From Osher Doctorow
I will use the word "Finiticize" to indicate going from infinity to
finite numbers of objects or just "finite scenarios" for simplicity.
Computer advocates of "finite" scenarios, who are becoming fewer and
fewer now that Chaitin recognizes infinity, ask how one constructs
infinity from the finite, and throw up their hands as they give up.
But what about the opposite direction? If we accepted infinity, how
would we "finiticize" it?
One way would be to replace real numbers (or as some theorists now
argue rational numbers or p-adic numbers) by integers. What happens?
The answer is: you get counting!
Mathematically:
1) G(n) = (n - 1)! = 1 times 2 times 3 times ... times (n - 1)
So we count from 1 to n - 1 (or 0 to n - 1 since 0! = 1), but the
multiplication is somewhat of a distraction so look at this:
2) C(n, r) = n!/[r!(n - r)!) where C(n, r) = number of combinations of
n things taken r at a time
3) P{n, r) = n!/(n - r)! where P(n, r) = number of permutations of n
things taken r at a time
Here r < = n and both are nonnegative integers. Now we are in "pure"
counting! But there's more! The hypergeometric probability
distribution, which is a discrete rather than continuous random
variable distribution, is the "sampling without replacement" analog of
the binomial and Bernoulli "sampling with replacement" distributions.
Its discrete density function or "probability mass function" fX(x) is:
4) fX(x) = C(r1, x)C(r - r1, n - x)/C(r, n)
where a population has r1 objects of type 1, r2 = r - r1 objects of
type 2, and a sample of size n is drawn from the population which has
total population size r. I'm using the notation of Hoel, Port, and
Stone's Introduction to Probability Theory, Houghton Mifflin: Boston
1971 here (the authors are/were from UCLA) with a few differences.
Not only does changing the argument a of the gamma function G(a) to an
integer yield (a - 1)!, and not only do we derive combinations and
permutations from this, but we can even sample without replacement
(without putting sampled objects back into the sample) from a finite
population and get the probable result from this.
Osher Doctorow
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