Re: Cantorian pseudomathematics


Jesse F. Hughes wrote:
> cbrown@xxxxxxxxxxxxxxxxx writes:
>
> > david petry wrote:
> >> Jesse F. Hughes wrote:
> >>
> >> > But existential claims are not falsifiable, so we should banish the
> >> > existential quantifier.
> >>
> >> Not quite. When we assert that something exists, we must also give some
> >> finite set in which it can be found. Thus ...
> >>
> >> > Suppose I prove: There exists n, P(n).
> >>
> >> We would have to prove: There exists n <= N, P(n), where N is
> >> explicitly given. That is a falsifiable statement.
> >>
> >
> > In Euclidean geometry, given two distinct points A, B, there exists an
> > equilateral triangle having AB as a side.
> >
> > What are "n" and "N" in this case? Is this a falsifiable statement? Is
> > there a constructive proof of the assertion?
>
> Worse, now he's clarified computational experiment thus:
>
> A statement of the form "Turing machine T, given input M, will (or
> will not) halt within N steps", describes a computational
> experiment.
>
> Euclidean geometry is pretty far from producing predictions of that
> sort.

That's completely false. When dealing with a continuum, the
predictions will be about approximations to the continuum.

Thus, for example, the theorem that the angle bisectors of a triangle
meet at a common point would say something like, "within the precision
of our experiment, the angle bisectors of a triange will meet at a
common point". That does correspond to an experiment as I have defined
it.




> "Demons were like genies or philosophy professors---if you didn't word
> things /exactly/ right, they delighted in giving you absolutely
> accurate and completely misleading answers."
> -- Terry Pratchett, /Wyrd Sisters/

.

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