More seriously, I would say, off part of the surface of my head, that one
can recognize some cultural artifact as being "mathematical" if it has to
do with numbers, or with constructs which have grown out of studies and
actions concerning numbers such as algebraic or formal systems, or it has
to do with shapes abstracted from those we see, or which we envision on the
bases of those we see, or it has to do with connections between numbers and
shapes.
I've always been fond of thinking that formal logic, as delineated by
Aristotle, grew in Aristotle's mind out of his study of the mathematics of
his time. One can also argue that formal logics can form a kind of
language in which mathematics can be to some extent communicated. One
might want to add, in the manner of Hilbert (I think it was, in his book
with Cohn-Vossen), that the roots of mathematics are intuitions of
mathematicians. I presume "intuitions" here is a translation of the German
word "Anschauungen", which to my mind carries with it shades of Kant's
philosophy, and also implications of geometric visualization, or at least a
kind of visualization of structures that, to repeat myself, have grown out
of studies and actions concerning numbers and shapes.
Of course, use of the word intuition brings to mind Brouwer's philosophical
attitudes, which I suppose are as forceful as any in opposing the view that
mathematics is somehow subsumed by logic, rather than, let's say, being one
of its tools. "Logic creates nothing," Brouwer was reputed as saying,
"logic destroys".
And of course Russell and Whitehead never produced the projected third
volume (I think it was to be the third) of their *Principia Mathematica*, a
work which they appear to have hoped would unilaterally and totally furnish
a language of a sort in which all the actions and ideas of mathematicians
could be formulated. Stated thus baldly and I hope not too simplistically,
this program seems infeasible if only because it leaves out mathematicians
as human creatures. You might say it tends to make them into computers.
And then of course, speaking formally, there is the Russell paradox.
And in conclusion, let me offer my private definition of the term
"function" as used by mathematicians, as an alternative to a certain kind
of subset of a Cartesian product, or some kind of abstract correspondence.
Definition: A *function* is something a mathematician does. :-)
And in conclusion (this is the second conclusion -- why not?) I'm fond of
the dictum of Poincare to the effect that set theory is a disease from
which mathematics will eventually recover. And so it has, to some extent,
though set theory remains a lovely tool in many situations --
"mathematical" situations, that is.
Gordon Fisher gfisher@shentel.net