[From Bill Powers (930113.0730)]
Martin Taylor (930112.1830) --
You have indeed managed to introduce a disturbance, in your paper
on Ockhams's Razor. I might entitle my comments, "A critique of
pure reason."
In principle, any hypothesis can be written as a linear
assemblage of words and symbols taken from a finite set of
possible symbols. Inasmuch as we can enumerate the letters used
in the words, each hypothesis can be given a number.
I have never understood the logic of this notion, which I, too,
saw a generation ago. When I first saw it I couldn't believe that
it was taken seriously. Most such finite sets of letters must be
gibberish. The number of non-gibberish sets must be enormously
greater than meaningful nongibberish sets, including as they do
all paragraphs from all works of literature that have yet to be
written in nonexistent languages, or backward, or from the inside
out, or in various substitution codes, or encrypted ciphers, and
so on.
Merely placing letters (or even whole propositions) in one-to-one
correspondence with qualitative "facts" that can be true or false
by no means exhausts the possibilities for forming propositions
as hypotheses about real or imagined nature. Propositions can
contain quantitative references, algebraic and differential
equations, and statements of quantitative inequality. Every
hypothesis containing a quantitative statement has as many
potential variants as there are real numbers (and they are quite
statable in a finite number of symbols: consider the Pytharogean
Theorem, which entails a trancendental number. Consider the
symbols "..." and their meaning).
Even within the realm of qualitative true-false propositions,
there are enormously more propositions that can be expressed in
statements of length L; far more than N^L. That number assumes a
set of propositions that could be expressed as a sequence of 1's
and 0's of length L. But that is not the only way to calculate
the number.
You may recall my proposal that a person is never aware OF the
level of control he/she is working FROM. Presumably, you and John
Gabriel, in reasoning about this subject, have been working FROM
the logic or reasoning level. And in line with my proposition,
you have neglected to take logic into account, and have dealt
only with binary sequences, the next level down.
How many different hypotheses can be formed as logical
propositions involving L variables, each capable of being only
true or false?
The answer is N = 2^(2^L). For only 8 logical variables, this
number is ten to the 77th power. You have therefore vastly
underestimated the number of possible hypotheses, and greatly
overestimated the a priori probability of truth of any one
hypothesis, even one so simple as to rest on only 8 propositions.
In your paper on Ockham's razor, containing about 13,000 letters,
I think that there is a reliance on at least 8 propositions
capable of being independently true or false. So the a priori
probability of this entire argument being meaningful must be
extremely if not vanishingly small.
If you take the point of view, as I do, that symbols are
arbitrary pointers to (mainly) nonverbal experiences, then we
simply assign symbols on the basis that they must differentiate
between experiences. There is no danger, obviously, of running
out of conveniently short symbols, particularly as we can re-use
the same symbols with different meanings in different contexts.
Neither are we likely to run out of ways of stringing symbols
into sequences, and it is utterly impossible that we or all of
humanity could run out of potential logical propositions
concerning nature or anything else, using a different 8-variable
proposition every nanosecond for the rest of the estimated
lifetime of the universe.
The real problem is to tie our logic to the logic of direct
experience. This can't be done by inventing arbitrary rules and
then playing out their consequences by symbol-manipulation. The
number of possible games that could be played this way is
staggeringly, unthinkably, large. The proportion of such games
that has something to do, even peripherally, with the universe of
experience is too small to estimate. There is essentially no
chance that a symbolic argument using variables that have no
demonstrated experimental or experiential meaning will ever
uncover a law of nature. To find logical propositions that relate
to natural law, we must first perceive the logical relationships
that actually occur, and then if we like, translate them into
symbolic language. Trying to do it the other way around is a
losing proposition.
Best,
Bill P.