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imaginatorium@xxxxxxxxxxxxx said:
> 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.
I think that may be a straw man. I can't recall Han specifying originally that
we were talking about a distribtution over the naturals, though that became a
topic of conversation. In fact, Han is really talking about arbitrarily large
and small finite values as infinite and infinitesimal, in a potential sense. I
don't think he's trying to use the aleph_0 naturals, persay. He basically said,
if you have some number of elements N, and each has equal probability, then
that probability is 1/N, and if N is infinite, then 1/N is infinitesimal. That
argument seems perfectly sound to me, as long as you're not dealing with the
naturals. If you are, you got the problems that you indicate.
>
> 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".
Since there is nothing in the Peano axioms that specifically states that the
values cannot be infinite, the imposition of finiteness is an additional
criterion, based on your desire for a minimal such set. Surely, you don't deny
that that is a property true for every natural in your set, and as such,
dsitinguishes them from the set of infinite naturals, if such a thing is
assumed to exist? The set of finite naturals has no specifiable upper limit,
and that's where you get into problems.
>
> 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.)
>
Right. I agree with that. But, if you have a specific infinite number of
possibilities then it's a different story, and as I see it, that's what Han
started out suggesting.
>
>
> > "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)...
Those words have meanings. Your set of finite naturals is "potentially" but not
"actually" infinite. Unusual? Me? They didn't break the mold, because they
never used one. This sucker here's hand-carved and painted! :)
>
> 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?
Of the pofnats, no average can be assigned, since as you correctly point out,
that would be equivalent to identifying the largest finite natural. Over the
entire range of Big'un, the average would tend to be about Big'un/2 as
expected, which is also a megabigulous number. Now, what proportion of those
million would be finite naturals? Uh, about 0% on average, since the set of
finite naturals is going to be some indeterminate infinitesimal portion of the
set from 1 to Big'un. The probability is basically equivalent to one of your
countable infinities divided by an uncountable infinity, or an infinitesimal
probability. It's not impossible for there to be finite value in the set of a
million random values between 0 and Big'un, but it's only infinitesimally
likely. Of course it's more likely than any particular pofnat, by a multiple
of....how ever many pofnats there are. Well, that's why it's an indefinite
infinitesimal. It's a very.....large infinitesimal. Aleph_0 is impossible to
calculate anything with. It's like saying, "What's the square root of a whole
lot? Like, pretty much?"
>
> Brian Chandler
> http://imaginatorium.org
>
>
--
Smiles,
Tony
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