Re: Calculus XOR Probability
- From: cbrown@xxxxxxxxxxxxxxxxx
- Date: 1 May 2006 20:19:39 -0700
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:
Tony Orlow wrote:
It's a bit different from y=x being different from 2y=2x. There is a real
difference between the staircase in the limit and the diagonal, as demonstrated
by your example.
It seems to me that saying "y = x" and "2y = 2x" are the "same thing"
can only mean that for all pairs p = (px,py), p satisfies the first
condition if and only if it satisfies the second condition. The set of
all pairs satisfying these condition is the line with slope 1 going
through the origin. The fact that the second equation contains a 2 and
the first does not is irrelevant.
It also seems to me that saying "y = 1 - x; x, y >= 0" and "the limit
of the staircases" are the "same thing" can only mean that for all
pairs p = (px, py), p satisifies the first condition if and only if it
satisfies the second condition. The set of all pairs satisfying these
conditions is the diagonal D. The fact that the second condition uses
the word "limit" while the first does not is irrelevant.
As you didn't respond to the above, can I assume that you agree with
it? Or do you disagree that figures in the x/y plane are not adequately
defined by sets of points (x,y) satisfying formulas such as y=f(x), or
f(x,y)=c for some constant c?
<snip>
Perhaps that more general statement about the infinite case has problems, if
one applies a deifnition of limit which doesn't preserve the measure one is
looking at in the limit.
The whole point of the staircase argument is to get you to consider
this fact, which I will repeat for emphasis:
The more general statement about the infinite case may have problems,
if one applies a definition of limit which doesn't preserve the measure
one is looking at in the limit.
It may, but doesn't necessarily. It depends.
On what, exactly?
<snip>
Given the standard notion of oo, it
would make sense that as n->oo, n*1/n should continue to equal 1. It simply
doesn't change, and there's no reason to think it should.
Indeed, no one knowing the usual definition of limit would disagree
with the statement "lim n->oo (n*(1/n)) = 1", when applying that
definition.
Just as no one knowing the usual definition of limit would disagree
with the statement "lim n->oo (n*(2/n)) = 2", when applying that
definition.
<snip>
Since you haven't given a definition of "in the infinite case", it is
pointless to argue with you regarding whether your statement is either
false or true.
The infinite case is where 1/n is no longer a measurable real value.
Presumably, the infinite case is also one where "1/n" is also not a
finite set of rational numbers; or a finite set of geometric figures.
We have both now described something that the infinite case /is not/;
but can you describe to me what the infinite case /is/?
<snip>
Perhaps your statement about the infinite case has problems. How does
you justify that, when you apply your definition of limit, it preserves
the measure you are looking at in the limit?
When dealing with a one dimensional set like the naturals, the only way to
confoud the measure is through rearangement...
When you assert that "the only way to confound...", why on earth should
I agree with you? By which I mean, what is your mathematical argument
that it /must/ be true, just as the sum of the naturals from 1 to n
/must/ be n*(n+1)/2?
So, I feel justified, until I see a good counterexample, that
equality between quantitative expresseions proven inductively hold in the
infinite case.
So, your idea of a mathematical argument is: n*(n+1)/2 is the sum of
the naturals from 1 to n, because you haven't seen a good
counter-example?
<snip>
Perhaps your definition of "infinite case" is not appropriate for the
measure (inductive proof of an equality) you are trying to get out of
it.
In what way? We're talking about individual probabilities in a set of
possibilities as the size of the set approaches and/or becomes oo.
Since I have no idea what you mean by "individual possibilities in a
set of possibilities as the size of the set approaches/becomes oo", I
can't argue that what you say is true or false. Instead, it is
meaningless.
<snip>
Is it supposed to imply that if I say "ok then, the piirjoon of ("0",
"00", "000", ...} is also N. Since "000" is just another way of writing
0, the piirjoon of {"0", "00", "000", ...} = 0, so therefore N = 0.
Therefore, with Han's result, 0*(1/0) = 1", that then is an equally
valid argument?
No, that was somewhat convoluted, since Han didn't claim that was N, but the
conclusion is correct, given equal 0's. :)
Wow. So all a mathematical argument is to you is just a bunch of
mathematical looking words, slung together with connectives like
"therefore" and "since", which is correct if the conclusion agrees with
your intuition, and incorrect if it does not?
Uh, no. I didn't claim that was proper logic. I just agreed with the
conclusion. That doesn't mean the argument is correct.
So, is the argument correct or incorrect? "Convoluted" merely means
"complicated". If it is incorrect, why is it incorrect?
<snip>
The example was brought in not to bear directly on his conclusion; it
was brought in to demonstrate the flaw in the /logic/ of his
/argument/.
Well, I don't think what you're pointing out is really a flaw. I think the
example you gave has the same stricture, but fails for a different reason. It's
not that the proof is wrong, but the interpretation of the limit as being the
same as the diagonal. We've discussed the problem with that assumption.
You have yet to demonstrate that either:
* the function that I defined as the limit of the staircase curves
fails to yield a set of points; or
* a figure in the x/y plane is anything other than a set of points
(e.g., those points p = (x,y) satisfying y=f(x) or f(x,y) = c).
Equality between sets of points is based only on those sets of points
having the same membership. So I don't see why you say there's a
"problem".
It pertains to clarifying valid versus invalid methods of deduction in
a mathematical argument; which in turn pertains to whether or not your
or Han's arguments regarding "infinite probability distributions" are
valid or invalid.
I guess, except the problem you demonstrated was not one shared by the question
of probabilities in the infinite set of possibilities.
Until you define what you mean by statements like this, I cannot
express an opinion regarding whether such a statement is true or false;
instead it is meaningless.
Cheers - Chas
.
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