And that argument was already refuted. — Lionino

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

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

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

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. — Lionino

He is wrong. Please see the above figure.

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. — Lionino

I don't think that time ever comes.

"(◇¬p → ¬□p)". — Lionino

What does that mean?

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

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

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.

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. — Arne

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.

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

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

I don't think that time ever comes. — MoK

how newtonian of you. :-)

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

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.

how newtonian of you. :-) — Arne

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

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

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.

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

I have no doubt.

I was joking.

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

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

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

I agree.

Similarly, perhaps the whole is limited by time. — Arne

What do you mean?

I have no doubt.

I was joking. — Arne

I know. :wink:

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

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."

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." — Arne

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

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

No time, no whole.

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

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.

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

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

This means that W1 is bounded by something else — MoK

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?

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. — MoK

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?

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

Thank you very much for the reference to the book.

He is wrong. Please see the above figure. — MoK

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.

I don't think that time ever comes. — MoK

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.

What do you mean by "bounded by"? — Alkis Piskas

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.

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)? — Alkis Piskas

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 $W_1$ and the room is $B_1$.

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

OK, I try my best to give good examples.

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? — Alkis Piskas

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

I hope things are clear now.

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.

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'. — Philosophim

Cool, that works.

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.

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

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.

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

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.

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. — Lionino

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.

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

Cool. Bye.

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. — MoK

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.

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. — MoK

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.

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

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?

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. — MoK

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

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.)

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

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

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.

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. — Philosophim

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

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

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

Isn't hyperspace bound by its own self? — Philosophim

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

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

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

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? — Alkis Piskas

Yes.

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. — Alkis Piskas

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

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.) — Alkis Piskas

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.

OK. We agree then. :smile: