Re: Cantorian pseudomathematics


david petry wrote:
> Keith Ramsay wrote:
> > david petry wrote:
>
> > 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".
>
>
> But we can tell, just by looking at X, whether it is a constructively
> meaningful statement. Sometimes it is not so clear when the statement
> is made in informal natural language, though.
>
> When you write "we had a discussion...", are you talking about fifteen
> years ago? I vaguely recall such a discussion, but the problem was
> with the interpretation of X->Y. Usually we say that X->Y is equivalent
> to "I have a way to convert a proof of X into a proof of Y", in which
> case X must be a constructively meaningful statement (i.e. it must make
> predictions about the results of computational experiments). But it
> might also be interpreted as not(X) OR Y, in which case, not(X) must be
> constructively meaningful.
>
>
> >
> > |> 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
>
>
> Proving theorems like AC->X is not doing constructive mathematics, even
> in a "broad" sense.
>
>
> > (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.
>
>
> Actually, I can't really follow that paragraph. But the question of
> whether a statement has computational meaning is independent of whether
> it has been proven.

[snip]

Hi, David:

As I did earlier in the thread, I contend that your "computational
meaning" is only in the eye of the beholder. I am surprised now
by your reliance on "formal" statements, at least by deprecation
of "informal language". In any case you cannot escape the role
of interpretation by relying on formal language this way.

Perhaps you would give examples of formal sentences, eg.
in first order logic using the symbols of Peano arithmetic (or
such abbreviations as you feel may be formally well-defined)
which 1) have computational meaning, and 2) do not?

As I recall your informally stated criterion, it was that such
statements must "make a prediction" about computational
experiments. I believe if you examine this criterion more
rigorously, its apparent objectivity will vanish before you.

Previously you wrote: "The problem with the actual infinite is
that it is not observable. That is, we have no way to test
(falsify) statements about the actual infinite."

Is not this difficulty of a piece with the Pythagorean distaste
for incommensurates (what today we charmingly refer to as
"irrationals")? Point: if we cannot observe the entire decimal
expansion at once, or even with the lifetime of a man (or of
Mankind), do we entirely lack ways to falsify statements as
to whether or not the expansion is eventually periodic?

But the Pythagoreans did not have decimal arithmetic, you
will perhaps remind me. Just so. And there may be some
computational experiments which "falsify statements" that
have not yet occurred to anyone, and whose value is hence
not clear "just by looking at X."

I do not deny that the actual infinite poses difficulties which
the actual finite does not. However I'm with Bertrand Russell
when he lauds the 20th century's scholarly resolution of the
paradoxes of infinity as a great philosophical achievement.


regards, chip

.

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