[jeffreyH]

Take a sphere of set radial dimension. Draw two lines from the center of the sphere to its surface separated by a set angle. We can describe the arc that joins them at the surface by a mathematical formula. Now make the radius infinite and carry out the same procedure.

[PmbPhy]

It seems that not all mathematicians agree with this.

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

Nope. You're wrong. I believe that in your haste to attempt to prove me wrong you didn't take the time to completely absorb the material and as such you didn't understand that they said that there's no number which equals infinity. But that's not what I argued, Bill. Was it?

“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;”

You misread it. They are not talking about infinity as most other mathematicians do. It's a VERY poor webpage. Look at their proof at: https://www.math.toronto.edu/mathnet/answers/infnotnumber.html

No "infinity" concept exists in the context of any number system, if by number system one means a collection of concepts that have operations like addition and multiplication the way familiar numbers do, operations which obey the usual properties of arithmetic.

One way to see this is to think, what would infinity minus 1 be? It couldn't be a finite number, since no finite number plus 1 equals infinity. So it must be infinite, and this would mean

infinity - 1 = infinity

From this one can immediately see that the rules of arithmetic must be violated, since if they held one could subtract infinity from both sides to conclude that -1 = 0, which isn't true.

Did you see what they're talking about? They mean

...in which "infinity" would mean something one can treat like a number.

which is what nobody thinks of as being infinite. That's merely ONE instance/version in which your article speaks of infinity which doesn't exist. However they also give to other examples in which it does.
In this argument they're proving that there is no such thing as a number which equals infinity which is not what everyone else means by infinity. They claim that they're talking about the "concept" of infinity but they aren't. This is a really bad argument and is written extremely poorly.

Bill - I'm sorry but I can't respond to anymore of your responses on infinity unless I know that you took the time to read a section in a calculus text about it. You'd end up reading less than you do in all the internet articles that you've read in the past few days and you wouldn't keep making these errors. But I'm taking a lot of time out of my internet time to respond to your comments to help you. But you haven't been appreciative enough to take my advice and read what I suggested. I feel as if you're being disrespectful to me when the fact is that I'm being as respectful to you as I possibly can by doing a lot of work trying to help you understand this and all you do is skirt around my one single suggestion which will help you. I beg you to understand my point of view. It is not my intention to come across as a jerk to you by any means. The fact is that I consider you one of my best internet friends and an intelligent man to boot. I'm merely expressing my dissatisfaction with you not taking my word on this. You've known me for years now, right Bill? As such you should know that I don't take such strong positions unless I'm absolutely certain that what I'm suggesting will be helpful more than anything you're going to read from me or anybody else on the internet. So please don't get angry with me my friend. Okay?

See the attachment which completely defines the concept of infinity

 [ Invalid Attachment ]

Do you see this, Bill? This is all that I was asking you to read all these days. Do you see how small it actually is? And it completely covers the concept and shows that "becomes infinite" is identical to "increases without bound" by definition.

[JohnDuffield]

It seems that not all mathematicians agree with this.

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

I beg to differ!

[dlorde]

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

I beg to differ!

[)] []

[phyti39]

Bill S:
1.  Is infinity a number?
No.
infinite: not limited, not measurable, not countable, without a specific value
By definition, infinity is not a number, but a state of indeterminacy.

2.  Is eternity a length of time?
No.
eternity: time without a beginning and an end
Its definition implies an infinite duration, and therefore cannot be measured.

3.  Is it possible to define Cantor’s “absolute infinity”?
No.
Cantor was a self appointed spokesmen who thought he could explain infinity, i.e. things without end, despite the fact that no human experience encounters it, thus  the mind has no ability to conceptualize it. If you study something to a greater degree than the masses, they will consider you an expert. This would require a separate topic.

4.  If there had ever been (absolutely) nothing, could there be something now?   
Yes.
There is something now, and something cannot come into existence without a cause/reason.

5.  Could there be change without time?
Yes.
Time does not cause changes. As far as known, energy does. Time is like an accounting system, recording events of interest to a standard (clock) event.
This was done long before philosophy was augmented with measurement to become science.
In the quantum world, particles change states, A to B, and B to A. The physicist observes both while his clock accumulates ticks. One transition is the reverse of the other, not one moving backward in time. Change of states for basic entities is  acceptable behavior, but a shattered glass reassembling itself off the floor, and moving to the top of the table would be suspicious. To maintain perspective, running the film backwards (in real time) is not equivalent to an actual reconstruction of a complex object.
Ponder this.
If time is a causal factor, why are quantum predictions expressed as probabilities?
For a 24 hr period, people enter a restaurant off the interstate, eat and leave. Their circumstances vary, so there is no causal chain of events, i.e. their visits are random. Time did not cause their visits. It's memory that allows the mind to make fictitious connections for a sequence of events.
While watching animations on the display, are there really 'moving pictures'?
While watching a plane fly toward the horizon, is it the plane or its image, getting smaller?

[Bill S (OP)]

Pete, please don’t think that I lack appreciation for the time and patience you and other experts expend trying to help “hitch-hikers” like me.  As the main carer for two disabled family members I have little time to spare, and I have to say that when I hind a few minutes I tend to spend it interacting with others on line rather than trying to find the specifics of calculus or complex numbers that might, or might not, help. 

Thanks for your attached explanation; it is perhaps the best I’ve seen to date.  If this is what we have been discussing, and coming close to falling out about, we have certainly been at cross-purposes.  As far as I understand it, in my mathematically challenged way, I have absolutely no problem with it.

Some time ago I downloaded a calculus course which has remained unstarted somewhere on my external HD; maybe I should find it and  have a go.  []
 

[Bill S (OP)]

Phyti39; you raise some interesting points, some of which I would like to return to when time permits.  In the meantime, this puzzles me:

4.  If there had ever been (absolutely) nothing, could there be something now?   
Yes.
There is something now, and something cannot come into existence without a cause/reason.

Shouldn't that have been "No"?

[Ethos_]

Ponder this.
If time is a causal factor, why are quantum predictions expressed as probabilities?

Time is an abstract mental picture we humans define as change in progress. Time does not cause change, change is the evidence that time has passed.

While watching animations on the display, are there really 'moving pictures'?

No, but photons are moving between the screen and your eyes. And pixels have also changed places and colors.

While watching a plane fly toward the horizon, is it the plane or its image, getting smaller?

This question is a bit silly. Hold your finger up very close to your eyes, observe it's apparent size then move it to arms length. If you'll notice, it now appears smaller compared to when it was very close to your eyes. Are you prepared to suggest that just because your finger now appears smaller than when very close to your eyes that is has somehow shrunken?

The divergence of light is the reason for the phenomenon of perspective, not because the object has reduced in size.

[PmbPhy]

As the main carer for two disabled family members I have little time to spare, ..

Yes. I know that and that's why I wanted you to trust me and read what I suggested to you what I did. I.e. because you would have spent a great deal of time less in reading and discussing the point otherwise then actually reading the very small amount on the definition. And when I say that it'd be less reading I mean that'd it take less than one minute!

You saw and read the attachment that I created and inserted into my last post. Thus you saw now how little there actually was to read, didn't you? The reason I asked you to take my advice and my word and read it was because I would have had to do less work creating that attachment because I couldn't easily type it out since limit notation is hard to do by typing. I told you many times how little where was to read and you kept thinking that it would take too long. Do you now understand how little there was to actually read? Will you please take my word for it next time?

..when I hind a few minutes I tend to spend it interacting with others on line rather than trying to find the specifics of calculus or complex numbers that might, or might not, help.

You mean to tell me that when I (or someone else) posts a URL to something like a Wikipedia article of a webpage of mine that you've never looked at it? If you did then why would you think that reading a paragraph or two in a text would be any different or require more work? I would never repeatedly suggest reading a portion of a text it wasn't actually a mere paragraph or two.  I just wish you would have trusted me on this.

Some time ago I downloaded a calculus course which has remained unstarted somewhere on my external HD; maybe I should find it and  have a go.  []

That'd be a great idea.

[PmbPhy]

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

I beg to differ!

[)] []

My sentiments exactly, dlorde.

[phyti39]

Ponder this.
If time is a causal factor, why are quantum predictions expressed as probabilities?

Time is an abstract mental picture we humans define as change in progress. Time does not cause change, change is the evidence that time has passed.

While watching animations on the display, are there really 'moving pictures'?

No, but photons are moving between the screen and your eyes. And pixels have also changed places and colors.

While watching a plane fly toward the horizon, is it the plane or its image, getting smaller?

This question is a bit silly. Hold your finger up very close to your eyes, observe it's apparent size then move it to arms length. If you'll notice, it now appears smaller compared to when it was very close to your eyes. Are you prepared to suggest that just because your finger now appears smaller than when very close to your eyes that is has somehow shrunken?

The divergence of light is the reason for the phenomenon of perspective, not because the object has reduced in size.

I think you get the point. Perception is not what it appears to be. Time does not cause change is my argument against "change cannot happen without time". People still cling to this idea that there is something invisible orchestrating the sequence of events. Maybe it's some form of security blanket in a world where things are so temporary, including life.
Nevertheless, the history of time as a concept and as applied science, shows it as an alias for distance. Look at a Minkowski diagram and notice the vertical axis is ct.

[phyti39]

Bill S #51

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

I hope you don't believe in fatalism.

[phyti39]

Phyti39; you raise some interesting points, some of which I would like to return to when time permits.  In the meantime, this puzzles me:

4.  If there had ever been (absolutely) nothing, could there be something now?   
Yes.
There is something now, and something cannot come into existence without a cause/reason.

Shouldn't that have been "No"?

There was nothing (a physical universe), then one came into existence. Since the elements (energy or matter) had no prior existence, they can't be used to bootstrap themselves into exixtence.

[dlorde]

There was nothing (a physical universe), then one came into existence. Since the elements (energy or matter) had no prior existence, they can't be used to bootstrap themselves into exixtence.

When you say the bolded above, do you mean there was a physical universe? because it seems to me a physical universe isn't nothing.

If you mean instead that there wasn't a physical universe (i.e. there wasn't anything at all, so no causes or reasons), then you seem to be contradicting yourself - if something cannot come into existence without a cause/reason (i.e. the elements (energy or matter)... can't be used to bootstrap themselves into existence), and there is something now, then it follows that there can't have been nothing (no cause or reason).

I'm puzzled...

[dlorde]

Time does not cause change is my argument against "change cannot happen without time".

It's not really an argument - "change cannot happen without time" doesn't imply that time causes change. It just says time is a necessary condition for change.

Having said that, there are good arguments for the inverse dependency, "there's no time without change", which does have more of a causal flavour...

[PmbPhy]

There was nothing (a physical universe), then one came into existence.

That's not known at this point. The Big Bang theory cannot be used to trace back to t = 0 but to only a short time after that. Therefore we cannot say what came before that time. There is a theory called the Pre-Big Bang theory which uses string theory to address some of those scenarios.

Since the elements (energy or matter) had no prior existence, they can't be used to bootstrap themselves into exixtence.

We don't know that either. There are particles which do "bootstrap themselves into existence". In fact many particles do that. There's a whole slew of them in particle physics which have been seen in the lab merely "popping into existence" from the inertial energy which is already there.

[Bill S (OP)]

There's a whole slew of them in particle physics which have been seen in the lab merely "popping into existence" from the inertial energy which is already there.

This is one of the fascinating things about science; scientists seem to be able to claim that the "inertial energy which is already there" is nothing, yet maintain that it can give rise to something. 

I'm willing to believe that I have missed something; but I wish someone would explain what it is. My suspicion is that "nothing" is used in different ways, with different meanings, in different contexts, and that the trouble comes when the contexts get crossed over.

[Bill S (OP)]

BTW. If anyone is interested in seeing how the discussion, starting with the same OP, developed on another forum, you might be interested in looking at SAGG.  Just for fun.

http://www.scienceagogo.com/forum/ubbthreads.php?ubb=showflat&Number=53488&page=1

[dlorde]

This is one of the fascinating things about science; scientists seem to be able to claim that the "inertial energy which is already there" is nothing, yet maintain that it can give rise to something. 

I'm willing to believe that I have missed something; but I wish someone would explain what it is. My suspicion is that "nothing" is used in different ways, with different meanings, in different contexts, and that the trouble comes when the contexts get crossed over.

Yes; the articles I've seen by Krauss, Carroll, and others usually clarify what they mean by 'nothing' - and it's generally not the conceptual 'absolutely nothing' we've been kicking around here; for example:

To a physicist, the first version of nothing of is simply empty space with nothing in it. You wouldn' t have any particles, all the radiation and so, there's literally nothing in it. But that nothing is actually quite complicated because of quantum mechanics and relativity. It turns out empty space is a boiling, bubbling brew of virtual particles, popping in and out of existence in a time so short that you can' t even measure them.

So clearly it is important to specify exactly what you mean; and, to your credit, you did, when you talked of 'absolutely nothing'; albeit in my view it isn't meaningfully applicable to the physical world.

[Bill S (OP)]

My favourite  quote from Lawrence Krauss is:

“By nothing, I do not mean nothing…..”

OK, it's a bit unfair, but it makes a good quote.