FOM: rigor and intuition
V.Sazonov at csc.liv.ac.uk
Wed Feb 13 08:48:20 EST 2002
Gordon Fisher wrote:
> Vladimir Sazonov wrote:
> > Peter Schuster wrote:
> > > >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".
> > >
> > > How can you call a principle "quite intuitive" among whose consequences
> > > there are some which are commonly considered to be contra-intuitive?
> > Who knows in advance which consequences some "quite intuitive"
> > axiom can have. Getting these consequences we could start think
> > more about this axiom. But, e.g. AC seems to me (and seemingly
> > to most of mathematicians), nevertheless, sufficiently intuitive.
> Intuitive with respect to countable infinities, not so intuitive with
> respect to uncountable infinities?
Yes, "we could start think more about this axiom" and open a
way to new, more delicate intuitions around it. But, why ZFC,
not its part ZF or even ZF + not AC, is so widely adopted by
seemingly overwhelming majority of mathematicians
(consciously or not)?
I should note, that I am not a specialist in this very topic.
I just tried to reflect some general opinions. I also wanted
only to tell that some conflict between an intuition and
its formalization is usually inevitable, and this is a normal
thing which we should take into account when formalizing
("putting in a Procrustean bed") any idea. This contradicts
to the opinion (seemingly assumed in other postings)
that there should be complete coherence between intuition
and its formalization. Mathematics demonstrates just the
converse. The discussion started with some assertions on
some counter-intuitiveness of (expressing of) some formal
logical rules using "arbitrary objects". My reaction is:
1) this is understandable quite intuitively and straightforwardly
in some reasonable terms and 2) in any case, a small discomfort,
which still may exist, is a normal thing. Of course, if there is
****sufficiently strong reason*** for reconsidering logical
rules and their semantics by some developing the concept of
"arbitrary objects", it should be done. I did not read
the book of Kit Fine to judge. The very starting point
(something on incomplete coherence with intuition) is not
sufficiently convincing for me.
> Gordon Fisher gfisher at shentel.net
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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