Re: Calculus XOR Probability

Matt Gutting said:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:

<snip>

Han seems unbothered by the fact that 1 is not the value you get from
the sum of a countable number of 0's; he simply "wants" it to be 1. And
lo and behold, that is what his argument affirms, just as mine affirms
that the length of the diagonal is exactly what I "want" it to be: 2.

Tony:

No, he doesn't just "want" it to be 1. That is the probability of something
which is definitely true. If one of the elements is to be chosen from the set,
then the probability of one being chosen is 1. If there are multiple choices,
then that 1 representing the fact that one of those will be chosen is defined
to be the sum of the probabilities of each. That's how probability works,

Only if you can define a probability distribution, which is the assumption in
question here.

Not true. If you can establish a uniform probability distribution, then you can
say each possibility has the same probability. If you can formulate the
distribution at all, it will allow you to calculate which fraction of the
overall probability is assigned to each possiblity. But whether you can
determine what the distribution is or not, the overall probability that one
will be chosen, given the fact that you're choosing one, is 1, and the
probabilities of each mutually exclusive alternative contributes to that sum
equal to 1. That's a very basic concept that must hold, even if we don't know
what probability distribution we're looking at. If the probabiliies of all the
mutually exclusive alternatives in a set sums to zero, then what that says is
that no element from the set is selected.


Tony:
and
the question is why the standard system can't accomodate the infinite case,
with an infinite number of equally likely outcomes, one of which will
definitely happen. The probability of each is 0 in standard analysis,

Not precisely. It's probably more correct to say that there's no number which
can be assigned to that probability. Should there be? I don't see a problem with
answering "I can't tell you" to a question of probabilities.

Yes, precisely, the probability assigned to one of a set of equally likely
mutually exclusive alternatives is 0% in standard analysis, which is what this
is all about. That's not absolute zero, but some infinitesimal value which is
not handled by the standard system. So, you don't see a problem with not being
able to say, but is that what you prefer?


Tony:
and yet
each has a chance of being chosen, and so the probability is really nonzero.
It's only zero in standard finite analysis, but where you are talking about an
infinite set of possibilities, you are outside of standard territory, and the
answer is nonstandard: each possiblity has an infinitesimal probability, and
that infinity of infinitesimal values sums to 1 as expected. So, n*1/n=1 holds
in the infinite case, and there is no reason to expect it to fail. Is there?


Problem here is that you haven't really defined "infinitesimal" in any rigorous
and non-circular way.

Immaterial. If n is infinite, then 1/n is infinitesimal. It's smaller than any
finite value and yet nonzero, just like the probability we're discussing.


Charles said:

Similarly, if the sum of all steps "should be" 2, that's what I get as
I travel along the tiny stair steps. So, it's GOOD that n*2/n equals 2
for all (finite) n; in fact, I find it extremely good evidence that in
the limit, it will be 2 as well.


Tony:
If you construct a stairway there, well, it will be 2. That's not a diagonal
line. So, fine, your derivation is correct. No matter what size stairs you use,
or how many, it will always add up to 2. Nice proof. It just has nothing to do
with a diagonal line. The sum of the probabilities of all mutually exclusive
possibilities summing to 1, on the other hand, is a cornerstone of probability
theory, and something that should be considered always true.

If one can find a probability distribution for the elements, yes. Otherwise
all bets are off (pun intended).


Whether one can determine a distribution or not, there is never 110% chance of
something happening. If you have a set of mutually exclusive possiblities, and
one of them is going to happen, the probabilities of each happening must sum to
1. If it sums to less than 1, then you have left out some possibliites, and if
it sums to greater than 1, then some possiblities are not mutually exclusive.


Charles:


If I travel along the stair steps, no matter how small, then sum of all
the stair steps "should" sum to 2. The fact you seem to think that the
distance in the infinite case is actually sqrt(2) indicates an error.
The sum is correctly 2 for all finite n. The sum is correctly 2 for
infinite n as well, but the length of each stair step is 2/n, which in
the infinite case is an infinitesimal hodon [*], not absolute 0.

Yeah, sure. Like I said, it's not the diagonal line, even though it starts to
look like it. You're talking about a fractal dimension on the line, basically.
What does that have to do with probability? It's just an attempt to change the
subject into something ridiculous and trying to say it's the same argument.

Is the parallel making sense to you yet? Do you see /any/ difference in
these /arguments/, besides the fact that one gives an answer you "want"
or "should get", whereas the other doesn't?

Tony:

Yes, I do. The difference is that you're trying to say your example is about a
diagonal line, and it's not, whereas Han's trying to say his example is about
probability in an infinite set, which it is, to whatever extent he actually
means "infinite". ;-)

Both Han's proof and Charles' depend on reasoning from the behavior of an
increasing, but always finite, number of elements to an infinite number of
elements. To that extent, the proofs expose the same types of reasoning
flaws.

There's really no flaw in proving an equality between expressions in the
infinite case. Inequalities can be invalid in the infinite case, where the
difference causing the inequality has a limit of 0 as n->oo. Here, the flaw was
simply that the staircase never was a diagonal line, even if it appears so
visually as n->oo. Chas mentioned approximating lengths of curves with
infinitesimals, but those are always parallel to the curve, while his are
always at a 45 degree angle, causing the sqrt(2) error by the sine and cosine
of that angle.


Charles:

* No one is claiming that for finite n, n*1/n equals something other
than 1, anymore than than any one would claim that n*2/n equals
something other than 2.


Tony:
Good, so no one's totally insane....


Charles:
So there's no reason to repeat these assertions; they are accepted.

Each of the claims you have made so far for the correctness of Han's
argument can be mirrored as an equally valid claim for correctness of
my argument; just substitute "2" for "1" and "sqrt(2)" for "0", and
"finite number of stair steps" for "finite set of outcomes".


Tony:
Great, so it's true for the infinite staircase. And?


Charles:
But my /conclusion/ is so obviously false (by appeal to Pythagoras),
there must be something wrong with my /argument/.


Tony:
Yes, it has nothing to do with the diagonal line. Your line is still vertical
and horizontal.

And Han's increasingly large finite sets are still finite, whereas his
conclusion is drawn about something infinite. What's the difference between
using finite sums of lengths to draw conclusions about an "infinite" (in
some sense) sum of lengths, and using finite sets of equiprobable elements
to draw conclusions about an infinite set of elements?

No difference whatsoever, and Chas' conclusion that the infinitely fine
staircase still requires a line of length 2 is correct, but it's not a diagonal
line. The only mistake is thinking it's a diagonal line. the length is 2, and
the sum of the probabilities is 1, by proper inductive proof of equality
holding for the infinite case. Bith proofs are valid in my opinion.



I claim that if you can figure out why the /argument/ is wrong (not
just the conclusion), then you will also see why Han's argument is
wrong (above and beyond the fact that his conclusion is also
independently false, although perhaps not as obviously, by appeal to
Archimedes).

But it's not. Archimedean principle is mitigated at the infinitesimal level by
the fact that neighboring infinitesimals do not constitute distinct standard
reals, and so on the finite level, any discreteness of the infinitesimals, if
they are taken to be discrete, does not violate Archimedean principle on that
level. Further, on the infintiesimal level, Archimedean principle CAN be
preserved by defining midpoints in the infinitesimal intervals, because at that
level, the endpoint ARE distinct values in their nonstandard sense.

Again, you haven't defined infinitesimals rigorously and non-circularly,
so referring to them in an argument can't lead to valid conclusions.

Waaahhh...

Smaller than any finite but non zero. The multiplicative inverse of an
infinity.



Why is your conclusion wrong? It's not, if you're talking about a line broken
into vertical and horizontal elements. It will always be equal to the sum of
the vertical and horizontal distances traveled, because it's NOT the diagonal
between the starting and ending points.

What gives you the idea that one can "sum" the probabilities of an infinite
number of points to get the whole (1) any more than one can add up infinite
numbers of vertical and horizontal lengths to get a diagonal length?

Because they add linearly, without the need to be parallel to what they
"approximate", since the values are scalar and cannot be anything BUT parallel,
as opposed to the vectors that the treads and risers represent. If all your
infinitesimal segments were PARALLEL to the diagonal, then indeed they would
sum to, guess what, sqrt(2)!



If you have n possibilities all mutually exclusive and equally likely, and one
of them must occur, then the chances that any given one will occur is 1/n, so
that the probabilities of all will sum to 1, as expected.
If you have n steps in the diagonal, the length of each tread/riser is
2/n, so that the total length will sum to 2, as expected.
So, you EXPECT the diagonal to be equal to 2? Whatever. You're not discussing
things constructively at all.

This is much closer to the problem:

Why would you think that the answer is anything BUT 2? In fact, the
proof shows it /must/ be 2!

Equally, why would the sum of a countable number of equal reals not
possibly be 1? In fact, the proof shows it /must/ be 1!

I'm not especially comfortable with the use of the word "countable" in this
context. We are not talking about a uniform distribution over the standard
naturals. We've already agreed that doesn't exist, since it depends directly on
the contradictory "largest finite". What Han is suggesting is that the sum of
the probabilities of mutually exclusive events, one of which must happen, is
equal to 1, and that this rule should be preserved in all cases, including the
infinite case. He is pointing out that standard analysis fails in this regard,
and seems amenable to the notion that infinitesimal probabilties solve the
problem. The question is, why aren't you?

See earlier comments on "infinitesimal". I'm still awaiting the axiomatization
of "infinite" and "infinitesimal" which you said in another thread you were
working on. Until I can see that, the answer to your question must be "Because
I don't know that it makes sense."

How on Earth can it be that n*1/n<>1? How can that make sense?


<snip>

My argument claims that the right value is 2, and
that sqrt(2) is wrong (and therefore Euclidean geometry is flawed). Han
claims the right value is 1, and that 0 is wrong (and therefore, the
standard reals are flawed). We both maintain our respective right
values, all the way out to the limit; keeping continuity between the
finite and the "potential infinite".

Oh, that's a belch of hot air, Chas. How can you deny that the laws of
probability define 1 as the probability of something that is definitely true?
If we are choosing 1 of a set of elements, the probability that one from that
set will be chosen is 1. Do you disagree yet?

I don't know about him, but I don't. Yet.


Can you deny that the laws of probability say that the probability of one of a
set of mutually exclusive possibilities happening is equal to the sum of each
of them happening?

As phrased, yes, I can and do deny it.

What in the world could you be objecting to? Say we have a set of 3 mutually
exclusive possibilities, A, B and C. Is the probability P(A or B or C) not
equal to P(A)+P(B)+P(C)? At least in the finite case, you must agree.



Do you disagree that if there is a uniform probability distribution, that each
possiblity has an equal probability?

No, I don't disagree. But I thought that we were agreed there was no probability
distribution?

Forget the naturals already. That wasn't Han's original idea. Set theorists
brought that in to confound the topic. Drop it.



Do you agree that there may be an infinite number of possiblities in some
cases?

Yes.


Does it make sense to you that in an infinite set of possibilities, at least
SOME of them must have a probability of 0, even if they have a chance of
occurring? Does this 0 mean "no chance at all", or is it simply 0 for lack of a
standard treatment of the infinitesimals?

It's not 0, as I understand it. It's simply that there's no well-defined way
to talk about a number having "a chance of occurring". I don't have a problem
with that. Should I?

You can accept that as alright for your purposes, but that doesn't mean it's
wrong to consider. If you don't care about how to talk about it, then don't.
But don't sit around telling people they are being dumb for discussing it.
Infinitesimal probabilities are the way to discuss it. If you don't like that,
don't discuss it. Personally, I think it's crucial for the next step in the
study of infinity and the development of the foundations of math. So, if you
don't mind, I think I shall engage in such conversation to see what actual
problems it causes. So far, I don't see any.


<snip>

Consider: isn't this "exactly how" how we are taught that the length of
a continuous curve can be approximated by "infinitesimally small" line
segments?

That doesn't work if you use infinitely small line segements that aren't
PARALLEL with the curve you are approximating, now, does it? Your example is
diagonal-free. Move on. Consider the fact that n*1/n=1 for specific infinite n,
and its specific infinitesimal multiplicative inverse.

See above comments on infinitesimals.

Yeah, I saw.

"I don't know what you mean by 1/n being infinitesimal for infinite n because
you haven't rigorously developed a theory I can microscopically analyze, and I
can't understand simple concepts like lim(x->oo: 1/x)->0."


Matt
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--
Smiles,

Tony
.

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