Re: Calculus XOR Probability

Matt Gutting said:
Tony Orlow wrote:
Matt Gutting said:
Tony Orlow wrote:
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.
It must hold, even if we don't know what probability distribution we're talking
about, *assuming* that we're looking at a probability distribution. As I said,
the question is whether one can be defined here. You appear to be assuming it
can. I'm not.

Somethign tells me you're still trying to deal with a uniform probability
distribution over tha naturals. That's impossible, but given any set of n
equally likely outcomes, if exactly one of them will occur, each has a 1/n
chance of being the one to occur. To assume you have a set of possibilities
WITHOUT a probability distribution is just to say you have no idea what any of
the individual probabilities is. That has no effect on the fact that if exactly
one is to occur, the probabilities of each sum to that 1.

I'm not assuming a *uniform* probability distribution over the naturals. I'm
saying that what you are asserting is only true when one can define *some*
probability distribution over the naturals. In effect, I'm not saying that I
"have no idea what any of the individual probabilities is"; instead, I'm saying
that referring to "individual probabilities" is meaningless in this case,
so that saying "the individual probabilities sum to 1" is likewise meaningless.

First of all, discussing the naturals is a digression and a straw man. We have
n, the upper bound of the set. There are n elements. The probability of each
being chosen has meaning if we are choosing one. If we know how many we have,
if we can calculate and average of n/2, then we can assign a probability,
albeit infinitesimal. With the naturals, you can't. We agree. Forget the
naturals.



The difficulty here is that you haven't really shown an explicit and consistent
way of defining and working with infinitesimals. That must come first, before
you begin using them to explain things.

What do you want, a number system? By all means, use the T-riffics. If we have
1:000...000 equally likely possibilities one of which must happen, each has a
1/1:000...000=0:000...001 chance of being chosen.

After reading through numerous posts on the subject, I still don't see that this
is a consistent treatment of infinitesimals.

What inconsistencies do you detect?


<snip>

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.

That's still circular. If you haven't defined infinitesimal carefully, and only
in terms of other well-defined concepts (and "infinite" isn't such a concept,
because you haven't really defined it well except in relation to
infinitesimals), then you can't use the term to explain or define anything else.

If we are discussing an infinite set of possiblities, and I define
infinitesimal in terms of that infinity, it's not circular. If you want to have
a rigorous notion, start with infinity, but I rather doubt your "injections
into a proper subset" definition is going to help a whole lot here. This isn't
rocket science. n*1/n=1 and lim(n->oo: n*1/n)=1 and lim(n->oo:n)*lim(n->oo:
1/n)=1. All is One, Cricket.

The equation (lim x->:a f(x)g(x)) = (lim x->a:f(x))(lim x->a:g(x)) is only true
when both limits exist. It's explicitly proved that way. You can't use that
theorem here, since (lim n->oo:n) does not exist.

Sure, I know, "undefined". So far, undefined. Undefinable? Is sqrt(-1)
undefinable? It used to be undefined. What happened? Did we define it and find
it useful? Gee, could that happen here? Sure, that's what's going on. (lim n->
oo:n)=oo. Now, if you used a SPECIFIC oo, you'd have a specific answer,
wouldn't you?


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.

Again, this is only true if there exists a probability distribution (determined
or not) on the set. If there doesn't exist a probability distribution, and there
seems to be disagreement on this point, then what you say is simply not the
case.

We're no longer talking about the problematic set of naturals, as far as I'm
concerned.

It doesn't matter what set we're talking about, as long as it's infinite (or,
if you prefer, unbounded).

No, it must be infinite, yet bounded, to be able to define the probability of
an individual element. The problem with the naturals is a lack of defined
range, which of course has been a sticky issue when I've been trying to put
forth the Inverse Function Rule.


<snip>

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.
Could you explain what it means for an infinitesimal (sc. "infinitesimally
long line segment", I suppose) to be "parallel" to a curve?

Do you need this explained? It means that the segment ((x1, y1),(x2, y2)) is
parallel to the curve if there can be defined a point on the curve
perpendicular to a point on the segment where points (x3, y3) and (x4, y4)
arbitrarily close to that point on either side have the property that (x3-x4)/
(y3-y4)=(x1-x2)/(y1-y2), or at leats the difference is arbitrarily close to 0.
You'll probably have a correction regarding this, but then, why did you ask?
You know what parallel means, and it doesn't mean at a 45 degree angle.


You're being, at best, very loose with language here: one can't have a point
(on the curve or anywhere else) perpendicular to another point (on the segment
or anywhere else). I *think* you mean that one can find a secant line "nearby"
the curve (and yes, I'm aware that I'm using that loosely; I can formalize
it if you'd like) which is parallel to a secant line on the other curve.

Sure, whatever. Do you see that the angle at which your elements intersect the
diagonal directly accounts for the error of sqrt(2)? If not, I went into
greater detail in an earlier post to Chas.


<snip>

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.
Neither is true, because you're using induction on finite quantities
to assert results about infinite quantities.

That is not a violation, as I see it, as long as one proves an equality.

See above. You appear to be misunderstanding the limits of inductive arguments.

I appear to be of a different opinion regarding this matter. You can chalk it
up to misunderstanding if that makes you feel better, but my arguments in this
area have all held, as far as I've been able to tell. COunterxamples like this
are easily explained, so my point remains unrefuted.



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...
That's how reasoning works. One may arrive at a true conclusion using
ill-defined terms, but the argument leading to the conclusion will be an
invalid one.

Like your ill defined diagonal as the limit of the staircase as n->oo.

That's exactly the point. That argument was intentionally posed as a flawed
argument, to point out the flaws in Han's argument.

But, the flaw in that argument is not what makes it similar to han's argument.
The only flaw is in equating the fractal diagonal with the normal self-parallel
diagonal. They're different animals, despite the fact that your microscope is
too feeble to see the difference.



Smaller than any finite but non zero. The multiplicative inverse of an
infinity.
That's circular, because you haven't come up with a coherent definition of
infinity, nor any reason to believe that the inverse of "an infinity"
will be non-zero.

First of all, if you have an objection regarding the definition of infinity,
you should have brought that up weeks ago. Secondly, the argument regarding the
chance of one of an infinite set being chosen being smaller than any finite
value, and yet nonzero, has been exhaustively discussed without any valid
objection. Each has a probability of 0% in the standard world, but is not
without a chance in reality. Been there, done that. Each of n equally likely
events has probability 1/n, and if n is infinite, as put forth to begin with,
this probability is infinitesimal.

I believe I did bring up objections regarding your usage of infinity; we have
discussed elsewhere the need for you to formalize and axiomatize your notions
of infinity, and you agreed that it was necessary and important. I haven't seen
any development in this area.

I haven't been doing that all over the newsgroup. Just getting basic notions
like the fractal natural of Chas' line is difficult enough. When I state a rule
clearly, I get complaints about every term. I think the theory ends up putting
forth infinite sets in the context of order, such that:

x<y<z -> x<z Order
x<z -> x<y<z Internal Infinity
y -> x<y<z External Infinity

I gave a bunch of this to Tribble a couple days ago, and went into discussion
of how both use the same formula for generating elements in opposite
directions, at least in one sense. That needs to be expanded to include another
interpretation of successor that produces the H-riffics. In any case, can you
think of a sense of infinity where a nonzero reciprocal could be finite, or
anything but infinitesimal?


The fact that you have seen no valid objection to an argument is not
necessarily an indicator that there is no valid objection; it just means that
you haven't seen one. As I said above, the assumption you're making here (one
of the assumptions, at any rate) is that it makes sense, when talking about
infinite sets, to consider the phrase "equally likely" as meaningful. It's not.

Why not? It's impossible to define a uniform probability distribution on the
naturals due to its unboundedness, but given an infinite upper bound of n, it's
not impossible to define the likelihood of individual elements. It's just not
part of standard mathematics.



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)!
But how do you know that your probabilities add? Without a probability
distribution (known or unknown), they don't.

Sum(x=1->n: 1/n)=1. We already DEFINED the probability distribution to be
uniform, not over the naturals, but from 1 through n.

That works when n is finite; that is, it works for any set 1 through n.

It works for any set from 1 through n, whether n is finite or infinite.

But none of these sets is the natural numbers.

Good. To hell with the naturals. They make a crappy domain.

If you're not talking about an infinite
set, we have no argument. But if you are, then what infinite set are you
intending to discuss? The same objection holds for any infinite set: the
concept of "probability of an individual event occurring" is meaningless in
such a context.

That's opinion, not logic. If you throw a dart at a board randomly, every point
has a chance of being hit, but there are an infinity of points, so that chance
is effectively 0, but not really 0. Is it meaningless to talk about the chance
of a single point being hit? That idea has meaning for me.

Flip a coin aleph_1 times and generate bits to find a point in [0,1). Is there
not a uniform probability distribution among the points? Throw your balls all
in a vase by noon, shake vigorously an infinite number of times, and pick. Is
one ball more likely than any other?


<snip>

How on Earth can it be that n*1/n<>1? How can that make sense?
It doesn't make sense, if n exists and if one can do ordinary arithmetic with n.
What I'm saying is that, absent an axiomatization such as I mentioned, I don't
know that either of these premises is true.

And you don't know they're not. You have certainly heard enough
"circumstantial" evidence that points to a strong possiblity, but your
assessment of that possibility appears to be under the influence of other
factors.

If you don't have an axiomatization, it is meaningless to say that they are
true, just as it is meaningless to say that they are false.


<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.
That's because, in the finite case, one can always arrive at a probability
distribution over the set.

And if one postulates an infinite set with a uniform probability distribution,
can one assume that it has a uniform probability distribution? (sigh)

Only if it turns out to be logically consistent for an infinite set to have
a uniform probability distribution. You can't work with sets of postulates that
are logically inconsistent.


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.
It's not the naturals; it's any infinite set.

If n is infinite, and there are n equally likely events, and one is going to
happen, how can you argue there is no uniform probability distribution? There
is by definition.

Again, this depends on the phrase "equally likely events" having meaning in
this context. I contend that there is no logically consistent meaning that can
be assigned to the phrase in this context.

<snip>

Infinitesimal probabilities are dependent on the existence and putative nature
of infinitesimals. So far, I don't see that you've developed an explicit theory
governing their existence and nature.

Is that grounds to dismiss the notion entirely? It's obvious that in the case
Han suggests, the individual probabilities are indeed infinitesimal, whether
you have a formal system of representation, calculation, or anything else for
those infinitesimal values. But still, don't forget the T-riffics.

Nothing is "obvious". (I noticed that most of all in math textbook proofs.
Whenever they said "it is clear/obvious/trivial that..." I knew I was in for
the hard work trying to figure out *why* it was so.) I'm not worried about a
formal system of representing or calculating with them. I'm worried about
a formal system of *defining* them: as you've mentioned elsewhere, you're
apparently working on one, and you seemed to have accepted my statement that
until you did so, I couldn't accept any statement you made about them.


I'm not saying that you're dumb for discussing it. I'm saying that without a
rigorous understanding of your concepts of "infinite" and "infinitesimal,"
it's simply impossible to draw any certain conclusions about these sorts
of probabilities. And I do see a problem with attempting to draw conclusions
before you know that your arguments are valid.

These arguments are so basic, there is no reasonable doubt in my mind that what
we are discussing is entire reasonable. It is clear that extensions to digital
number systems can handle such cases, and that this is the only solution to the
question. So, I dunno, I feel rather confident that Han is being argued with
over nothing more than the suggestion that standards need to be expanded.


Again, this is not "clear" to me, nor to others.


"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."
If you haven't developed a theory that I can analyze (or any theory even vaguely
formal), then I'm perfectly justified in questioning any conclusions you claim
to be based on concepts of that "theory". And I can understand simple concepts
like "the limit as x increases without bound of 1/x is zero"; I just don't see
where infinity comes into the sentence.

Matt

Yes, well, with the concerted effort in the last cetury plus to all but
eliminate "infinity" from the vocabulary and compromise on a system the
obfuscates the whole notion in favor of finite "rigor", it's not surprising.
But, I'll remind you that "infinity" came into the "sentence" when Han
postulated an infinite set with a uniform probability distribution.


And he based his conclusions on analogies to finite cases, thus eliminating
"infinity" from the sentence again. The counter-arguments were simply that
it isn't valid to reason from those finite cases to infinite ones. Those
arguments still stand.

Matt
*** Posted via a free Usenet account from http://www.teranews.com ***


--
Smiles,

Tony
.

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