[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.