FOM: rigor and intuition

[Gordon Fisher]

The recent message by Vladimir Sazonov has made me
recall a comment made to me some years ago by William
(Bill) Duren, once head of mathematics at Tulane, and later
a dean at University of Virginia.  He remarked that he had
noticed that as mathematicians get older and older, there
writings contained less and less formal and symbolic language,
and more and more ordinary language.

This led me to speculate that when mathematicians are young,
they need to rely on formalization to make accurate use of
some material which has been developed up to their time in
the course of people doing mathematics, and perhaps also
to make use of ways of achieving a kind of certainty based
on past formulations of the intuitions (imaginations) of
mathematicians.  As they age and get more and more experience
with mathematics, and with one or more ordinary languages,
mathematicians may get better and better at  expressing their ideas
in ordinary languages.  Well, many of them, anyway.

Logicians, on the other hand, may follow some other paths.  I know
of one who did a PhD in logic in a philosophy department who told
me he was led to this by trying to understand infinity as it had been
presented to him in elementary calculus.  He later became a professor
of computer science.  I have a feeling, based on his dissertation
topic
(something about ordinary mathematical induction) and the way he
thought about so-called infinite loops in computer programs, that he
never did make his peace with, for example, the idea that the natural
numbers go on forever -- a situation that may lead mathematicians in
one sort of ways, logicians in another sort, philosophers
(non-logicians)
in another, theologians in still another, and who knows where people
who are none of these may be led by the notion of infinity?

Gordon Fisher     gfisher at shentel.net