Re: Church-Turing compared to Zuse-Fredkin thesis (two new papers)

From: Arthur J. O'Dwyer (ajo_at_nospam.andrew.cmu.edu)
Date: 02/17/04 

Date: Tue, 17 Feb 2004 15:15:59 -0500 (EST)

[I've dropped comp.theory.cell-automata from the F-up list, since
neither of us posts from it and this thread really isn't topical
in any newsgroup.]

On Tue, 17 Feb 2004, Lester Zick in comp.theory wrote:
>
> <ajo@nospam.andrew.cmu.edu> in comp.ai.philosophy wrote:
> >On Tue, 17 Feb 2004, Stephen Harris wrote:
> >> "Lester Zick" <lesterDELzick@worldnet.att.net> wrote...
> >> > On Mon, 16 Feb 2004 20:55:52 GMT, "r.e.s." wrote:
> >> > >
> >> > > According to http://mathworld.wolfram.com/ConstructibleNumber.html
> >> > > a number must be algebraic (i.e. not transcendental) in order to be
> >> > > "constructible" in the sense you describe.

> >> > I'm curious about this. Basically what you're saying is that
> >> > constructible real numbers are turing computable
> >
> > All constructible numbers are algebraic.
> > All algebraic numbers are computable.
> > All constructible numbers are computable.
> >
> >> > even though irrational.
> >
> > All rational numbers are constructible.
> > Some constructible numbers are rational.
> > Some constructible numbers are irrational.
>
> Would you not say here that all rational and irrational squares

  First of all, what do you mean "rational and irrational squares"?
Did you mean to write "square roots"? (A "square" number is a number
X such that X = Y^2 for some Y. A "square root" is a number X such
that X^2 = Y for some Y. In both cases, you need to specify the
range of X and Y in order to make any sense -- *every* positive real
number is the square of another real number, and every real number
is the square root of another positive real number.)

> correspond to an infinite number of differences between irrational
> squares

  What do you mean by "correspond to"?

> and that all those differences are constructible?

  It is trivially true that for all integers X, Y, a, b:

    a \sqrt{X} + b \sqrt{Y}

is constructible (and also of course computable). This includes
the special case a=1, b=-1.

> >> http://members.ispwest.com/r-logan/glossary.html
<snip>
> > This isn't really true either, although I think here the guy had
> >at least an inkling of the correct statement. See the definition
> >at MathWorld, which incidentally is a great place to find this sort
> >of thing, at least if you can stand waiting for the images to load
> >(if you're on a dialup connection).
>
> Well, I'm not trying to make an issue of this. I know that among
> irrationals we have constructibles that appear to me to represent
> differences between irrational squares

  Irrational square roots?

> corresponding to steady state solutions as well as to non steady
> state turing computability.

  That doesn't make sense. What do you mean by "corresponding to,"
"steady state solutions," "non-steady-state Turing computability"?

> And we
> also have non constructible irrationals referred to as transcendentals
> that are also classed as irrationals although I find the terminology
> confusing.

  Perhaps the diagram here will help.
http://www.contrib.andrew.cmu.edu/~ajo/disseminate/numbers.jpg

  You see the box marked "Rationals"? Well, everything outside that
box (but within the box marked "Real Numbers") is /not rational/ --
that is to say, /irrational/. The word /irrational/ is just a
mathematician's way of saying /not rational/ -- /not expressible
as a ratio of two integers/.

> I once suggested to a mathematician that the term irrational be used
> to describe constructible numbers only

  Now you see why that's foolish: /irrational/ means /not rational/,
so you can't re-define it to mean anything else. Unless you're
proposing to re-define /rational/ to mean /not constructible/... but
that's even sillier! You'd do better to try to understand the logic
behind the existing terminology before inventing your own.

> and the term transcendental be
> used to describe non constructible numbers.

  So you would make "the cube root of 2" a transcendental number?
Or would you use a different definition of "constructible" also?

> The idea was that the
> squares of all constructible numbers both rational and irrational
> correspond to the difference between irrational squares. But obviously
> the idea fell on deaf ears.

  Not /deaf/, just /reasonable/.

BTW, have you visited MathWorld yet?

> >http://mathworld.wolfram.com/ConstructibleNumber.html

-Arthur

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