Re: use of real numbers in mathematics and physics

Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:
: On 12/2/2004 1:14 PM, Arnold Neumaier wrote:
: > Alfred Einstead wrote:
: >> baez@xxxxxxxxxxxxxx (John Baez) wrote:
: >> The root of the matter, addressing the subject header itself, is that
: >> a real number conveys an infinite amount of information. [...]
: >>
: >> In quantum physics 1/2 of the problem is already gone, since the
: >> p's and q's in nature no longer contain an infinite amount of
: >> information when taken together. The pure state then, too, have
: >> an inherent fuzziness associated with them.
: >
: > But whether or not a state is pure, all expectations are exact
: > real numbers, with an infinite amount of information.
: > And this will be so in any reasonable form of physics.
: > Cast out continuity, and you lose physics.
:
: Having pondered a lot about how my ideas on the subject of real numbers
: can be agreed with G. Cantor's theory, I have to apologize for my
: hopefully not yet too late response supporting you.
: Please find my pertaining reasoning at
: http://iesk.et.uni-magdeburg.de/~blumsche/M280.html
: and do not take it an April joke.
: Eckard Blumschein

Besides the Cantorian approach, the notions of Kolmogorov
complexity and computable real numbers demonstrate that
most (if not all) the real numbers used in physics do _not_
contain infinite information.
The Kolmogorov complexity of a string (which may be a
sequence of digits) is, simply, the length of the shortest
computer program which emits that string. So, for example,
pi contains only a little information since a relatively
short program can emit it. Of course, it'll take quite
a while for that program to finish, and I won't go into
space (i.e., memory requirements) versus time (number of
instructions executed) complexity.
Physical behavior is often modelled as differential
equations or as other mathematical objects that can be
programmed. So all the resulting real numbers are
computable, and hence carry only finite information.
"Kolmogorov complexity" (also called "algorithmic
information theory") is easily googleable, see, e.g.,
http://en.wikipedia.org/wiki/Algorithmic_information_theory
The standard text is
http://homepages.cwi.nl/~paulv/kolmogorov.html
(which is a bit dumbed down from its more monograph-like
first edition).
"Computable reals" or "computable real numbers" is
also pretty well represented on google. I'm not sure if
there's a standard text, but you might try "Computability
in Analysis and Physics", Marian B. Pour-El and
Jonathan I. Richards, Springer 1989, ISBN 0-387-50035-9.
--
John Forkosh ( mailto: j@xxxxx where j=john and f=forkosh )

.

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