FOM: rigor and intuition
Vladimir Sazonov
V.Sazonov at csc.liv.ac.uk
Wed Feb 13 08:44:36 EST 2002
Gordon Fisher wrote:
>
> 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.
Only one note: an automatic (implicit unconscious) using of
formalism is, nevertheless, - using a formalism. I never asserted
the necessity for working mathematicians of COMPLETE formalizations.
However, in some cases this issue can arise. Also, we should think
more carefully on POTENTIAL FORMALIZABILITY. It is sufficiently
delicate question what does it mean. There is one proof, due to
Orevkov (used as a *part* of his demonstration that cut elimination
is non elementary), whose potential formalizability might be questioned
due to its non-acceptable length - just imaginary "finite" proof.
(Of course, his whole proof on complexity of cut elimination is
doubtless!)
> 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?
Yes, I believe this is a very important issue - reconsidering the
concepts of finity/infinity. I mean here not large cardinals,
whichever this topic might be important and interesting.
I mean that intuitive infinity IN natural numbers starts much
earlier, before it is usually considered. I wrote about this many
times to FOM.
>
> Gordon Fisher gfisher at shentel.net
--
Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Department of Computer Science tel: (+44) 0151 794-6792
University of Liverpool fax: (+44) 0151 794 3715
Liverpool L69 7ZF, U.K. http://www.csc.liv.ac.uk/~sazonov
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