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

From: Arthur J. O'Dwyer (ajo_at_nospam.andrew.cmu.edu)
Date: 02/20/04

  • Next message: Tao Zhang: "The complexity of a piece of code" 
    Date: Thu, 19 Feb 2004 20:15:47 -0500 (EST)

    On Tue, 17 Feb 2004, Lester Zick wrote:
    >
    > On Tue, 17 Feb 2004 15:15:59 -0500 (EST), "Arthur J. O'Dwyer"
    > >On Tue, 17 Feb 2004, Lester Zick in comp.theory wrote:
    > >>
    > >> 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.)
    >
    > The squares of rational and irrational numbers.

      I.e., all real numbers. Okay, but that seems kind of pointless.

    > >> correspond to an infinite number of differences between irrational
    > >> squares
    > >
    > > What do you mean by "correspond to"?
    >
    > Are the result of.

      So that leaves us with: "All real numbers are the result of an
    infinite number of differences between irrational squares." And
    that looks trivially true. [But see below.] What did you mean
    to write?

    > >> 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?
    >
    > No, irrational squares. Squares having sides of irrational numbers.

      Okay. That's an interesting concept -- "irrational squares."
    You mean "all numbers X, which may be rational or not, such that
    X is the square of an irrational number." There are |R| of these
    numbers, where |R| is the cardinality of the reals. I don't know
    if anyone's looked at them before; but then I don't see any really
    interesting properties of them, either.

    > >> [...] 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/.
    >
    > Sure. Except that you wind up with various kinds of numbers sharing
    > the same non rational character. Bit confusing I would think.

      Not at all. All those non-rational numbers *do* have the same
    non-rational character -- none of them are rational! Isn't this
    obvious?

    > >> 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.
    >
    > I think I do understand the rationale of not rational numbers. I just
    > prefer to classify numbers according to their most prominent trait.
    > Many if not most numbers are irrational. Some are constructable,
    > some are transcendental, and there are probably others. So calling
    > numbers irrational doesn't really clarify analysis very much if the
    > causes of the irrationality are different for different types.

      So you propose calling all numbers that are divisible by sqrt(2)
    "Type S2" numbers, all numbers that are prime "Type P" numbers, all
    numbers that are the number of the taxi you just drove up in "Type
    T" numbers, and so on? The point is that numbers (and things in
    general, actually) can have *multiple* *useful* *characteristics*!
    It makes no sense at all to arbitrarily classify them into mutually
    exclusive sets, any more than it would make sense for a library
    catalog to list "Hamlet" only under "One Word Play Titles" and not
    under "Shakespeare," "Tragedy" and "H"!

    > >> 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?
    >
    > Hard to say whether the cube root of 2 is merely an irrational
    > constructible or some form of transcendental non constructible
    > without first understanding how it is arrived at. Do you know of any
    > construction for the cube root of 2?

      None exists with paper, straight-edge and compass. Of course there's
    a trivial construction of the Nth root of M in N-space, but the ancient
    Greeks didn't care about that -- all they had was paper and pencil
    (and usually not even that).

    > >> 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?
    >
    > Not yet.

      Well, do so!

    -Arthur

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