[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.)
Gabriel
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