[FOM] intuitions of logic in Chicago and Cambridge

Gabriel Stolzenberg gstolzen at math.bu.edu
Sun Feb 26 20:08:36 EST 2006

   In "intuitions of intuition in Chicago and Cambridge" (submitted
February 26), I quote Bill Tait as saying,

>   In the first sense, one could say intuition takes over where
> logic can't go....

   Here I want to add the following comment to my reply.

   How does this play out for the question of the validity of the
law of excluded middle?  Can logic go there?  Unless I very badly
misunderstand, it can't go as an arbiter.  Does this mean that the
question is supposed to be settled by our intuitions?  I think a
lot of classical mathematicians think so.

   In fact, classical mathematicians  sometimes use their logical
intuitions to "prove" the law of excluded middle.  Although they
don't realize it, they use excluded middle reasoning to prove the
statement of the law of excluded middle.

   It goes like this.  I say something like, "How do you know that
'P or not P' is true?"  The immediate response is, "Well, suppose
not.  Then we'd have a contradiction.  So it's true."

   Actually, they don't even say, "So it's true," because in ordinary
mathematical practice that part goes without saying---because the law
of excluded middle goes without saying!

   (This "proof" of the law of excluded middle is reminiscent of the
following "proof" of the consistency of mathematics.  "Suppose not.
Then we have a contradiction."  Except that the latter is not meant
to be taken too seriously.)

    Gabriel Stolzenberg

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