Re: Shakespeare - Probability
From: Barnaby Finch (barnabyfinch_at_verizon.net)
Date: 12/30/04
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Date: Thu, 30 Dec 2004 20:50:21 GMT
In any infinite random sequence, any given string of digits, no matter how
long, will certainly occur, and, in fact, will occur infinitely often. So,
somewhere in the digits of Pi, not only does the complete works of
Shakespeare occur, but it occurs infinitely often, as does slight
variations. So the complete works of Shakespeare, except that one play is
entitled "King Aardvark", also occurs infinitely often.
In the Mathworld.com entry under "real numbers" it states, "Almost all real
numbers are lexicons, meaning that they do not obey probability laws such as
the law of large numbers (Gruber 1991; Calude and Zamfirescu 1998; Trott
2004, p. 69)."
See C. Calude, T. Zamfirescu. "The typical number is a lexicon", New Zealand
Journal of Mathematics, 1996.
The following quotes are from a paper on the lawlessness of the universe, by
Cristian Calude, at
http://tph.tuwien.ac.at/~svozil/publ/m.txt
It states, "Chaitin, Calude, prove that every random sequence is a _lexicon_
, i.e., in every random sequence every word---of any length---appears
infinitely many times. The fact that the first billion digits of a random
sequence are perfectly lawful, for instance by being exactly the first
digits of the decimal expansion of Pi, does not modify in any way the global
property of randomness. But, to perceive this _global randomness_ we have to
go beyond the finite; we have to access infinity, which is not physically
possible!
Also -- "A lexicon is a number which is disjunctive in any base. A lexicon
contains all writings, which have been or will be ever written, in any
possible language. A lexicon expresses the strongest qualitative idea of
randomness; pure randomness is a much more demanding property, but the
natural, typical realization of a lexicon is by tossing a pure coin, i.e.,
with probability one a lexicon real is purely random. In fact, the situation
is more dramatic, as by Gcs' theorem, (see Calude, p. 155-165), every real
is effectively reducible to a random one.
And -- "Actually, the above relations show how un-ordered, chaotic, most
numbers are, when one adopts the point of view of topology rather than
measure theory. As a consequence, one constructively shows that the typical
number is a lexicon (the class of lexicons is larger than a set which is
residual and of measure-one). Consequently:
"Constructively most numbers do not obey any probability laws.
This result (from which we can immediately deduce the Oxtoby and Ulam's
theorem stating that the law of large numbers is false in the sense of
category) says that:
The system of real numbers, our very basic language of expressing the
natural laws, is fully contaminated by randomness.
In Chaitin's words, "God not only plays with dice in quantum mechanics, but
even with the whole numbers"."
Tipler p. 78 - " ....the Universe must go through a calculable number of
combinations in the great game of chance which constitutes its existence. In
infinity, at some moment or other, every possible combination must once have
been realized; not only this, but it must once have been realized an
infinite number of times. Simply tossing a fair coin is all that is required
to produce everything, to produce as outcome the entire Universe. Everything
is there, on the lexicon. All Shakespeare, every galaxy, each human
brain..."
Calude concludes, "Quantum mechanics and relativity have shown that it is
impossible to know everything: quantum mechanics argues that we cannot know
because we are part of the system and relativity suggests that the Universe
is too big to be known as we cannot have enough energy. Mathematics teaches
us that, with extremely rare exceptions, any real number representing the
outcome of the measurements is a lexicon. And a lexicon, although we can
define it, although we understand what we mean by that mathematical
expression, is way beyond the capabilities of the human mind because the
bits in the sequence of a lexicon are devoid of any order, any law, even the
law of large numbers. Thus we assert that the impossibility of the human
mind to fully understand the world of mathematics must necessarily imply
the impossibility of the human mind to completely understand the Universe as
Universal natural laws do not exist. Or, to put it differently, it
necessarily implies that the Universe, defined as the set of measures
describing it, is devoid of any overall lawfulness, of any order, and,
consequently, the Universe can be the outcome of the most elementary random
process, it the toss of a fair coin."
Fascinating!
Barnaby Finch
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