FOM: Canonical non-computable number?
Kanovei
kanovei at wmwap1.math.uni-wuppertal.de
Thu Sep 13 11:58:57 EDT 2001
> Date: Fri, 7 Sep 2001 17:31:24 -0400 (EDT)
> From: William Calhoun <wcalhoun at planetx.bloomu.edu>
> I think the definition of "canonical" needs to be considered. To think
> this through, I'm going to consider an analogous question: What is a
> canonical irrational number?
This does not seem to be a meaningful question
(neither does a version for non-algebraic numbers).
Indeed, the word *canonical* is used in mathematics mostly as
*canonical form of such-and-such equation* etc,
meaning that any equation (or other object considered) can be
transformed to a canonical equation so that something essential
is preserved.
An assertion like "$\pi$ (or sqrt(2)) is a canonical irrational"
just cannot be thought of this way.
The simplest known, perhaps, but not canonical.
> Finally, why is pi a canonical irrational? It is canonical in the sense
> that it is a constant that naturally appears everywhere
If we really have in mind extraterrestrials, *everywhere* is too a
strong description, because in spaces with very different geometry
(in vicinity of a black hole, for instance)
\pi may naturally appear only as one of dosens funny constants
in depths of advanced papers.
V.Kanovei
PS. I add a note of condolence to FOM colleagues in NYC and Washington
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