From Osher Doctorow
COPYRIGHT NOTICE
Discrete Probability Dangers
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
The Loop Quantum Gravity (LQG) school, partly following Sir Michael
Atiyah's algebraic topology studies, has found itself arguing for a
discrete universe based at the quantum level - a far cry from its
claimed position as "heir" to Einstein's GR.
Discrete probability, contrary to many physicists and even a number of
mathematicians' opinions, is not merely an approximation to continuous
probability distributions, but can be totally unrelated to the latter.
This brings about many dangers since it can go off in almost any
direction like algebra itself - without a clue as to where it is going
other than fads.
For physicists to have intuitions about physics problems, they should
arguably be able visualize them at least in their own minds. Yet we
live in the macroscopic "phase" or world, and things mostly look
continuous to us at this level and much more rarely discrete. It is
true that chemistry has a lot of discrete objects, but chemistry is at
a primitive level mathematically in its own field, and mostly when
chemistry interacts with physics do chemists have to learn some very
advanced mathematical and physics techniques.
To especially exemplify the problem, let's toss a coin three times and
write down the sequence of results in terms of H (heads, which means
when the coin lands you see a head - in the USA, the head of a
President) and T (tails, which means that when that coin lands you
don't see the head of a President but the other side of the coin). So
for example:
1) HTT
means that the first toss of the coin lands Heads, the second lands
Tails, and the third lands Tails. We toss a coin each time in the
usual way, without looking at it or otherwise interfering with the way
it lands, so coin tosses are statistically independent, which means
roughly that they don't depend on each other. When objects or events
or processes are independent, they follow multiplicative laws, so for
example since it is well known that a fair (non-loaded or defective)
coin has a probability of 1/2 of landing on H and a probability of 1/2
of landing on T, we get:
2) P(HTT) = (1/2)(1/2)(1/2) = 1/8
Moreover, this is the same probability as that of HTH, HHT, THT, etc.
The binomial distribution counts the probability of k heads occurring
out of n tosses, where 0 < = k < = n and n is an integer > = 1. Its
graph typically looks like a rough approximation to the normal/Gaussian
bell-shaped curve, the approximation improving as n increases. This is
a very popular distribution in mathematical statistics.
However, even in this "close to continuous" example (at least for large
n, and improving as n gets larger), if you study the problem you will
notice that the deeper elements like HTT are very far from what we are
interested in when we study most causation. H and T in HTT don't
depend on each other, so "causation" is, if not lacking, at least
toward the lower end of the scale of what most causation is about
intuitively.
I have to leave for a while, but in the meantime you can see that even
in the "best possible case" scenarios, our intuition seems to get
really defective for discrete events.
Osher Doctorow
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