Re: Calculus XOR Probability

Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:
Han de Bruijn wrote:
cbrown@xxxxxxxxxxxxxxxxx wrote:

What does surprise me is that you don't see the obvious parallel
between my reasoning that since lim (n*(2/n))=2, therefore the diagonal
has length 2; and your reasoning that since the lim (n*(1/n))=1, that
therefore there must exist a uniform distribution on the naturals.


<snip>

If you use 2 steps, you still get 2.

If you use a 2 element set, you still get a total probability of 1.

Uh, yeah, except that 1 is the value you want for the probability of the entire
sample. 2 is not the value you want for the diagonal.

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.



As you use more and more
steps, it looks more and more like a diagonal line, but the length stays the
same as when it did not.

As you use more and more elements, it "looks" more and more like a
uniform distribution on the naturals, but the total probability stays
the same as when it did not (sic; i.e., whatever that means).

The sum of all outcomes should be 1 if they are mutually exclusive, equally
probable, and one must be chosen. So, it's GOOD that n*1/n equals 1 for all
(finite) n.

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.



You are not approximating the length of the diagonal. That sum is never
equal nor getting closer to the answer.

You are not approximating a uniform distribution on the naturals. That
sum (1) is never equal to or getting closer to "the answer" (which is
either 0, since the sum of a countable number of 0's is 0; or infinite
by the archimedean property of the reals).

Uh, no. If one of the outcomes is to be selected, then the probability of all
outcomes should sum to 1. The fact that you get a sum of 0 in the infinite case
indicates an error. The sum is correctly 1 for all finite n. The sum is
correctly 1 for infinite n as well, but the the probability of each outcome is
1/n, which in the infinite case is infinitesimal, not absolute zero.


By exactly the same logic:

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.

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?

[*] e.g., http://plato.stanford.edu/entries/geometry-finitism/


So, the question here
is, why would you think it gives a correct answer at oo, if it gives an equally
incorrect answer for all finite nnumber of steps? The limit of the error is not
0.

So, the question here is, why would you think it gives a correct answer
at oo, if it gives an equally incorrect answer for all finite number of
steps? The limit of the error is not 0.

Uh, excuse me. Maybe I'm getting confused, but maybe not. If there is a set of
n equally likely outcomes, and one is picked, is not the chance of one being
picked equal to 1, and is this not the sum of n individual probabilities of
1/n? Is the chance of picking any given natural from 1 to a million 1
millionth? What error is there in "n*1/n=1" for any finite n? None, so what is
the parrot act all about? Is this what you call analysis? Please try to answer
the question, sincerely.


Alright, I'll stop - but I encourage you to do it for yourself: Just
substitute in your argument "trillions of tiny steps" and "What error
is there in "n*2/n" for any finite n?"

Note that:

* No one is claiming anything other than that, in a finite set of n
equal outcomes, each outcome has probability 1/n; just as no one would
claim anything other than that if there are (finite) n steps, each
tread and riser has length 2/n.

* 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.

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".

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

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).



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!


As n increases
without bund, this relationship is preserved and functions without issue, given
expected results.

As n increases without bund (sic), this relationship is preserved and
functions without issue, given expected results.

So, you DO think the diagonal of a unit square is 2. I see. And you think Han
and I are cranks.....


No, I don't think that the diagonal is "really" 2. But the logic is
/exactly/ the same.

Han uses this logic to conclude that there is a uniform distribution on
the naturals. Someone might claim that Han's conclusion is obviously
wrong, because the Archimedean Principle /clearly/ implies that the sum
of a countable number of equal real numbers must either be 0 or
infinite. Han replies that the Archimedean Principle therefore
indicates a flaw in the real numbers.

I uses the same logic to conclude that the diagonal of a unit square
has length 2. Someone might respond that my conclusion is obviously
wrong, because the Pythagorean Theorem /clearly/ implies that the
diagonal is sqrt(2). I reply that the Pythagorean Theorem therefore
indicates a flaw in Euclidean geometry.

But there' no need to go off on a tangent into some kind of bizzare
alternative mathematics here. Our grandiose counterclaims are both just
/hot air/. Neither reply is valid; because in both cases, the
conclusions /don't follow/ from the arguments to start with.


So, the question here is, why would you expect this
relationship between n 1/n's summing to 1 to break down at n=oo?

So the question here is, why would you expect this relationship between
n 2/n's summing to 2 to break down at n=oo?

I wouldn't, but if you're using some technique to approaximate the diagonal, I
would expect it to get closer with successive iterations, not maintain the same
wrong value.

What "wrong value"? 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".

But you're getting warmer. Consider your use of "technique to
approximate". Where in my argument do I actually /justify/ my
assertion that the stairstep really is a valid "technique of
approximating" the length of the diagonal?

Now apply this same question to Han's argument regarding a
distribution:

Where is it actually /justified/ that "approximating" a uniform
distribution on the naturals can be accomplished by looking at uniform
distributions on finite sets of naturals?

What does it mean, mathematically, to "get closer with successive
iterations" to a uniform distribution on the naturals? Exactly how "far
away" is a uniform distribution on {1..10} from a uniform distribution
on the naturals?

This requires /at least/ as much solid justification as claiming that
the stair-step approach "approximates" the diagonal in the limit.

And such justifications must be done carefully. For example, I claim
that the stairstep approach approximates the diagonal because the error
here can be defined as the total area of difference between the
stairstep curve and the diagonal. This area certainly approaches 0 as n
approaches infinity. So the stairstep curve, in the limit, will have a
0 area difference with the diagonal, and therefore is the same curve;
thus they must have the same length, which by Han's logic must be 2;
since "Nature abhors skipping about like a little girl": Glibnits. QED.

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?

I don't think you even thought for a second before writing. Your loss.

I didn't realize thinking was required in this newsgroup.

Cheers - Chas

.

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