined as the study of formal structures, or the
like, or that it be confined to "proving", is to be too narrow.
Certainly these activities are vital in mathematics, and are
effected in various ways.  I won't say, however, that they
are "central" to mathematics, which I think would be misleading,
because of the way people tend to think geometrically, so many
might take this to mean that I don't think that the thinking of humans,
as they interact with their environments, in the ways they have
come into being in the world, individually and collectively,
is also vital to mathematics, and incidentally preceded in historical
time such activities as "proving" and contructing "formal structures".

I note in conclusion, before I run on too long, and wear out such
welcome as I may have on this list, that some of these thoughts
owe something to the philosophical school known as phenomenology,
and especially to work of Edmund Husserl, who started out as a
mathematician.  However, my career has been as a mathematician,
more "intuitively" than "formally" inclined, although to be sure, I
have spent a lot of time coming to terms with a plethora of formal
systems, from algebraic structures as used by non-philosophical
mathematicians, to formal structures as used by logicians who
are mathematically inclined.

I remember once Hans Samelson saying years ago at the
University of Michigan (mid-1950s or so), as I remember
it now, that one might think of mathematicians as distributed
along a line segment with geometers at one end and
algebraists at the other.  Those around the middle of the
segment could be regarded as having a generous dose
of both points of view.  The degrees of these abilities
in a given person would have to be attached in an
auxiliary way to points on the segment, and I suppose
more than one person could occupy the same "point"
on the segment.  One might include "formal systems"
and "formal logics" as kinds of "algebras" (or vice versa!)

Gordon Fisher     gfisher at shentel.net


>
>
> Name: Peter M. Schuster
> Instituition: University of Munich, Mathematics Department
> Research interest: constructive mathematics
> http://www.mathematik.uni-muenchen.de/~pschust