The Finite Universe and the Fallacy of Composition


Dennis Hardin

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One of the major conundrums in the history of philosophy is the issue of whether or not the universe is spatially finite. In other words, does it have boundaries?

This is the first of the 'Antimonies of Pure Reason' described by Immanuel Kant in The Critique of Pure Reason. Kant argues that reason is inadequate here (as with the other antimonies) because it can establish with equal cogency contradictory answers to the question. In other words, we can establish logically that it must be both finite and infinite.

The standard or “orthodox” Objectivist answer is that the universe is temporally infinite but spatially finite. Leonard Peikoff, in a podcast dated 12-29-08, said that “you can’t go outside the universe because the universe is finite and there is no out there.” He is answering a question about what happens when you reach the boundary of the universe, which he says is identical to a question he asked Ayn Rand during the first year that he knew her. The question is: If you keep going forever, wouldn’t you eventually reach the boundary, and then what happens? Peikoff clearly implies that Ayn Rand agreed with the premise that the universe does have boundaries (although she apparently gasped when he asked the question).

Peikoff’s answer is to state that the whole problem derives from attempting to visualize the universe perceptually. When we do that, we are, in effect, looking at the universe from the outside, but there is no outside. It is impossible to reduce the universe to the perceptual level in this way, because perception only enables one to see objects in relation to other objects. Peikoff then goes on to say that, if your spaceship kept traveling forever, it would eventually move in a trajectory that would prevent it from reaching the so-called boundary. While the perceptual point-of-view argument has merit, the speculation about such a hypothetical trajectory strikes me as incoherent.

I think that the proper Objectivist answer is agnosticism on this issue. We simply do not know the answer. But to argue that, because every particular thing that exists is finite, therefore the universe must be finite is the fallacy of composition; i.e., trying to infer that something is true of the whole from the fact that it is true of some part of the whole.

Here is how the question was answered by scientist David Ross in the April, 1995 issue of Full Context:

“I don’t see why there has to be some definite number of avocados in the universe. I don’t see why there can’t be endless avocados. I’m not claiming that there are. I’m trying to be careful to leave the burden of proof on the finitists, where it belongs. But from the fact that each avocado has to have a specific identity, I don’t see how I can infer that there’s some definite number of them in existence.”

“….I don’t see why [Objectivists] demand that the world be spatially finite, when they allow for it to eternal. It seems to me that just as the universe has no age, it has no size or weight.”

I consider that a much more rational approach for Objectivists to take on this issue.

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One of the major conundrums in the history of philosophy is the issue of whether or not the universe is spatially finite. In other words, does it have boundaries?

This is the first of the 'Antimonies of Pure Reason' described by Immanuel Kant in The Critique of Pure Reason. Kant argues that reason is inadequate here (as with the other antimonies) because it can establish with equal cogency contradictory answers to the question. In other words, we can establish logically that it must be both finite and infinite.

The standard or "orthodox" Objectivist answer is that the universe is temporally infinite but spatially finite. Leonard Peikoff, in a podcast dated 12-29-08, said that "you can't go outside the universe because the universe is finite and there is no out there." He is answering a question about what happens when you reach the boundary of the universe, which he says is identical to a question he asked Ayn Rand during the first year that he knew her. The question is: If you keep going forever, wouldn't you eventually reach the boundary, and then what happens? Peikoff clearly implies that Ayn Rand agreed with the premise that the universe does have boundaries (although she apparently gasped when he asked the question).

Peikoff's answer is to state that the whole problem derives from attempting to visualize the universe perceptually. When we do that, we are, in effect, looking at the universe from the outside, but there is no outside. It is impossible to reduce the universe to the perceptual level in this way, because perception only enables one to see objects in relation to other objects. Peikoff then goes on to say that, if your spaceship kept traveling forever, it would eventually move in a trajectory that would prevent it from reaching the so-called boundary. While the perceptual point-of-view argument has merit, the speculation about such a hypothetical trajectory strikes me as incoherent.

I think that the proper Objectivist answer is agnosticism on this issue. We simply do not know the answer. But to argue that, because every particular thing that exists is finite, therefore the universe must be finite is the fallacy of composition; i.e., trying to infer that something is true of the whole from the fact that it is true of some part of the whole.

Here is how the question was answered by scientist David Ross in the April, 1995 issue of Full Context:

"I don't see why there has to be some definite number of avocados in the universe. I don't see why there can't be endless avocados. I'm not claiming that there are. I'm trying to be careful to leave the burden of proof on the finitists, where it belongs. But from the fact that each avocado has to have a specific identity, I don't see how I can infer that there's some definite number of them in existence."

"….I don't see why [Objectivists] demand that the world be spatially finite, when they allow for it to eternal. It seems to me that just as the universe has no age, it has no size or weight."

I consider that a much more rational approach for Objectivists to take on this issue.

The question of boundary and finiteness are separate. A topological metric space can be finite (upper bound of the distance between pairs of points) but still have no boundary points. Example: The surface of sphere taken as topological space in and of itself is metrically finite but has no boundary points.

If the physical universe has positive curvature and is closed it is possible that it has no boundary points. If the physical universe is open it may be finite and bounded or infinite (in the metric sense) and unbounded.

Ba'al Chatzaf

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One of the major conundrums in the history of philosophy is the issue of whether or not the universe is spatially finite. In other words, does it have boundaries?

This is the first of the 'Antimonies of Pure Reason' described by Immanuel Kant in The Critique of Pure Reason. Kant argues that reason is inadequate here (as with the other antimonies) because it can establish with equal cogency contradictory answers to the question. In other words, we can establish logically that it must be both finite and infinite.

The standard or "orthodox" Objectivist answer is that the universe is temporally infinite but spatially finite. Leonard Peikoff, in a podcast dated 12-29-08, said that "you can't go outside the universe because the universe is finite and there is no out there." He is answering a question about what happens when you reach the boundary of the universe, which he says is identical to a question he asked Ayn Rand during the first year that he knew her. The question is: If you keep going forever, wouldn't you eventually reach the boundary, and then what happens? Peikoff clearly implies that Ayn Rand agreed with the premise that the universe does have boundaries (although she apparently gasped when he asked the question).

Peikoff's answer is to state that the whole problem derives from attempting to visualize the universe perceptually. When we do that, we are, in effect, looking at the universe from the outside, but there is no outside. It is impossible to reduce the universe to the perceptual level in this way, because perception only enables one to see objects in relation to other objects. Peikoff then goes on to say that, if your spaceship kept traveling forever, it would eventually move in a trajectory that would prevent it from reaching the so-called boundary. While the perceptual point-of-view argument has merit, the speculation about such a hypothetical trajectory strikes me as incoherent.

I think that the proper Objectivist answer is agnosticism on this issue. We simply do not know the answer. But to argue that, because every particular thing that exists is finite, therefore the universe must be finite is the fallacy of composition; i.e., trying to infer that something is true of the whole from the fact that it is true of some part of the whole.

Here is how the question was answered by scientist David Ross in the April, 1995 issue of Full Context:

"I don't see why there has to be some definite number of avocados in the universe. I don't see why there can't be endless avocados. I'm not claiming that there are. I'm trying to be careful to leave the burden of proof on the finitists, where it belongs. But from the fact that each avocado has to have a specific identity, I don't see how I can infer that there's some definite number of them in existence."

"….I don't see why [Objectivists] demand that the world be spatially finite, when they allow for it to eternal. It seems to me that just as the universe has no age, it has no size or weight."

I consider that a much more rational approach for Objectivists to take on this issue.

This is a false dichotomy, but a tricky one. The universe can be finite and unbounded or "self bounded." Understanding this requires understanding of higher dimensional thinking. I have tried to explain this several times on several threads. First, the notion of an actually infinite universe is incoherent. No ratio can be asserted between a finite body and an infinite one, meaning that no mathematical equation about an infinite universe can be expressed in terms of any observed object. An infinite universe could not be imagined, described in real terms, or observationally confirmed. It could only be defined negatively, and in terms of ignorance. A temporally infinite universe can be spoken of as a potentiality only. But upon heat death it would reach a changeless and timeless state.

The simple explanation is this. Just as a two dimensional being living on the surface of an immensely large but finite sphere could travel as far as he liked in any direction without reaching a boundary, a being in our universe can travel as far as he likes without reaching a boundary because space curves back on itself in a hypersphere. That hypersphere is, of course, billions of light years in diameter. So one could not actually do it within the current age of the universe, but if it were possible to travel at almost infinite speed one could leave in a straight line in one direction, circumnavigate the universe like someone walking straight east at the Equator, and end up back where one began. The universe is huge, but finite in expanse, yet has no edge.

The issue remains an aporia only for those who lack the necessary concepts. The same "dilemmas" faced the falt earthers, who expected to reach an edge of the world, not realizing that it was a sphere. Peikoff's ignorance on this issue is startling, yet, unfortunately, not unique. There is no reason for Objectivists who are not familiar with the finite yet unbounded model to assert their own confusion as a principle. Rather than assert their agnosticism or faith on the issue, they should study the necessary concepts. There is no cosmo-epistemological dispensation for untutored Objectivists.

The finite yet unbounded universe is the standard cosmological model. Hawking explains it in his Universe in a Nutshell. I also recommend Geometry, Relativity and the Fourth Dimension by Rudolph Rucker.

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The question of boundary and finiteness are separate. A topological metric space can be finite (upper bound of the distance between pairs of points) but still have no boundary points. Example: The surface of sphere taken as topological space in and of itself is metrically finite but has no boundary points.

If the physical universe has positive curvature and is closed it is possible that it has no boundary points. If the physical universe is open it may be finite and bounded or infinite (in the metric sense) and unbounded.

Bob, can you find anything helpful to explain this on the internet? The article closed universe at wikipedia is unfortunately one of the poorest I have ever read, and it does not clearly distinguish between closure in space and in time. Once you have the concept the philosophical "dilemma" disappears but I despair of being able to explain it without visual aids and in a live forum.

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Subject: The legitimate concept of infinity -- unboundedness in space and in time, but not necessarily in matter

> a being in our universe can travel as far as he likes without reaching a boundary because space curves back on itself in a hypersphere. [Ted]

Why would it be a contradiction to say that one can get in a space ship and keep going endlessly in the same direction? Even if there is a finite amount of matter in space, and one's space ship billions of years from now were to be the furthest piece of matter from earth and still getting further. I wouldn't use the term 'infinite' for this, but would say 'unbounded' instead.

It is analogous to the way we say the number line is 'infinite' in mathematics. We mean that one can always come up with a larger number. Or in the case of subdividing the number line. You can always break apart any two numbers by finding a number in between them. "Infinitely" many times. [i will leave out Cantor and levels of infinity.]

And in the same way that time is unbounded, because no matter how many quazillions of years in the future, you can always add a later time.

Edited by Philip Coates
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Subject: The legitimate concept of infinity -- unboundedness in space and in time, but not necessarily in matter

> a being in our universe can travel as far as he likes without reaching a boundary because space curves back on itself in a hypersphere. [Ted]

Why would it be a contradiction to say that one can get in a space ship and keep going endlessly in the same direction? Even if there is a finite amount of matter in space, and one's space ship billions of years from now were to be the furthest piece of matter from earth and still getting further. I would use the term 'infinite' for this but I would say 'unbounded'.

Did you mean to say that you wouldn't use the term infinite?

It is analogous to the way we say the number line is 'infinite' in mathematics. We mean that one can always come up with a larger number. Or in the case of subdividing the number line. You can always break apart any two numbers by finding a number in between them. "Infinitely" many times. [i will leave out Cantor and levels of infinity.]

And in the same way that time is unbounded, because no matter how many quazillions of years in the future, you can always add a later time.

The idea that there is some region of empty space beyond matter filled space is contradicted by observation, which shows that space is uniform in all directions (yet "younger" the further away in light years) and is based on a lack of understanding of the curvature of space in four dimensions.

Imagine if you were a miniscule, two-dimensional being living on the surface of the earth. You could, assuming you have a boat, potentially travel in a straight line east for as long as you like. If you lived long enough, you would come back to your beginning. At no point would you reach the end of the earth. You could also travel north, or in any other direction, and come back where you started. You would not be able to visualize this directly, since the earth is curved in three dimensions, and you can only visualize east and west, or north and south, but not up and down, as if the universe were some giant flat National Geographical map. You might even think that a flat universe would either have to have an edge, or go on for ever. If some crazy mathematician were to tell you he had dreamt up a "three dimensional" object with a finite yet unbounded surface, you might think he was crazy, or that he was talking about some map that stretches as you reached its edge.

Well, the same model extended to a three dimensional "surface" of a fourth-dimensional hypersphere, in which one can travel as far as one likes in any of the three dimensions, and, after travelling for at least 26 billion light years in any one direction, and return to where one began (although the universe would have expanded yet more) explains the current thinking about the physical shape of the universe.

No faith, stretching, agnosticism, boundaries, edges, empty space, or inexpressible infinities are needed.

Edited by Ted Keer
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Why does such conjecturing have anything to do with Objectivism as oposed to Conjecturism?

--Brant

Well, for one thing, some Objectivists would like to have a coherent answer to Kant. Emphasis on the word coherent.

So far, based on the above, I'd have to say we are doing a piss poor job of it. Agnosticism makes a lot more sense to me than any of the other responses.

how do finite things get turned into an infinite something?

If you began adding up all the finite things in the universe, how do you know you would ever reach the point when there were no more things to be counted?

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If you began adding up all the finite things in the universe, how do you know you would ever reach the point when there were no more things to be counted?

I know I'd never; my time is finite and all my resources are finite. The only thing infinite is nothing--or agnosticism. How do I know there's no God? Ignorance is not knowledge.

--Brant

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If you began adding up all the finite things in the universe, how do you know you would ever reach the point when there were no more things to be counted?

I know I'd never; my time is finite and all my resources are finite. The only thing infinite is nothing--or agnosticism. How do I know there's no God? Ignorance is not knowledge.

--Brant

Taking the position that we don’t have all the answers at present is not equivalent to nothingness; awareness of our cognitive limitations is a stage of awareness. The concept of God is inherently incoherent and self-contradictory. God explains nothing as a principle and there is zero evidence of his existence. That’s how I know there is no God.

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If you began adding up all the finite things in the universe, how do you know you would ever reach the point when there were no more things to be counted?

I know I'd never; my time is finite and all my resources are finite. The only thing infinite is nothing--or agnosticism. How do I know there's no God? Ignorance is not knowledge.

--Brant

Taking the position that we don't have all the answers at present is not equivalent to nothingness; awareness of our cognitive limitations is a stage of awareness. The concept of God is inherently incoherent and self-contradictory. God explains nothing as a principle and there is zero evidence of his existence. That's how I know there is no God.

There is zero evidence for the existence of infinity. You can substitute infinity for God. No matter how many things are counted you'll never find infinity; it's an idea only. Things are finite, ideas are infinite. You cannot churn metaphysical butter out of epistemological cream.

--Brant

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> Did you mean to say that you wouldn't use the term infinite? {Ted]

You're right. I went back and cleaned up that sentence.

> The idea that there is some region of empty space beyond matter filled space is contradicted by observation, which shows that space is uniform in all directions

What makes you think our telescopes actually see to the end of space, that we're not only looking at a region of space?

> a fourth-dimensional hypersphere

Please! Nobody knows what that is. It's a theoretical construct. Speaking of three dimensions of space and a 'fourth dimension' of time is not a real -physical- dimension in the same way three-dimensional space is. I don't want to get into the 'curvature' of space again (at best another metaphor) -- nor do I want to keep pointing out things like when a sun's gravity curves light, that is not the same as saying space itself is curved.

> There is zero evidence for the existence of infinity. [brant]

Please reread post #6. I gave some examples of where it exists (i) in the physical world, if you change the concept to unboundedness; (ii) in the mathematical world.

,,,,,,,,

PS: It is frustrating when I makes several arguments in a careful post and, instead of dealing with the actual arguments, a poster or two simply -repeats- the point my arguments were intended to refute. Without dealing with those arguments and as if they did not recognize that an argument had actually been made and needs to be dealt with.

Edited by Philip Coates
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Please! Nobody knows what that is. It's a theoretical construct. Speaking of three dimensions of space and a 'fourth dimension' of time is not a real -physical- dimension in the same way three-dimensional space is. I don't want to get into the 'curvature' of space again (at best another metaphor) -- nor do I want to keep pointing out things like when a sun's gravity curves light, that is not the same as saying space itself is curved.

If you accept that light follows the path the minimizes a certain action in a differentiable manifold, it is the same as saying that the manifold is curved. The evidence for Einstein's General Theory of Relativity is rather impressive. Everything from gravitational lensing, to gravitational red-shift and working GPS systems. So I take curved space-time as seriously as my GPS device. How much evidence do you require? Einstein's theory does probably does not hold in Black Holes but there is no way (that anyone knows) to test the theory in a Black Hole as no information comes out.

I am willing to bet that you think space-time is a flat (0 curvature) Euclidean space (the say Newton thought about it). The physical evidence indicates otherwise.

Just a word here: Curvature is not a simple scalar quantity like length or temperature. In physical space it is a fourth order tensor.

Ba'al Chatzaf

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Taking the position that we don't have all the answers at present is not equivalent to nothingness; awareness of our cognitive limitations is a stage of awareness. The concept of God is inherently incoherent and self-contradictory. God explains nothing as a principle and there is zero evidence of his existence. That's how I know there is no God.

Anything that "explains" everything explains nothing.

Ba'al Chatzaf

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> Did you mean to say that you wouldn't use the term infinite? {Ted]

You're right. I went back and cleaned up that sentence.

> The idea that there is some region of empty space beyond matter filled space is contradicted by observation, which shows that space is uniform in all directions

What makes you think our telescopes actually see to the end of space, that we're not only looking at a region of space?

> a fourth-dimensional hypersphere

Please! Nobody knows what that is. It's a theoretical construct. Speaking of three dimensions of space and a 'fourth dimension' of time is not a real -physical- dimension in the same way three-dimensional space is. I don't want to get into the 'curvature' of space again (at best another metaphor) -- nor do I want to keep pointing out things like when a sun's gravity curves light, that is not the same as saying space itself is curved.

> There is zero evidence for the existence of infinity. [brant]

Please reread post #6. I gave some examples of where it exists (i) in the physical world, if you change the concept to unboundedness; (ii) in the mathematical world.

,,,,,,,,

PS: It is frustrating when I makes several arguments in a careful post and, instead of dealing with the actual arguments, a poster or two simply -repeats- the point my arguments were intended to refute. Without dealing with those arguments and as if they did not recognize that an argument had actually been made and needs to be dealt with.

I dealt with your "unboundedness" a while back by agreeing with it--that is, the universe is expanding into nothing.

As for the rest of your epistemological gobbledygook, it's not metaphysics.

--Brant

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I did a search of "infinity" on the Objectivism Research CD. It yielded only three significant hits -- two by Rand and one by Peikoff. I post them without comment.

Rand, ITOE, p. 17.

An arithmetical sequence extends into infinity, without implying that infinity actually exists; such extension means only that whatever number of units does exist, it is to be included in the same sequence. the same principle applies to concepts: the concept "man" does not (and need not) specify what number of men will ultimately have existed—it specifies only the characteristics of man, and means that any number of entities possessing these characteristics is to be identified as "men."

ITOE, p. 148.

The concept of "infinity" has a very definite purpose in mathematical calculation, and there it is a concept of method. But that isn't what is meant by the term "infinity" as such. "Infinity" in the metaphysical sense, as something existing in reality, is another invalid concept. The concept "infinity," in that sense, means something without identity, something not limited by anything, not definable.

Peikoff, OPAR, p,. 31-32.

As Aristotle was the first to observe, the concept of "infinity" denotes merely a potentiality of indefinite addition or subdivision. For example, one can continually subdivide a line; but however many segments one has reached at a given point, there are only that many and no more. The actual is always finite.

Ghs

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Dennis, while you have chosen to begin a thread on this topic which was discussed at length in the Harriman thread, you have not once commented on the finite yet unbounded model, or referred directly to the prior discussion. Do you find the prior discussion unhelpful?

I am not sure who you are addressing, Brant, but in any case, the universe is most definitely not expanding into "nothing." There is no nothing to expand into. Again, this is an intuitive yet ultimately incoherent Newtonian notion of an infinite, open three dimensional space.

Phil, are you familiar with the Cosmic Microwave Background Radiation? It is the glow left over from the time when the universe had cooled and expanded enough that rather than all radiation being absorbed immediately by surrounding atoms, it was able to escape into free space. It is as far as can be seen into space and back in time (which are the same thing, due to the constancy of the speed of light) and what it shows is not that the big bang happened in one point in some far corner of the universe, but that it exists in every direction you look. Paradoxically, all of space is inside the Big Bang. There is no empty space off in one direction, and big bang in another direction. The big bang surrounds us because the universe is self bounded. This is direct observational confirmation of the finite yet unbounded model. The paradox is soluble only if one comprehends the fourth dimensional reality.

This really is not a subject for debate, gentleman. The scientific models are fully consistent with a spatially finite closed universe with no edge and no empty space. Understanding this requires possession of the proper concepts, which do exist in rigorous form, although they may be difficult to comprehend. The finite yet unbounded model of the universe has the strange properties of being, (1) mathematically coherent, (2) consistent with observation, and (3) in total agreement with Rand on the finity of existence. If you want to criticize that theory, you need to comprehend it first.

Again, I recommend Hawking's Universe in a Nutshell.

Edited by Ted Keer
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Don't take my word for it.

Here is Carl Sagan explaining the finite yet unbounded model of the universe in episode 10 of Cosmos.

The concepts are fully illustrated with diagrams and animations.

Watch both clips, because the explanation depends upon comprehension of the more basic concepts introduced in the beginning of the first clip, each in total about 10 minutes long.

http://www.youtube.com/watch?v=H7YWW0-DCew

http://www.youtube.com/watch?v=-YbZgh7PV3Q&feature=related

Edited by Ted Keer
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Just a word here: Curvature is not a simple scalar quantity like length or temperature. In physical space it is a fourth order tensor.

In other words, when physicists speak of the curvature of space, the don't mean by this what non-physicists mean by "curvature." Physicists use the word in a technical sense. Correct?

Ghs

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I am not sure who you are addressing, Brant, but in any case, the universe is most definitely not expanding into "nothing." There is no nothing to expand into.

Again, I recommend Hawking's Universe in a Nutshell.

Since it's not expanding into something there is nothing to expand into so it expands into nothing.

--Brant

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And this is why we have mathematics because the human language, as great as it is, is not precise enough to express certain concepts.

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Don't take my word for it.

Here is Carl Sagan explaining the finite yet unbounded model of the universe in episode 10 of Cosmos.

The concepts are fully illustrated with diagrams and animations.

Watch both clips, because the explanation depends upon comprehension of the more basic concepts introduced in the beginning of the first clip, each in total about 10 minutes long.

[videos deleted]

I watched both video clips. No offense, Ted, but I thought Sagan's explanation was poorly done -- and I'm generally a fan of Sagan. His Flatlander example is misleading and confusing. For one thing, are the Flatlanders capable of looking up? If so, what will they see? Or are their necks locked into one position?

Moreover, the Flatlanders would supposedly see nothing but flat lines. So do these these lines have any height? If so, then the Flatlanders can see a third dimension. If not -- if these lines are the abstract lines of Euclidean geometry -- then there is literally nothing to see.

Note what happens when the two-dimensional plane of the Flatlanders is morphed into a three-dimensional sphere. There is still space outside that sphere -- a blue background in the video. Sagan also talks about the "surface" of the curved universe. Okay, if one is on this surface and looks up, what will one see? Or are our necks, like the necks of Flatlanders, also locked in place?

When Sagan goes into his routine about a fourth physical dimension, he claims this is something we can talk about but not experience. We cannot even imagine a fourth dimension, according to Sagan. The main problem here is that a curved universe does not require positing a mysterious fourth dimension. The spherical model only requires three dimensions.

It also requires a conception of "space" as something rather than nothing -- a specific type of existent that has properties (e.g., curvature) and that can be affected by gravitation. I am still attempting to understand what kind of existent space is supposed to be. What are its constituent elements? Energy? Subatomic particles?

Ghs

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I am not sure who you are addressing, Brant, but in any case, the universe is most definitely not expanding into "nothing." There is no nothing to expand into.

Again, I recommend Hawking's Universe in a Nutshell.

Since it's not expanding into something there is nothing to expand into so it expands into nothing.

--Brant

No, not really. The meaning of the claim that the universe is expanding is that on average, all bodies are moving away from each other, not that they are moving into previously unfilled empty space. In actuality, this means that galaxies not close enough to be gravitationally bound to each other are moving apart. The space between galaxies is increasing. The total volume of the universe in ratio to any set unit such as the volume of an atom is increasing over time. There is no empty space for it to expand into. "Moving into" is an inapplicable notion in this case.

I refer you to the clips from Cosmos that I just posted.

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Just a word here: Curvature is not a simple scalar quantity like length or temperature. In physical space it is a fourth order tensor.

In other words, when physicists speak of the curvature of space, the don't mean by this what non-physicists mean by "curvature." Physicists use the word in a technical sense. Correct?

Ghs

The physical definition of curvature on a manifold is the same as the mathematical definition. It all started with Gauss and Riemann.

Ba'al Chatzaf

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Don't take my word for it.

Here is Carl Sagan explaining the finite yet unbounded model of the universe in episode 10 of Cosmos.

The concepts are fully illustrated with diagrams and animations.

Watch both clips, because the explanation depends upon comprehension of the more basic concepts introduced in the beginning of the first clip, each in total about 10 minutes long.

[videos deleted]

I watched both video clips. No offense, Ted, but I thought Sagan's explanation was poorly done -- and I'm generally a fan of Sagan. His Flatlander example is misleading and confusing. For one thing, are the Flatlanders capable of looking up? If so, what will they see? Or are their necks locked into one position?

Moreover, the Flatlanders would supposedly see nothing but flat lines. So do these these lines have any height? If so, then the Flatlanders can see a third dimension. If not -- if these lines are the abstract lines of Euclidean geometry -- then there is literally nothing to see.

Note what happens when the two-dimensional plane of the Flatlanders is morphed into a three-dimensional sphere. There is still space outside that sphere -- a blue background in the video. Sagan also talks about the "surface" of the curved universe. Okay, if one is on this surface and looks up, what will one see? Or are our necks, like the necks of Flatlanders, also locked in place?

When Sagan goes into his routine about a fourth physical dimension, he claims this is something we can talk about but not experience. We cannot even imagine a fourth dimension, according to Sagan. The main problem here is that a curved universe does not require positing a mysterious fourth dimension. The spherical model only requires three dimensions.

It also requires a conception of "space" as something rather than nothing -- a specific type of existent that has properties (e.g., curvature) and that can be affected by gravitation. I am still attempting to understand what kind of existent space is supposed to be. What are its constituent elements? Energy? Subatomic particles?

Ghs

I am not certainly offended. I am trying to explain, not argue. I understand both the truth of what is being said, and the difficulty of understanding that truth.

There is nothing wrong with Sagan's presentation. He cannot do the presentation except in three dimensions. This leads to the same distortions as drawing a cube does in two dimensions. Of course he shows the sphere as floating in a blue field. That is a requirement of our way of seeing.. No, the flatlanders cannot "lift" their heads out of their two dimensions. For them the surface they live on is the entirety of existence. While we, as three dimensional beings imagine it, for them there is no up. You are failing apply the analogy strictly, and are applying third dimensional preconceptions in both the case of the flatlanders and the fourth dimension. I fully understand it, but don't have any easy way to help you overcome that.

The notion that something always has to be inside something else is a limitation of our imagination given the scale we live on. It applies to everything on the scale of our direct perception. (There have been recent reports that people actually can, contra Sagan, learn to visualize the fourth dimension using practice with virtual reality programs.) But just as we imagine solidity as applying to entities, although the concept doesn't apply to objects at the quantum level, so our ideas of space as Newtonian is inapplicable on the cosmological scale.

It should strike you as odd that whatever direction we look, we see the radiation from the big bang surrounding us. This is indeed a fourth dimensional phenomenon. If the big bang were a three dimensional sphere, it would lie at some location within space. Instead, it lies in every direction we look, as if it "surrounds" space. For the flatlander this would be like standing at the North pole and looking south to see the South pole in any direction he looked. How, he would think, could he be "surrounded" by a point?

I suggest you try to imagine the universe as a large cubical room with a door set in the center of each wall. If you exit the north door, you enter the south door. If you exit the east door, you come back in through the west door. If you exit the top door, you enter the bottom door. That would be a self-sufficient finite space with nothing existing outside of it. The universe is the same, just with the "doors" expanded so that there are no walls left, and the volume of the room expanded to billions of lightyears in diameter.

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