[FOM] on bill tait's answers to my questions (I)
Gabriel Stolzenberg
gstolzen at math.bu.edu
Tue Mar 21 23:54:15 EST 2006
This is the first part of my reply to Bill Tait's answers of
March 16 to my questions of March 15.
> Gabriel, Your subject line borders, at least to my ear, on insulting.
> I hope that you did not intend it to be.
How so? It was meant as a pun on Dr. Phil, the guy on Oprah. Is
there something objectionable about him? I thought he was supposed to
be a good guy. And an authority!
Anyway, I'm very sorry for upsetting you. Things are complicated
(and painful) enough as they are!
> Even in cases like the intermediate value theorem, which is not
> constructive, the technique used to prove a constructively valid weaker
> form is a refinement of Bolzano's original nested intervals argument.
It also is used to prove, under a transversality assumption, a
constructively valid *stronger* form that has important applications.
> My last remark is motivated by a consideration that your question
> may be ignoring: The agenda of classical mathematics is not
> restricted to finding constructive bounds or to proving existential
> propositions only when constructive bounds exist.
My question has nothing to do with your concern. I asked because
an affirmative answer to it is part of the self-image of classical
mathematics. Also, it is not just that the agenda is "not restricted"
to such things. They are not on it, nor should they be. This is
classical mathematics, where constructive considerations are, as I
can report from personal experience, tedious and irrelevant.
Re whether a "pure" existence proof can be an annoying distraction
from getting a construction," some classical mathematicians find the
following theorem fascinating.
Theorem. There exist irrational numbers, b and e, such that b
to the e is rational.
Proof. Try letting both b and e be sqrt(2). If b to the e is
rational, we're done. If it's irrational, let it be b and let e
be sqrt(2) again. This gives 2, so in both cases we're done.
But we don't know whether the b that works is sqrt(2) or (sqrt(2)
to the sqrt(2)). So we've proved that it exists without knowing how
to find it.
For a construction, this example doesn't seem to help. But if
one thinks instead about the general phenomenon, one may experiment
with something like "no power of 2 is a power of 3," which makes
log(3) to the base 2 irrational, which makes 2 times it irrational.
So, if b = sqrt(2) and e = (2 times log(3) to the base 2), the result
is 3. And we are done.
Also, Bishop gave a method, using the Baire category theorem, for
constructing zillions of examples.
To be continued.
With best regards,
Gabriel
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