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

Gabriel Stolzenberg gstolzen at math.bu.edu
Fri Mar 24 00:25:24 EST 2006

   This is the fourth part of my response to Bill Tait's answers
of March 16 to my questions of March 15.

   I first want to clarify two things in my previous message, the
first one minor, the second one not.

   In that message, I remarked,

>  Shortly after Bishop died, Paul Halmos, who had been Bishop's
>  thesis advisor at Chicago, told Fred Richman, "Errett and I had
>  a disagreement.  Now he knows that I'm right."  How would you
>  parse this one?

   However, I forgot to add, "Yes, it's a joke.  But sometimes a joke
is not only a joke."

   Secondly, in response to Bill's assertion,

> > Proof is defined by means of axioms definitions and rules of
> > inference.  This is what is objective and independent of your
> > intuitions, or whatever, and mine,

I replied,

> The second sentence is wrong.  Indeed, it follows from the first
> that proof is dependent on our intuitions about axioms, definitions,
> rules of inference and many other things.

   However, in borrowing Bill's word "intuitions," as I did above, I
may have given the impression that I was using it in the ordinary way.
Bill was, I wasn't.  I was thinking "mindsets" but writing "intuitions,"
which may be too narrow.  E.g., I don't know how comfortable people
would be calling habits of thought "intuitions."  But it doesn't matter.
They are features of a mindset.

   To continue, here is Bill quoting one of my questions and then
answering it.

> >   Why do you dislike the interpretation of classical math as
> >  the part of constructive math in which we investigate how it
> >  helps to be omniscient?

> I don't really dislike it; it is amusing.

   Amusing?  Bill, as someone who published a paper with the subtitle
"constructive mathematics is part of classical mathematics," I think
you have an obligation to go further and explain that, in fact, each
is a part of the other.

   The belief that constructive math is limited *because* it is a
restricted part of classical math has had significant consequences
for mathematics.  Yet, although it is equally obvious that classical
math is a restricted part of constructive math, one never hears it
said that *therefore* classical math is limited.  Nor should it be
said.  Neither system is limited, even though each can be represented,
in a natural way, as a restricted subsystem of the other.

   (As for what I mean by "in a natural way," note that, if I prove
that LEM implies the Riemann hypothesis, classical mathematicians
won't tell me to come back when I've gotten rid of that assumption.)

   To be continued.  (One more to go.)


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