[Bill S (OP)]

No offence, but I'm not sure whether you're being deliberately obtuse, or you're unwilling or unable to articulate the point you want to make.

Earlier this evening when I was tending to the "calls of domestic duty" I was thinking that I would have apologise for appearing obtuse and repetitive.  You, and others, are probably well aware that I have been involved in discussions about infinity with various people in different threads and forums for quite some time.  I have found, from experience, that it pays to be absolutely sure what other people are saying before drawing conclusions. 

If that makes little sense, blame the single malt.   []

I'll come back to it tomorrow when I can think clearly.   

[alancalverd]

It's the smallest vector between two somethings.

In order to avoid being “something”, would it not have to be infinitely small?

Depends how far apart the somethings are. The definition holds for all values of any dimension.

[Bill S (OP)]

Depends how far apart the somethings are. The definition holds for all values of any dimension.

You’ve lost me there, Alan.  Could you give me an example in which (say) the somethings are 1m apart?

[alancalverd]

Okay, way out in deep space there are two hydrogen atoms a meter apart, with no other entities between them. Then we have a meter of nothing in at least one dimension, and quite possibly a lot more nothing elsewhere.

You might argue that photons and particles of stuff are passing through the gap in any observable part of the universe, but these transits are momentary, interspersed by periods of nothing. Which is just as well, since if there were something, the particles wouldn't be able to transit without interacting.

Which leads me to a whole new train of thought. I have just defined absolute nothing, but there is probably an infinity of relative nothings, defined as the absence of anything significant betwen two points. This is for instance a pleasant aspect of aviation, when I request a zone transit, say of a military range, on a specific course, and get "nothing known to affect": i.e. plenty of air, but no chunks of metal about to intersect my intended vector.

 

[dlorde]

Okay, way out in deep space there are two hydrogen atoms a meter apart, with no other entities between them. Then we have a meter of nothing in at least one dimension, and quite possibly a lot more nothing elsewhere.

You might argue that photons and particles of stuff are passing through the gap in any observable part of the universe, but these transits are momentary, interspersed by periods of nothing. Which is just as well, since if there were something, the particles wouldn't be able to transit without interacting.

As I understand it, quantum field theory says that the gap contains continuous fields, all of overall zero energy, except the Higgs field. If this isn't enough 'something', the zero-energy fields are only zero on average - uncertainty means that they oscillate, continually generating excitations, virtual pairs of particles that quickly annihilate. However one interprets these quantum oscillations, they too are something rather than nothing. I suspect it could be said that spacetime is these oscillating fields.

[Bill S (OP)]

Which leads me to a whole new train of thought. I have just defined absolute nothing, but there is probably an infinity of relative nothings, defined as the absence of anything significant betwen two points

Thanks for the explanation, Alan.  Dlorde is ahead of me with the sort of questions I would want to ask, so I'll sit back for a while to see what happens.   

[Bill S (OP)]

Yor_on, I’ve been giving some consideration to your post #4 and I have some questions.

 

Also the idea of many paths co-existing, as well as its opposite, the idea of a infinite

In what way do you consider the “many paths” to be the opposite of infinity?

1. is meaningless, to me that is,

Is it meaningless because you need a definition of infinity before you can decide if it is a number, or not?

2. Eternity? As ill defined as 'infinity' to me, doesn't tell me a thing. The 'instant in between' before that outcome, is that existing a 'eternity'?

Can you be equating an instant with eternity?  I’ve been down that road myself, I would be interested to know where it leads you.

Then you will get both your 'infinity', as well as 'eternity' on one side, the arrow on the other.

I always have infinity and eternity on the same side; definitely no arrow (of time) associated with either.  How could it be any other way?

[PmbPhy]

Which leads me to a whole new train of thought. I have just defined absolute nothing, but there is probably an infinity of relative nothings, defined as the absence of anything significant betwen two points

Thanks for the explanation, Alan.  Dlorde is ahead of me with the sort of questions I would want to ask, so I'll sit back for a while to see what happens.

It's stuff like this that made me tired of conversations about infinity. The concept is simply to someone who's taken their first year of calculus or taken a real analysis course.

[Bill S (OP)]

It's stuff like this that made me tired of conversations about infinity. The concept is simply to someone who's taken their first year of calculus or taken a real analysis course.

Perhaps those who find these things simple (I assume you meant simple) should post simple explanations for the benefit of lesser mortals, instead of bald statements that are presumably meant to be taken on faith.

For example, you have still not explained how you believe something can emerge from nothing. 

[JohnDuffield]

I'd like to hear about that one too Bill.

Okay, way out in deep space there are two hydrogen atoms a meter apart, with no other entities between them. Then we have a meter of nothing in at least one dimension, and quite possibly a lot more nothing elsewhere. You might argue that photons and particles of stuff are passing through the gap in any observable part of the universe, but these transits are momentary, interspersed by periods of nothing. Which is just as well, since if there were something, the particles wouldn't be able to transit without interacting.

Cough. Space isn't nothing. When a seismic wave moves through the ground, the ground waves. When an ocean wave moves through the sea, the sea waves. When a gravitational wave or an electromagnetic wave moves through space, space waves. And then you can perform pair production and make matter out of one of these waves. Don't think of matter as something and space as nothing, because matter and space aren't all that different. Have a read of Does Matter Differ from Vacuum? by Christoph Schiller.   



Which leads me to a whole new train of thought. I have just defined absolute nothing, but there is probably an infinity of relative nothings, defined as the absence of anything significant betwen two points. This is for instance a pleasant aspect of aviation, when I request a zone transit, say of a military range, on a specific course, and get "nothing known to affect": i.e. plenty of air, but no chunks of metal about to intersect my intended vector.
[/quote]

[Bill S (OP)]

I'd like to hear about that one too Bill.

Sorry John, I'm not sure what you want to hear more about.  Could be I'm having a "senior moment".  []

[Bill S (OP)]

Perhaps it is time to look at some of the things I might be able to take from this thread so far.

I am aware of the fact that scientific veracity does not depend on democratic vote, but “hitch-hikers” like me do well to seek and heed the opinions of the more knowledgeable.

I have often been asked why I think there can be no change, or meaningful division in infinity.  Answers like “Because it makes sense to me”, or “because infinity has a lot in common with Barbour’s Platonia”, are obviously not of the best, so I try appealing to such reasoning as “time and eternity are entirely different concepts”, and my answers always seem to be countered by Cantor-type responses which do little more than muddy the waters.

Let’s see if the responses to these questions help the situation.  The majority view seems to be:
Is infinity a number?      No.
Is eternity a length of time?     No.
Could there be change without time?     No.

A sequence is a number of things/concepts/etc.
Infinity is not a number; therefore it cannot be a sequence.
An “infinite sequence”, such as the natural numbers, is not truly infinite.  Boundless – yes; infinite – no.

Eternity is not a length of time. It is not time at all.  Without time there can be no change, therefore there can be no change in eternity/infinity.

Now I need to square that with the idea that the cosmos is infinite, and our Universe is “part” of the cosmos.

There are many scientists who believe the Universe is finite.  Many of those who opt for an infinite Universe see it as being infinite in the way that they see the sequence of numbers as being infinite.  It really makes little difference which one believes.

We exist in an infinite cosmos in which there is no change or differentiation.  Every “part” is the whole.  Nothing happens, everything just “is”.

We are conscious of time as being linear, and experience our Universe as passing through time; but that is because we are restricted to 3+1 dimensions, we cannot see or experience infinity, and in order to survive in this “illusion” must be able to make sense of our world.

I this science? Yes. 

If there had ever been nothing, there could be nothing now.  There must always have been something.  That something must be eternal/infinite, and therefore changeless.

Will we, as individuals ever experience/understand the infinite cosmos?  The answer to that is not science, it may be philosophy, theology or plain old guesswork, but not science.   

[PmbPhy]

An “infinite sequence”, such as the natural numbers, is not truly infinite.  Boundless – yes; infinite – no.

This is incorrect. There is such things as infinite sequences. Being an infinite sequence means that the value of the partial sequence has no bound. For example: the sequence

1, 2, 3, .... , etc.

1, 2², 3², .... etc.

are both infinite sequences. Each has no bound. The sequence

1, 2, 1, 2, 1, 2, ..., etc.

is an infinite sequence as well but this one is bounded.

Bill - Have you ever considered picking up a book on calculus and read the first couple of chapters? You can do it in a day or two and after that you'll have answers to all the questions you've asked over the last month or so. You'll do much much less work reading too.

[dlorde]

A sequence is a number of things/concepts/etc.
Infinity is not a number; therefore it cannot be a sequence.

That needs some explanation - it makes no sense to me as it stands. A sequence is an ordered list; there's no requirement I'm aware of that it must be finite. (it can also have zero elements)

An “infinite sequence”, such as the natural numbers, is not truly infinite.  Boundless – yes; infinite – no.

What is the justification for this? What is the difference between 'infinite' and 'truly infinite' ?

Eternity is not a length of time. It is not time at all.  Without time there can be no change, therefore there can be no change in eternity/infinity.

Eternity is defined as endless time, time without end, infinite time, time of indefinite or infinite extent, etc. So whatever it isn't, it is some sort of time; that's what the word means.

[PmbPhy]

That needs some explanation - it makes no sense to me as it stands.

Same here.

Bill - It's best to back up assertions like this when you make them, otherwise they're not of much use.

[JohnDuffield]

I'd like to hear about that one too Bill.

Sorry John, I'm not sure what you want to hear more about.  Could be I'm having a "senior moment".  []

How you get something from nothing. By the way, I was reading your post, and I spotted this:

"Without time there can be no change, therefore there can be no change in eternity/infinity."

Thar's your problem! You got it back to front. It should be without change there can be no time. Or motion of you prefer. A clock clocks up some kind of regular cyclical motion and shows you some cumulative display that you call the time. But it doesn't literally "measure the flow of time". It isn't some kind of cosmic gas meter.   

[Bill S (OP)]

This is incorrect. There is such things as infinite sequences. Being an infinite sequence means that the value of the partial sequence has no bound. For example: the sequence

1, 2, 3, .... , etc.

1, 22, 32, .... etc.

“…the value of the partial sequence has no bound.”

Precisely! It is unbounded, you could never establish that it was infinite, other than in principle.

In the first half of the 1980s when I started a science based degree course with the Open University, I dabbled in calculus.  Unfortunately, a major career change, and the learning curve that accompanied that prevented any further study, and 30+ years on I have suffered from the “use it or lose it” syndrome, but I recall nothing that would influence my thoughts about non-mathematical infinity.  Mathematical infinities, as I have said before, are not a matter of contention to me.

[PmbPhy]

“…the value of the partial sequence has no bound.”

Precisely! It is unbounded, you could never establish that it was infinite, other than in principle.

I was talking about the following statement

An “infinite sequence”, such as the natural numbers, is not truly infinite.  Boundless – yes; infinite – no.

and not whether someone could sit down in a lab with pen and paper and do it. Those are two entirely different things. You keep talking about boundless as being different than infinite when in actuality they are synonyms. To be exact - Something is infinite when it has no limit. When someone writes 1, 2, 3, 4, .... the "..." actually is defined to mean that it has no limit, that it keeps going and never ends. It is therefore infinite.

You really should study this again. It's not as if it'd be a waste of time for you because you'll spend less time talking about it here then actual study time. It's only a few pages long and doesn't require much math background at all because this is the foundation of math. Just a bit of friendly advice from your buddy Pete!

[Bill S (OP)]

There is such things as infinite sequences. Being an infinite sequence means that the value of the partial sequence has no bound. For example: the sequence

1, 2, 3, .... , etc.

1, 22, 32, .... etc.

are both infinite sequences.

It seems that not all mathematicians agree with this.

https://www.math.toronto.edu/mathnet/answers/infinity.html

“Number systems come in many sizes. There is the "natural number system", which is just the set of numbers used in counting: 1, 2, 3, 4, and so on. Or, one can expand this number system to include additional concepts, such as negative numbers, fractions, even the so-called "imaginary" numbers (which are not really imaginary at all). Each of these concepts exists provided we look for it in the context of a large enough number system.

Now the question is, does infinity exist in the same way that these concepts (negative numbers, fractions, etc.) do?
In other words, does there exist any number system which, as well as including the familiar numbers we are used to, also includes an "infinity" concept?

The answer is no;”

[dlorde]

It seems that not all mathematicians agree with this.

I suspect there's no field of human knowledge where everyone agrees about everything.