The whole is limitless

Reply to MoK
If the whole is limitless it has no bounds. Then you introduce bounds and impose limits. What?
Maybe I can get it on a second try but why?

Nope, not getting it. W1 is no longer the whole you started with.

You are proving an unlimited whole is not affected by bounds....okay.

Quoting Mark Nyquist

If the whole is limitless it has no bounds. Then you introduce bounds and impose limits. What?
Maybe I can get it on a second try but why?

Nope, not getting it. W1 is no longer the whole you started with.

What I am trying to show is that there are two cases where what you consider as the whole is either limited or limitless. If [math]W_1[/math] is limitless we reach our goal otherwise [math]W_1[/math] is limited and it is bounded by something else, let's call it [math]B_1[/math]. Etc.

Reply to MoK
Okay. Worth exploring. It takes time to understand it.

Reply to MoK
The problem I'm seeing with your approach is that you don't identify the whole as a concept. Its origin is a mental abstraction. Limits are mental definitions. If you apply limits to infinity you no longer have infinity.

I covered this in my Universal Form post not long ago if you want to understand why I object to your method.

Quoting MoK

To show this let's assume that the whole is limited, let's call the whole W1. This means that W1 is bounded by something else, let's call this B1.

This doesn't follow, unless by "limited" you mean just that: being limited by something else. But that would make your argument a simple and uninteresting tautology.

Quoting Mark Nyquist

The problem I'm seeing with your approach is that you don't identify the whole as a concept. Its origin is a mental abstraction.

By the whole, I mean whatever that exists, spacetime, material, etc.

Quoting Mark Nyquist

Limits are mental definitions.

No, it can be physical as well.

Quoting Mark Nyquist

If you apply limits to infinity you no longer have infinity.

Correct. But what is the implication of this to my argument?

Quoting Mark Nyquist

I covered this in my Universal Form post not long ago if you want to understand why I object to your method.

Could I have a link to your post?

Quoting SophistiCat

This doesn't follow, unless by "limited" you mean just that: being limited by something else.

By limited I mean restricted in size. Think of spacetime for example. If spacetime is restricted in size then we can reach its edges by moving in straight lines (of course if spacetime is not a closed manifold). The problem is what is beyond the edges. It cannot be nothing since nothing does not have any geometry and occupies no room. So, whatever is the beyond edges of spacetime is something. Therefore, what I said follows.

Reply to MoK
I'm not good at links on my smart phone but go to the main page here and it is listed as:

Is Universal Form a good tool?

The last entry as of now.

Reply to Mark Nyquist
Ok, I found the thread. I am however confused with what you are trying to say. I will read it more and post my opinion in the thread if I understand something.

Quoting MoK

By limited I mean restricted in size. Think of spacetime for example. If spacetime is restricted in size then we can reach its edges by moving in straight lines (of course if spacetime is not a closed manifold). The problem is what is beyond the edges. It cannot be nothing since nothing does not have any geometry and occupies no room. So, whatever is the beyond edges of spacetime is something. Therefore, what I said follows.

OK, so you are basically reprising Lucretius' javelin argument:

For whatever bounds it, that thing must itself be bounded likewise; and to this bounding thing there must be a bound again, and so on for ever and ever throughout all immensity. Suppose, however, for a moment, all existing space to be bounded, and that a man runs forward to the uttermost borders, and stands upon the last verge of things, and then hurls forward a winged javelin,— suppose you that the dart, when hurled by the vivid force, shall take its way to the point the darter aimed at, or that something will take its stand in the path of its flight, and arrest it? For one or other of these things must happen. There is a dilemma here that you never can escape from.

Reply to SophistiCat
Oh, I didn't know about him and his argument. I am glad you mentioned it.

Quoting MoK

By limited I mean restricted in size. Think of spacetime for example. If spacetime is restricted in size then we can reach its edges by moving in straight lines (of course if spacetime is not a closed manifold). The problem is what is beyond the edges. It cannot be nothing since nothing does not have any geometry and occupies no room. So, whatever is the beyond edges of spacetime is something. Therefore, what I said follows.

What if space is closed? As in, a loop where going in a certain direction for enough time sends you back to where you started. The world would then not be limitless, but still unbounded.

Quoting https://www.astronomy.com/science/what-shape-is-the-universe/

A mind-boggling property of this universe is that it is finite, yet it has no bounds.

Edit: You might reply that it is limitless because it does not have an edge. But then your original argument is that there is a W=W1+B1+B2+ that goes to infinity. That would be untrue if the universe is closed, as you don't need to add anything for it to be limited. Yet, if you redefine limitless to mean edgeless, your argument becomes either tautological (the universe with edges is surrounded by something) or a discussion on nothingness (the universe with edges is surrounded by nothing).

Quoting Lionino

What if space is closed? As in, a loop where going in a certain direction for enough time sends you back to where you started. The world would then not be limitless, but still unbounded.

A mind-boggling property of this universe is that it is finite, yet it has no bounds.
— https://www.astronomy.com/science/what-shape-is-the-universe/

I think if space is closed then it is embedded in a hyperspace.

Reply to MoK I don't know what hyperspace is, neither how closedness of the universe implies one.

Quoting Lionino

I don't know what hyperspace is, neither how closedness of the universe implies one.

By hyperspace I mean a space of more than three dimensions. Why does the closedness of the space imply a hyperspace? Because any closed manifold has a local curvature. To help the imagination think of a closed 2D space, a sphere for example, instead of a 3D one. Each point on this sphere has a curvature at any given point which means that the surface of the sphere bends at each location on the sphere. The fact that the sphere bends at each location on its surface requires a higher dimension space, in this case minimally 3D space, where the sphere is embedded within otherwise the sphere cannot bend at each location on its surface and we cannot have a curvature.

Reply to MoK What you are saying makes sense, however:
Quoting Does space curvature automatically imply extra dimensions?

No, general relativity is based on something called "intrinsic curvature", which is related to how much parallel lines deviate towards or away from each other. It doesn't require embedding space-time in a higher dimensional structure to work.

Also

Nope, spacetime curvature says nothing about the dimensionality. Your intuition here is probably wrong because human imagination needs 'some dimension to bend into' in order for something to be curved (i.e. an embedding in a higher-dimensional space). This is just our lack of imagination showing, though.

Quoting Lionino

What you are saying makes sense, however:
No, general relativity is based on something called "intrinsic curvature", which is related to how much parallel lines deviate towards or away from each other. It doesn't require embedding space-time in a higher dimensional structure to work.
— Does space curvature automatically imply extra dimensions?

Here I am not talking about intrinsic curvature in spacetime that is caused by a massive object locally but extrinsic curvature which tells us what is the global geometry of space.

Quoting MoK

Here I am not talking about intrinsic curvature in spacetime that is caused by a massive object locally but extrinsic curvature which tells us what is the global geometry of space.

The closedness of spacetime does not require extrinsic curvature. If anything, your argument would require infinite dimensions, as each time you evaluate the extrinsic curvature of a dimension another one would be in order.

Quoting Lionino

The closedness of spacetime does not require extrinsic curvature.

It requires as I illustrated.

Quoting Lionino

If anything, your argument would require infinite dimensions, as each time you evaluate the extrinsic curvature of a dimension another one would be in order.

Not all hyperspaces that I am talking about are necessarily closed so we could deal with finite hyperspace dimensions which accommodate everything. You are however right that we need infinite dimensions if all hyperspaces are closed. I don't see any problem with hyperspace which has infinite dimensions though.

Quoting MoK

It requires as I illustrated.

Where did you show that a closed universe requires extrinsic curvature?

Quoting MoK

Not all hyperspaces that I am talking about are necessarily closed so we could deal with finite hyperspace dimensions which accommodate everything. You are however right that we need infinite dimensions if all hyperspaces are closed.

Whether something is closed refers to intrinsic curvature.

Quoting MoK

I don't see any problem with hyperspace which has infinite dimensions though.

It is a physically-unfounded belief about the empirical world arrived at using a priori syllogisms in a natural language.

Reply to MoK Seems to me you have reinvented the universal set. As such what you have shown is that there is nothing outside of W, not that W is limitless.

This user has been deleted and all their posts removed.

Reply to MoK
I'm still having trouble understanding what you mean by the whole. Is it a philosophy term? As the whole is the sum of its parts, the whole universe in the physical sense, mathematics or what I thought at first, a concept of the whole being something limitless or infinite.

I can make some progress on your argument but then the conversation goes in another direction such as the physical universe which was never stated.

I'm thinking if it's the physical universe we can't impose our own mathematical model on it without knowing what it is.

Quoting Lionino

Where did you show that a closed universe requires extrinsic curvature?

Here. Moreover, considering that you return to the point you started (if you move on a straight line closed manifold) requires extrinsic curvature. How could you return to the point you started if the global geometry of space was flat? You get a square if you cut a balloon and put it on a flat surface. You reach a dead end if you move in a straight line on the square. So a square is not a proper representation of a sphere.

Quoting Lionino

Whether something is closed refers to intrinsic curvature.

Actually, after some thought, I realized that even intrinsic curvature also requires a higher dimension. How the lines could deviate towards or away from each other if the geometry of space is flat. This figure is very illustrative:

https://en.wikipedia.org/wiki/File:Cassini-science-br.jpg

Quoting Lionino

It is a physically-unfounded belief about the empirical world arrived at using a priori syllogisms in a natural language.

It is physically necessary if it is logically necessary. Moreover, we don't know what is the right curvature of spacetime. Spacetime could simply be limitless if its geometry is flat globally.

Quoting Banno

Seems to me you have reinvented the universal set. As such what you have shown is that there is nothing outside of W, not that W is limitless.

But W could not be limited as I argued.

Quoting Mark Nyquist

I'm still having trouble understanding what you mean by the whole. Is it a philosophy term? As the whole is the sum of its parts, the whole universe in the physical sense, mathematics or what I thought at first, a concept of the whole being something limitless or infinite.

I can make some progress on your argument but then the conversation goes in another direction such as the physical universe which was never stated.

I'm thinking if it's the physical universe we can't impose our own mathematical model on it without knowing what it is.

By whole I mean whatever that physically exist.

Reply to MoK Hello again MoK!

Let me take your abstract into a thought example for a minute. Lets say that in the universe, only a single grain of sand exists. Now we claim that is the whole, but what is the definition of the whole? Usually 'the whole' is seen as 'everything'. But then you add in something outside of the whole as binding the whole. I'm confused here. What is outside of the grain of sand that is binding the sand?
It would seem that the bind to me is the internal limitation of the sand's matter.

But let me explore your other line of thinking and be charitable where possible. Lets say that the grain of sand is actually bound by 'nothing'. You then note that this binding plus the original whole creates a secondary whole. This doesn't quite work in your variable setup, as W1 and W2 are clearly different concepts here. While a whole indicates 'totality', these are obviously different totalities. So how do I see fixing this?

Perhaps what would make more sense is that some 'thing' is bounded and has limitations where there is 'nothing'. 'Nothing' may bind 'something', but 'nothing' has no limits. Is that more along the line of what you were thinking of?

Quoting MoK

Here.

And that argument was already refuted.

Quoting MoK

How the lines could deviate towards or away from each other if the geometry of space is flat.

The counter-example I am giving is exactly where space is not flat, where it has positive curvature.

Quoting MoK

Actually, after some thought, I realized that even intrinsic curvature also requires a higher dimension.

It doesn't.

No, general relativity is based on something called "intrinsic curvature", which is related to how much parallel lines deviate towards or away from each other. It doesn't require embedding space-time in a higher dimensional structure to work.

Quoting physicsforums

In summary, it is important to distinguish between extrinsic curvature, which involves bending through an additional dimension, and intrinsic curvature, which is directly visible on a surface without reference to an extra dimension.

For example, if you draw a triangle on the surface of the paper, the sum of the interior angles of the triangle will be 180 degrees. When you bend the paper or even roll it up into a cylinder nothing will change and the angles will still add to 180.

In order to have intrinsic curvature, you have to look at a manifold with at least two dimensions--for example, a 2-sphere [aka circle]. Then your question can be rephrased as: how is it possible to tell that a 2-sphere is curved, without making any use of an embedding of it into a space with more than 2 dimensions? The answer to that is, by looking at geodesic deviation, which can be measured purely within the surface.

Quoting MoK

It is physically necessary if it is logically necessary.

Until physics tells us that our human-made logic is not absolute and that we may have to reframe, as modern physics may make us do.

Quoting MoK

Spacetime could simply be limitless if its geometry is flat globally.

It could be, some scientists even believe that. My counterexample is to show that your idea is not logically necessary by showing the possibility of the contrary (?¬p ? ¬?p).

Quoting MoK

What I am trying to show is that there are two cases where what you consider as the whole is either limited or limitless

And therein is your flaw. Considering the whole to be limited is simply a mistake in logic. And we already knew that.

Quoting Philosophim

Hello again MoK!

Hello Philosophim!

Quoting Philosophim

Let me take your abstract into a thought example for a minute. Lets say that in the universe, only a single grain of sand exists. Now we claim that is the whole, but what is the definition of the whole? Usually 'the whole' is seen as 'everything'. But then you add in something outside of the whole as binding the whole. I'm confused here. What is outside of the grain of sand that is binding the sand?
It would seem that the bind to me is the internal limitation of the sand's matter.

First, you need a space as large as the size of the sand to embed the sand within. Now, the question of what is outside of the space is valid.

Quoting Philosophim

But let me explore your other line of thinking and be charitable where possible. Lets say that the grain of sand is actually bound by 'nothing'. You then note that this binding plus the original whole creates a secondary whole. This doesn't quite work in your variable setup, as W1 and W2 are clearly different concepts here. While a whole indicates 'totality', these are obviously different totalities. So how do I see fixing this?

Perhaps what would make more sense is that some 'thing' is bounded and has limitations where there is 'nothing'. 'Nothing' may bind 'something', but 'nothing' has no limits. Is that more along the line of what you were thinking of?

Nothing cannot have any geometry or occupy any room so nothing cannot bind the space that the sand is within.

Quoting Lionino

And that argument was already refuted.

Where? I already argue that a flat manifold with a boundary cannot represent a closed manifold.

Quoting Lionino

The counter-example I am giving is exactly where space is not flat, where it has positive curvature.

How the lines could deviate towards or away from each other if the space does not bend? Here is the figure that illustrates what I am trying to say:

https://en.wikipedia.org/wiki/File:Cassini-science-br.jpg

Quoting Lionino

It doesn't.

No, general relativity is based on something called "intrinsic curvature", which is related to how much parallel lines deviate towards or away from each other. It doesn't require embedding space-time in a higher dimensional structure to work.

In summary, it is important to distinguish between extrinsic curvature, which involves bending through an additional dimension, and intrinsic curvature, which is directly visible on a surface without reference to an extra dimension.

He is wrong. Please see the above figure.

Quoting Lionino

Until physics tells us that our human-made logic is not absolute and that we may have to reframe, as modern physics may make us do.

I don't think that time ever comes.

Quoting Lionino

"(?¬p ? ¬?p)".

What does that mean?

Quoting Arne

And therein is your flaw. Considering the whole to be limited is simply a mistake in logic. And we already knew that.

Lionino does not think so. He thinks that the whole can be limited. Please see my discussion with him.

Reply to MoK Exactly. The flaw in the notion of a limited whole is our obsession with "thingness". By definition, a limit to the whole cannot be a "thing" or it would be included in the whole.

If there is a whole, then it includes all. If it does not include all, then it is not the whole. Beyond that is philosophy as industry.

Quoting Arne

Exactly. The flaw in the notion of a limited whole is our obsession with "thingness. By definition, a limit to the whole cannot be a "thing" or it would be included in the whole.

If there is a whole, then it includes all. If it does not include all, then it is not the whole. Beyond that is philosophy as industry.

That was my reply to Philosophim. Lionino thinks that the whole could be a closed manifold. I am trying to show that any closed manifold is embedded in hyperspace. He does not agree so the discussion is ongoing.

By the way, glad to see that you agree that the whole is limitless.

Quoting MoK

glad to see that you agree that the whole is limitless.

I state unequivocally that the whole cannot be limited by "thingness." How you interpret my statement is on you.

Quoting MoK

I don't think that time ever comes.

how newtonian of you. :-)

Quoting Arne

I state unequivocally that the whole cannot be limited by "thingness." How you interpret my statement is on you.

I didn't say that your statement is on me. I mean, we both conclude that the whole is limitless so we agree on the conclusion.

Quoting Arne

how newtonian of you. :-)

Actually, I am very open to changing my mind if I am shown to be wrong. :wink:

Quoting MoK

I didn't say that your statement is on me. I mean, we both conclude that the whole is limitless

I agree that the whole includes "all". I neither agree nor disagree that the whole is "limitless."

I suspect "all" and "limitless" have different implications regarding ideas such as finite/infinite.

Similarly, perhaps the whole is limited by time.

Quoting MoK

Actually, I am very open to changing my mind if I am shown to be wrong. :wink:

I have no doubt.

I was joking.

Quoting Arne

I agree that the whole includes "all". I neither agree nor disagree that the whole is "limitless."

Well, if by "all" you mean "everything that exists" then following my OP I can show that the whole is limitless.

Quoting Arne

I suspect "all" and "limitless" have different implications regarding ideas such as finite/infinite.

I agree.

Quoting Arne

Similarly, perhaps the whole is limited by time.

What do you mean?

Quoting Arne

I have no doubt.

I was joking.

I know. :wink:

Quoting MoK

Similarly, perhaps the whole is limited by time.
— Arne
What do you mean?

By saying "perhaps the whole is limited by time" I mean perhaps the whole is limited by time. It is an idea that emerged shortly before I said it and I suspect I am not the first person to consider something to that effect. I have not thought it through to the point of making it a proposition. Thus the word "perhaps."

Quoting Arne

By saying "perhaps the whole is limited by time" I mean perhaps the whole is limited by time. It is an idea that emerged shortly before I said it and I suspect I am not the first person to consider something to that effect. I have not thought it through to the point of making it a proposition. Thus the word "perhaps."

What do you mean by "limited" in this case?

Quoting MoK

What do you mean by "limited" in this case?

No time, no whole.

Quoting Arne

Nonetheless, my primary area of philosophical interest is ontology (the nature of being) and my primary ontological disposition is Heideggerian.

Cool. I didn't know about him. I am not a philosopher by education. Good to know about him because I have something extra to read.

Quoting MoK

Good to know about him because I have something extra to read.

Being and Time is his most noted work. I highly recommend.

Quoting MoK

This means that W1 is bounded by something else

What do you mean by "bounded by"?
It normally means having something as its edge or simply an edge around something. What edge do you have in mind? And why that edge is part of W1, in a way that W1 is actually W1 + edge (B1)?

All this is too abstract. In such cases it is always recommended to give some example(s).

Maybe you can use this: Water in a glass. W1 is the whole (quantity of) water. And we have two kinds of "edges" or boundaries: the glass --around and at the bottom of the water (B1)-- and the air above the water (B2). According to your argument, W1 is actually W1+B1+B2. Right? Can this be considered a valid case?

Quoting MoK

First, you need a space as large as the size of the sand to embed the sand within. Now, the question of what is outside of the space is valid.

Ok, I see. So if I have your idea right, you believe that space is a thing. If this is the case, and space is an actual thing, then just replace my example of 'a grain of sand' with 'a section of space'. Once again, wouldn't the bounds of space be the internal limitations of space itself? I agree with you that nothing cannot bind space, but if space is limited, how is it bound by something outside of itself then? How is the limitation of space not bound by its own internal volume?

Quoting Arne

Being and Time is his most noted work. I highly recommend.

Thank you very much for the reference to the book.

Quoting MoK

He is wrong. Please see the above figure.

The figure is a 2d representation of space-time distortion. These several people I quoted are not wrong about physics, you are.

Quoting https://physics.stackexchange.com/questions/547140/what-is-intrinsic-curvature

So how do you discover intrinsic geometry empirically? You measure angles, you measure dot products and you see what the values are. If those values are what you'd get with flat space, you're in a flat space. If they're what you'd get in curved space, well, you're in a curved space. You can consider this the definition of a curved space. You don't have to envision space bending into some other space. Just that in our space, we measure dot products of basis vectors to have some non-zero value.

We call it "curvature" because it works exactly like curvature. Angles and distances measured are exactly what they would be if the space was curved. We don't assume an embedding space because we don't need to to get the right answers. So why add something to the theory that cannot be observed?

Unless you can observe the embedding space, then no, you cannot deduce that you exist embedded in a higher space. That's an assumption that cannot be tested.

If your reply does not address these quotes directly, I will move on.

Quoting MoK

I don't think that time ever comes.

Yeah, because you did not fully read the first answer in the link. I won't quote it, it is right there in the beginning.

Quoting Alkis Piskas

What do you mean by "bounded by"?

To explain that I need to explain what I mean by limited. By limited I mean restricted in size. By "bounded by" then I mean that there exists something that surrounds the limited thing.

Quoting Alkis Piskas

It normally means having something as its edge or simply an edge around something. What edge do you have in mind? And why that edge is part of W1, in a way that W1 is actually W1 + edge (B1)?

Think of a ball in a room for example. The ball is restricted in size so the room surrounds it. So in this case the ball is [math]W_1[/math] and the room is [math]B_1[/math].

Quoting Alkis Piskas

All this is too abstract. In such cases it is always recommended to give some example(s).

OK, I try my best to give good examples.

Quoting Alkis Piskas

Maybe you can use this: Water in a glass. W1 is the whole (quantity of) water. And we have two kinds of "edges" or boundaries: the glass --around and at the bottom of the water (B1)-- and the air above the water (B2). According to your argument, W1 is actually W1+B1+B2. Right? Can this be considered a valid case?

No, [math]B_2[/math] is what surrounds [math]W_2[/math] where [math]W_2=W_1+B1[/math]. In the example of the ball, [math]W_1[/math] is the ball, and [math]B_1[/math] is the room. [math]B_2[/math] is what surrounds the room, the rest of the building for example. So, [math]W_3[/math]=[math]W_1[/math]+[math]B_1[/math]+[math]B_2[/math]. [math]B_3[/math] then is for example the city which surrounds the ball [math]W_1[/math], the room [math]B_1[/math], and the building [math]B_2[/math]. So, [math]W_4[/math]=[math]W_1[/math]+[math]B_1[/math]+[math]B_2[/math]+[math]B_3[/math]. Etc. This chain as you can see is ongoing unless there exists [math]B_i[/math] such that [math]B_i[/math] is limitless. Either way, the chain where all [math]B_i[/math]s are limited is limitless because there is no end for the chain or there exists a [math]B_i[/math] that is limitless which makes the whole limitless.

I hope things are clear now.

Quoting Philosophim

Ok, I see. So if I have your idea right, you believe that space is a thing.

If by "space is a thing" you mean that space is a substance then that is still the subject of debate. If by space you mean a continuous area that is unoccupied then we are into business.

Quoting Philosophim

If this is the case, and space is an actual thing, then just replace my example of 'a grain of sand' with 'a section of space'.

Cool, that works.

Quoting Philosophim

Once again, wouldn't the bounds of space be the internal limitations of space itself?

Space in principle could be limitless. A section of it is however limited.

Quoting Philosophim

I agree with you that nothing cannot bind space, but if space is limited, how is it bound by something outside of itself then?

Space in principle could be limitless if it is flat. Space however could be a closed manifold. In this case, the space is limited but it is surrounded by something else, let's call it hyperspace.

Quoting Philosophim

How is the limitation of space not bound by its own internal volume?

Space is bounded by its own volume which is limitless if it is flat otherwise it is limited. Space then is surrounded by something else in the second case so-called hyperspace.

Quoting Lionino

The figure is a 2d representation of space-time distortion. These several people I quoted are not wrong about physics, you are.

So how do you discover intrinsic geometry empirically? You measure angles, you measure dot products and you see what the values are. If those values are what you'd get with flat space, you're in a flat space. If they're what you'd get in curved space, well, you're in a curved space. You can consider this the definition of a curved space. You don't have to envision space bending into some other space. Just that in our space, we measure dot products of basis vectors to have some non-zero value.
— https://physics.stackexchange.com/questions/547140/what-is-intrinsic-curvature

We call it "curvature" because it works exactly like curvature. Angles and distances measured are exactly what they would be if the space was curved. We don't assume an embedding space because we don't need to to get the right answers. So why add something to the theory that cannot be observed?

Unless you can observe the embedding space, then no, you cannot deduce that you exist embedded in a higher space. That's an assumption that cannot be tested.

If your reply does not address these quotes directly, I will move on.

Ok, so I have to draw a figure to show what I mean. Here is the figure: https://ibb.co/09dXsQH. As you can see the curvature on each point of this surface is zero. Why? Because the angle between each pair of immediate lines is constant regardless of whether you are close to the center or not. If you however draw the same picture on the surface of a balloon then you observe that the angle between lines changes as you get close to the center. If you don't have a ball or balloon then please check this figure:
https://en.wikipedia.org/wiki/File:Cassini-science-br.jpg.
It is clear in this figure that the angle between each pair of immediate lines increases as you get closer to the center therefore you have a positive curvature.

So, to sum it up, the existence of a curvature in space means that the space must bend inside a hyperspace as the 2D space in the figure of wiki bends toward the third direction.

Quoting Lionino

If your reply does not address these quotes directly, I will move on.

Cool. Bye.

Quoting MoK

Ok, I see. So if I have your idea right, you believe that space is a thing.
— Philosophim
If by "space is a thing" you mean that space is a substance then that is still the subject of debate. If by space you mean a continuous area that is unoccupied then we are into business.

Wouldn't a continuous area that is unoccupied be 'nothing' though? I am ok with the idea of simply stating, "space is a substance" as a start.

Quoting MoK

Once again, wouldn't the bounds of space be the internal limitations of space itself?
— Philosophim
Space in principle could be limitless. A section of it is however limited.

In principle, perhaps. But the entire point you're trying to make is that the whole is limitless. If space is the whole, we have to prove that, not declare it. If I'm trying to prove that cheese is a moon rock, I can't just say, "Cheese is a moonrock" as one of the arguments. This is a 'begging the question' fallacy.

Quoting MoK

In this case, the space is limited but it is surrounded by something else, let's call it hyperspace.

Alright, lets look at hyperspace then. Doesn't he same question about space and the grain of sand apply here as well? Isn't hyperspace bound by its own self?

Quoting MoK

Space is bounded by its own volume which is limitless if it is flat otherwise it is limited. Space then is surrounded by something else in the second case so-called hyperspace.

This is a contradiction though. Something cannot be both limitless and limited.

Reply to MoK
I think I see what you mean here. There is always a whole larger than and surrounding sub-holes. E.g. W2 = W1+B1. And that this goes ad infinitum. Right? Well, I'll talk about this later.
But this is not exactly what you said in the OP. You said that "This means that the whole is not W1 but W2 where W2=W1+B1". But W2 is simply a different, larger whole, including W1. W1 is still W1. It has not changed. It has not become W2. This is what I discussed in my previous comment.

Now, about wholes --or the whole as you say-- going ad infinitum, i.s. being limitless.
If this were the case, then the Universe itself --which includes all the "wholes"-- should be also surrounded by something larger than it. E.g. there could be another Universe, larger than our known Universe .But we don't and can't know that. Or there can be the case of parallel universes. Which remains still to be proved. With our present knowledge the Universe includes everything. (Except if this knowledge has changed and I don't know it.)

Quoting Philosophim

Wouldn't a continuous area that is unoccupied be 'nothing' though?

No, nothing is the absence of space, physical objects, etc.

Quoting Philosophim

I am ok with the idea of simply stating, "space is a substance" as a start.

That is alright. Saying that space is a substance does not resolve any issue here nor it helps us to prove the argument.

Quoting Philosophim

In principle, perhaps. But the entire point you're trying to make is that the whole is limitless. If space is the whole, we have to prove that, not declare it. If I'm trying to prove that cheese is a moon rock, I can't just say, "Cheese is a moonrock" as one of the arguments. This is a 'begging the question' fallacy.

That is what I am trying to show in OP. [math]W_1[/math] is either limited or limitless. If it is limitless then we reach the conclusion otherwise it is surrounded by something else, [math]B_1[/math]. Then the whole is [math]W_2[/math]=[math]W_1[/math]+[math]B_1[/math]. [math]W_2[/math] again is either limited or limitless. Etc.

Quoting Philosophim

Alright, lets look at hyperspace then. Doesn't he same question about space and the grain of sand apply here as well?

You mean hyperspace? Hyperspace is either closed which mean it is limited or it is open which means that it is limitless.

Quoting Philosophim

Isn't hyperspace bound by its own self?

What do you mean by is bounded by its own self?

Quoting Philosophim

This is a contradiction though. Something cannot be both limitless and limited.

I mean if space is open is limitless otherwise it is closed which means that it is limited.

Quoting Alkis Piskas

I think I see what you mean here. There is always a whole larger than and surrounding sub-wholes. E.g. W2 = W1+B1. And that this goes ad infinitum. Right?

Yes.

Quoting Alkis Piskas

But this is not exactly what you said in the OP. You said that "This means that the whole is not W1 but W2 where W2=W1+B1". But W2 is simply a different, larger whole, including W1. W1 is still W1. It has not changed. It has not become W2. This is what I discussed in my previous comment.

Yes, [math]W_1[/math] stays the same and does not change. I mean what we consider as the whole [math]W_1[/math] which is limited is not the whole but something bigger [math]W_2[/math] where [math]W_2[/math]=[math]W_1[/math]+[math]B_1[/math].

Quoting Alkis Piskas

Now, about wholes --or the whole as you say-- going ad infinitum, i.s. being limitless.
If this were the case, then the Universe itself --which includes all the "wholes"-- should be also surrounded by something larger than it. E.g. there could be another Universe, larger than our known Universe .But we don't and can't know that. Or there can be the case of parallel universes. Which remains still to be proved. With our present knowledge the Universe includes everything. (Except if this knowledge has changed and I don't know it.)

Yes, we cannot have physical access to the whole so we can never physically confirm that is limitless. But my argument shows that the whole is limitless.

Reply to MoK
OK. We agree then. :smile:

Reply to MoK
I'm not getting this at all. Whether the universe is limited or unlimited is a matter of physical state. If we conclude that its state is unknown then this discussion is just an attempt at a mental overlay that has no bearing whatsoever. Seems like any mental model we can contrive would be the same. Just a speculation.

So the best we can do is examine the universe we do know and base our models on the known. That could lead to reasonable projections of some of unknowns but still would have a physical basis and not mental abstractions.

Quoting MoK

Wouldn't a continuous area that is unoccupied be 'nothing' though?
— Philosophim
No, nothing is the absence of space, physical objects, etc.

You need to redefine space as being something then. An 'unoccupied' area is seen as 'nothing'. Things occupy. Nothing does not.

Quoting MoK

I am ok with the idea of simply stating, "space is a substance" as a start.
— Philosophim
That is alright. Saying that space is a substance does not resolve any issue here nor it helps us to prove the argument.

Its fairly important here because most people see space as 'nothing'. There is an old term for the idea that there really is no emptiness, and that all of space, or nothingness, is filled by a substance called "Aether". Aether was eventually debunked by science, but for your purposes the idea of space being 'something' instead of nothing, can be helpful here.

Quoting MoK

That is what I am trying to show in OP. is either limited or limitless. W1
is either limited or limitless. If it is limitless then we reach the conclusion otherwise it is surrounded by something else, B1. Then the whole is W2 =W1+B1. W2 again is either limited or limitless. Etc.

Oh, I see what you're doing here now! Clever! The only problem is you have necessitated that something always be bounded by something else, when it is commonly known that things are not bound by other substances, but the mass of their own matter. So while clever if things were bound by other things, its just not the case that they are. Further, that's not really the definition of "the whole" but really, 'a thing'. The whole is generally considered 'everything' which of course is bound by the entirety of its internal parts, and can have no other thing outside of itself.

Quoting MoK

I mean if space is open is limitless otherwise it is closed which means that it is limited.

No disagreement here, you just have to demonstrate that space is limitless or limited.

Quoting Mark Nyquist

I'm not getting this at all. Whether the universe is limited or unlimited is a matter of physical state. If we conclude that its state is unknown then this discussion is just an attempt at a mental overlay that has no bearing whatsoever. Seems like any mental model we can contrive would be the same. Just a speculation.

So the best we can do is examine the universe we do know and base our models on the known. That could lead to reasonable projections of some of unknowns but still would have a physical basis and not mental abstractions.

As I discussed we cannot have physical access to the whole but a very small part of it. So the only way to understand what its size is is through reason.

Quoting Philosophim

You need to redefine space as being something then. An 'unoccupied' area is seen as 'nothing'. Things occupy. Nothing does not.

Well, that is a matter of definition of things. Could we please agree that the condition in which there is no thing, namely no space, no material objects,.... is nothing?

Quoting Philosophim

Its fairly important here because most people see space as 'nothing'. There is an old term for the idea that there really is no emptiness, and that all of space, or nothingness, is filled by a substance called "Aether". Aether was eventually debunked by science, but for your purposes the idea of space being 'something' instead of nothing, can be helpful here.

I still think that that is irrelevant but we can think of space as substance if you wish.

Quoting Philosophim

Oh, I see what you're doing here now! Clever! The only problem is you have necessitated that something always be bounded by something else, when it is commonly known that things are not bound by other substances, but the mass of their own matter.

Well, if that was the case, namely if the whole was limited, then it has an edge or it is closed. We already discussed the case the whole is closed. The question which is relevant then is what is beyond the edge if the whole is open. What is beyond the edge cannot be nothing as we discussed so it is something. This means that what we call the whole is not whole but something else.

Quoting Philosophim

So while clever if things were bound by other things, its just not the case that they are. Further, that's not really the definition of "the whole" but really, 'a thing'. The whole is generally considered 'everything' which of course is bound by the entirety of its internal parts, and can have no other thing outside of itself.

The whole does not have an outside.

Quoting MoK

Well, that is a matter of definition of things. Could we please agree that the condition in which there is no thing, namely no space, no material objects,.... is nothing?

As long as you view space as a substance, this is fine. This is why it is not irrelevant. If space is not a substance, it is usually synonymous with 'nothing'.

Quoting MoK

The question which is relevant then is what is beyond the edge if the whole is open. What is beyond the edge cannot be nothing as we discussed so it is something. This means that what we call the whole is not whole but something else.

I still don't see why there cannot be nothing beyond the edge of something. I get that you want to define the whole as bounded by something else, but you've given no reason why that necessarily must be. Try to disprove the scenario I'm going to put in front of you. Referring earlier, I have a grain of sand with nothing else in the universe existing around it. Why is that a contradiction under your viewpoint?

Quoting MoK

The whole does not have an outside.

I also do not understand this. Are you saying that the whole is infinite? That seems to be the conclusion, so once again we're begging the question. I think what would really help to flesh out your definition of the 'whole' is to give an example of what that would be.

Quoting Philosophim

As long as you view space as a substance, this is fine. This is why it is not irrelevant. If space is not a substance, it is usually synonymous with 'nothing'.

I don't agree with you that space is synonymous with nothing but for the sake of argument, we can assume that space is a substance. One problem is resolved.

Quoting Philosophim

I still don't see why there cannot be nothing beyond the edge of something. I get that you want to define the whole as bounded by something else, but you've given no reason why that necessarily must be. Try to disprove the scenario I'm going to put in front of you. Referring earlier, I have a grain of sand with nothing else in the universe existing around it. Why is that a contradiction under your viewpoint?

Well, this we discussed it. Nothing has no geometry nor can occupy a room therefore nothing cannot surround a thing.

Quoting Philosophim

I also do not understand this. Are you saying that the whole is infinite?

The whole is larger than any infinity that you can imagine.

Quoting Philosophim

That seems to be the conclusion, so once again we're begging the question.

It is not the begging the question. If the whole has an outside then there is something outside of it therefore what we consider as the whole with an outside is not the whole.

Reply to MoK Hey Mok, been away a few days. :)

Quoting MoK

I don't agree with you that space is synonymous with nothing but for the sake of argument, we can assume that space is a substance.

As long as we're identifying space as 'something', that's fine by me for this argument.

Quoting MoK

Well, this we discussed it. Nothing has no geometry nor can occupy a room therefore nothing cannot surround a thing.

Lets make sure we're not making 'vocabulary reality', a common thing we can do in philosophy. Vocabulary is used to describe reality, it does not create reality.

Nothing does not 'surround' anything in a substantive sense. But if there is a limit to something, does nothing surround it in a directional sense? Yes. Its just words to describe the idea that beyond something, there is nothing. The only way this cannot be is if the entire universe is a thing without limits. This is what we're trying to prove by your philosophy, so it cannot be part of the premises.

Quoting MoK

I also do not understand this. Are you saying that the whole is infinite?
— Philosophim
The whole is larger than any infinity that you can imagine.

This doesn't make any sense. Infinity means 'uncountable', or 'without end'. How can something be larger than something without end?

Quoting MoK

That seems to be the conclusion, so once again we're begging the question.
— Philosophim
It is not the begging the question. If the whole has an outside then there is something outside of it therefore what we consider as the whole with an outside is not the whole.

No, if the whole has an outside, that outside can be something, or it can be nothing. I get the feeling what you really want to prove here is "Nothing is impossible". Maybe that would be a better tactic?

Quoting Philosophim

Lets make sure we're not making 'vocabulary reality', a common thing we can do in philosophy. Vocabulary is used to describe reality, it does not create reality.

Nothing does not 'surround' anything in a substantive sense. But if there is a limit to something, does nothing surround it in a directional sense? Yes. Its just words to describe the idea that beyond something, there is nothing. The only way this cannot be is if the entire universe is a thing without limits. This is what we're trying to prove by your philosophy, so it cannot be part of the premises.

What do you mean by directional sense?

Quoting Philosophim

This doesn't make any sense. Infinity means 'uncountable', or 'without end'. How can something be larger than something without end?

Georg Cantor showed that there is an infinity of infinities.

Reply to Philosophim
Good point about not making vocabulary reality.
There are also infinities and mathematical models that are not physical objects but only mental objects.

The thing is..... physical and mental are both handled with our brains/minds so they get commingled.

Quoting MoK

What do you mean by directional sense?

Imagine a grain of sand. Outside is nothing. "Outside" is the direction.

Quoting MoK

This doesn't make any sense. Infinity means 'uncountable', or 'without end'. How can something be larger than something without end?
— Philosophim
Georg Cantor showed that there is an infinity of infinities.

I think you need to go into the specifics of how Cantor's theorem applies to the argument. This doesn't explain anything by itself.

Quoting Philosophim

Imagine a grain of sand. Outside is nothing. "Outside" is the direction.

Please check the following figure:

https://ibb.co/nj94LtJ

If by nothing you mean the black area then that cannot be nothing since nothing cannot have a geometry, property, and occupy room.

Quoting Philosophim

I think you need to go into the specifics of how Cantor's theorem applies to the argument. This doesn't explain anything by itself.

This was an answer to you when you asked whether the whole is infinite. I answered that the whole is bigger than any infinity you can imagine.

Quoting MoK

If by nothing you mean the black area then that cannot be nothing since nothing cannot have a geometry, property, and occupy room.

No, I'm not saying there exists a black area, I'm saying there's nothing. It is the logical consequence of there being a limit. To state there is a limit means there is an end. What is beyond the end? Nothing. The only way to avoid this is to state that the whole is limitless. But this has to be proven, and I'm not seeing anything but a conjecture here.

Quoting MoK

This was an answer to you when you asked whether the whole is infinite. I answered that the whole is bigger than any infinity you can imagine.

I understood that was your answer, but your answer doesn't explain itself well. I am familiar with Cantor's theory and I still don't see how this applies to what you stated.

Quoting Philosophim

No, I'm not saying there exists a black area, I'm saying there's nothing. It is the logical consequence of there being a limit. To state there is a limit means there is an end. What is beyond the end? Nothing. The only way to avoid this is to state that the whole is limitless. But this has to be proven, and I'm not seeing anything but a conjecture here.

I know but the very existence of a limit means that there is nothing beyond it! What is beyond the end? It is either something or nothing. Take your pick.

Quoting MoK

I know but the very existence of a limit means that there is nothing beyond it! What is beyond the end? It is either something or nothing. Take your pick.

If 'the whole' is everything and the whole has a limit, then by consequence there is nothing past that limit. If the whole is limitless, then there is no end, thus 'nothing' cannot exist. But one has to prove that the whole of existence is limitless, which we cannot do.

Quoting Philosophim

If 'the whole' is everything and the whole has a limit, then by consequence there is nothing past that limit. If the whole is limitless, then there is no end, thus 'nothing' cannot exist. But one has to prove that the whole of existence is limitless, which we cannot do.

You cannot draw a figure in which the whole has a limit and there is nothing beyond its limit.

Quoting MoK

You cannot draw a figure in which the whole has a limit and there is nothing beyond its limit.

Correct, because you cannot draw 'nothing'. This doesn't negate what I've stated. If you have limits, nothing must be beyond those limits. The only way to avoid there being 'nothing' is if everything is infinite and eternal.

Quoting Philosophim

Correct, because you cannot draw 'nothing'. This doesn't negate what I've stated. If you have limits, nothing must be beyond those limits. The only way to avoid there being 'nothing' is if everything is infinite and eternal.

It negates what you have said. I am afraid that I don't see any point to repeat myself.

Quoting MoK

Correct, because you cannot draw 'nothing'. This doesn't negate what I've stated. If you have limits, nothing must be beyond those limits. The only way to avoid there being 'nothing' is if everything is infinite and eternal.
— Philosophim
It negates what you have said. I am afraid that I don't see any point to repeat myself.

I disagree, but we've both said our piece now. :) Good chatting with you again MoK,