FOM: rigor and intuition

[Vladimir Sazonov]

Matthew Frank wrote:
> 
> I said:
> 
> > rigor and intuition need not be in conflict.
> 
> and Vladimir Sazonov responded
> 
> > Some conflict is inevitable, as it is shown by the example of quite
> > intuitive Axiom of Choice leading to non measurable sets and other
> > "paradoxes".
> 
> I would not call this a case of conflict between intuition and rigor.  It
> is a conflict between some intuitions and some formal systems, 

Yes! Have you any doubts that formalizability = rigor? 
Do you really believe that it is always possible to formalize 
anything as deep as set theory without conflicts with the intuition? 

> or between
> intuitions about sets and intuitions about volumes.

Intuition about sets has no immediate conflict with the intuition 
about volumes. This conflict arises only trough a formalism. Intuitions 
themselves are so vague things that sometimes they are unable to 
"contact" one with another without some help. You should first prove 
the existence of non-measurable sets. And this is a rather formal 
enterprise. 

> 
> Sazonov also said:
> 
> > without rigor (formalisms) there is no mathematics
> 
> I would say that rigor and formalisms provide a needed basis for
> understanding among mathematicians.  If all geometers treated rigor and
> formalisms in the way that Gromov and Thurston do, I think that subject
> would collapse in misunderstandings among its researchers.   Gromov and
> Thurston's ideas have been incorporated into the mainstream of geometry
> largely because other mathematicians have taken the time to work them out
> rigorously. 

Yes, because otherwise this would not be considered as a mathematics. 
You just confirm what I say. Any ideas, preliminary considerations, 
informal proofs, whichever important they are, should be formalized 
if they pretend to be mathematical. Other sciences, like physics, 
have different criteria. Note, that I say nothing against using 
intuitions in mathematics which is impossible both without intuitions 
and formalisms (supporting and reflecting these intuitions). I say 
only about relationship between intuition and rigor. 

> 
> It seems that the mathematical community is willing to tolerate low
> standards of rigor from people whose intuitions are unusually helpful, but
> holds most of us to higher standards of rigor.  This seems to be a
> productive approach.

Who doubts? 

I conclude that the only point where we seemingly disagree 
is your hope that appropriate conflictless formalisms always 
exist. I think that we always should (or can) try, but realize 
that in general this is impossible. Only something ESSENTIAL  
in our intuition and corresponding formalism may be in a 
coherence. That is quite enough to develop mathematics successfully. 

> 
> --Matt

-- 
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