Re: Calculus XOR Probability

Tony Orlow 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.


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.

Right, so you agree that on selecting say 1 million natural numbers at
random, using the uniform distribution you're providing, the mean value
of this million natural numbers will be found to be something. My
question is: what is that something? What will the mean value of these
numbers be?

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.

Uh, about three mistakes. First of all, it would help if you understood
that what mathematicians are saying is impossible is a uniform
probability distribution over the pofnats. You may even have a uniform
probability distribution over something else that you have invented,
but this is rather obviously not a counterexample to the claim that the
mathematicians are making.

Secondly, no, the set of pofnats is not defined "using finiteness of
value as a property". Show me a definition of the pofnats (which will
be referred to by their normal name, you know what that is) by a
mathematician that mentions a "property of finiteness of value".

Thirdly, even though there is no referent for "it", there is also
absolutely no "amorphous upper boundary". There is the total absence of
any upper boundary at all. (That's the point, here, of course: I would
expect the mean value of this million numbers to be roughly "half-way"
to a right-hand boundary if one existed. Since one does no exist, I
conclude that the mean value is not obviously going to exist either.)



"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?

I don't discuss the i-word with you, since it's a waste of time (though
I must say it's unusual to meet someone who apparently "believes" in a
"completed" infinity but not a "potential" one, not that those terms
have any very clear meaning to me)...

Anyway, are you saying that of the million numbers, about half will be
more than "Big'un/2"? I ask again: what proportion of the million
numbers do you think might look like pofnats? Typically just a smidgen?
Or quite a few? What do you think would be the expected mean value of
those of the numbers picked from your distribution that _did_ look like
pofnats? About 42, perhaps?

Brian Chandler
http://imaginatorium.org

.

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