Re: Calculus XOR Probability

In article <MPG.1e9ed912ca0ce11598abf2@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

imaginatorium@xxxxxxxxxxxxx said:

Tony Orlow wrote:

<snip>

Han's original point is that calculus is very precise and works, whereas
the
case where you have a uniform probability distribution over an infinite
set of
possibilities is not well handled by the classical notion that all
individual
probabilities sum to 1, because the individual probabilities are
considered
equal to zero. It is the opinion of both of us that this can be resolved,
among
other ways, by assigning infinitesimal nonzero probabilities to each
possibility, leaving intact the notion that the sum of the individual
probabilities sums to 1. I am not sure why this is roundly rejected. Can
you
address that?

Um, I'm not a probability expert, but isn't it the case that given a
distribution, you can say what the _expected value_ of a random
selection from it is. In other words, if you choose say a million
values from this distribution, and take their mean, you will get
roughly the same answer each time. What would you expect this value to
be if you are "selecting natural numbers at random"? Recall that this
"natural number" is that of normal maths, which I've called "pofnats"
specially for you. I think normal people will reasonably expect that at
least a goodly proportion of the numbers chosen from your alleged
uniform distribution will be recognisably ordinary pofnats, and not
just stuff like 3*Big'un^(-oo) if I may anticipate you slightly.

Brian Chandler
http://imaginatorium.org



Yes, I think that what you say about an expected value would be true of any
well defined set of possible quantities that could be selected. Given a set
of
n equally spaced possible outcomes, the average should be the smallest plus
n/2
times the spacing difference, or the middle element of the set. So, I think
it's reasonable to expect that. And, I think that expectation is satisfied by
the notion that any infinite number n of equally likely outcomes has a
probability for each of 1/n, and an average value given by the sum of the
values divided by n. Now, the problem comes in when you start trying to
address
the pofnats, because that set is defined using finiteness of value as a
property, which property leads to a totally amorphous upper boundary. "Not
infinite" isn't a specific value comparable to any others. To know the
average
value of the naturals would be to know the largest natural, since it would be
half that value, so it obviously leads to a contradiction. So, the problem,
probabilistically speaking, is that you don't have a well defined value
range.
The same is true of n*1/n, insofar as n is an amorphous value as well. If,
however, you specify a particular value as a formula on Big'un, the unit
infinity, then you can treat it as a specific value, and the probability as a
specific infinitesimal. Can you see the difference bwteen using a "completed"
infinity like Big'un and a potential infinity like aleph_0?

The standard treatment is based on an available system of axioms whereas
TO's "competed infinity" is not.
.

Relevant Pages