## Dimensionless points

What is the basis for reason? And mathematics?

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Philosophy Explorer
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### Dimensionless points

It's been argued before that dimensionless points can't physically exist to which I disagree. I want to add an additional argument.

Scientists say that empty space can't exist due to QM which spreads mass throughout the universe. With QM the dimensions can be 0 which is supported by science. Therefore you can have dimensionless points, objectively speaking.

Also, from Quora, it was asked whether dimensionless points make sense. This answer was posted:

"This is actually a pretty subtle question, mathematically. The word , as used in the question, only has meaning relative to some conceptual space, i.e. metric space, topological space, etc. Conceptually, dimension is defined using vectors, hence, is really only meaningful in . Vector spaces are defined over and every field has a metric defined on it (generally the ), hence, every vector space is a . In general metric spaces a “dimensionless point” does make sense; it is most often represented by the .

Some mathematicians may argue that every “point” is dimensionless but this is not really true. In the context of formal mathematical construction, “point” is a primitive term, which means it is left undefined and the definition is induced by the construction. So, technically speaking, the null vector is the only dimensionless point since every other point in the space needs at least one non-zero vector, hence, at least one dimension to define it. The dimension of each point in the space, then, is a function of its relation to the .

I believe this gets at the heart of your question "

PhilX
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### Re: Dimensionless points

Phil, It's not possible to argue against some random Quora thread that you didn't link, and that is ambiguous at best if not flat out wrong.

Of course it's not true that a point in the plane, say (a 2-vector) has dimension. Take the point (1,1). As a vector it has length sqrt(2). But as a point, it's just a dimensionless point sitting at the intersection of the lines y = 1 and x = 1.

You (or the Quora author) are equivocating the idea of the distance between (1,1) and the origin; and the point nature of (1,1) itself.

And nobody can argue with your anonymous Quora link since you quoted parts of it out of context and did not trouble yourself to provide the actual link.

One could spend their entire life quoting parts of random Quora links and spouting bullshit about them. Have you got a ... point?
Philosophy Explorer
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### Re: Dimensionless points

wtf wrote: Sun Jul 01, 2018 9:47 pm Phil, It's not possible to argue against some random Quora thread that you didn't link, and that is ambiguous at best if not flat out wrong.

Of course it's not true that a point in the plane, say (a 2-vector) has dimension. Take the point (1,1). As a vector it has length sqrt(2). But as a point, it's just a dimensionless point sitting at the intersection of the lines y = 1 and x = 1.

You (or the Quora author) are equivocating the idea of the distance between (1,1) and the origin; and the point nature of (1,1) itself.

And nobody can argue with your anonymous Quora link since you quoted parts of it out of context and did not trouble yourself to provide the actual link.

One could spend their entire life quoting parts of random Quora links and spouting bullshit about them. Have you got a ... point?
Here you're wrong in saying I quoted in parts (unless you're counting the author's name which is irrelevant here) as this link proves: https://www.quora.com/Is-a-dimensionles ... make-sense

QM says every point in space covers all of space, therefore every part (dimensionless point) in space must exist for every point in space to cover, otherwise you wouldn't have QM.

PhilX
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### Re: Dimensionless points

Philosophy Explorer wrote: Sun Jul 01, 2018 11:47 pm Here you're wrong in saying I quoted in parts (unless you're counting the author's name which is irrelevant here) as this link proves: https://www.quora.com/Is-a-dimensionles ... make-sense
Thank you for the link. I note that the author lists his interests as, "Yoga and Meditation in the Tibetan tradition." Fine as far as it goes, but why should we take this person as an authority on math and physics? Of course credentials or not, the author could redeem him or herself by speaking with authority and obvious technical knowledge on mathematical concepts. This I did not see. I saw lot of buzzword dropping of terms the writer clearly doesn't understand, along with various errors of simple logic.

I found at least five substantive material mathematical errors in the two-paragraph post, and one major logic error. How many can you find?
wtf
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### Re: Dimensionless points

Q: Is a dimensionless point possible? Does a “dimensionless point” make sense?

Wes Hansen, Yoga and Meditation in the Tibetan tradition
Answered Oct 14, 2017
Wes Hansen wrote: This is actually a pretty subtle question, mathematically.
I'd say it's a subtle question physically. Nobody knows if there are "points" in the real world. But mathematicians perfectly well accept the existence of individual points, dimensionless entities in space. We start by identifying the real number line with the conceptual idea of Euclid's line. That's a philosophical leap that can be challenged on various grounds. But in terms of standard, accepted mathematics, nobody disagrees. The number 3 is a dimensionless point on the number line. So is the number pi. A real number is the address of a point on the real line. That is the convention everyone agrees to, whether they've ever given the matter any thought or not.

In the plane, a point is the intersection of two lines. Or, it's a pair (x,y) where x is a real number and y is a real number. In Euclidean n-space a point is just an n-tuple of real numbers.

When you do math, this is a given.

So as I say, the physics is philosophically murky. But the math is perfectly clear. Dimensionless points exist, because we say they do. Every real number and every n-tuple of real numbers is a dimensionless point in some n-space.

Wes Hansen wrote: The word point, as used in the question, only has meaning relative to some conceptual space, i.e. metric space, topological space, etc.
Name-dropping of the worst sort. Now we can't determine this just from what he wrote. He might redeem himself by explaining why he name-checked those particular mathematical topics. However reading ahead, he is not going to redeem himself.

Wes Hansen wrote: Conceptually, dimension is defined using vectors, hence, is really only meaningful in vector spaces.
Well, yes and no. These days of course, dimension is understood in terms of the general theory of fractal dimension. It's true that Euclidean n-space has dimension n, but we have a more general explanation for that now, if you think of it that way.

Of course we do traditionally define dimension for linear spaces (aka vector spaces). That is true. But it's not true to say that dimension is "only meaningful" (OP's words) in vector spaces. So the OP is revealing not only his level of knowledge, but his Dunning-Krugeristic hubris of mansplaining to us how smart he is about math. Oooh, vector spaces. Ooooh, metric spaces. Now I wouldn't give him a hard time unless I'd been reading ahead and determined that this name-dropping of things he doesn't understand is only going to get worse.

Wes Hansen wrote: Vector spaces are defined over fields
Yes. This is the first mathematically accurate thing he's said. Let me explain briefly. A field in math is a system of numbers in which you can add, subtract, multiply, and divide (except by 0). Familiar examples are the real numbers and the rational numbers. Less familiar examples include the various finite fields such as the integers mod 5, which is the symbols {0, 1, 2, 4} with addition defined "mod 5," ie by only looking at remainders after dividing by 5.

Call the underlying field F.

A vector space consists essentially of the set of all n-tuples of elements of F. Well there are infinite-dimensional vector spaces but nevermind them. Examples are the real numbers as a 1-dimensional vector space over itself; the the plane consisting of all pairs (x,y) where x and y are real numbers, etc. This is just familiar stuff.

Wes Hansen wrote: and every field has a metric defined on it (generally the usual metric),
Now this is where I KNOW that the author has a very modest mathematical understanding, and is dropping buzzwords with little or no understanding.

I mentioned earlier that there are finite fields. Consider the integers mod 5. If you add any two you get another one, and the addition is commutative, and the multiplication distributes over the addition. That makes Z_5, as we'll call it, a ring.

It's also a field, because we can divide in Z_5. This is not immediately obvious. Let's look at some examples.

First, can we invert nonzero elements? That is, since 2 is an element of Z_2, what is 1/2? Now this is a really good question. What does the notation 1/2 mean, anyway? In the real numbers, 1/2 is the number that we multiply 1/2 by, to get 1. And the answer is 2, right?

So we use the same definition in Z_5. Is there an element of Z_5 we can multiply by 2 and get 1? Yes there is: It's 3. Because 2 x 3 = 1 in Z_5. We only care about the remainder mod 5.

If you work out all the other possibilities, you'll see that every nonzero element of Z_5 has a multiplicative inverse. So F_5 is a field.

HOWEVER!!!! Z_5 does not have a "usual" order, nor can it be ordered in any way consistent with its arithmetic.

In other words we could simply declare that 0 < 1 < 2 < ... < 4 by the usual everyday meaning of those symbols. But then what if I take 4 and add 2? I'll end up at 1.

So this is an example of a field that has no "natural" concept of order.

Now Z_5 is one of the first examples they'd show you when they defined fields, just to make sure you get a sense of the generality of the field concept.

So I suspect our revered OP is skimming Wikipedia and picking up terminology he does not understand.

I am not being unfair in any way. If you think vector spaces have a built-in metric structure just because the real numbers and Euclidean n-space do; that means you made it through a sophomore math class in linear algebra, but no more. Or you've been skimming Wikipedia. That's that, Phil.

However I shall soldier on.

Wes Hansen wrote: hence, every vector space is a metric space.
Ye Gods!! But this is just a consequence of his earlier error. He doesn't realize that fields are much more general than the Euclidean ones he knows about. How would you put a metric or topological structure on Z_5?

So I can't ADD obliquy for this remark alone. It is in fact a correct logical conaquence of his premises. I'll grant him that.

Wes Hansen wrote: In general metric spaces a “dimensionless point” does make sense; it is most often represented by the null vector.
Good God. This is really dumb. Now I'm going to assume that since he's been name-checking concepts in linear algebra, that he learned that dimension applies to a vector space as a whole, not the individual vectors. Individual vectors don't have a notion of dimension in linear algebra.

I'm downgrading my assessment. I don't think he took sophomore linear algebra. I think he took high school linear algebra. Or else skimmed the Wiki article on it.

Phil you have not been very discerning about this particular selection of source material.

It's true that the zero vector (NOT the "null vector," I've never heard that terminology used. Zero has a perfectly well defined meaning in a vector space) is itself a zero-dimensional subspace. The only one. Which is fine. But you can linearly translate it anywhere in the vector space to have a subset, namely the set containing some particular vector and nothing else; that, if translated to the origin, would be a subpace of dimension 1.

The OP simply doesn't know any math. Which is fine. But the OP is also making shit up as if he did know something about math. And that, I object to.

But Phil. Quora? That site has a few brilliant contributors but quite a lot of garbage.

Wes Hansen wrote: Some mathematicians may argue that every “point” is dimensionless but this is not really true.
Which mathematicians would argue otherwise? He's making this up. Nobody argues that a real number is a dimensionless point. It's baked into every aspect of modern mathematics. Even if it's wrong in some way -- it's UNIVERSALLY BELIEVED. So he's just making up this "controversy" in his own head.

Now if he says it's not "really" true, well then I perfectly well agree with him! Mathematics doesn't claim to be real. If the physicists find math useful, more the better. But math is done for the sake of math. When it comes to whether there are "really" dimensionless points in the physical world ... well personally I don't think there are, but it's something that nobody knows.

So we may have here a standard case of confusing math with physics. In math, points are dimensionless. In physics, who knows. Points could be quantum thingies or infinitesimal lumpy things or who the hell knows. Math isn't physics.

Wes Hansen wrote: In the context of formal mathematical construction, “point” is a primitive term, which means it is left undefined and the definition is induced by the construction.
By calling a point "undefined," our friend the OP is of course referring to Euclid's conception of point, line, and plane being undefined terms.

That's all well and good, but we're not in Greece 2300 years ago and we're also not in high school geometry.

Today, a point is an n-tuple of elements of a field in some vector space over that field. If the field is the real numbers, each n-tuple is geometrically zero-dimensional. If you turned it into a little vector space it would be the zero-vector. The OP thinks that WHERE a point is changes the nature of the point. That's not true.

There's a more general use of "point" in a function space, where a "point" of our space is some function on some other space. But that doesn't concern us now.

Wes Hansen wrote: So, technically speaking, the null vector is the only dimensionless point since every other point in the space needs at least one non-zero vector,
No it doesn't. I have no idea what he's saying. The point (0,0) in the plane is just the address of that point in the plane. The point (1,1) is some other point. As points they are identical.

It's true that (0,0) is special algebraically. It's the additive identity of the vector space. But its topological and geometric properties have nothing to do with that. (0,0) is just a geometric point in the plane like any other. If you renamed it (5,17) you'd just adjust all your equations by this "change of basis" and your laws of physics would come out the same and so would the math. A coordinate system is just a thing you impose on the plane. There are other coordinate systems. The nature of the underlying points doesn't change just because you rename them all.

Wes Hansen wrote: hence, at least one dimension to define it.
He's really confused. He thinks that the vector (1,1) needs to be defined in terms of (0,0) and that makes it "one dimensional." But he doesn't understand what dimension is. The plane, which is a vector space over the reals, is 2-dimensional. But linear algebra doesn't assign any notion of dimension to points at all. One point, or vector, is exactly like another. OP is just hallucinating this line of thought.

Wes Hansen wrote: The dimension of each point in the space, then, is a function of its relation to the basis.
By basis I assume he means origin. Else he's using the word basis wrong. "Each point in the space" DOES NOT HAVE A DIMENSION. It's the space itself that has a dimension. And he says the dimension of a point -- which is something that's not defined in linear algebra -- "is a function of is relation to the basis." This is just something he made up out of a half-heard lecture or Wiki page. The coordinates of a point with respect to a particular basis are indeed a function of the basis. But there's no concept of dimension. The dimension is just defined as the number of elements in some basis, after you prove the theorem that all bases have the same number of elements. (Otherwise the definition wouldn't make sense).

I think the OP did take college linear algebra -- but while he was on acid. He came away with a hallucinatory version of the subject in his head.

Wes Hansen wrote: I believe this gets at the heart of your question . . .
Well Phil, the question is why you posted this drek?
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### Re: Dimensionless points

wtf wrote: Mon Jul 02, 2018 2:45 am
Philosophy Explorer wrote: Sun Jul 01, 2018 11:47 pm Here you're wrong in saying I quoted in parts (unless you're counting the author's name which is irrelevant here) as this link proves: https://www.quora.com/Is-a-dimensionles ... make-sense
Thank you for the link. I note that the author lists his interests as, "Yoga and Meditation in the Tibetan tradition." Fine as far as it goes, but why should we take this person as an authority on math and physics? Of course credentials or not, the author could redeem him or herself by speaking with authority and obvious technical knowledge on mathematical concepts. This I did not see. I saw lot of buzzword dropping of terms the writer clearly doesn't understand, along with various errors of simple logic.

I found at least five substantive material mathematical errors in the two-paragraph post, and one major logic error. How many can you find?
I'm addressing some points you raised in this post. Just because someone believes in something (currently) known to be unscientific doesn't mean you should disregard everything he says. Should I disbelieve what Isaac Newton said in his theories about light and gravity due to his belief in alchemy? Should I disbelieve everything you have said because you incorrectly said I quoted in parts?

I have taken an introductory course in linear algebra so I know about vectors, but other terminology is too technical for me to follow. What I do know is that QM says that every particle in space occupies all of space, but there must be dimensionless points for those particles to go into. I also gave the example of a record player before. While not proof, it does suggest that dimensionless points do exist since theoretically they would form the center of record players.

PhilX
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### Re: Dimensionless points

Philosophy Explorer wrote: Mon Jul 02, 2018 5:25 amShould I disbelieve everything you have said because you incorrectly said I quoted in parts?
You seem unduly hung up on my having supposedly accused you of something. Surely if you posted some words from a Quora link but did not post the link, I'm entitled to wonder whether the quote is represented accurately in context. But if you feel I unjustly maligned you, I apologize. It seems an inconsequential point.

Philosophy Explorer wrote: Mon Jul 02, 2018 5:25 am I have taken an introductory course in linear algebra so I know about vectors,
Then you know that the concept of the dimension of an individual point is not defined or discussed in any way in linear algebra. This should have served as a clue to you that the OP doesn't know what he's talking about, making your choice of source material all that much more inexplicable. If you said you were IGNORANT of linear algebra that would make more sense! But if you learned that the dimension of a vector space is the number of vectors in any given basis, then you already know the OP is confused. So in theory that would send you looking for a better quote to reproduce. But it didn't.

Philosophy Explorer wrote: Mon Jul 02, 2018 5:25 am but other terminology is too technical for me to follow.
I can assure you that as someone who has studied fields and metric spaces and topological spaces and dimension, the OP is flat out making up everything he said. He's "not even wrong." He just has a mishmash of barely-understood memories of linear algebra and a penchant for bullshitting about stuff he doesn't understand. You don't need to have known that, but now you do. I gave the detailed descriptions of the technical terms so that any reader can in principle verify that what I say is right.

Philosophy Explorer wrote: Mon Jul 02, 2018 5:25 am What I do know is that QM says that every particle in space occupies all of space, but there must be dimensionless points for those particles to go into.
I did say I have no idea what the ultimate nature of physical space is, or even whether the question is meaningful. I'm not even arguing about whether there are dimensionless points. I'm wondering why you posted that nonsense from Quora as if it supports ANY point of view. It's so wrong on virtually everything that the OP has nothing sensible to say at all.

Philosophy Explorer wrote: Mon Jul 02, 2018 5:25 am I also gave the example of a record player before.
I didn't see that, but I'm not really arguing whether there are dimensionless points in the world. I'm simply noting that the Quora post you linked was garbage; and that mathematically, there is not the slightest doubt that there are dimensionless points in Euclidean space. It comes with the definition of points as n-tuples of real numbers.
Philosophy Explorer wrote: Mon Jul 02, 2018 5:25 am While not proof, it does suggest that dimensionless points do exist since theoretically they would form the center of record players.
Well if you haven't got a proof; and nobody else has a proof; then what I said is correct. Regarding the physics, nobody knows. Regarding math, there are lots of dimensionless points.

ps -- I wouldn't mind hearing about some of the evidence that dimensionless points exist physically. I'm just saying that this particular Quora post is a very poor starting point for any sensible discussion of anything.
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### Re: Dimensionless points

wtf wrote: Mon Jul 02, 2018 6:04 am
Philosophy Explorer wrote: Mon Jul 02, 2018 5:25 amShould I disbelieve everything you have said because you incorrectly said I quoted in parts?
You seem unduly hung up on my having supposedly accused you of something. Surely if you posted some words from a Quora link but did not post the link, I'm entitled to wonder whether the quote is represented accurately in context. But if you feel I unjustly maligned you, I apologize. It seems an inconsequential point.

Philosophy Explorer wrote: Mon Jul 02, 2018 5:25 am I have taken an introductory course in linear algebra so I know about vectors,
Then you know that the concept of the dimension of an individual point is not defined or discussed in any way in linear algebra. This should have served as a clue to you that the OP doesn't know what he's talking about, making your choice of source material all that much more inexplicable. If you said you were IGNORANT of linear algebra that would make more sense! But if you learned that the dimension of a vector space is the number of vectors in any given basis, then you already know the OP is confused. So in theory that would send you looking for a better quote to reproduce. But it didn't.

Philosophy Explorer wrote: Mon Jul 02, 2018 5:25 am but other terminology is too technical for me to follow.
I can assure you that as someone who has studied fields and metric spaces and topological spaces and dimension, the OP is flat out making up everything he said. He's "not even wrong." He just has a mishmash of barely-understood memories of linear algebra and a penchant for bullshitting about stuff he doesn't understand. You don't need to have known that, but now you do. I gave the detailed descriptions of the technical terms so that any reader can in principle verify that what I say is right.

Philosophy Explorer wrote: Mon Jul 02, 2018 5:25 am What I do know is that QM says that every particle in space occupies all of space, but there must be dimensionless points for those particles to go into.
I did say I have no idea what the ultimate nature of physical space is, or even whether the question is meaningful. I'm not even arguing about whether there are dimensionless points. I'm wondering why you posted that nonsense from Quora as if it supports ANY point of view. It's so wrong on virtually everything that the OP has nothing sensible to say at all.

Philosophy Explorer wrote: Mon Jul 02, 2018 5:25 am I also gave the example of a record player before.
I didn't see that, but I'm not really arguing whether there are dimensionless points in the world. I'm simply noting that the Quora post you linked was garbage; and that mathematically, there is not the slightest doubt that there are dimensionless points in Euclidean space. It comes with the definition of points as n-tuples of real numbers.
Philosophy Explorer wrote: Mon Jul 02, 2018 5:25 am While not proof, it does suggest that dimensionless points do exist since theoretically they would form the center of record players.
Well if you haven't got a proof; and nobody else has a proof; then what I said is correct. Regarding the physics, nobody knows. Regarding math, there are lots of dimensionless points.

ps -- I wouldn't mind hearing about some of the evidence that dimensionless points exist. I'm just saying that this particular Quora post is a very poor starting point for any sensible discussion of anything.
Clearly the QM statement I made is evidence which is supported by physicists that dimensionless points do exist, otherwise we wouldn't have QM (which someone has said is one of physics strongest theories).

PhilX
wtf
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### Re: Dimensionless points

Philosophy Explorer wrote: Mon Jul 02, 2018 6:31 am Clearly the QM statement I made is evidence which is supported by physicists that dimensionless points do exist, otherwise we wouldn't have QM (which someone has said is one of physics strongest theories).
So we agree that there's evidence but no proof. I can live with that, especially since I don't know much physics.

I tend to doubt that any physical theory is "true" in the sense of being exactly correct about physical reality. Every scientific theory is a mathematical model and one should not confuse the map with the territory. But I don't know enough about the specifics of QM to know whether it predicts dimensionless points. IMO if it does that's an argument against QM; or at least against the idea that QM won't someday be generalized or modified just as relativity extended and generalized Newtonian gravity.
Arising_uk
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### Re: Dimensionless points

Fucks sake! There are no 'dimensionless points' in reality, if by this you mean a physical 'thing', as its a contradiction of terms!!
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### Re: Dimensionless points

Arising_uk wrote: Mon Jul 02, 2018 10:54 am Fucks sake! There are no 'dimensionless points' in reality, if by this you mean a physical 'thing', as its a contradiction of terms!!
Prove it. You already claimed you can't see it, but not seeing it is the same as not existing.

PhilX
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### Re: Dimensionless points

Philosophy Explorer wrote: Prove it. You already claimed you can't see it, but not seeing it is the same as not existing.

PhilX
I do not say it's about 'seeing' it, I say it is about logical contradiction which makes a statement always false. So show me a square circle? A round cube? A hexagonal triangle? A married bachelor? Etc, etc.
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### Re: Dimensionless points

Arising_uk wrote: Mon Jul 02, 2018 11:13 am
Philosophy Explorer wrote: Prove it. You already claimed you can't see it, but not seeing it is the same as not existing.

PhilX
I do not say it's about 'seeing' it, I say it is about logical contradiction which makes a statement always false. So show me a square circle? A round cube? A hexagonal triangle? A married bachelor? Etc, etc.
I already logically proved it when I talked about QM earlier which says that particles occupy all of space. I'll be awaiting your response before I respond further.

PhilX
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### Re: Dimensionless points

Philosophy Explorer wrote: Mon Jul 02, 2018 11:20 am I already logically proved it when I talked about QM earlier which says that particles occupy all of space.
Didn't you say earlier that you haven't got proof? That you have "evidence" in the form of a claim (unsourced, unlinked, unsupported) that QM implies dimensionless points? How did you get from admitting you haven't got proof to claiming you do?

ps here it is ...
Philosophy Explorer wrote: Mon Jul 02, 2018 5:25 amWhile not proof, it does suggest that dimensionless points do exist since theoretically they would form the center of record players.
How did you get from "while not a proof" to claiming you have proof? And center of record players? What if God has an iPod? I don't see anything here about record players. WTF are you talking about Phil?

pps -- Do you have a cat? Sometimes people's cats type things when their humans aren't looking. If you left your computer on and turned your back, your cat might have typed claims your posts don't support.
Arising_uk
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### Re: Dimensionless points

Philosophy Explorer wrote: I already logically proved it when I talked about QM earlier which says that particles occupy all of space. I'll be awaiting your response before I respond further.

PhilX
If 'particles' occupy all of Space then there is no 'Space' and if they do what need 'dimensionless points'?