The Finite Universe and the Fallacy of Composition


Dennis Hardin

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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).

Here are some of the conceptual problems I have with the question Is the universe infinite?

First, what do we mean by "the universe"? Well, when I use this term, I mean "everything that exists." So the original question becomes Is everything that exists infinite?

I can't honestly say that I even know what the question is supposed to mean. The most obvious interpretation translates into the question Is there an infinite number of existents in the universe? The problem here is that "infinite," as used in this context, is not a number. On the contrary, it signifies that no specific number can be assigned. And if this is the case, then it makes no sense to speak of "everything" that exists, because there is no "every" or "all" of which we can speak.

Second, when most philosophers have spoken of an "infinite universe," they have meant that nonexistence (i.e., empty space) has no limitations. The term "space," when used in this sense , is not a thing with specific properties. It is nothing -- and nothing has no characteristics and therefore no limitations.

Confusion inevitably arises when we reify nonexistence and speak of it as if it were a type of existence. This most often occurs when we speak of space -- again, in the sense of empty space --as being "infinite." This can give the thoroughly misleading impression that "infinity" is an attribute of an existent known as "space" -- whereas all we really mean is that nonexistence has no boundaries. We can no more limit nonexistence than we can lasso nonexistence or cook nonexistence for dinner.

Ghs

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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

The simple answer to your question, George, is yes.

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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.

Drop the condescending attitude, pronto, or we are going to have some major problems. I have listened to too many Christians over the years tell me that they understand the truth of their "born again" religious experiences, and that they sympathize with my difficulty in understanding them, to take any similar bullshit from someone on OL.

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.

By definition, one cannot draw a cube, even a distorted one, in two dimensions. This is a logical impossibility.

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.

No, you do not, because you cannot, fully understand a two-dimensional universe. The example itself rests on all kinds of equivocations, such as one I mentioned earlier, namely: How could the Flatlanders even perceive a line unless it had the third dimensional attribute of height? An abstract line without three dimensions would literally be imperceptible.

A two-dimensional universe of Euclidian lines and planes is an abstraction, one based on our three-dimensional universe. There is no such thing as a two-dimensional universe, and we cannot even conceive of one consistently. It is a sloppy, ill-formed, and thoroughly misleading analogy, and there is no way to apply such an analogy "strictly."

I understand this, but I don't have any easy way to help you overcome your difficulties in understanding it. (If you don't like this kind of crap from me, then stop dishing it out.)

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?

Instead of positing a mysterious fourth dimension that even Sagan claims we cannot imagine, has it occurred to you to question the conventional version of Big Bang theory? If you don't want to do this, then perhaps we can discuss the possibility that angels have pushed things around into their present configuration. They, too, would "save the appearances."

Ghs

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The only thing infinite is nothing--or agnosticism. How do I know there's no God? Ignorance is not knowledge.

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.

The burden of proof falls on the person who affirms the truth of a proposition, and here we have one of the (not so frequent) cases of an atheist actually claiming knowledge of non-existence of a god (not just expressing his/her absence of belief in transcendence).

(In Why Atheism, p. 23, Ghs quotes the atheist G. W. Foote who challenged a critic: "to refer me to one Atheist who denies the existence of God.")

One can reject concepts of god as inherently contradictory, like e. g. the characteristics 'all powerful' plus 'all good' don't match; scientific findings contradict what is said about creation in the Bible, etc.

But isn't it a non-sequitur to conclude from this: "I know there is no God." (?)

As for not having evidence: "Failure to prove a positive does not establish the negative." (Ghs, Why Atheism, p. 21)

Absence of evidence (of a god) is not necessarily evidence of absence (of a god).

Edited by Xray
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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.

Drop the condescending attitude, pronto, or we are going to have some major problems. I have listened to too many Christians over the years tell me that they understand the truth of their "born again" religious experiences, and that they sympathize with my difficulty in understanding them, to take any similar bullshit from someone on OL.

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.

By definition, one cannot draw a cube, even a distorted one, in two dimensions. This is a logical impossibility.

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.

No, you do not, because you cannot, fully understand a two-dimensional universe. The example itself rests on all kinds of equivocations, such as one I mentioned earlier, namely: How could the Flatlanders even perceive a line unless it had the third dimensional attribute of height? An abstract line without three dimensions would literally be imperceptible.

A two-dimensional universe of Euclidian lines and planes is an abstraction, one based on our three-dimensional universe. There is no such thing as a two-dimensional universe, and we cannot even conceive of one consistently. It is a sloppy, ill-formed, and thoroughly misleading analogy, and there is no way to apply such an analogy "strictly."

I understand this, but I don't have any easy way to help you overcome your difficulties in understanding it. (If you don't like this kind of crap from me, then stop dishing it out.)

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?

Instead of positing a mysterious fourth dimension that even Sagan claims we cannot imagine, has it occurred to you to question the conventional version of Big Bang theory? If you don't want to do this, then perhaps we can discuss the possibility that angels have pushed things around into their present configuration. They, too, would "save the appearances."

Ghs

If I wanted to insult you, George, I would have just done so. You brought up the issue of not wanting to offend, don't be upset if I tell you why I am not offended.

The two dimensional world is not posited as real, it is brought up to make the idea of a fourth dimension clear through analogy. Objecting that a two-dimensional being couldn't see is likely true but not relevant to the analogy.

The forurth dimension is not something "mysterious," not some mystical other world with mysterious properties. It is no more mysterious than the fact that the locally flat appearing surface of the earth is actually curved into a sphere. There is nothing outside of our three dimensional universe necessarily posited by a finite yet unbound universe. The fact that you see the big bang in every direction you look is direct confirmation of the fact that space bounds itself.

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Space may not be curved and things may not come back to their starting points eventually, but that's certainly not true for the several threads on this topic. :rolleyes:

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The two dimensional world is not posited as real, it is brought up to make the idea of a fourth dimension clear through analogy. Objecting that a two-dimensional being couldn't see is likely true but not relevant to the analogy.

In order for an analogy to work, the analogues must be similar in relevant respects. To posit an inconceivable two-dimensional world does not illustrate or clarify anything -- unless you want to say: "Here's a two-dimensional world that makes no sense, and it is analogous to a four-dimensional world that makes no sense, either." But if this were the point of the analogy, one could as easily posit the world of Toon Town in Who Framed Roger Rabbit instead of Flatland. Any impossible hypothetical world suffice.

The point of Sagan's analogy is supposed to be this: A fourth dimension is to us as a third dimension is to Flatlanders. But at least we can see in three dimensions. The Flatlanders, in contrast and by your own admission, could not see anything, not even in two dimensions. So the comparison becomes one between an impossible world where the inhabitants would have no awareness of anything versus our world where we are aware of a three dimensional universe.

Sagan's analogy is very similar to an analogy commonly use by mystics. I am using "mystics" in the strict sense to denote those who claim that they can directly experience God, or the divine nature, without the aid of the senses. In defense of their claims, mystics sometimes invoke the analogy of a man blind from birth who has no conception of colors. To make this parallel to Sagan's analogy more exact, we may posit a world of blind people who have never encountered a sighted person and who have no idea of what "vision" means.

Now, suppose a sighted person were to enter this world of the blind, rather like Sagan's 3-D apple enters the 2-D world of the Flatlanders, and suppose this intruder tries to explain his perceptions of color to the blind. Mystics claim that even though they won't understand what the sighted person is talking about, his perceptions are veridical nevertheless.

From here the mystic pushes his analogy home to its predetermined destination, namely, that though he cannot explain his direct experiences with God to people who are spiritually blind, his experiences are veridical nonetheless.

Despite the many holes in the mystic's analogy, it is actually better than Sagan's. This is so because the mystic's hypothetical analogue (a world of the blind) at least makes sense on its own terms. Sagan's Flatland, in contrast, is unintelligible.

The forurth dimension is not something "mysterious," not some mystical other world with mysterious properties....

Sagan expressly says that we cannot experience the fourth dimension or even imagine what it is. If this doesn't qualify as mysterious, I don't know what would.

Ghs

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The two dimensional world is not posited as real, it is brought up to make the idea of a fourth dimension clear through analogy. Objecting that a two-dimensional being couldn't see is likely true but not relevant to the analogy.

In order for an analogy to work, the analogues must be similar in relevant respects. To posit an inconceivable two-dimensional world does not illustrate or clarify anything -- unless you want to say: "Here's a two-dimensional world that makes no sense, and it is analogous to a four-dimensional world that makes no sense, either." But if this were the point of the analogy, one could as easily posit the world of Toon Town in Roger Rabbit instead of Flatland. Any impossible hypothetical world would suffice.

The point of Sagan's analogy is supposed to be this: A fourth dimension is to us as a third dimension is to Flatlanders. But at least we can see in three dimensions. The Flatlanders, in contrast and by your own admission, could not see anything, not even in two dimensions. So the comparison becomes one between an impossible world where the inhabitants would have no awareness of anything versus our world where we are aware of a three dimensional universe.

Sagan's analogy is very similar to an analogy commonly use by mystics. I am using "mystics" in the strict sense to denote those who claim that they can directly experience God, or the divine nature, without the aid of the senses. In defense of their claims, mystics sometimes invoke the analogy of a man blind from birth who has no conception of colors. To make this parallel to Sagan's analogy more exact, we may posit a world of blind people who have never encountered a sighted person and who have no idea of what "vision" means.

Now, suppose a sighted person were to enter this world of the blind, rather like Sagan's 3-D apple enters the 2-D world of the Flatlanders, and suppose this intruder tries to explain his perceptions of color to the blind. Mystics claim that even though they won't understand what the sighted person is talking about, his perceptions are veridical nevertheless.

From here the mystic pushes his analogy home to its predetermined destination, namely, that though he cannot explain his direct experiences with God to people who are spiritually blind, his experiences are veridical nonetheless.

Despite the many holes in the mystic's analogy, it is actually better than Sagan's. This is so because the mystic's hypothetical analogue at least makes sense on its own terms. Sagan's Flatland, in contrast, is unintelligible.

The forurth dimension is not something "mysterious," not some mystical other world with mysterious properties....

Sagan expressly says that we cannot experience the fourth dimension or even imagine what it is. If this doesn't qualify as mysterious, I don't know what would.

Ghs

No, the only relevant part of the analogy is that just as two dimensional beings might imagine that they must either live on a flat map, which has an edge, or on an infinite map, with no edge, they might actually live on a finite surface with no edge, such as a sphere, the same applies to us. We might think that we live in a finite space with edges, or an infinite space, but we could live in a finite space with no edge, the surface of a hypersphere. There is absolutely no need to assume the biological feasibility of the creatures of flatland. All that is relevant is the dimensional analogy, not the cute anthropomorphization.

In saying that we cannot imagine (i.e., form an image of) such a thing, Sagan is simply saying that we cannot directly visualize it without distortion. Not that we cannot conceive of it. We cannot strictly imagine quantuum phenomena either.

Your repeated comparison of the with religion is, frankly, unhelpful. While the concept may be hard to grasp, no one asks you take the finite yet unbounded model on faith. You won't find an actual physicist or mathematician who claims it is self-contradictory, incoherent, or otherwise invalid. Carl Sagan, Stephen Hawking, and Martin Gardner are hardly mystics. Nor do I think your salvation rests on your understanding the model. It is simply a nice thing to understand because it is entirely consistent with observation, physical and mathematical models, and Rand's Aristotelian insistence on the finity of all actual existents.

The objection that space cannot not bent is based on two related errors. First, space is neither emptiness nor an abstraction. It is simply the sum field of all bodies. The concept of space can have a sense in which we omit the specific locations and trajectories, and extensions, and so forth of bodies. That is an abstraction. But space(time) itself is the sum of the fields of all physical entities. Space is indeed a real thing, if not a classical Newtonian solid body. I am no mathematical expert, but I would refer you to http://en.wikipedia..../Field_(physics)

The second problem is that Harriman objects to the bending of space because he wrongly analyzes space ontologically as a relation. But, strictly speaking, while things like location, or extension are relations, per se, space is not a relation, no more than a magnetic field is a relation. Saying that space is bent is perfectly meaningful in its sense.

Finally, you might also consider black holes as an aid toward considering the fourth dimension. Black holes are often imagined as funnel shaped punctures in a two dimensional representation of space. But one does not fall "down" a black hole as if there were some single down direction in outer space. Instead, from whatever direction one falls into a black hole, one falls forever into its center, withouit ever reaching it. Black holes stretch space in a fourth dimension, so that as one falls into one side of a black hole, one does not fall towards the other side, but away from all sides in a fourth spacial dimension.

Edited by Ted Keer
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No, the only relevant part of the analogy is that just as two dimensional beings might imagine that they must either live on a flat map, which has an edge, or on an infinite map, with no edge, they might actually live on a finite surface with no edge, such as a sphere, the same applies to us. We might think that we live in a finite space with edges, or an infinite space, but we could live in a finite space with no edge, the surface of a hypersphere. There is absolutely no need to assume the biological feasibility of the creatures of flatland. All that is relevant is the dimensional analogy, not the cute anthropomorphization.

If flatlanders can imagine a map of any kind, including a flat map, then they can imagine three dimensions. If you are going to use an analogy, then the analogue has to make sense. It has to be coherent on its own terms. The Flatlander analogue is not, and you cannot get around this problem by merely dismissing the "cute anthropomorphism." This problem has nothing do with with anthropomorphism per se; it has to do with the incoherence of the analogue itself. The points that Sagan wishes to illustrate simply cannot be illustrated with a 2-D model. He might as well use the analogue of a one-dimensional world, or a zero-dimensional world.

Moreover, it is not necessary to invoke a fourth dimension in order to imagine or explain a finite but boundless universe. The model of a three-dimensional sphere works just fine, and I have no problem with it in terms of its intelligibility. (Whether it is true or not is separate issue.)

Finally, you might also consider black holes as an aid toward considering the fourth dimension. Black holes are often imagined as funnel shaped punctures in a two dimensional representation of space. But one does not fall "down" a black hole as if there were some single down direction in outer space. Instead, from whatever direction one falls into a black hole, one falls forever into its center, without ever reaching it. Black holes stretch space in a fourth dimension, so that as one falls into one side of a black hole, one does not fall towards the other side, but away from all sides in a fourth spacial dimension.

Unless you can specify empirical tests by which your theory of black holes can be verified or falsified, then you have not presented a scientific theory at all. All you have given us is cosmological speculation, and highly questionable speculation to boot.

Ghs

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Your repeated comparison of the with religion is, frankly, unhelpful. While the concept may be hard to grasp, no one asks you take the finite yet unbounded model on faith.

The mystic need not ask us to accept on faith his direct and non-sensory experience of God. The point of his analogue of a land of the blind is to show how we cannot reject his claim out of hand as impossible merely because it does not comport with our normal experiences. Sagan invokes the Flatland analogy for the same purpose.

Even you have talked about us being trapped in a three-dimensional view of the universe (or something to this effect). This refers to the limitations of our sensory perceptions. The mystic says exactly the same thing. He argues that we should not assume that our senses can reveal everything that can be known about the universe.

As for corroborating evidence, the mystic will call our attention to the millions of people who have had similar experiences, including those born-again Christians who claim to have a personal relationship with Jesus.

I have no doubt that you find such comparisons "unhelpful." You might find them more helpful if you were willing to exercise the same skepticism about "scientific" claims as you exercise in the realm of religion.

Ghs

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No, the only relevant part of the analogy is that just as two dimensional beings might imagine that they must either live on a flat map, which has an edge, or on an infinite map, with no edge, they might actually live on a finite surface with no edge, such as a sphere, the same applies to us. We might think that we live in a finite space with edges, or an infinite space, but we could live in a finite space with no edge, the surface of a hypersphere. There is absolutely no need to assume the biological feasibility of the creatures of flatland. All that is relevant is the dimensional analogy, not the cute anthropomorphization.

If flatlanders can imagine a map of any kind, including a flat map, then they can imagine three dimensions. If you are going to use an analogy, then the analogue has to make sense. It has to be coherent on its own terms. The Flatlander analogue is not, and you cannot get around this problem by merely dismissing the "cute anthropomorphism." This problem has nothing do with with anthropomorphism per se; it has to do with the incoherence of the analogue itself. The points that Sagan wishes to illustrate simply cannot be illustrated with a 2-D model. He might as well use the analogue of a one-dimensional world, or a zero-dimensional world.

Moreover, it is not necessary to invoke a fourth dimension in order to imagine or explain a finite but boundless universe. The model of a three-dimensional sphere works just fine, and I have no problem with it in terms of its intelligibility. (Whether it is true or not is separate issue.)

Finally, you might also consider black holes as an aid toward considering the fourth dimension. Black holes are often imagined as funnel shaped punctures in a two dimensional representation of space. But one does not fall "down" a black hole as if there were some single down direction in outer space. Instead, from whatever direction one falls into a black hole, one falls forever into its center, without ever reaching it. Black holes stretch space in a fourth dimension, so that as one falls into one side of a black hole, one does not fall towards the other side, but away from all sides in a fourth spacial dimension.

Unless you can specify empirical tests by which your theory of black holes can be verified or falsified, then you have not presented a scientific theory at all. All you have given us is cosmological speculation, and highly questionable speculation to boot.

Ghs

You can remove the flatlanders entirely from the picture. All that is necessary is the analogy that, as a two dimensional surface can be finite yet have no edge if it is closed in on itself by being curved into a sphere in three dimensions, so, likewise, a three dimensional volume can be finite yet have no edge if it is closed in upon itself in a fourth spatial dimension. The relevant analogy is purely geometrical.

A simple spherical universe would either exist in an infinite empty space or be somehow bounded. The infinite space is incoherent. It remains for you to explain the nature of the boundary if you do not accept my solution, that three dimensional space bounds itself due to its being curved into the surface of a hypersphere.

Neither is a simple three dimensional spherical universe consistent with observation. Explain, if the universe is spherical, how we see the big bang surrounding us in every direction we look, while we are simultaneously moving away from it.

As for the black hole, are you telling me that unless I can prove to you that matter can fall into them from any direction, i.e., that their gravitational effect extends around them in all three dimensions, you are going to claim the privilege of assuming that they are flat holes in a two-dimensional poster universe? That sounds absurd, but I am not sure what else you might mean. My description of the black hole is not my speculation, it is the standard model.

I don't see the point as open to argument or in need of support. It is a matter of understanding the concept or not. It is not a question of values or other matters where human choice is involved. Its simply a matter of higher mathematics. If you have questions that would make the concept of a finite yet unbounded universe comprehensible to you, I will try to answer them. Or, if you do feel you can explain the finite yet unbounded theory in its strongest form, yet still disprove it, I would be interested in hearing an affirmative disproof.

I despair of being able to clarify the issue much further on line. Maybe a phone conversation would work, although I think it would really be necessary to have pen and paper at hand.

Edited by Ted Keer
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As for the black hole, are you telling me that unless I can prove to you that matter can fall into them from any direction, i.e., that their gravitational effect extends around them in all three dimensions, you are going to claim the privilege of assuming that they are flat holes in a two-dimensional poster universe? That sounds absurd, but I am not sure what else you might mean. My description of the black hole is not my speculation, it is the standard model.

I said nothing about two-dimensional black holes. What I said was very clear. I said that unless you can specify empirical tests by which we can verify or falsify the standard model of black holes, then the theory is not a scientific theory. It is cosmological speculation.

And you said a lot more about black holes than you indicate here. In an earlier post you wrote: "Black holes stretch space in a fourth dimension, so that as one falls into one side of a black hole, one does not fall towards the other side, but away from all sides in a fourth spacial dimension." Okay, what empirical tests have been devised to verify or falsify this theory? Is it even falsifiable in principle? If so, what would falsify it?

I don't see the point as open to argument or in need of support. It is a matter of understanding the concept or not. It is not a question of values or other matters where human choice is involved. Its simply a matter of higher mathematics.

Exactly. You are speaking of mathematical models, and a mathematical model doesn't necessarily correspond to something in the real world. You need empirical evidence to establish a correlation. So what is the empirical evidence that when something falls into a black hole it falls away from all sides into a fourth spacial dimension? Given the density of black holes, such that not even light can escape, how do we know what happens inside of one?

Cosmological speculation is fine -- it can be very entertaining, provoke people to think, and provide good material for science fiction movies -- but let's not confuse speculation with science.

I could explain all this more easily over the phone, but you would need to take notes.

Ghs

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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 light-years in diameter.

Ted,

Thanks for taking the time to explain your viewpoint, but I'm afraid I just do not find what you say terribly helpful.

I took a look at Hawking’s Universe in a Nutshell at Amazon, and I would have to say I am less than impressed. Hawking endorses the Big Bang theory and speaks of notions such as “boundaries of space-time.” He obviously endorses the idea that existence suddenly appeared out of nonexistence at the point where time began. I don’t think I am going to find that line of reasoning helpful.

And I also agree with George that Sagan’s explanations of “finite but unbounded’ make no sense to me whatever.

The problem seems to be that, once we start talking about the dimensions of the universe--as with your above example of "billions of light years in diameter"--then we are talking about boundaries. If curved space somehow translates to a constantly expanding universe, we are still talking about boundaries--boundaries which constantly expand.

And once we posit a boundary, we have to ask--what is beyond the boundary? Well, obviously nothing is beyond the boundary. But nothing does not exist. That's why I contend that it makes more sense to leave open the possibility that--just as time is eternal--the universe is, in fact, infinite, even though we have difficulty conceiving of what that means.

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You can remove the flatlanders entirely from the picture. All that is necessary is the analogy that, as a two dimensional surface can be finite yet have no edge if it is closed in on itself by being curved into a sphere in three dimensions....

I hereby pronounce Sagan's Flatlander analogy officially dead. After a long and fitful illness, it finally succumbed to the Death of a Thousand Qualifications. :)

(This expression was coined by Antony Flew, btw.)

Ghs

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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).

Here are some of the conceptual problems I have with the question Is the universe infinite?

First, what do we mean by "the universe"? Well, when I use this term, I mean "everything that exists." So the original question becomes Is everything that exists infinite?

I can't honestly say that I even know what the question is supposed to mean. The most obvious interpretation translates into the question Is there an infinite number of existents in the universe? The problem here is that "infinite," as used in this context, is not a number. On the contrary, it signifies that no specific number can be assigned. And if this is the case, then it makes no sense to speak of "everything" that exists, because there is no "every" or "all" of which we can speak.

Second, when most philosophers have spoken of an "infinite universe," they have meant that nonexistence (i.e., empty space) has no limitations. The term "space," when used in this sense , is not a thing with specific properties. It is nothing -- and nothing has no characteristics and therefore no limitations.

Confusion inevitably arises when we reify nonexistence and speak of it as if it were a type of existence. This most often occurs when we speak of space -- again, in the sense of empty space --as being "infinite." This can give the thoroughly misleading impression that "infinity" is an attribute of an existent known as "space" -- whereas all we really mean is that nonexistence has no boundaries. We can no more limit nonexistence than we can lasso nonexistence or cook nonexistence for dinner.

Ghs

George,

You argue that infinity “signifies that no specific number can be assigned. And if this is the case, then it makes no sense to speak of 'everything' that exists, because there is no 'every' or 'all' of which we can speak.”

This strikes me as confusing mathematics with physical reality. If everything means the “totality” of everything, then there should be a total we can theoretically point to. But suppose that mathematical concept (total) is not applicable to the universe as a whole, or everything which exists. Because it applies to all the things we’re familiar with, does not mean it necessarily applies to existence, as such. Again, it's the fallacy of composition.

I’m not sure what you mean when you say “nonexistence has no boundaries” or “limiting nonexistence.” I may have misunderstood you, but this seems to amount to “speaking of nonexistence as if it were a type of existence.” I have a hunch this may just be my misunderstanding of your views. In any case, let me state clearly my own view that you cannot do anything with nothing. Nonexistence does not exist. Period. And there is no such thing as empty space, because that would amount to nothing having the quality of existence—a contradiction in terms.

It is because nothing cannot exist that the idea of the universe being finite seems absurd to me. And “finite but unbounded” is equally absurd (and, based on the Sagan excerpts, apparently defies coherent explanation). If the universe is finite, it has dimensions. If it has dimensions, it has to have boundaries. And if it has boundaries, then something has to exist on both sides of those boundaries—but nothing does not exist.

Repeating the last paragraph of my response to Ted:

And once we posit [or logically imply] a boundary, we have to ask--what is beyond the boundary? Well, obviously nothing is beyond the boundary. But nothing does not exist. That's why I contend that it makes more sense to leave open the possibility that--just as time is eternal--the universe is, in fact, infinite, even though we have difficulty conceiving of what that means.

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Endless Existence

To affirm that existence exists is to imply the reality of act, the act of recognition. To affirm that existence exists implies that existence endures. Duration is inseparable from concrete existence (to borrow from Lambert).

To affirm that existence is identity is to affirm reality of both the actual and the potential in existence. Therefore, again, duration is inseparable from concrete existence. Notice also that the fundamental principle for the comprehension of existence, the principle of non-contradiction, requires a temporal clause.

In the present stage of advancement of our physics, all physical existents possess mass-energy, stand in relations of spacetime, and have associated De Broglie wave features. Every body (gas, liquid, solid, . . .) possessing rest mass affords a physical frame of reference for clocking a time known as its proper time. All other physical existents stand in spacetime relations to any such clock. This much physics and the metaphysics of the previous two paragraphs are in tune.

To say that any and all knowledge is an affirmation, at least implicitly, of the axiom existence exists is to decide as false any conclusion affirming a concrete neither enduring nor standing in concrete relations to enduring concretes. Metaphysics can say a little beyond that concerning existence and time: Existence exists. Existence is identity. If no existence at all, no identity at all. Then if all existence came into existence, it could not do so in a specific way. Such a coming into existence would be without identity, without existence.

That argument of mine (with help from Parmenides) shows only that past time of existence is endless. Perhaps someone here has a metaphysical argument to the conclusion that future time of existence is endless. (One approach would be to try to demonstrate that the possibility of an end to future time of existence implies the possibility of a beginning of past time of existence. The latter possibility being false, so would be the former. I suggest, however, that before rushing to press with any kind of metaphysical argument, it would be wise to assimilate the article “Could Time End?” by George Musser in the September issue of Scientific American.)

Rand did not commit to print much about space. She wrote in “For the New Intellectual” that space and time and existence were among basic concepts arising from experience; she repudiated Kant’s view that their referents are not in and from reality, that they are a priori forms of perception (space and time) or a priori categories of the understanding (existence). In Atlas Shrugged, she had repudiated mystical claims of a mode of being superior to existence on earth, a mode of being they call “‘another dimension’, which consists of denying dimensions” (1035). Rand did maintain in print the thesis that existence always has existed and always will exist (MvMM 25). She had earlier implied the eternity of existence, in the Anthem line “I know not if this earth on which I stand is the core of the universe or if it is but a speck of dust lost in eternity.”

From Between Metaphysics and Science:

There were supposedly irresolvable conflicting dogmatic, reasoned answers on what Kant took to be questions for reason alone, with no possible settlement by empirical confrontation. Some of these are proving to be partly susceptible to modern science after all. I am referring to Kant’s Antinomies of Pure Reason * in the Dialectic part of Critique of Pure Reason (A420–61 B448–89; Bennett 1974, 114–227; Grier 2001, 172–229; Abela 2002, 217–30). Under Rand’s philosophy, part of the first antinomy (whether the world had a beginning and whether space is endless) can be settled partly by metaphysics and partly by science.

Dennis opened the present thread this way:

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.

Here is the argument, from the First Antinomy, that Kant presents in support of the proposition that the world (all existence) has no beginning.

For suppose it has a beginning. In that case, since the beginning is an existence preceded by a time wherein the thing is not, a time must have preceded wherein the world was not, i.e., an empty time. In an empty time, however, no arising of any thing is possible; for no part of such a time has, in preference to another part, any distinguishing condition of existence rather than non-existence (whether one assumes that the world arises of itself or through another cause). Hence although many a series of things can begin in the world, the world itself cannot have a beginning and hence is infinite with regard to past time. (A427 B455)

This argument can join in a friendly handshake with the one I gave along lines of Rand’s metaphysics. This argument that Kant has set up for his critique is good in its affirmation that time is not apart from existence. Indeed, I’ll say this argument is just fine. The way in which it relies on Leibniz is fine by me. There is reliance on the principle of mathematical induction in this argument, but that is fine too.

What does Kant think of this argument? He thinks it prima facie unobjectionable, but stresses that it has some sense only as it pertains to the phenomenal world (mundus phenomenon), not to the contrasting intelligible world (mundus intelligible) (A431–33 B459–61). In Rand’s metaphysics, the world is the world given in perception, and all intelligibility and intelligible structure stands in definite derivative relations to that one world.

Kant would dispute the soundness of the argument and the truth of its conclusion as concerns such a single world. His method of proof in the Antinomies is the indirect method: assume the denial of the thesis, and show this leads to contradiction. His explanation for how both ~A can be self-contradictory while also its antithesis ~~A can be self-contradictory is by his claim that the object world (all existence) is not a possible object of experience. Both A and ~A are false in the final analysis because they and their proofs attempt to go beyond the bounds within which non-contradiction and logical bivalence limit possibility (further, Abela 2002, 217–27).

Objectivism diverges from Kant in that analysis. Objectivism must show the argument for the contrary thesis—the world has a beginning in time—to be incorrect. The argument Kant sets up is as follows:

For assume that the world has no beginning as regards time. In that case, up to every given point in time an eternity has elapsed and hence an infinite series of successive states of things in the world has gone by. However, the infinity of a series consists precisely in the fact that it can never be completed by successive synthesis. Therefore an infinite bygone world series is impossible, and hence a beginning of the world is a necessary condition of the world’s existence. (A426 B454)

This argument is no good. There can be an infinite set (say, on the order of the rational line Q) of possible planes parallel to and in between the window to my right and the one to my left. My inability to conclude a counting of such planes does not show they cannot exist. Likewise, my inability to complete counting of past times does not show they cannot exist. So the real reason Kant was able to mount the contradiction “‘~A is self-contradictory’ and ‘~~A is self-contradictory’” is because ~A is not really self-contradictory. (See further, Grier 2001, 183–94.)

References

Abela, P. 2002. Kant’s Empirical Realism. Oxford.

Bennett, J. 1974. Kant’s Dialectic. Cambridge.

Grier, M. 2001. Kant’s Doctrine of Transcendental Illusion. Cambridge.

Kant, I. 1781, 1787. Critique of Pure Reason. W.S. Pluhar, translator. Hackett.

~~~~~~~~~~~~~~~~

Ted, the last couple of decades, the consensus in physics has been that the large-scale structure of the universe is flat, which is open. As you likely know, the mass-energy of the universe is thought to be finite, indeed constant, forward and backward in time.

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Ted, the last couple of decades, the consensus in physics has been that the large-scale structure of the universe is flat, which is open. As you likely know, the mass-energy of the universe is thought to be finite, indeed constant, forward and backward in time.

You are not distinguishing between closure in time and closure in space. At any time in the history of the universe since the big bang, space has been closed and finite. Time is also finite, and closed in the past. It had been expected that gravity would be strong enough to end the expansion of the universe at some point in the future, causing it to shrink back in a big crunch. This would be considered a positive curvature of time in spacetime, It is now believed that space will continue to expand without end, at least until the heat death of the universe. This makes spacetime a cone (or similar shape) open and flat in the time dimension, but curved and closed in the space dimension. Even with an open universe, time will end, will in effect "evaporate" once all the protons have decayed and change ceases.

In the image below, time is open and flows from the big bang on the left smoothly on into the foreseeable future. Space is closed and finite, represented by the circumference of the cone at any given time. Spacetime is flat in the time dimension, but closed at any actual instant. in time.

While the finite yet unbounded model can be reduced to its perceptual underpinnings, and is consistent with both observation and logic, infinite universe models are based on negative definitions and floating abstractions. Infinite models of actual reality are unobservable, incoherent, unconfirmable, and undefined.

CMB_Timeline300.jpg

Edited by Ted Keer
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Ted, the last couple of decades, the consensus in physics has been that the large-scale structure of the universe is flat, which is open. As you likely know, the mass-energy of the universe is thought to be finite, indeed constant, forward and backward in time.

You are not distinguishing between closure in time and closure in space.

CMB_Timeline300.jpg

Space-time is a total manifold. Space and time are not separate. The space-time manifold cannot be reduced to the cartesian product of a spatial manifold and a linear time space.

Ba'al Chatzaf

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Ted, the last couple of decades, the consensus in physics has been that the large-scale structure of the universe is flat, which is open. As you likely know, the mass-energy of the universe is thought to be finite, indeed constant, forward and backward in time.

You are not distinguishing between closure in time and closure in space.

CMB_Timeline300.jpg

Space-time is a total manifold. Space and time are not separate. The space-time manifold cannot be reduced to the cartesian product of a spatial manifold and a linear time space.

Ba'al Chatzaf

You can distinguish between spacelike and timelike events. Yes, you cannot actually measure all of space instantaneously, because it takes time for what is happening elsewhere to arrive at the observer's location. Hence one cannot measure space without implicitly making a statement about time.

But the question is this: Is a finite unbounded model of the universe mathematically coherent and scientifically meaningful, and is it consistent with the observations?

What do you say, Bob?

Edited by Ted Keer
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Endless Existence

Rand did not commit to print much about space. She wrote in “For the New Intellectual” that space and time and existence were among basic concepts arising from experience; she repudiated Kant’s view that their referents are not in and from reality, that they are a priori forms of perception (space and time) or a priori categories of the understanding (existence). In Atlas Shrugged, she had repudiated mystical claims of a mode of being superior to existence on earth, a mode of being they call “‘another dimension’, which consists of denying dimensions” (1035). Rand did maintain in print the thesis that existence always has existed and always will exist (MvMM 25). She had earlier implied the eternity of existence, in the Anthem line “I know not if this earth on which I stand is the core of the universe or if it is but a speck of dust lost in eternity.”

Stephen,

In the podcast referenced in my initial post, Peikoff clearly does state that Ayn Rand believed the physical universe does have boundaries. You wrote a very interesting piece here, but I am not clear if you agreed or disagreed with the premise that the universe has such boundaries, or how you wound answer questions about the implications of such boundaries for empty space (i.e., nothingness).

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What do you say, Bob?

If the average curvature of the Cosmic space-time manifold is positive then it is finite in both its spatial dimensions and temporal dimension. Rather like the surface of a sphere which as constant positive curvature. However, there is strong empirical evidence that the curvature of the Cosmos is slightly negative or tending to zero (this is rather hard to measure) and so is probably infinite spatially and temporally in the limit. My guess is that the spacial part is growing without bound and at an ever accelerating pace. The the spatial limit in infinite time is infinite. I.E. the spatial projection will grow without bound which is bad news since there seems only to be a finite amount of energy in the Cosmos. That means the Cosmos will get thinner and more feeble as time passes. The lights will go out and it will become cold and dead. Enjoy it while you can.

Ba'al Chatzaf

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Endless Existence

Rand did not commit to print much about space. She wrote in “For the New Intellectual” that space and time and existence were among basic concepts arising from experience; she repudiated Kant’s view that their referents are not in and from reality, that they are a priori forms of perception (space and time) or a priori categories of the understanding (existence). In Atlas Shrugged, she had repudiated mystical claims of a mode of being superior to existence on earth, a mode of being they call “‘another dimension’, which consists of denying dimensions” (1035). Rand did maintain in print the thesis that existence always has existed and always will exist (MvMM 25). She had earlier implied the eternity of existence, in the Anthem line “I know not if this earth on which I stand is the core of the universe or if it is but a speck of dust lost in eternity.”

Stephen,

In the podcast referenced in my initial post, Peikoff clearly does state that Ayn Rand believed the physical universe does have boundaries. You wrote a very interesting piece here, but I am not clear if you agreed or disagreed with the premise that the universe has such boundaries, or how you wound answer questions about the implications of such boundaries for empty space (i.e., nothingness).

Dennis,

I do reject the idea that Rand rejected, the idea that there could be empty time during which the universe did not exist and then at some time it came into existence. No time in which time was the only existent. This is a matter settled by metaphysics, as in the piece I wrote for you. This is part and parcel of Rand’s metaphysics.

That is not to say there could not be a first instant of time and that in that instant the universe with all its mass-energy existed. It would still be the case that there was no time at which the universe did not exist, no time that existence in time did not exist. And no time devoid of existence.

The reason we think seriously about the possibility in the preceding paragraph—a cosmological singularity—is due to physics, specifically classical GR (with possible upset from future quantum-gravity understanding of energy and spacetime near the initial singularity left suspended). Those are matters for physics, not metaphysics. They originated in physics and will be settled by physics.

In her periodicals, Ayn Rand was able to publish any of her ideas she pleased. What she was only willing to say, but not commit to print, should be taken as unsettled opinion. If it can be shown that some unpublished stand on an issue follows logically and uniquely from her metaphysics, then the view is part of that metaphysics.

I do not agree with the idea that an empty space would be nothing. Whether there can be regions of space that are absolutely empty is a matter for physics, not metaphysics.

I did not take up questions on unboundedness and finitude of space and spacetime as it would take at least several days to refresh my GR and modern cosmology to give clear, informative answers. I cannot take that up at this time. It is possible in the distant future that I will take up writing that final section of “Space, Rotation, and Relativity” that I never delivered in Objectivity—the section on GR—and that the modern cosmological models can be discussed therein.

In the year that Einstein published his theory of general relativity, he also published the popular account Relativity: The Special and General Theory. That was in 1916. An English translation, with some additions, appeared in 1920. On the centenary of Einstein’s 1905 inauguration of relativity theory, a special reissue of this work was produced.* (It is the 1920 version, not the later, expanded edition of 1954.) In this work, Einstein gives the generally educated reader chapter 31 “The Possibility of a ‘Finite’ and Yet ‘Unbounded’ Universe.” In the commentary for this special edition, written by Robert Geroch, you will find in the section “Cosmology” an outline of how finite-but-unbounded models have become contenders, noncontenders, and back with advancement of GR data and development, with advancement of Einstein’s light.

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I do not agree with the idea that an empty space would be nothing. Whether there can be regions of space that are absolutely empty is a matter for physics, not metaphysics.

Well, regardless of whether it is technically a philosophical issue, it was definitely an issue to which Ayn Rand devoted some attention. I saw Nathaniel Branden deliver an NBI lecture in New York in 1967, and in the Q & A period he made the statement that “I can prove, and Ayn Rand can prove, that there is no such thing as empty space, or a void.” And, as I recall, he equated such a void with nothingness. He declined to offer the proof at the time.

At a Jefferson School conference in the late 80s, Harry Binswanger attempted to offer such a proof, or at least a validation. It was essentially this: The whole idea of space, as most people understand it, is invalid. The concepts of place, position and region are all entity-based. Even on a subatomic level, the space between particles or quantums is a field—not empty space. The place of something is defined by the entities which share its boundaries. In the alleged void, where there is nothing, there can be no distinguishable “places.” If you say that the void exists between point A and point B, the distance between those points can have no identity, and therefore there is no way to say that point A is at a different place from point B.

That’s a summary based on my notes. One of these days, I will take the time to listen to the lectures again to see if I missed something. Unless, of course, everyone at OL takes one look at my present explanation and is so impressed that no further debate is needed. :P

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That's a summary based on my notes. One of these days, I will take the time to listen to the lectures again to see if I missed something. Unless, of course, everyone at OL takes one look at my present explanation and is so impressed that no further debate is needed. :P

Put a mirror at B. Shine a light from A to B and record the time until the reflection is detected. This time divided by the speed of light is the distance between A and B. Light does not need a medium to get from here to there so the space could be empty.

There is no aether that fills space. This has been known since 1887 when the negative result of the Michelson-Morley interferometer experiment indicated that there is no medium to carry light. This experiment has been done year after for 123 years with ever improving technology and the same negative result. No aether. Space is mostly empty.

You will rely on philosophy. I will rely on careful experimentation and measurement. Who do you think will profit the most?

Ba'al Chatzaf

Edited by BaalChatzaf
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