[FOM] on bill tait's answers to my questions (II)
Gabriel Stolzenberg
gstolzen at math.bu.edu
Wed Mar 22 13:22:25 EST 2006
This is the second part of my reply to Bill Tait's answers of
March 16 to my questions of March 15.
It begins with Bill quoting me and then doubting what I say.
> > Among classical mathematicians, constructive math is the study of
> > what can be proved without omniscience."
> I doubt that you are right about what classical mathematicians would
say.
Trust me. These were my people. And in a sense they still are.
Did you really never hear things like, "I want to do God's mathematics"?
> Classical math admits the logical axiom 'A or not-A'. You may parse
> this as an expression of omniscience---as did Bishop. But I don't see
> it that way: Doesn't my view get to count? The talk about omniscience
is metaphorical and perhaps good for pep-talks in favor of constructive
math; but what is its real content?
Yes, talk about omniscience is metaphorical, just in the way you
say it is. A metaphor for LEM. But how dare you then assume that,
when we (Bishop and I) use it, we do not mean it that way? I find
that outrageous.
By the way, this is probably the first time I've ever used the
word "omniscience" in writing. God seems to be punishing me for it!
As for the idea that talk about omniscience may be good for pep
talks in favor of constructive math, I suppose some of the polemical
comments in Bishop's book can be read this way. E.g., "If God has
mathematics that needs to be done, let him do it himself." But do
you really think such talk was more than metaphorical?
Also, as I indicated above, some classical mathematicians like
the idea that they are doing "God's mathematics," no matter that
this may be metaphorical. 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?
Finally, you make it sound as if an assumption of omniscience is
bad. But if, as you say, it is merely a metaphor for LEM, surely you
can't mean this. Maybe you mean that it's a bad metaphor. Maybe it
is. But so is "choice" in "the axiom of choice." That's a really
bad metaphor. (So bad that even "omniscient choice" would be better,
wouldn't it?) Do you propose that we get rid of it?
> 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.
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. Put away the Tractatus; we
should be guided here by the rule following considerations. And ask
yourelf why you believe that you can distinguish "It is not, so far
as I see, dependent on our intuitions" from "It is independent of our
intuitions."
To be continued.
With best regards,
Gabriel
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