FOM: rigor and intuition

charles silver silver_1 at
Wed Feb 13 07:29:40 EST 2002

Vladimir Sazonov writes:
----- > > 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.
Vladimir Sazonov:
> 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.

    I don't want to be seen as constantly defending Kit Fine, but I do
believe Fine clearly has a point here that's being neglected, and I think
it's a rather important point--also one that's in line with Martin Davis's
comments about Robinson's formalization of infinitesimals.   Suppose a
person has an intuition--a mathematical intuition--that flies in the face of
available formalism.   Take the quotation I cited, where Kit Fine has an
"intuition" in favor of "arbitrary objects," but he knows that the available
formalism does not support this intuition.  What does he do?   As indicated
by the quotation, he first presents the material to his students (in this
case, universal generalization) in terms of the available formalism, *not*
in terms of his intuition.  Why?  I think obviously because he knows that
his intuition could be *wrong*.   He wants to check his intuition, make sure
he hasn't been led astray, and also to see whether there may be something
more to that intuition deserving of greater attention.  I admit that now I'm
on the shaky ground of psychological interpretation since I have no real
knowledge of what was in his head at the time besides his words in the
quoted part of the Preface to his book.  So, I am interpreting, or perhaps
merely *guessing*.   At any rate, I therefore *guess* that after having
thought for a while about his intuition, Fine decided that it's a valuable
one and that he can make it quite rigorous.  But to make this intuition
rigorous required an entirely new formalism, one that no one else seems to
have previously thought of.   In fact, he tells us that this was how his
book originated.

    Let us speculate on what A. Robinson was thinking?   What was in his
head at the time?   I have no idea.   If anyone can comment knowledgeably
about what Robinson thought about *prior* to his discovery and subsequent
development of non-standard analysis, those comments would be valuable and
much appreciated.  Admittedly, it is possible that Robinson simply stumbled
upon how to formalize infinitesimals, but I would suppose that he had a
notion, an idea, an *intuition*, that, contra. Berkeley and others who had
ridiculed infinitesimals, they really did (or *could*) make sense.   And, I
am further supposing that he developed this "intuition" and made it formal.
And viola, that's why we have non-standard analysis today!

    However, there can also be some interplay between intuition and
formalism in the discovery of new ideas.   One can be formally working on
some matter or other and while working on it suddenly realize that something
completely different can be shown.   In this case, there's the formalism
plus the flash of insight (the "intuition") that together lead to a new
idea.   And, of course, there are much more nuanced scenarios.

    My only point is that intuition apart from formalism should not be
discounted, but examined, which is what I think Kit Fine did.   *After*
examining it, he decided that his intuition was valuable and could lead to
an entirely different formal development.   My appreciation of this point by
no means indicates that I thereby consider "arbitrary objects" to
exist--whatever that means.


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