FOM: rigor and intuition

Vladimir Sazonov V.Sazonov at
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 

> 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

Vladimir Sazonov                        V.Sazonov at 
Department of Computer Science          tel: (+44) 0151 794-6792
University of Liverpool                 fax: (+44) 0151 794 3715
Liverpool L69 7ZF, U.K.

More information about the FOM mailing list