In the beginning ...

[Andrew D]

I am still awash in an ocean of paper. I have only a little time at the moment.

The thing that interests me is the claim that if the past is infinite, every possible thing must necessarily have already occurred. I understand both General Meade and thestoat to be making that claim.

It seems to me that even if the past is infinite, the future could also be infinite. Even if the set of all odd integers is infinite, there can still be even integers that are not within the set of all odd integers, and the set of all even integers can also be infinite. Even if the set of all negative numbers is infinite, there can still be positive numbers that are not within the set of all negative numbers, and the set of all positive numbers can still be infinite. Etc.

So how does one explain, in English, how it is that if the past is infinite, every possible thing must necessarily have already occurred, even though there is an infinite (or potentially infinite) future in which those possible things which have not yet occurred might still occur? It seems to me that an infinite (or potentially infinite) future leaves open the possibility that a possible thing which has not yet occurred, even in an infinite past, might still occur in an infinite (or potentially infinite) future. In English, why am I wrong?

Is that question clear? I hope so, because it is the question that has interested me all along.

[thestoat]

Very easy to answer questions - a little tougher to answer them. And so much easier to participate in a discussion where you only ask the questions but never answer any. For the record, I don't actually believe that an infinite amount of time has occurred since time is bound with the fabric of space, and the age of the universe is not itself infinite. However, yes, if we had already had an infinite amount of time then anything that could occur would have occurred. This was explained previously with the roll of dice analogy.

The problem is that infinity is a mathematical concept - not a reality. I cannot think of anything that is infinite (anyone?). So you are asking to translate from a language in which these things may be described into a language where they really cannot (they can be defined - but not described).

So how does one explain, in English, how it is that if the past is infinite, every possible thing must necessarily have already occurred, even though there is an infinite (or potentially infinite) future in which those possible things which have not yet occurred might still occur? It seems to me that an infinite (or potentially infinite) future leaves open the possibility that a possible thing which has not yet occurred, even in an infinite past, might still occur in an infinite (or potentially infinite) future. In English, why am I wrong?

You are not wrong. In fact, if the past was infinite then everything possible would not only have happened - they would have happened an infinite number of times ... and if the future was to be infinite then everything possible would be due to happen again an infinite number of times.

This is a description though, not an explanation - for that you need the language to maths and to fully understand the dice roll analogy.

Having said all that, it could be in discussing infinite pasts and futures that we are moving away from countable infinities to higher order infinities (I don't think we are, yet, since I am pretty sure infinity squared is still just infinity - but infinity to the power infinity is larger)

[Andrew D]

The roll-of-the-dice analogy did not explain the matter at all. 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.

This:

The problem is that infinity is a mathematical concept - not a reality. I cannot think of anything that is infinite (anyone?). So you are asking to translate from a language in which these things may be described into a language where they really cannot (they can be defined - but not described).

and this:

This is a description though, not an explanation - for that you need the language to maths and to fully understand the dice roll analogy.

amount to statements that you cannot explain your claim in English. I can only wish that you had said so earlier.

[Andrew D]

By the way, I have been asking people to demonstrate the truth of claims they have made. That inquiry does not require me to demonstrate anything or to answer any questions (except questions seeking my clarification of what what I perceive to be the claims whose truth I am asking to be demonstrated). The people making the claim that if the past is infinite, then every possible thing must necessarily have already occurred, even if there is an infinite (or potentially infinite) future in which they might yet occur bear the burden of demonstrating the truth of that claim.

It's a bitch, I know, but by making a claim, one voluntarily undertakes the burden of demonstrating its truth. Or acknowledging one's inability to demonstrate its truth. Which appears by your own words to be where you are now.

[Andrew D]

Even so, what questions -- other than the snide "do you understand it?" questions in which the "it" remains conspicuously unexplained -- have I not answered? "See above" in a twenty-two-page thread is not helpful.

[thestoat]

As for the questions you are avoiding responding to ... here is the first question:
You wrote:

Once it is acknowledged that the statement “the distinction is important in more complex cases relating to some sort of infinity” does apply to “the simple infinity we are discussing,” the contrary statement that “999999999999999999999999 (which we could approximate to an infinite time)” should be withdrawn.

I wrote:

Not at all, and I find it truly bizarre that you keep coming back to this point that has been explained so many times. Here I shall repeat it ...

Approximations are used all the time in many branches of physics and maths. They have to be since no measurement is perfect. Now, 9999999999999999999999999 millennia might not be a good approximation of infinity when dealing with the age of the universe, but it would be a good approximation when dealing with the lifespan of a ladybird.

Or consider rays of light from the sun. We consider them to be parallel. They are not parallel, but near as dammit when making calculations. They would only be parallel if the sun was infinitely far away, which of course it isn't.


Let's think of 2 of these rays of light falling on a book you are reading (let's just consider 2 rays, one on one extremity of the book, the other on the other). The rays of light can be considered to form an isosceles triangle. One side of this triangle is about 6 inches long. The other 2 sides are 93 million miles long. Question - what are the internal angles of the triangle?

Any normal person with a reasonable understanding of maths would say "hmmm. Let's pretend that this 93 million miles is infinity." When you do that, the answer becomes 90 degrees, 90 degrees and 0 degrees. That answer is normally good enough. But if you insist that you can never approximate infinity they you will say

"Ah ha - wrong answer. The actual answer is ...".

I wonder - can you tell me? My calculator can't calculate such accurate numbers. It will be something like
0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 degrees and
2 lots of
89.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999998

Most people would, I suspect, be happy to say "well, call it two lots of 90 and 1 lot of zero degrees", which is what you will get if you assume, in this case, 93 million miles is infinity.

Perhaps you can see that approximating large numbers to infinity can actually be very useful, if you understand the maths. Or perhaps you still don't agree?

Second question. You wrote:

How can one rationally agree with a proposition which one finds inexplicable?

I asked:

Then please explain to me the nature of quantum mechanics - or do you disagree with QM?

I also asked a similar question

Unfortunately I can not claim the credit for the proposition - far cleverer mathematical minds than mine did that. I have merely learned and agree with them. But, again (I do seem to have to repeat myself a lot), Quantum mechanics does not stand logical scrutiny. Is that also false?

So - two questions you have avoided so far. And these I have asked several times. So, to be nice and simple ...

Q1. Let's think of 2 of these rays of light falling on a book you are reading (let's just consider 2 rays, one on one extremity of the book, the other on the other). The rays of light can be considered to form an isosceles triangle. One side of this triangle is about 6 inches long. The other 2 sides are 93 million miles long. Question - what are the internal angles of the triangle?
Q2. Since you ask "How can one rationally agree with a proposition which one finds inexplicable?", I ask you to either deny quantumn mechanics works or explain it.

statements that you cannot explain your claim in English. I can only wish that you had said so earlier.

Errr - you aren't concentrating.

In February I said

Infinity is a mathematical concept - not a physical thing (something you clearly have yet to grasp). It cannot be explained in layman's terms and more than the square root of minus 1 (which, as noted previously, is also very important in mathematics).

I also said

I am not running away - I just can't explain it

So, quite clearly, I had said so earlier.

The roll-of-the-dice analogy did not explain the matter at all. 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.

[thestoat]

(I was going to answer the finally quoted text in my post above, but then decided to leave it for now pending Andrew's answers)

[Sean]

The roll-of-the-dice analogy did not explain the matter at all. 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.

I believe that the dice analogy was originally mine Andrew and as the original point was exactly what you have posted above then I think it explained the matter rather well.

[quaddriver]

Let's think of 2 of these rays of light falling on a book you are reading (let's just consider 2 rays, one on one extremity of the book, the other on the other). The rays of light can be considered to form an isosceles triangle. One side of this triangle is about 6 inches long. The other 2 sides are 93 million miles long. Question - what are the internal angles of the triangle?

Im of the opinion that the point angle is about .00000000007 degrees

Now recompute for an observer on regula-9

[Andrew D]

I am well aware that in mathematical operations (including mathematical operations in physics) in which the difference between almost surely and surely is not significant, one number can be approximated to another. But as I previously observed, with a link, the difference between almost surely and surely is significant when dealing with questions of infinity, which means that such approximations in dealing with questions of infinity lead to invalid operations.

The correct answer to your question about the smallest of the three angles formed by two rays of sunlight and a book is, of course, some minute fraction of a degree. But it is not zero.

If your assertion about the rays of light were correct, it would play havoc with modern astronomy. We examine things that are light years away precisely by, among other things, measuring minute differences in angles. A single light year is approximately five-trillion eight-hundred seventy-eight billion four-hundred ninety-nine million eight-hundred ten thousand (5,878,499,810,000) miles. Therefore, if I have done the arithmetic correctly, the angle formed by light rays from that distance is more than sixty-three thousand times smaller than the one in your example. But astronomers do not approximate that number to zero.

Any normal person with a reasonable understanding of maths would say "hmmm. Let's pretend that this 93 million miles is infinity." When you do that, the answer becomes 90 degrees, 90 degrees and 0 degrees. That answer is normally good enough. But if you insist that you can never approximate infinity they you will say

"Ah ha - wrong answer. The actual answer is ...".

* * *

Most people would, I suspect, be happy to say "well, call it two lots of 90 and 1 lot of zero degrees", which is what you will get if you assume, in this case, 93 million miles is infinity.

Exactly. People say "call it two lots of 90 and 1 lot of zero degrees" precisely because they recognize that it actually is not two lots of 90 and 1 lot of zero degrees.

For many operations, such approximations are useful, because -- for what feels like the umpteenth time -- in such operations, the distinction between almost surely and surely is not significant. But in dealing with questions involving infinity, the distinction between almost surely and surely is significant, and I see nothing in what you have written that explains why we should ignore that important distinction.

(Your insistence on ignoring that distinction is also rather odd, because you have said (or agreed with someone who said) that I am treating an infinite number as if it were simply a very large number. And yet here you are, doing exactly that.)

For purposes of discussion, I am willing to accept (at least in part because it does not matter) that you have demonstrated that over an infinite past, the likelihood that not every possible thing has already happened is very small; that almost surely, every possible thing has already happened. But you have not -- most emphatically not -- (and this is the part which does matter) demonstrated that over an infinite past, every possible thing absolutely must have already happened; that surely, every possible thing has already happened.

The latter claim is the one that has interested me from the beginning of this. The problem is not my failure to understand what you have written. The problem is that what you have written does not support your position.

[Andrew D]

Q2. Since you ask "How can one rationally agree with a proposition which one finds inexplicable?", I ask you to either deny quantumn mechanics works or explain it.

That is a false dilemma. Just as I cannot rationally agree with a proposition which I find inexplicable, I also cannot rationally disagree with such a proposition.

I am told by an authority I consider reliable (Hawking) that the mathematics of quantum physics are necessary for the operation of various kinds of technology. So I suppose that it "works". But because I do not understand the explanations I have encountered of it, I have no way of knowing whether those explanations are correct.

[thestoat]

Q2. Since you ask "How can one rationally agree with a proposition which one finds inexplicable?", I ask you to either deny quantumn mechanics works or explain it.

That is a false dilemma. Just as I cannot rationally agree with a proposition which I find inexplicable, I also cannot rationally disagree with such a proposition.

I am told by an authority I consider reliable (Hawking) that the mathematics of quantum physics are necessary for the operation of various kinds of technology. So I suppose that it "works". But because I do not understand the explanations I have encountered of it, I have no way of knowing whether those explanations are correct.

But in your quote you suggest you cannot rationally agree with a proposition that you find inexplicable. Yet here you are agreeing with the inexplicable QM. If you are now also saying that you cannot rationally disagree with a proposal that you find inexplicable then are you not therefore irrationally disagreeing with the concepts of infinity?

[thestoat]

The correct answer to your question about the smallest of the three angles formed by two rays of sunlight and a book is, of course, some minute fraction of a degree. But it is not zero.

Of course it is not zero - I have never said it is.

But astronomers do not approximate that number to zero.

Of course they do not - you approximate large distances to infinity when it makes sense - not all the time.

People say "call it two lots of 90 and 1 lot of zero degrees" precisely because they recognize that it actually is not two lots of 90 and 1 lot of zero degrees
...
But in dealing with questions involving infinity, the distinction between almost surely and surely is significant, and I see nothing in what you have written that explains why we should ignore that important distinction.

Again, I have never said the answer is zero. But calculations can assume it is zero when convenient to do so - and assuming zero implicitly assumes the distance from the earth to the sun is infinity.

You need to remember that you were accusing me of "reasoning about infinity by imagining a vast but finite number" - I have merely here given you an actual example of where that is a valid thing to do.

Now - I accused you of the same (I believe I made the first accusation that you then invalidly (as just demonstrated) turned on its head) because you cannot get your head round the rather simple equation

60 X ∞ = ∞

If you treat infinity as infinity, it makes sense
If you treat infinity as a large number, then of course it does not make sense

This is where I believe you are missing the point.

For purposes of discussion, I am willing to accept (at least in part because it does not matter) that you have demonstrated that over an infinite past, the likelihood that not every possible thing has already happened is very small; that almost surely, every possible thing has already happened. But you have not -- most emphatically not -- (and this is the part which does matter) demonstrated that over an infinite past, every possible thing absolutely must have already happened; that surely, every possible thing has already happened.

Wrong, I have shown that the probability that everything that could happen has happened is definite. I have done this using some basic mathematical equations and references to mathematicians specialising in infinity. Your response has been to suggest maths is wrong and you are right. If you will not accept a mathematics answer (or answers from mathematicians) for a mathematical concept then I cannot think of anything else I can say to help you understand.

[thestoat]

Im of the opinion that the point angle is about .00000000007 degrees

Now recompute for an observer on regula-9

I found a calculator that could help me ... the answer is approx 2.9 x 10^-11 - so pretty close to your answer

Where is regula-9 then? Never heard of it

[Andrew D]

Wrong, I have shown that the probability that everything that could happen has happened is definite.

You have done no such thing. You have done nothing more than assert that almost surely and surely are the same for present purposes, consistently ignoring the fact that in problems dealing with infinite numbers, the distinction between almost surely and surely is important.

In order to prove that it is absolutely certain that in an infinite past, everything that could have happened has happened, you need to demonstrate that that is so without using any approximations. Anything short of that is merely a demonstration that it is very likely that in an finite past, everything that could have happened has happened. And that simply does not cut it.

You have done nothing more than show that in an infinite past, everything that could have happened almost surely has happened -- if you have even done that; I am stipulating arguendo that you have, because it does not matter -- and asserted that everything that could have happened surely has happened. That is, you have done nothing but substitute surely for almost surely because that substitution suits your predetermined conclusion.

If you care to demonstrate that in an infinite past, everything that could have happened absolutely must have happened -- not just almost surely have happened, which is all (if even that) that you have managed thus far -- feel free to have at it. If you care only to insist that when it comes to the idea of an infinite past, almost surely is the same as surely, you can expect nothing but my continuing to point out that you are making the same error over and over and over ... ad infinitum.

[thestoat]

Wow, you find one reference that talks about surely and almost surely in the context of infinity and hide behind that. I have given links to mathematical forums discussing this very thing where they agree with me (actually, more accurately, I agree with them). So when you say I have not shown this you are either very forgetful or ignoring the posts.

If you care to demonstrate that in an infinite past, everything that could have happened absolutely must have happened -- not just almost surely have happened, which is all (if even that) that you have managed thus far -- feel free to have at it.

Again, I have done this using some basic mathematical equations and references to mathematicians specialising in infinity. Your response has been to suggest maths is wrong and you are right, and then to apparently forget that I referred to those texts. If you will not accept a mathematics answer (or answers from mathematicians) for a mathematical concept then I cannot think of anything else I can say to help you understand.

[Andrew D]

It is certainly true that you cannot help me understand something which you either cannot or will not explain.

This is a twenty-two page thread with a multitude of links. I do not recall a link discussing the precise claim at issue -- that in an infinite past, every possible thing absolutely must have happened -- but if you repost it, I will look at it.

Again, I have done this using some basic mathematical equations and references to mathematicians specialising in infinity.

If you mean that you have demonstrated, using some basic mathematical equations, the truth of your position on the precise claim at issue (and there are mathematicians who concur in your demonstration), I do not recall the alleged demonstration that you have in mind.

If you mean that you have used some basic mathematical equations to demonstrate something other than your position on the precise claim at issue, and there are mathematicians who agree with your position on the precise claim at issue, so what? That is nothing but an argument from authority: Mathematicians say it is so; therefore, it is so.

I have repeatedly identified what I am looking for: an explanation, in English, of why the existence of an infinite past means that everything which could have happened absolutely must have happened -- as distinct from almost surely has happened, a very different proposition which, in the present context, is of no significance. I have not seen such an explanation, from you or from anyone else except, perhaps, General Meade, whose latest posting on the subject I am still considering.

[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 D]

The correct answer to your question about the smallest of the three angles formed by two rays of sunlight and a book is, of course, some minute fraction of a degree. But it is not zero.

Of course it is not zero - I have never said it is.

But you did say that it is reasonable to treat that number as if it were zero, even though it is not:

Most people would, I suspect, be happy to say "well, call it two lots of 90 and 1 lot of zero degrees", which is what you will get if you assume, in this case, 93 million miles is infinity.

The precise claim at issue is that in an infinite past, everything that could have occurred absolutely must have occurred. You may "be happy to say" that that claim is true, but I am not. I will be happy to say that it is true only when it has been shown to be true.

But astronomers do not approximate that number to zero.

Of course they do not - you approximate large distances to infinity when it makes sense - not all the time.

People say "call it two lots of 90 and 1 lot of zero degrees" precisely because they recognize that it actually is not two lots of 90 and 1 lot of zero degrees
...
But in dealing with questions involving infinity, the distinction between almost surely and surely is significant, and I see nothing in what you have written that explains why we should ignore that important distinction.

Again, I have never said the answer is zero. But calculations can assume it is zero when convenient to do so - and assuming zero implicitly assumes the distance from the earth to the sun is infinity.

I can readily see how you find your equation of "very probably has happened" with "absolutely must have happened" to be "convenient". Whether it "makes sense," however, is an entirely different matter.

So explain: In the context of the precise claim at issue -- that in an infinite past, everything that could have occurred absolutely must have occurred -- why does it make sense to equate a very high probability with an absolute certainty.

You need to remember that you were accusing me of "reasoning about infinity by imagining a vast but finite number" - I have merely here given you an actual example of where that is a valid thing to do.

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.

Now - I accused you of the same (I believe I made the first accusation that you then invalidly (as just demonstrated) turned on its head) [actually, it has not been demonstrated] because you cannot get your head round the rather simple equation

60 X ∞ = ∞

If you treat infinity as infinity, it makes sense
If you treat infinity as a large number, then of course it does not make sense

This is where I believe you are missing the point.

No, you are missing the point. 60 X ∞ = ∞ does not make sense. 60 X ∞ = ∞ is babbling nonsense (unless one takes seriously the proposition that a minute is equal to an hour). What makes sense is this:

Sixty times one infinite number is another infinite number which is sixty times greater than the first infinite number.

I do not see why this is even problematic. We know that with finite numbers, 60 times y equals 60y. But when it comes to infinite numbers, magic occurs, and I am expected to accept, without the benefit of an explanation in English, a proposition whose logical outcomes are babbling nonsense. Why should I -- why would any rational person -- do that?

[thestoat]

But you did say that it is reasonable to treat that number as if it were zero, even though it is not:

It IS reasonable to treat it as zero. It is called an approximation. I thought (from a previous post) you understood that!

So explain: In the context of the precise claim at issue -- that in an infinite past, everything that could have occurred absolutely must have occurred -- why does it make sense to equate a very high probability with an absolute certainty.

I shall try - as per my last post, taking things 1 step at a time.

Whether it "makes sense," however, is an entirely different matter.

Whether it "makes sense to you," however, is an entirely different matter.

So explain: In the context of the precise claim at issue -- that in an infinite past, everything that could have occurred absolutely must have occurred -- why does it make sense to equate a very high probability with an absolute certainty.

It is only "a very high probability" if you treat infinity as a very large number. If you treat infinity as infinity then it becomes absolutely certain by the very definition of infinity. This is what infinity means. It is a mathematical concept to be able to produce results.

No, you are missing the point. 60 X ∞ = ∞ does not make sense. 60 X ∞ = ∞ is babbling nonsense

This is where you once again state that maths is wrong and you are right. Can you explain to me why you believe maths is wrong and you are right about its own concept? "Common sense" just doesn't cut it - to a mathematitian, 60 X ∞ = ∞ IS common sense, and it is their concept. I repeat, since historically you have not been good at answering questions (though very good at asking them - a lawyers skill?)

Can you explain to me why you believe maths is wrong and you are right about its own concept?

I do not see why this is even problematic. We know that with finite numbers, 60 times y equals 60y. But when it comes to infinite numbers, magic occurs, and I am expected to accept, without the benefit of an explanation in English, a proposition whose logical outcomes are babbling nonsense. Why should I -- why would any rational person -- do that?

I do actually vaguely see why this troubles you - if you treat infinity as a large number. It is a concept. It doesn't exist in the real world. How about I define infinite as

"some concept that, when multiplied by a real number does not change".

Does that help?