Re: Cantorian pseudomathematics

david petry wrote:
|Keith Ramsay wrote:
|> david petry wrote:
|
|> |But wouldn't you agree that a tremendous amount of the mathematics
of
|> |the transfinite will never have a constructive counterpart?
|
|> An expert in constructivism once said to me that there
|> seemed to be areas in classical mathematics that had no real
|> corresponding area in constructive mathematics, but I don't
|> think he meant it in the sense of individual results, since
|> he'd recently been explaining that for each theorem X of
|> classical mathematics, there was a corresponding theorem
|> AC->X of constructive mathematics, where AC is the axiom of
|> choice.
|
|I just don't see how AC->X could be considered to be part of
|constructive mathematics.
|
|> When the axiom of choice isn't used, one could
|> consider LEM->X instead, where LEM is the law of excluded
|> middle. (The axiom of chioce implies the law of excluded
|> middle.)
|
|Again, if you put the nonconstructive assumptions back in, you are no
|longer doing constructive mathematics.

>>From the point of view of constructive mathematics, AC is
just yet another curious statement that turns out on further
investigation to imply the law of excluded middle. The fact
that it's one of the main axioms used in mainstream
mathematics is irrelevant.

I don't see how one can practically speaking prevent oneself,
as one is doing constructive mathematics, from dealing
sometimes with statements of the form X->Y where it turns
out afterward that (sometimes) X is not constructively valid,
but is nonconstructively provable.

We had a discussion of statements of the form X->Y where X
is false, and one of my points there was that we need to be
able to proceed in a way that doesn't require us to declare
things to be nonsensical retroactively. If I can't tell, just
by looking at X, whether it's "okay" to use it as a premise,
even just hypothetically, then there's something wrong with
your notion of "okay".

|> And also it's not clear that every
|> constructive theorem in the sense he and I were considering
|> is constructive in the sense you and I are considering.
|
|I would say not.

So, I was pointing out the contrast between your idea and
the one I more often consider.

In the context of explaining why it's not clear to me that
a "lot" of "Cantorian mathematics" will never have a
constructive counterpart, in your sense, the point is (1) it
always has a constructive counterpart in a broad sense, and
(2) I don't see where, within mathematics seen from a
constructive point of view, it is supposed to be that it
departs from having computational meaning. We could consider
cases where someone was investigation a question of the form
X->Y and eventually found that X is false, that there were
no examples of the kind of thing they were investigating, and
things like this, but as long as it's decided after the fact,
and there isn't a criterion we can use in advance to see that
the question lacked content, I don't think it counts.

|> Some results like Cantor's theorem are perfectly natural for
|> a constructive mathematician like Bishop.
|
|You might say that, but the constructive version of Cantor's theorem
is
|not the same theorem as the classical version.

I don't know in what relevant sense you consider them not
the same. Cantor proved that for every function N->R there
exists an element of R not in the image. Bishop has a theorem
in his analysis textbook of the fact that in any interval
[x0,y0] where x0<y0, there is an element different from every
element in the image. His proof is essentially the same as
Cantor's first one.

|> The general theory
|> of cardinality as pursued by classical mathematicians seems
|> a bit curious from a constructive point of view. The original
|> definition of cardinality is equivalent to asking for various
|> sets X and Y whether there is an injection from X to Y. When
|> one proves constructively that there is one, in many cases
|> that means one has constructed a continuous, computable
|> injection from X to Y. So it seems to me that constructively,
|> this question has something in common with topology.
|
|That's right, sort of.

Oh, you are too kind.

|One way to think about it, although it's not
|exactly the right way, is to think that an infinite set is
|constructively defined by (constructively) defining a countable
|neighborhood base for the set.

In constructive mathematics, there is a field investigating
properties of spaces in relation to continuous maps, and I
think it makes sense to call it "topology", even though one
doesn't usually try to treat it using the family-of-open-sets
approach preferred by most classical mathematicians.

Let X be the unit circle. Nonconstructively, we say that
(0,1), [0,1], R, R^2, X, and lots of other spaces all have
the same cardinality. But cardinality is by definition a
question of the existence of injections from one space to
another. Typically, when a classical mathematician proves
the existence of a discontinuous function defined on all of
a space, it's nonconstructive. The usual injection from X
to [0,1] for example has a single discontinuity, and can be
computed only for points where we know either that the point
is separate from the discontinuity, or where we know which
side of the discontinuity the point is on.

Constructively, we can show the existence of injections from
(0,1) to [0,1] to R to X to R^2, and from R to [0,1] to (0,1),
which we could denote by |(0,1)|=|[0,1]|=c <= |X| <= c^2.
But the existence of injections from R^2 to X or from X to
R are nonconstructive (requiring a nonconstructive form of
the trichotomy principle) for basically topological reasons.
>>From certain Brouwerian assumptions it follows that
c < |X| < c^2, but only some people take those as axiomatic.

There are also times when surjections matter more. For Cantor
these were the same questions, since he believe in the axiom
of choice, which implies that if X is nonempty, then the
existence of an injection from X to Y is equivalent to the
existence of a surjection from Y to X. The theorem he proved
about N and R is the nonexistence of a surjection from N to R,
which seems to me more apt to be interesting than proving the
nonexistence of an injection from R to N.

Keith Ramsay

.

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