[FOM] on bill tait's answers to my questions (V)

Gabriel Stolzenberg gstolzen at math.bu.edu
Fri Mar 24 00:46:21 EST 2006

   This completes my response to Bill Tait's answers of March 16
to my questions of March 15.

   As part of his answer to one of the questions, Bill remarked,

> Of course, [Bolzano's] proof [of the intermediate value theorem]
> was not well-known, but Cauchy's proof some four years later seems
to have been widely known.

   Cauchy's Cours d'Analyse is one of my all-time favorite math books.
(I named my own analysis book, "A new course of analysis," in homage
to it.)  If I remember correctly, the IVT is "proved" by an appeal to
intuition like the one in "If a man goes up a mountain on one day and
comes down the next, there is a time of day at which he was at the
same position on both days.  Proof.  Suppose it is two men on one day,
one going up, the other going down.  Then they will meet."

   The error here is to mistake an intuition of a position at which
the men will meet for one of a time at which they are both at that

   The next statement is about Kronecker.

>  It is also worth mentioning that for him your constructivism
> is too profligate. He rejected objects not representable by integers
> and properties which are not algorithmic. So it wasn't for him a
> matter of rejecting excluded middle; rather mathematics should not be
> concerned with propositions for which EM is not provable.

   Bill, if you found these views worth mentioning, I wish you had
continued into less well charted territory and considered how they
bear on the question of what, if anything, set talk is about.  And
whether it matters.

   Also, at least in this context and probably always, there is no
such thing as rejecting an object.  It's rather that one does not
accept and maybe even rejects the idea that there are such objects.
Think of the case of ghosts.

   Finally, Bill puts some questions to me.

> Now let me ask a question: Why is it so important to you to wage war
> with classical mathematics?

   War?  That's a new one.  I love classical mathematics.  How not?
It's one of the greatest intellectual achievements of mankind.  And
I've had the privilege to be part of it.  This is where I start from.
You must have someone else in mind.  As you seem to have had during
most of our exchange.

  In Bill's next question, "its" refers to classical mathematics.

> Why should its existence (indeed, thriving existence) threaten the
> pursuit of constructive math?

   I'll be glad to tell you.  But if you really want to know, you'll
have to start taking seriously the existence of the two mindsets.  It
can't be explained without them.

> The bible-thumping seems completely inappropriate to me.

    Bill, after finding so much of what I've said here to be so far
off-base and even perverse, aren't you at all curious about what may
actually be going on?

   Gabriel Stolzenberg

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