FOM: rigor and intuition

[Matthew Frank]

In response to the quote from Kit Fine that

> > when there is a clash between intuition and rigour,
> > when one's sense of rigour prevents one from saying
> > what, from an intuitive point of view, it seems that one can say,
> > then it is rigour and not intuition that should give way.

Arnon Avron responded that

> Both logic and experience have taught me that whenever there is
> a clash between intuition and rigour it means that something is
> wrong with the intuition. Rigour simply cant be wrong.


I think both of the above misidentify the conflict.  Rigor and intuition
need not be in conflict; one should not evaluate either rigor or intuition
as right or wrong.

Rather, given a rigorous formal system and some intuitions, one should ask
whether the formal system aritculates or accords well with the intuitions,
and whether the intuitions are useful to us in finding proofs in the
formal system.

Martin Davis provided two useful examples, things about which some people
have intuitions that are not well articulated in standard rigorous systems
but which can be well articulated in other rigorous systems:  arbitrary
objects, with Fine's new rigorous system for them; and infinitesimals,
with Robinson's new rigorous system for them.  (I would also emphasize
that Robinson's systems articulate only some intuitions about
infinitesimals, and other rigorous formal systems articulate others.)
Neither of these is an example of rigor giving way.  


There is also a question about rigor and intuition in proofs, somewhat
different from the question previously being discussed.  Proofs may be
more or less rigorous and more or less intuitive.  Some prominent
mathematicians (notably the geometers Gromov and Thurston) have 
preferred to find intuitive proofs and present their proofs intuitively,
and think there are more important things to do than worry about how to
present the proofs rigorously.  This is perhaps a more interesting example
of rigor giving way.


We--and especially we who are interested in foundations of math--should
nurture our intuitions.  This could be an important function for
foundations of math:  to help articulate various mathematical intuitions
in such a way as to make them more useful in mathematical practice.

--Matt