31 Jan 16
Originally posted by humyYou would be wrong. There is a significant difference between open and closed sets. The distinction is a fundamental part of set theory and affects all of mathematics beyond the trivial. In fact the moment you leave the finite realm it becomes important.
If we are dealing just with pure mathematics, I assume making a special distinction between an unreachable limit and a reachable limit would generally be petty superfluous.
But, if we are dealing not with just pure mathematics but specifically dealing with probabilities relevant in the physical world,
I am unconvinced. I believe your probability curve does not correspond to anything in the real world. You even suggested at the start that you had 'defined is that way'. Why did you not say you 'discovered it that way'?
But what does that really mean for what is in the REAL physical world? Does it really make sense for the REAL world?"
The real physical world has quantum dynamics which makes your scenario impossible.
31 Jan 16
5 edits
Originally posted by twhiteheadI made a poor choice of words that didn't express what I meant. I didn't mean to imply that there is no important difference between open or closed sets -of course there is an important difference!
You would be wrong. There is a significant difference between open and closed sets. ...
revising what I said, I should have said that something more like:
"....If we are dealing with the output of a limit of mathematical functions for just pure mathematics, I assume making a special distinction between the outputted numerical value of an unreachable limit and that of a reachable limit, and make that distinction even if those two numerical values are the same by still categorizing them as a different kind of numerical value, would generally be petty superfluous thing to do.
But, if we are dealing not with just pure mathematics but specifically dealing with probabilities relevant in the external world, then making such a distinction between the numerical values of unreachable limit and reachable limit may be necessary to avoid certain non-mathematical epistemological contradictions even though absolutely no purely mathematical contradiction or any other kind of purely mathematical 'problem' may result from failure to make such a distinction.
...."
31 Jan 16
7 edits
Originally posted by humyedit error
I could be wrong but my current thinking is:
If we are dealing just with pure mathematics, I assume making a special distinction between an unreachable limit and a reachable limit would generally be petty superfluous. But, if we are dealing not with just pure mathematics but specifically dealing with probabilities relevant in the physical world, then the dis ...[text shortened]... ally mean for what is in the REAL physical world? Does it really make sense for the REAL world?"
I said that second paragraph all totally wrong and the third paragraph slightly wrong making it nonsense. That should be said more like:
"....If we are dealing with the output of a limit of mathematical functions for just pure mathematics, I assume making a special distinction between the outputted numerical value of an unreachable limit and that of a reachable limit, and make that distinction even if those two numerical values are the same by still categorizing them as a different kind of numerical value, would generally be petty superfluous thing to do.
But, if we are dealing not with just pure mathematics but specifically dealing with probabilities relevant in the external world, then making such a distinction between the numerical values of unreachable limit and reachable limit may be necessary to avoid certain non-mathematical epistemological contradictions even though absolutely no purely mathematical contradiction or any other kind of mathematical 'problem' may result from failure to make such a distinction.
...."
31 Jan 16
Originally posted by humyAnd I still disagree (although in doing so I am contradicting what I said earlier in the thread about not having a special name for a mode that crossed that boundary). The fact that mathematicians often do not make the distinction leads to much confusion. As in the case of Zeno's paradox or more interestingly this one:
"....If we are dealing with the output of a limit of mathematical functions for just pure mathematics, I assume making a special distinction between the outputted numerical value of an unreachable limit and that of a reachable limit, and make that distinction even if those two numerical values are the same by still categorizing them as a different kind of numerical value, would generally be petty superfluous thing to do.
https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF
where the sum of the series 1+2+3+4+ .... comes to -1/12
But, if we are dealing not with just pure mathematics but specifically dealing with probabilities relevant in the external world,"
I am not convinced that such open sets can exist in the real world. Can you give an example of a real-world situation where something is probable infinitely close to a value but can not happen at the value?
31 Jan 16
1 edit
Originally posted by twhiteheadDistinction between what? I am no longer exactly sure what you are referring to here but I implied in my last post that mathematicians MUST make the distinction between open sets and closed sets so, if you meant that, we are in total agreement!
And I still disagree (although in doing so I am contradicting what I said earlier in the thread about not having a special name for a mode that crossed that boundary). The fact that mathematicians often do not make the distinction leads to much confusion.
31 Jan 16
Originally posted by twhiteheadI am going a bit off-topic here but I just like to point out I made a great big thread earlier on this forum about that 1+2+3+4+ .... = -1/12
.... As in the case of Zeno's paradox or more interestingly this one:
https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF
where the sum of the series 1+2+3+4+ .... comes to -1/12
...
and both I and some others here came to the conclusion that that is rubbish! and in fact:
1+2+3+4+ .... = infinity
Here is my last post on the thread of this subject:
"...
I have been mulling over some of the links I have been given and finally noticed that one clearly implies that the infinite series:
1 + 2 + 3 + 4 …
cannot possibly equal -1/12 !
http://en.wikipedia.org/wiki/Divergent_series
“...In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. ...”
Thus, if the above statement is accurate and always true, the infinite series:
1 + 2 + 3 + 4 …
cannot possibly converge to -1/12 or anything finite since it is clear that the individual terms do not approach zero.
If the above statement in that link is accurate and always true, the same goes for the infinite series
1 – 1 + 1 – 1 + 1...
but, in this case, if I am interpreting the link correctly and that link is correct, it doesn't equal infinity but rather simply doesn't have a sum!
Thus, if I am interpreting the link correctly and that link is correct, the whole premise on that video link
that claims to prove that the infinite sum 1 + 2 + 3 + 4 … equals -1/12 must be wrong (and is all a bit of nonsense ) because that video link says the infinite series 1 – 1 + 1 – 1 + 1... converges on ½ while the http://en.wikipedia.org/wiki/Divergent_series link implies it cannot converge because the individual turns don't tend to zero.
..."
31 Jan 16
12 edits
Originally posted by twhiteheadIf I take you above statement completely literally, that is a contradiction you ask for (assuming you mean that "something" is BOTH something in the real world and infinitely close to a value but cannot happen. Not sure if I got that meaning right ). But I don't claim any such contradiction can exist in the real world.
... Can you give an example of a real-world situation where something is probable infinitely close to a value but can not happen at the value?
I hope what you might be trying to ask is something vaguely along the lines of;
Can you give an example of a real-world situation where a mode of a probability distribution of a random variable X in the real-world has a value that is completely impossible for any instance of X to be in the real world?
-or at least I hope you want to ask something like that.
If so, here is an example of that:
The mode of a probability density distribution strictly and specifically of the difference in speed of an overtaking car (no particular car, just any one randomly sampled out of the set of all overtaking cars) and the thing it is overtaking where that distribution is such that its probability density approaches some finite maximum value as you approach (but I would personally claim never reach ) zero difference in speed.
At zero speed difference, the car isn't overtaking thus this distribution isn't for that. You are only allowed overtaking cars in the sample space for this distribution because this distribution is only defined for overtaking cars.
31 Jan 16
Originally posted by humyI am well aware of the thread and was a participant. I had thought though that I was the first to bring it up in response to a Zeno's paradox thread.
I am going a bit off-topic here but I just like to point out I made a great big thread earlier on this forum
Anyway, my point is that the 'sum' in that case is defined differently than the sum in normal finite mathematics. The fact is that the standard definition of 'sum' really only applies to finite sets and cannot be used. But other definitions suitable for infinite sets exist and can be used.
Given that the mode we are talking about is defined for infinite sets we should respect the rules of infinite sets.
31 Jan 16
Originally posted by humySo zero is defined out of the sample space. I think this is different from your OP example where zero was in the sample space but had a probability of zero.
You are only allowed overtaking cars in the sample space for this distribution because this distribution is only defined for overtaking cars.
There supposedly could of course be real world cars of a certain frequency that match the zero. Or could there? Quantum mechanics would suggest not.
Even if it took a billion billion years to overtake you would not say the speed difference is zero. Yet you were quite comfortable earlier calling infinitesimals 'zero'.
It is an interesting concept though: simply defining out certain numbers.
31 Jan 16
4 edits
Originally posted by twhiteheadNo, zero is defined out the physical sample space of my OP example just like this 'car' one.
So zero is defined out of the sample space. I think this is different from your OP example where zero was in the sample space but had a probability of zero....
But zero is allowed in the domain for the mathematical equation for it just like zero is allowed in the domain for the mathematical equation for this 'car' one. Both have a probability of zero.
Even if it took a billion billion years to overtake you would not say the speed difference is zero. Yet you were quite comfortable earlier calling infinitesimals 'zero'.
As far as I can recall, I never called infinitesimals 'zero'. I assume that infinitesimals being 'zero' is a contradiction given the meaning I assume most people naturally give the word 'infinitesimal' which would be something vaguely like "extremely close to zero but NOT zero".
31 Jan 16
9 edits
Originally posted by twhitehead-and I say a former i.e. 'zero'. I am very surprised you would say the latter.
No, both are undefined at that point.
If I asked you 'what is the probability of flipping tails on a two headed coin, would you say 'zero' or 'undefined'? I say the latter.
I wonder if you might be defining "undefined" in an eccentric way I don't understand. I wonder; is this is the real source of all the confusion on this thread?
ANYONE;
If I asked you 'what is the probability of flipping tails on a two headed coin, would you say 'zero' or 'undefined'?
( personally I would say 'zero' )
31 Jan 16
1 edit
The definitions would seem to imply that an event must be possible before a probability is assigned.
https://en.wikipedia.org/wiki/Sample_space
https://en.wikipedia.org/wiki/Probability_space#Complete_probability_space
Also see:
https://en.wikipedia.org/wiki/Event_(probability_theory)
Under this definition, any subset of the sample space that is not an element of the σ-algebra is not an event, and does not have a probability.
31 Jan 16
Originally posted by humyYou can assign values to non-absolutely convergent series which are reasonable for some purposes. Bear in mind that physical quantities in quantum field theories are calculated from integrals that are formally divergent and one consequence of this, which cannot be explained any other way that I know of, is that the strength of the coupling depends on the energy scale. These kooky looking results represent a way of regulating formally divergent series so that in theories where they arise physical information can be extracted from them.
I am going a bit off-topic here but I just like to point out I made a great big thread earlier on this forum about that 1+2+3+4+ .... = -1/12
and both I and some others here came to the conclusion that that is rubbish! and in fact:
1+2+3+4+ .... = infinity
Here is my last post on the thread of this subject:
"...
I have been mulling over some of the ...[text shortened]... nt_series link implies it cannot converge because the individual turns don't tend to zero.
..."