FOM: rigor and intuition

[Matthew Frank]

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, or between
intuitions about sets and intuitions about volumes.


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.

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.
 
--Matt