Re: Calculus XOR Probability
- From: Virgil <vmhjr2@xxxxxxxxxxx>
- Date: Mon, 01 May 2006 16:36:49 -0600
In article <MPG.1ec02da4e7a7ccc98ac9b@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
cbrown@xxxxxxxxxxxxxxxxx said:
Keeping in mind what was just stated: Perhaps Han's statement about
the infinite case has problems. How does Han justify that, when he
applies his definition of limit, it preserves the measure he is
looking at in the limit?
Well, the idea that n equally likely events each have a chance of 1/n
holds for all n, including the arbitrarily large. 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.
Nor any to think it should not change. Finite arithmetic does not carry
over uniformly to non-finite cases.
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.
Nor is n, nor is n*(1/n)
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.
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.
TO has repeatedly proven that he will disagree with the conclusions of
perfectly valid proofs when his "intuition" tells him to. So that the
guiding principle of TOmatics is "Intuition rules!"
My own intuition tells me that won't work in mathematics.
Almost as nonsensical as bringing in a curve measure problem with
nonparallel elements when we are discussing a probability
distribution over an infinite set.
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.
<snip>
Do you have to agree that my definition of "limit" is useful in
every, or even any, application? Not at all; but at least you
can figure out /exactly what I mean/.
Sure, the only problem is it doesn't really pertain to the
problem of infinite probability distributions.
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.
But it WAS one of whether a property holding in all finite cases need
hold in the limit case, which is spot on the issue, and showed that such
a property need NOT always hold in the limit case.
That being so, when one wants to claim that a property of any nature
carries over from a sequence of finite cases to a limiting "infinite"
case, one must support that claim with valid proof.
.
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