In the beginning ...

[Andrew D]

So when you claim that it is reasonable / it makes sense to approximate a finite number to an infinite number (as in the case of light rays coming from the sun), that approximation is reasonable / makes sense. And when you concede that it is not reasonable / it does not make sense to approximate a finite number to an infinite number (as in the case of light rays coming from a light year or more away), that approximation is not reasonable / does not make sense.

Surely you can see why that line of argument lacks persuasive value.

Why is the approximation reasonable in the former case but not the latter? Why does the approximation make sense in the former case but not the latter? And most crucially, why is the precise claim at issue -- that in an infinite past, everything that could have occurred absolutely must have occurred -- like the former rather than like the latter?

As to my asking questions without answering questions -- besides the fact that I have answered questions -- you made the claim (by way of agreeing with General Meade's assertion of it) that in an infinite past, everything that could have occurred absolutely must have occurred. It's a bitch, I know, but the burden is on you to demonstrate the truth of the claim that you made.

[Andrew D]

Let us suppose that you are correct. Let us suppose that ∞ X 60 = ∞ (read as "an infinite number multiplied by sixty is that same infinite number").

How does that prove that in an infinite past, everything that could have occurred absolutely must have occurred?

(One of many invaluable lawyer's skills -- separating those questions which actually bear on the ultimate issue from those which do not.)

[MajGenl.Meade]

Let us suppose that you are correct. Let us suppose that ∞ X 60 = ∞ (read as "an infinite number multiplied by sixty is that same infinite number").
How does that prove that in an infinite past, everything that could have occurred absolutely must have occurred?

you made the claim (by way of agreeing with General Meade's assertion of it) that in an infinite past, everything that could have occurred absolutely must have occurred. It's a bitch, I know, but the burden is on you to demonstrate the truth of the claim that you made

Asked and answered. counsellor! Back on page 19 or something or other - my last sensible post, assuming a first of course. Aftert having proven that "past" infinity is no larger and no smaller than "all infinity" and despatching your odd/even number argument (coup de grass):

The absolute equality of infinity as a set – that the infinite number of moments in the “past” is the same “amount” as the infinite number of moments in the “future” – is critical. In infinite time, all potentials will be actualized – otherwise they were not potentials. If the “amount” of time “past” is infinite, it is exactly the same “size” or “amount” as all time. Therefore in that identical infinity of time (“past” for clarity) all potentialities must have been actualized. Since they have not acualized, time “past” is not infinite.

And I think you never got around to responding to my largish post (which I interpreted as victory) but instead got diverted into some obscure mathematics argument which I do not pretend to understand

Meade

[Sean]

Let's take this one step at a time. I believe it was Sean who tried to distil the subject down to the probability of rolling a dice. In an infinite number of rolls a 6 will definitely be rolled. Not probably. Not most likely. Not almost surely. Will definitely. Do you agree with this?

I don't!

The probability of a six being rolled is so close to 1 as to be virtually indistinguishable and generally approximated as 1 in mathematics. I have never known a mathematician to regard this probability as a guarantee with regards to the concept of infinity. My learning however is far from complete (in fact the probability of me knowing everything about the concept of infinity is approaching 0...) and I would be grateful for the additional knowledge if I am mistaken.

[Andrew D]

Sean: Bingo!

General Meade: As I posted above, I still have your then-latest response on the subject under consideration. Please do not take my having responded to others while holding your posting in abeyance as a deflection. My responses to some postings come readily to mind; my responses to others do not. Your then-latest posting is among the latter.

[thestoat]

Ah, sorry Sean - thought it was you. For your reference, the "evidence" I pointed Andrew to is here http://mathforum.org/library/drmath/view/55871.html. Unfortunately, in Andrew's mind, maths is wrong and he is right.

In this article they state:

To make a long story short, if you have only a finite number of
outcomes and you take an infinite number of trials, you will end up
getting the outcome you are looking for.

Why is the approximation reasonable in the former case but not the latter? Why does the approximation make sense in the former case but not the latter? And most crucially, why is the precise claim at issue -- that in an infinite past, everything that could have occurred absolutely must have occurred -- like the former rather than like the latter?

Ok. I have already answered these several times before, but I know, you will say "22 page thread ... blah blah ... can't be bothered ... blah blah". So here it is once again.

Q1. Approximations are made to simplify the mathematics where the result of making the approximation is good enough. Thus it doesn't matter if the angle of the light rays is 0 degrees is 0.00000000007 degrees. 0 is good enough for the calculation. If it isn't good enough and more accuracy is required then you cannot approximate 93 million miles with infinity.
I am stunned you cannot understand that.
Q2. The approximation does not hold when the number in question NEEDS to be infinity. In the case of this angle, (or the dice rolls) if I were to say "the angle is zero" you would say "no it isn't" and then you would be right. It would only be exactly zero "AT INFINITY", and in that situation you cannot approximate infinity with anything other than infinity because the result is not being used for a calculation, it is being used in a discussion about infinity.
I am stunned you cannot understand that.
Q3. Because in an infinite past you *must* treat infinite time as infinite time and NOT as a very large number of time. If you treat infinite time as infinite time then the probability becomes 1. If infinite time is simply a large number of time then the probability tends to 1, but doesn't actually reach 1.

the burden is on you to demonstrate the truth of the claim that you made

I have explained (several times, most recently in this very post) why this is the case. I have also provided reference to an article also explaining it. Again, if you will not take mathematics' word for it, there is nothing else I can do

Let us suppose that you are correct. Let us suppose that ∞ X 60 = ∞ (read as "an infinite number multiplied by sixty is that same infinite number").

Ooohhh - there's a though. For a laugh let's make believe that maths is right about maths. How novel.

How does that prove that in an infinite past, everything that could have occurred absolutely must have occurred?

It doesn't. It is a mathematical formula that tried to educate you about infinity when you asked

If you have doubts about that, just ask yourself: "How many minutes are there in an infinite number of hours?"

and then got all confused when you didn't understand the answer. Thus yes, it is a side issue of sorts, though I still believe you really need to understand infinity before arguing about it, and your lack of knowledge about basic infinity mathematics clearly shows you do not.

[thestoat]

And I think you never got around to responding to my largish post (which I interpreted as victory) but instead got diverted into some obscure mathematics argument which I do not pretend to understand

Ah, Meade, there is where you differ from Andrew, since he does pretend to understand.

[thestoat]

Let's take this one step at a time. I believe it was Sean who tried to distil the subject down to the probability of rolling a dice. In an infinite number of rolls a 6 will definitely be rolled. Not probably. Not most likely. Not almost surely. Will definitely. Do you agree with this?

Andrew - was "Bingo" your answer? Does that mean you disagree with the above?

If you don't disagree with the above, we can move on ... if you do disagree with the above then my question now becomes "Will you continue to disagree if I bring in other mathematical references?"

[Sean]

Thanks for that link Stoat... interesting reading indeed.

One rather glaring flaw jumps out though...

An easier way to think about this is picking lottery numbers. Imagine
you are filthy rich and decide to buy a bunch of lottery tickets in an
effort to win Powerball. Since you are filthy rich, you can afford to
buy six jillion lottery tickets with every possible combination of
numbers that could come up, and thus you would be guaranteed to win
the lottery. It's the same concept with monkeys typing.

Here he is comparing deliberately choosing every possible permutation of lottery numbers with random typing. This ridiculous flaw in his logic leads me to believe that 'Dr Math' is to mathematics what 'Dr Phil' is to medicine...

Particularly as elsewhere on the same site he states this:

The study of probability helps us figure out the likelihood of something happening. For instance, when you roll a pair of dice, you might ask how likely you are to roll a seven. In math, we call the "something happening" an "event."

The probability of the occurrence of an event can be expressed as a fraction or a decimal from 0 to 1. Events that are unlikely will have a probability near 0, and events that are likely to happen have probabilities near 1.*

In any probability problem, it is very important to identify all the different outcomes that could occur. For instance, in the question about the dice, you must figure out all the different ways the dice could land, and all the different ways you could roll a seven.

* Note that when you're dealing with an infinite number of possible events, an event that could conceivably happen might have probability zero. Consider the example of picking a random number between 1 and 10 - what is the probability that you'll pick 5.0724? It's zero, but it could happen.

Likewise, when dealing with infinities, a probability of 1 doesn't guarantee the event: when choosing a random number between 1 and 10, what is the probability that you'll choose a number other than 5.0724? It's 1.

http://mathforum.org/dr.math/faq/faq.prob.intro.html

Your shot!

[thestoat]

Here he is comparing deliberately choosing every possible permutation of lottery numbers with random typing

I think what he is doing here is suggesting the analogy that even though something is incredibly unlikely (winning the lottery), with enough combinations it will happen.

And ... he we go ... the elsewhere statement is correct (though it had me stumped for a while). I think the statement that you don't approve of is

Note that when you're dealing with an infinite number of possible events, an event that could conceivably happen might have probability zero. Consider the example of picking a random number between 1 and 10 - what is the probability that you'll pick 5.0724? It's zero, but it could happen.

Now if we just consider integers, then the probability of picking 5 is 10% - or 1 / 10.
If we just consider all numbers to 1 decimal place from 1 to 10, there are 100 of them and the probability of picking 5.0 is 1% - or 1 / 100.
If we consider all numbers from 1 to 10, there are infinitely many of them and the probability of picking 5.0724 is 0% - or 1 / infinity.

It is, unfortunately, one of the problems in dealing with infinity that these paradoxes arise.

[thestoat]

This is interesting. The probability of rolling a 6 in x dice rolls is calculated as

1 - (5/6)^x

Now I just typed this into Wolfram alpha that will aid with computational analysis, with x=infinity.

The answer?

1.

[Sean]

I think what he is doing here is suggesting the analogy that even though something is incredibly unlikely (winning the lottery), with enough combinations it will happen.

If you deliberately pick every possible permutation of six numbers then winning the lottery is not at all unlikely... in fact it is a guarantee! This is why his analogy doesn't hold water. The possible permutations for the lottery numbers (if deliberately chosen as he has stated) are finite. Random permutations of monkey typing are infinite in as much as they can repeat. The two do not stand up to comparison.

Let me put it another way (back to the damned dice again...) if you roll a die six times a '6' is not a guarantee. If however you place the die down six times and deliberately choose which number is turned up each time (1 then 2 then 3...etc) then I think you'll be fairly certain of being able to make that '6' happen.

And ... he we go ... the elsewhere statement is correct (though it had me stumped for a while). I think the statement that you don't approve of is

Note that when you're dealing with an infinite number of possible events, an event that could conceivably happen might have probability zero. Consider the example of picking a random number between 1 and 10 - what is the probability that you'll pick 5.0724? It's zero, but it could happen.

Not at all. In fact the statement I was interested in was one that I fully approve of... the one I put in bold and italics...

when dealing with infinities, a probability of 1 doesn't guarantee the event

This is the statement that fully supports my assertation that

I have never known a mathematician to regard this probability as a guarantee with regards to the concept of infinity

whilst leaving your statement that

In an infinite number of rolls a 6 will definitely be rolled. Not probably. Not most likely. Not almost surely. Will definitely

as yet without any backing.

[Sean]

This is interesting. The probability of rolling a 6 in x dice rolls is calculated as

1 - (5/6)^x

Now I just typed this into Wolfram alpha that will aid with computational analysis, with x=infinity.

The answer?

1.

I do not doubt that answer for a moment Stoat but as your chosen expert on the subject says:

when dealing with infinities, a probability of 1 doesn't guarantee the event

[thestoat]

Not at all. In fact the statement I was interested in was one that I fully approve of... the one I put in bold and italics...

Ah, sorry - I must be going blind. I was most interested in the one I explained since there was a star next to it. Expanding what he says ...

an event may have a probability of zero but still happen
conversely
an event may have a probability of 1 but still can't not happen

I can see what he means (certainly no Dr Phil imo). This is a problem with infinity.

What he is saying is one minus zero does not equal one.

Mathematically I can't square that

[Andrew D]

I understand this:

Approximations are made to simplify the mathematics where the result of making the approximation is good enough. Thus it doesn't matter if the angle of the light rays is 0 degrees is 0.00000000007 degrees. 0 is good enough for the calculation. If it isn't good enough and more accuracy is required then you cannot approximate 93 million miles with infinity.

perfectly well. In fact, I have already addressed it:

You have given an example in which the difference between very high probability and absolute certainty is of no significance. But the question before is not whether, in an infinite past, there is a very high probability that everything that could have occurred has occurred. The question before us is whether, in an infinite past, it is absolutely certain that everything that could have occurred has occurred. The distinction between very high probability and absolute certainty -- between almost surely and surely -- is crucial to the question before us, precisely because the question is about absolute certainty, not about very high probability.

For the precise claim at issue, no approximation is good enough. The claim at issue deals with absolute certainty. Any approximation, no matter how small, is necessarily not absolute certainty. No approximation can produce a valid explanation of the claim at issue; producing such an explanation by way of approximation is absolutely impossible. The whole business of approximations is entirely irrelevant, and we should just set it aside and move on.

Which also pretty much disposes of this:

The approximation does not hold when the number in question NEEDS to be infinity. In the case of this angle, (or the dice rolls) if I were to say "the angle is zero" you would say "no it isn't" and then you would be right. It would only be exactly zero "AT INFINITY", and in that situation you cannot approximate infinity with anything other than infinity because the result is not being used for a calculation, it is being used in a discussion about infinity.

Exactly. With respect to the precise claim at issue, we cannot approximate an infinite past (i.e., the infinite number of moments in an infinite past) to any finite number. The number of moments in an infinite past NEEDS to be an infinite number for the self-evident reason that we are talking about an infinite past, not a finite past. Again, the whole business of approximations is entirely irrelevant, so I do not see how discussing it further can serve any useful purpose.

This:

Because in an infinite past you *must* treat infinite time as infinite time and NOT as a very large number of time. If you treat infinite time as infinite time then the probability becomes 1. If infinite time is simply a large number of time then the probability tends to 1, but doesn't actually reach 1.

is, as Sean has pointed out, directly contradicted -- or, more precisely, the conclusion you have drawn that the probability that every possible thing has occurred in an infinite past is directly contradicted -- by one of your own sources: When dealing with infinities, a probability of 1 does not guarantee the event. So even if the probability that every possible event has occurred is 1, the infinitude of that past means that the occurrence during that infinite past of every possible thing is not guaranteed. So much for the claim that if the past is infinite, every possible thing must already have occurred.

[Andrew D]

A while back, I posted:

If it is true that when one rolls six dice, any combination of the numbers 1 through 6 is possible in a single roll, then it is true that any combination of sequences is possible; if the combination 3-2-5-4-6-5 is possible, then the combination of sequences 3-2-5-4-6-5, 3-2-5-4-6-5 is possible. If the combination of sequences 3-2-5-4-6-5, 3-2-5-4-6-5 is possible, then the set of combinations {3-2-5-4-6-5, 3-2-5-4-6-5, 3-2-5-4-6-5 ... } is possible. And that means that it is possible that even in an infinite number of rolls of the dice, 1 will never occur.

It seems to me that if I am correct, then it follows that even in an infinite past, not everything that could have occurred absolutely must have occurred. Has someone posted an explanation of why I am wrong (if I am)?

[Sean]

an event may have a probability of zero but still happen
conversely
an event may have a probability of 1 but still can't not happen

In fact that's not what he says at all Stoat. Removing your double negative (smokescreen? ) your second statement translates* as:

an event may have a probability of 1 but still must happen

He (in plain English with no explanation or expansion necessary) actually states quite categorically that:

when dealing with infinities, a probability of 1 doesn't guarantee the event

*If you meant something different by that statement I apologise for my presumptive reverse engineering of it and look forward to the King James version.

[Sean]

A while back, I posted:

If it is true that when one rolls six dice, any combination of the numbers 1 through 6 is possible in a single roll, then it is true that any combination of sequences is possible; if the combination 3-2-5-4-6-5 is possible, then the combination of sequences 3-2-5-4-6-5, 3-2-5-4-6-5 is possible. If the combination of sequences 3-2-5-4-6-5, 3-2-5-4-6-5 is possible, then the set of combinations {3-2-5-4-6-5, 3-2-5-4-6-5, 3-2-5-4-6-5 ... } is possible. And that means that it is possible that even in an infinite number of rolls of the dice, 1 will never occur.

It seems to me that if I am correct, then it follows that even in an infinite past, not everything that could have occurred absolutely must have occurred. Has someone posted an explanation of why I am wrong (if I am)?

It is my (and Dr Math's) firm belief that you are indeed correct Andrew. I would still of course welcome and accept evidence to the contrary...

[thestoat]

I understand this [...] perfectly well. In fact, I have already addressed it

Then why ask the question

Why is the approximation reasonable in the former case but not the latter?

That was the question I was explicitly answering with my comment.

For the precise claim at issue, no approximation is good enough

Of course it isn't. I have stipulated that many times. It is why I expressly said

The approximation does not hold when the number in question NEEDS to be infinity

The whole business of approximations is entirely irrelevant

It was brought about because you did not understand how large numbers could be approximated to infinity for some calculations. But yes, let's move on.

is, as Sean has pointed out, directly contradicted -- or, more precisely, the conclusion you have drawn that the probability that every possible thing has occurred in an infinite past is directly contradicted -- by one of your own sources: When dealing with infinities, a probability of 1 does not guarantee the event. So even if the probability that every possible event has occurred is 1, the infinitude of that past means that the occurrence during that infinite past of every possible thing is not guaranteed. So much for the claim that if the past is infinite, every possible thing must already have occurred.

I agree that the source does appear to contradict itself (so interesting that this is the first time you even acknowledge the source - before this you had metaphorical fingers in your ears saying "nah nah I can't hear you"). But I actually thin they are trying to be clever with maths, to sort of point holes in the infinity concept and also enthuse the reader. You are forgetting (or ignoring) in your haste to conclude that the source I quoted explicitly says

To make a long story short, if you have only a finite number of
outcomes and you take an infinite number of trials, you will end up
getting the outcome you are looking for.

I suspect we won't agree. There is conflicting evidence on the web and indeed within the mathematics. I would love someone with a deep knowledge of maths to explain one way or another but it does appear that even maths is split on this issue.

[Sean]

To make a long story short, if you have only a finite number of
outcomes and you take an infinite number of trials, you will end up
getting the outcome you are looking for.

While this is a true statement it is not applicable to a situation where the outcome is purely random (ie monkeys or dice). In a completely random situation infinite repeats of particular outcomes are possible and therefore infinite outcomes are possible.

Going back to the lottery analogy if you choose (given a 6 number draw from a pool of 49 numbers for argument's sake) outcomes starting with 1,1,1,1,1,1, 1,1,1,1,1,2, 1,1,1,1,1,3 and so forth you will eventually reach the selection 49,49,49,49,49,49. This represents a finite set. If however repeats are allowed (and using random selections they have a probability of 1 but are not of course guaranteed ) then an infinite set of outcomes is possible.

QED*



*Which as any fule no means Quod Erat Demonstrandum**



**Which is Latin for "So don't argue with me you bastard!"