FOM: rigor and intuition

[Vladimir Sazonov]

Matthew Frank wrote:
> 
> 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.


I cannot agree. Without rigor there is no mathematics at all.


> 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.

Completely agree! Formalisms (which embody mathematical rigor) 
are our instruments, mechanisms (for thought) which cannot be 
wrong in principle. They may be only useful, convenient, intuitive 
or not. 

> 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.


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". 

> 
> 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.

We always should try, but, actually, there is no hope for a 
complete harmony between Procrustean bed of a formalism and 
the intuition which we try to formalize. Moreover, after 
formalizing, the initial intuition may change radically. 
We even can forget what was the initial intuition before 
formalization because it is usually disappearing and after 
formalization can exists only TOGETHER or DUE TO 
formalization. Which was our intuition on natural numbers 
before studying Peano arithmetic? Who can recall? Did we think 
from the beginning on potential infinity or we imagined a 
biggest natural number before a teacher told us (being only 
obedient children) how we MUST think? Vice versa, any reasonable 
mathematical formalism always has some "underlying" intuition 
without which it would be impossible to use it. 


> 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.

I agree with Vladik Kreinovich (Hi, Vladik!) who already replied 
to this. I could only repeat that without rigor (formalisms) 
there is no mathematics. 

> 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.

Yes, of course! But always remembering that mathematical 
intuition cannot exist in a pure form, only with and due to 
formalisms. Otherwise it is not a mathematical intuition. 

> 
> --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