Re: use of real numbers in mathematics and physics
- From: John Forkosh <forkosh@xxxxxxxxxxxxxxxx>
- Date: Fri, 6 May 2005 19:36:44 +0000 (UTC)
Oz <Oz@xxxxxxxxxxxxxxxxxxxx> wrote:
John F <john@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
: 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 by a finite-length program, 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 )
:
:
: Perhaps sliding into philosophy as an alternative to mathematically
: describing precisely what one is saying (due inability to do same):
Unfortunately, some mathematics is needed, at least to the extent
that you precisely define what you mean by "information." Intuition
is indispensably essential while you're choosing definitions and axioms,
and while you're writing a motivating discussion to defend these choices.
But after you complete these essential first tasks (which you might
argue contain the bulk of the real physics), math is indispensable
to develop the quantitative consequences of your ideas and to compare
them with experiment.
Classical information is normally formulated in terms of
Shannon entropy, and quantum information in trems of von Neumann
entropy. Kolmogorov complexity, and its consequences I mentioned,
is based on the former. I didn't explicitly mention this since it's
discussed in the literature I cited. An easier introduction is the
reprint book Information Randomness and Incompleteness, 2nd ed,
G.J. Chaitin, World Scientific 1990, ISBN 981-02-0171-0 (pbk).
If you feel that pi (or even 1.0 from item (1) below) contain
infinite information, then I'd want you to define what you mean by
"information."
: 1) I don't think you can distinguish complexity in a 3+1D world by
: taking the complexity of the 3D bit (the program) and ignoring the
: length of the time bit (required to obtain the precise result). I would
: thus say that pi requires infinite information even if its isn't
: complex. After all the number 'one' is very simple yet implies and
: infinite precision. You can't even do pi in analogue form since this
: would require a known precisely flat space over an extended area, which
: itself requires similarly infinite precision.
Time complexity doesn't contribute to the information content (a la
Shannon entropy) of a real number, which is why I only mentioned it
briefly. Think of it like this. You're familiar with the zip and
unzip programs that compress and decompress files? Suppose you zip
a file. Does the compressed version contain the same information
content (a la Shannon entropy) as the original? The answer is, "Yes."
And that's despite the fact that the unzip program may have to execute
many instructions to reconstruct the original file.
Your remaining items, especially (5), seem to deal with
the information content characterizing physical systems
rather than the information content characterizing real
numbers. That's a whole different question.
: 2) I hadn't spotted JB's comment that p & q together define the
: fuzziness of space, but I like the idea. Its a similar argument
: to (1) above.
:
: 3) About the only thing one seems to be sure about is that particles
: come in chunks of one since nobody ever measured (say) half an electron.
: However I don't even think this is really true as all detectors have a
: finite detection efficiency and particles do have an ability to be
: elsewhere due tunnelling. It may be here today, but (a low probability)
: of gone tomorrow. So not-measuring an electron doesn't mean that there
: isn't one there (or even near).
:
: 4) Its agreed that there is no global definition for 'the energy of the
: universe' and I would bet similar arguments ought to apply to most other
: constants. How can one be sure, for example, that the observable
: universe is always uncharged?
:
: 5) One is thus drawn to the conclusion that any 4-volume of space has a
: maximum information content. From a distant memory of an aged 'This
: Weeks Finds...' I think this is the information content on the surface
: of a black hole of equivalent surface (I've probably got this wrong),
: which is indeed a rather large number. Its rather irritating that this
: large number does not seem to reflect the size of h.
: --
: Oz Use oz@xxxxxxxxxxxxxxxxxxxx
:
--
John Forkosh ( mailto: j@xxxxx where j=john and f=forkosh )
.
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