FOM: rigor and intuition
V.Sazonov at csc.liv.ac.uk
Wed Feb 13 08:53:56 EST 2002
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
> 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
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.
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.
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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