[FOM] end of my response to Bill Tait's reply to "reading the bible with Bill"
Gabriel Stolzenberg
gstolzen at math.bu.edu
Fri Mar 10 19:23:28 EST 2006
This completes my response to Bill Tait's reply (Feb 25) to my
"reading the bible with Bill."
> It is a good example, illustrating
> the general---and well-known---fact that general non-constructive
> existence proofs often lose the information in particular cases that
> there is an algorithm for determining the object. Agreed, agreed,
> agreed.
> But it doesn't justify a new metaphysics of mathematics. The
> objects that the classical proof fails to compute and the
> constructive one does are the same objects---reals, real functions,
> whatever.
A new metaphysics of mathematics? What is the old one? Is it the
less than coherent fragments of thought evoked by suggestive language
that I and my colleagues sometimes had when, in the course of learning
to be classical mathematicans, we stopped to reflect on the ontology
of what, if anything, we were proving theorems about? Fragments that
we could grasp no better than the proverbial bar of soap that, whenever
you reach for it, slips away. A graduate student once described it
to me in terms of her own experience. "Doing classical mathematics,"
she said, "is like walking along the edge of a cliff. If you turn
your head, you'll be staring into an abyss. So you learn never to
turn your head."
> The objects that the classical proof fails to compute and the
> constructive one does are the same objects---reals, real functions,
> whatever.
Bill, am I supposed to take your word for this? You state it
without explanation or justification. Also, on the only sense I
see how to make of it, it seems plainly false. (Except perhaps as
a classical perception of constructive mathematics.)
Let's take a standard example, the classical intermediate value
theorem. f(0) < 0 < f(1) => there is a p in (0,1) with f(p) = 0.
Now let's see what kind of object this p allegedly is, first
classically and then constructively. Well, recalling the world of
mathematics as seen in a classical mindset (I have to do this from
memory, it's a long time since I made the shift), I might first say
that p is a set (because everything is a set). But then I would
wish to explain that, in a classical mindset, we don't know how to
make sense of set talk as being literally about objects. Moreover,
we don't care. It doesn't matter!
Now let's see what kind of object our intermediate value, p, is
supposed to be constructively. On the face of it, we don't know that
there is any such p but we can easily define something that looks as
if it might anyway suit Bill. Namely, we define "a classically-real
number" to be a set of real numbers that LEM (and maybe AC) implies
is a singleton. Then, although the intermediate value, p, is not a
real number, it is at least classically-real.
Does this help? Classically-real numbers are not quite the same
as classical real numbers but there seems to be an affinity.
Bill, I wonder if you might do better to focus less on talk about
objects and more on information, viewing talk about objects, in both
classical and constructive mathematics, as one way of packaging the
information.
> and so, to the extent that these methods, with suitable coding,
> are also methods of classical mathematics, constructive mathematics
> can be regarded as part of classical mathematics.
This talk of "suitable coding," which I've never heard before,
suggests that you have in mind something different from the usual
imbedding of constructive mathematics in classical mathematics as
the part devoted to seeing what can be proved without using the law
of excluded middle. Is that right?
> If you really
> mean that there are 'constructive properties' not intelligible in
> a 'classical mindset', then we are really not on the same page.
How would you feel about having professors of Spanish who know
neither the language nor any of the Spanish speaking cultures but
have read the great works of Spanish literature in translation?
I ask because, to me, studying and teaching Spanish literature this
way seems a lot like studying and teaching constructive mathematics
only in a classical mindset. Or vice versa.
More to the point, when you say that constructive properties are
intelligible in a classical mindset, are you thinking of this like
reading "Don Quixote" in translation or do you mean something more
interesting?
> It is possible to understand both classical and constructive
> mathematics without wearing the blinkers that you call 'mindsets'.
As I see it, the only way to know whether your claim is true is
to check out the mindsets. Until you have lived a while in each,
you can't rule out the possibility of discovering that there is a
great deal more to understand than you now suppose.
As for blinkers, I would say that understanding either classical
or constructive mathematics means donning the appropriate blinkers
and then practicing the mathematics that seems most natural.
Gabriel
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