md65536 Posted May 2, 2011 Posted May 2, 2011 (edited) If we again consider the two dimensional, positive curvature surface of a sphere, I think we can both agree that even though two dimensional, it must curve in a higher dimension ( the third ) to reconnect to itself. Count me out of all this agreement. I don't think you need to add dimensions to figure any of this out. If we have a spherical shell in 3 dimensions, and we remove all distance, we get a degenerate sphere which is a single point in 3 dimensions. Isn't it also a 2D surface? Doesn't this still describe topologically a 2D manifold that reconnects to itself in 2 dimensions? You don't need a 3rd dimension to describe this. I might be wrong. ... Similarly, we can describe a closed infinite space without the need for a 4th dimension. How? Using spacetime curvature. Imagine we were able to arbitrarily curve spacetime, so that we could arbitrarily map distances in one view of the universe, to different distances in another view. Then suppose you have an infinite ruler in your view of space that counts distance from you in intervals of 1 meter (starting with 0, 1, 2... say). Suppose I map interval number n on that ruler, with a length of 1m in your view, to a length of 1/2^n in my view. The infinite ruler in your view will map to a length of 2 m in my view. I could fit your infinite universe into a bounded sphere and keep it in my garage. Then, you could travel indefinitely along this ruler, while I see you slowing and approaching 0 velocity as you approach the boundary of the sphere in my garage. If you think this "arbitrary mapping" has no connection to reality, let's toss something into a black hole and see what happens. (Empty space should curve in the opposite way so it may be a bad example.) ... But the thing is, you're not going to have a model of the universe that's completely flat, while I have one that is closed. We're all going to have equivalent models of the universe, but we're also all going to have different views of the universe, that appear differently curved depending on our location, velocity, gravity field, and whatever else. So as you travel along an imaginary ruler, distant space will not appear to stay flat while allowing an actual infinite ruler to remain infinitely long. I have a to get a bit vague cuz I've passed the limits of my knowledge, but I imagine that a part of space that appears to stretch to infinity (ie to the horizon) from one location, might appear arbitrarily small from another location. This means that instead of you having an infinite universe that can be curved to fit in my garage, rather you have an infinite universe that can fit in your own finite universe. As you move through this seemingly infinite space, it keeps changing its appearance, so that it can always appear infinite, and yet always fit within a finite 3 dimensional space. Another way to think about it is this: Take an infinite cosmological horizon, and imagine squeezing that into an arbitrarily small length. You have just imagined connecting space that is infinitely far away in one direction with space infinitely far away in an opposite direction, into a closed manifold, without resorting to adding extra dimensions. Instead of "curling" it into extra dimensions, just stretch across 3 dimensions. If this is too vague, it's because I don't really know what I'm talking about! But I hope that I sound like I do , enough that you can imagine and contemplate. ... I don't mean to derail this thread with black holes or unnecessary complications, but my opinion is this: You don't need extra dimensions to curve an unbounded space and connect it to itself. You can do that by curving space in whatever dimensions you're already working with. ??? Edit: And now that I've said all that, I'm wondering... doesn't spacetime curvature not mean just stretching, but actually curving into the 4th dimension so that the reason a 3D distance looks stretched or compressed is that the distance includes an unseen time component? Which makes me kinda a little bit totally wrong? Edited May 2, 2011 by md65536
owl Posted May 2, 2011 Posted May 2, 2011 (edited) How about a little basic Euclidean geometry for openers into what space is, what "shape" it has (or not) and whether or not it is infinite. And before anyone blasts me with an anti-Euclidean, "we are past all that" lecture, please review the Ross paper on the Ontology and Cosmology of Non-Euclidean Geometry. at http://www.friesian.com/curved-1.htm... and my selected quotes from post 47. Before a "fourth spacial dimension" was invented, still with no ontological referent in the real world beyond math and abstract geometrical concepts, there were three spacial dimensions and one time "dimension" (which is simply event duration)... 1: the line (length with no area), 2: the plane (flat area) and 3: volume or space, whether defined with boundaries (all possible bounded volumes with whatever shapes) or endless, infinite space (without any boundary.) So, right off the bat, "flat space" makes no sense unless it is confined to a 2-D plane. Likewise "curved space" makes no sense in basic Euclidean unless describing a 3-D object like a curved surface, and the surface is curved, not "space." Review, Ross: What "curvature" would have meant to Euclid is now "extrinsic" curvature: that for a line or a plane or a space to be "curved" it must occupy a space of higher dimension, i.e. that a curved line requires a plane, a curved plane requires a volume, a curved volume requires some fourth dimension, etc. A 4th spacial dimension still makes no sense. Space, whether confined (bounded as defined geometric shapes) or infinite, is fully described by three axes. A fourth is superfluous... a product of imagination. I'll go to specific replies now. csmyth3025As to the final question, I think that it's already been pointed out that if the universe is infinite then for any two points (pieces of matter) that are distant from each other there will always be an infinite number of points even more distant. For all we know there might be a brick wall out there somewhere enclosing our universe. Don't try to trick me with your "...what lies beyond that wall..." question 'cause it's bricks all the way out from there! Chris To first comment: Absolutely! Second: Cute. A solid brick universe, or infinitely thick "wall!" MigLOwl, it seems like your objection to a finite but unbounded universe is due to the fact that there must still be an outside to it, since it needs to curve in on itself in another volume. No boundary (unbounded) means infinite, unless specially defined and ontologically ungrounded... like "4-d space." There is no limit to the space outside any defined sphere. What "curves" if space is just volume, regardless of what it contains or how empty it is? Stuff has shape, curved trajectories, etc., not empty space. If we again consider the two dimensional, positive curvature surface of a sphere, I think we can both agree that even though two dimensional, it must curve in a higher dimension ( the third ) to reconnect to itself. Similarily a three dimensional volume can also curve back in on itself and be bounded, but it must do so in a higher dimension, ie a fourth. See above simple Euclidean geometry. (Non-Euclidean is still math/model/conceptual, as per the Ross ontology.) A sphere is 3-D geometry. 4-D space is a pseudo-science myth. No agreement. Assume the two dimensional, flat ( euclidian ) space with no bounds analogy that you seem to prefer to the curved sphere, I know "flat space" is in common usage, but it invents a new meaning for "flat," as I explained above. The point is that your main objection, that there is an outside, is always there, whether dealing with 'flat' space or positive or negative curvature space. This outside is only in a higher dimension though and as such, is inconsequential. No. "It all" exists in 3-D space. No 4th required. As above, three axes describes all volume. You can add time, but to be clear, I would not call it a "dimension" but just the movement factor... that "it takes time" for all stuff to move around, and it is a dynamic, not a static snapshot universe. Back later for further replies. PS: Btw, needless to say, a point has no dimension, but is the 'starting point' for geometry. (A point can not "contain" anything... including Hawking's primordial 'singularity', which he said is "infinite mass density in a point of zero volume." Totally absurd of course, even if Hawking said it.) Edited May 2, 2011 by owl -1
Iggy Posted May 3, 2011 Posted May 3, 2011 (edited) A 4th spacial dimension still makes no sense. "intrinsic curvature" means curvature without a higher dimension. Space-time has intrinsic curvature and there is no 4th spatial dimension. Your own link tells you this. So, please, just stop Owl. You are not making a coherent argument. Edited May 3, 2011 by Iggy
owl Posted May 3, 2011 Posted May 3, 2011 Iggy: Iggy:Space-time has intrinsic curvature and there is no 4th spatial dimension. Similarly, we can describe a closed infinite space without the need for a 4th dimension. Space-time is now an established entity with intrinsic curvature? Inform the International Society for the Advanced Study of Spacetime! Write a paper on it and maybe you can deliver it at their next conference. ..."and there is no 4th spatial dimension." Didn't I just say that... more than once? Define "closed." (No, don't. Just explain what lies beyond the "closure" or "form" on whatever scale.) Again, I do not advocate (and never have) a 4th dimension. Still batting zero in communication here, as always with you. Why don't you just leave it alone if you can't understand it? -1
zapatos Posted May 3, 2011 Posted May 3, 2011 Perhaps nothing lies beyond the closure. Not the nothing like empty space, but the nothing of it being a meaningless question. Similar to the question of "What is in the basement of a house that has not yet been built?".
md65536 Posted May 3, 2011 Posted May 3, 2011 (edited) Perhaps nothing lies beyond the closure. Not the nothing like empty space, but the nothing of it being a meaningless question. Similar to the question of "What is in the basement of a house that has not yet been built?". Nothing must lie beyond the closure. Nothing that can be considered part of this universe. Otherwise, the closure doesn't enclose the universe, which means it's not a closure. Owl, you're the only one talking about space being an entity, as far as I can see. You're arguing against imaginary foes who, apparent only to you, are arguing that it is. Think of it only as a measure. If it is curving or being stretched etc, it is the measure that we're talking about. If matter and energy is in that space, the measure affects their shape and size, but that doesn't mean that the space itself is an entity. You can measure the distance (and the curvature) of the space between earth and moon but that doesn't imply there's an "entity" attaching one to the other. Disclaimer: I'm not a physicist. If the following doesn't make sense, it's not you, it's me. Imagine taking some infinitesimal distance, and stretching it to infinity. I think this is what you'd have beyond the "closure" of the universe. This would mean that there is no distance at all beyond the edge of the universe. The distance beyond it is ZERO. Also there should be no abrupt change between an "inside" and "outside", as in a membrane on one side of which there is distance and on the other side there is no distance. It would probably have to be a "soft" or continuous, gradual closure, which you could not possibly travel to. I think that this "no distance" or "infinitesimal distance stretching to infinity" describes infinitely open curved space, and if you approached a potential "edge of the universe" beyond which there is infinite open curvature, local space would still (as always) be flat, meaning that it would have to appear that space continues on for additional distance (smoothly going between your flat space and the infinitely open space beyond). By approaching the edge of space, you would necessarily move the edge of space. It only makes sense (to me at least!) because we're not talking about an "entity" here. Edited May 3, 2011 by md65536
MigL Posted May 3, 2011 Posted May 3, 2011 The fact that you find something absurd, or that Hawking said it, is not a valid argument against the existence fo higher dimensional topology. Otherwise you could disprove all of Quantum Mechanics.
DrRocket Posted May 3, 2011 Posted May 3, 2011 "intrinsic curvature" means curvature without a higher dimension. Space-time has intrinsic curvature and there is no 4th spatial dimension. Your own link tells you this. So, please, just stop Owl. You are not making a coherent argument. Herein "manifold" means differentiable manifold of class [math]C^\infty[/math]. Intrinsic curvature or just plain curvature, refers to any of several curvature tensors, most commonly the Riemannian curvature tensor, or in the case of GR the Einstein tensor, that are defined in terms of a connection or metric on the manifold without reference to any embedding of the manifold in any larger larger space (Euclidean space or vector space with metric of apppropriate signature). The key concept is that of a manifold, which is defined independently of any embedding space. There are (difficult) theorems that show that any Riemannian or pseudo-Riemannian manifold can be realized as an embedded manifold (Nash embedding theorem and later generalizations) in a space of suitably high dimension, but these theorems are not useful in GR and actually detract from a deep understanding. The general definition of a manifold is intrinsic, and that, rather than curvature, is what is really intrinsic. Cuirvature (perhaps 0 curvature) is just part of the package. Thinking of curvature in terms of an embedding takes you down the wrong path. It has nothing to do with the curvature of a manifold. For instance, all smooth curves (1-dimensional manifolds) are flat when arc length provides the metric.
owl Posted May 3, 2011 Posted May 3, 2011 Perhaps nothing lies beyond the closure. Not the nothing like empty space, but the nothing of it being a meaningless question. Similar to the question of "What is in the basement of a house that has not yet been built?". "No thing" means emptiness... empty space. Beyond any cosmic "structure," shape, or form you can imagine is more space, whether more stuff exists in it or or not. The finite mind has always had trouble "grasping" or comprehending infinity... endless space, but that doesn't make the universe finite just because our reach of mind may be finite and "create" a finite universe. md65536: Owl, you're the only one talking about space being an entity, as far as I can see. You're arguing against imaginary foes who, apparent only to you, are arguing that it is. Think of it only as a measure. If it is curving or being stretched etc, it is the measure that we're talking about. If matter and energy is in that space, the measure affects their shape and size, but that doesn't mean that the space itself is an entity. You can measure the distance (and the curvature) of the space between earth and moon but that doesn't imply there's an "entity" attaching one to the other. I have always argued that space (and time and spacetime) are not entities. Things are entities. Some argue that relationships between things are entities too... as in the substantive vs relational spacetime debate, which I am not going into here. If you posit that something is being curved or stretched (space in this case), it is up to you to explain what that something is in a coherent ontological argument. That is why the spacetime ontology argument is still ongoing with no consensus in sight. Imagine taking some infinitesimal distance, and stretching it to infinity. I think this is what you'd have beyond the "closure" of the universe. This would mean that there is no distance at all beyond the edge of the universe. The distance beyond it is ZERO. Distance usually means the linear measure of space between two points or objects. But does this mean that there is no space beyond things between which we can measure? No. As Chris astutely pointed out above: csmyth3025As to the final question, I think that it's already been pointed out that if the universe is infinite then for any two points (pieces of matter) that are distant from each other there will always be an infinite number of points even more distant. MigL: The fact that you find something absurd, or that Hawking said it, is not a valid argument against the existence fo higher dimensional topology. Otherwise you could disprove all of Quantum Mechanics. My "point" was that a geometric point is just a locus without dimension. Specifically a point has no volume, which is required to "contain" anything... like all the matter in the cosmos squeezed into such a point of zero volume, as per Hawking's absurd statement quoted above (re: the ultimate singularity.)
zapatos Posted May 3, 2011 Posted May 3, 2011 "No thing" means emptiness... empty space. Beyond any cosmic "structure," shape, or form you can imagine is more space, whether more stuff exists in it or or not. So if I ask what is in the basement of a house that has not yet been built, you believe 'empty space' to be the correct answer? Or are you just ignoring my suggestion that the question may be meaningless? The finite mind has always had trouble "grasping" or comprehending infinity... endless space, but that doesn't make the universe finite just because our reach of mind may be finite and "create" a finite universe. I am almost speechless. You are happy to use the argument on me that perhaps I just can't comprehend some things and therefore should not rule out the possibility that my idea is wrong. That I should keep an open mind. Yet when I suggested that to you previously, you barely took the time to ignore me. So back at you. Just because you cannot imagine finite space, that does not make it infinite.
md65536 Posted May 3, 2011 Posted May 3, 2011 I have always argued that space (and time and spacetime) are not entities. Things are entities. Some argue that relationships between things are entities too... as in the substantive vs relational spacetime debate, which I am not going into here. If you posit that something is being curved or stretched (space in this case), it is up to you to explain what that something is in a coherent ontological argument. That is why the spacetime ontology argument is still ongoing with no consensus in sight. No, we can speak about the properties of something (spacetime) without explaining anything more about what it is. That is EXACTLY what makes it not an entity. It doesn't have an existence independent of its properties. We don't need to discuss anything besides its relevant properties. Perhaps you falsely assume that if something has properties, then it is an entity. I think the ontology of spacetime is only relevant to this discussion if it IS an entity. Let us assume it is not, since the alternative would be a needless sidetrack to the discussion. Accept that spacetime can have properties (like length) without getting bogged down in the question of "what IS spacetime?" The most useful answer for this discussion is "nothing". Only its properties matter. Either that's a consensus, or a consensus on that point is not needed for the discussion to progress. My "point" was that a geometric point is just a locus without dimension. Specifically a point has no volume, which is required to "contain" anything... like all the matter in the cosmos squeezed into such a point of zero volume, as per Hawking's absurd statement quoted above (re: the ultimate singularity.) I don't see why something that has no volume can't contain other things that have no volume. To me this is key to understanding a 2-dimensional manifold without needing to envision it existing in a 3-dimensional volume (ie. "embedded in 3-space" if I understand DrRocket).
owl Posted May 4, 2011 Posted May 4, 2011 I'll start with your last post and then go to previous unanswered posts. name='md65536' timestamp='1304452939' post='605076']No, we can speak about the properties of something (spacetime) without explaining anything more about what it is. That is EXACTLY what makes it not an entity. It doesn't have an existence independent of its properties. We don't need to discuss anything besides its relevant properties. Perhaps you falsely assume that if something has properties, then it is an entity. You could benefit from a little study of ontology. When cosmologists claim that the Bang commenced a very rapid expansion of space, it is essential to understand what is said to be expanding. Likewise when every relativity text and website begins by "explaining" that "spacetime is curved" by the gravitational influence of mass, the question "what is curved?" is essential to understanding the theory. That is the focus of the several conferences over the years on the ontology of spacetime. Accept that spacetime can have properties (like length) without getting bogged down in the question of "what IS spacetime?" The most useful answer for this discussion is "nothing". So you assign "properties" to "nothing." Still, if there is nothing out there beyond what we can see, how can there be an end to space-as-emptiness, no-thing-ness? There can not. I don't see why something that has no volume can't contain other things that have no volume. You posit "things with no volume." I guess this is like "nothing" with "properties." You make no sense to me. Now to previous posts. Post 55, zapatos: Perhaps nothing lies beyond the closure. Not the nothing like empty space, but the nothing of it being a meaningless question. Similar to the question of "What is in the basement of a house that has not yet been built?". What does empty space mean if not absence of things, no -thing. Then you introduce a future house with a not yet existing basement as an argument against the infinity (endlessness) of space? It just does not fly... confused thinking, in my opinion. Post 56 by md65536: Nothing must lie beyond the closure. Nothing that can be considered part of this universe. Otherwise, the closure doesn't enclose the universe, which means it's not a closure. "The universe" means all there is (known and unknown) as contrasted with the visible cosmos. The question still remains, even with a supposed "finite universe," what lies beyond an "enclosed, finite universe?" Imagine taking some infinitesimal distance, and stretching it to infinity. I think this is what you'd have beyond the "closure" of the universe. This would mean that there is no distance at all beyond the edge of the universe. The distance beyond it is ZERO. See comment above. By approaching the edge of space, you would necessarily move the edge of space. An "edge of space" (as emptiness) makes no sense, again as above. It only makes sense (to me at least!) because we're not talking about an "entity" here. Right. No-thing-ness, emptiness, space... is not an entity, and there can be no end or edge, boundary, or wall-out-there to empty space. As I’ve repeated many times, if one posits such a boundary, what is beyond the boundary but more space, infinite space? zapatos, post 60, replying to my, The finite mind has always had trouble "grasping" or comprehending infinity... endless space, but that doesn't make the universe finite just because our reach of mind may be finite and "create" a finite universe. I am almost speechless. You are happy to use the argument on me that perhaps I just can't comprehend some things and therefore should not rule out the possibility that my idea is wrong. That I should keep an open mind. Yet when I suggested that to you previously, you barely took the time to ignore me. So back at you. Just because you cannot imagine finite space, that does not make it infinite. Finite space is easy to imagine. The space within any geometric form will do, or any form at all of finite size and the boundaries which de-fine it. Infinite means without boundary, edge, or end. My point was that just because you can not imagine endless space does not mean that there must be such an end. Your "logic" is backwards.
md65536 Posted May 4, 2011 Posted May 4, 2011 (edited) So you assign "properties" to "nothing." Exactly. Now you are starting to understand. Spacetime is nothing besides its properties (otherwise it would be an entity). Just like the basement of a house that hasn't been built can have a size and shape, it is still nothing. Finite space is easy to imagine. The space within any geometric form will do, or any form at all of finite size and the boundaries which de-fine it. This is such a bad example for this discussion, because it is hard not to imagine the geometric form existing in a larger volume. If you imagine only the simplest example of finite space, and then try to draw general conclusions from that example, you'll tend to draw false conclusions. Then you'll ask questions like "Even with a supposed 'finite universe,' what lies beyond an 'enclosed, finite universe?'" because you've falsely concluded, from your example, that there has to be something beyond. Infinite means without boundary, edge, or end. That's not what it means. The rational numbers between 0 and 1 are bounded by [0, 1] but they are infinite. They include "edges" at 0 and 1. If we allow infinite spatial curvature (such as with a black hole singularity), we might have infinite but bounded space. The singularity would also be an example of a space without volume, that contains other things. However, I think scientists expect black hole singularities to not physically exist. Edited May 4, 2011 by md65536
DrRocket Posted May 5, 2011 Posted May 5, 2011 Iggy: Define "closed." (No, don't. Just explain what lies beyond the "closure" or "form" on whatever scale.) Again, I do not advocate (and never have) a 4th dimension. Still batting zero in communication here, as always with you. Why don't you just leave it alone if you can't understand it? A closed manifold is a compact manifold without boundard. A compact space is one for which every open cover admits a finite subcover. If this is not clear please refer to any book on introductory topology. Lecture Notes on Elementary Topology and Geometry by Singer and Thorpe would be a good one.
mytechuniverse Posted May 5, 2011 Posted May 5, 2011 From my somewhat limited understanding, I believe the current hypothesis proposes that the universe is like the earth in a sense. When they discovered that the universe was expanding equally in all directions it was postulated that the universe is flat but it curved into a sphere. It is like the earth because to us it appears flat, but it has a horizon. The 3rd dimension is in a sense pushed out of a 2 dimensional universe from the curvature. In short, you can always measure the distance between two points in the universe and always have a number less than infinity. However, the universe has not ending point. There is no empirical starting point and stopping point. This means that the universe may have a finite spatial area, but it doesn't have start or end. I'm pretty sure newer theories have come up since Carl Sagan's Cosmos. I jus thought I would throw in my two cents.
michel123456 Posted May 5, 2011 Posted May 5, 2011 (edited) Owl must feel so alone. But somewhere he is right. IMHO. The Earth is not an analogy. The earth is curved, we are living on its curved surface, but we are 3D objects. We are not 2D individuals living in a 2D world. Accept that spacetime can have properties (like length) without getting bogged down in the question of "what IS spacetime?" The most useful answer for this discussion is "nothing". Exactly. Now you are starting to understand. This is insanity. Everybody here, except Owls, feels O.K. to discuss about "nothing" as it was 'something' with properties. But IF "nothing" has properties, then "nothing" is not nothing anymore, it becomes "something" mysterious. The ontologic problem that Owls tries to discuss is very well existing. No, we can speak about the properties of something (spacetime) without explaining anything more about what it is. This is pure imbecility IMHO. Perhaps you falsely assume that if something has properties, then it is an entity. It is an evidence, better say an axiom: if something (some-thing) has properties, it is an entity. It is Not a false assumption. Edited May 5, 2011 by michel123456
md65536 Posted May 5, 2011 Posted May 5, 2011 The Earth is not an analogy. The earth is curved, we are living on its curved surface, but we are 3D objects. We are not 2D individuals living in a 2D world. The Earth is an analogy, which means it is similar in some aspects but not all. The Earth is not equivalent. Moving upward through a 3rd dimension (whether you could do so infinitely or not) does not change the the fact that the 2D surface is closed. The possible existence of higher dimensions does not need to change the properties of the lower spatial dimensions. In general relativity spacetime is a 4-dimensional Lorentzian manifold. Under the assumptions of homogeneity and isotropy it can be decomposed as a one-parameter foliation of space-like 3-dimensional hypersurfaces (aka "slices"), without boundary. Those slices are spaces of constant curvature and are what is called "space" in cosmology. Homogeneous and globally isotropic spaces of constant curvature are of one of three types: the zero curvature case -- Euclidean 3-space, the positive curvature case -- the 3-sphere, and the negative curvature case -- hyperbolic space. Only the 3-sphere is compact, aka "finite". It is not known which, if any, case represents the physical universe. I think DrRocket's post back on page 1 is the best answer to the question of this thread. We don't know the answer, and we're not going to find it in an argument. The best we can do is to discuss what the alternatives really mean, so we can have a better understanding of what exactly we're discussing. For me, the most beneficial thing would be to stop discussing, and go read up on all of the topics that DrRocket has mentioned, which I don't understand. This is pure imbecility IMHO. It's hardly pure.
DrRocket Posted May 5, 2011 Posted May 5, 2011 Finite space is easy to imagine. The space within any geometric form will do, or any form at all of finite size and the boundaries which de-fine it. Infinite means without boundary, edge, or end. My point was that just because you can not imagine endless space does not mean that there must be such an end. Your "logic" is backwards. That's not what it means. The rational numbers between 0 and 1 are bounded by [0, 1] but they are infinite. They include "edges" at 0 and 1. You are correct in stating that owl is wrong. But your explanation is also wrong in this context. You are treating "infinite" as meaning an infinite point-set. In the context of a space-like slice of spacetime, aka "space", a manifold is "finite" if it is compact as a topological space -- or equivalently has finite volume. It is "infinite, or "open" otherwise. In either case there is no boundary (aka edge). The terms "infinite" and "finite" in cosmology are a bit unfortunate since they have other meanings in related disciplines.
owl Posted May 6, 2011 Posted May 6, 2011 Not much time tonight, just a quick reply. I wrote: Finite space is easy to imagine. The space within any geometric form will do, or any form at all of finite size and the boundaries which de-fine it. ... and md65536 replied: This is such a bad example for this discussion, because it is hard not to imagine the geometric form existing in a larger volume.If you imagine only the simplest example of finite space, and then try to draw general conclusions from that example, you'll tend to draw false conclusions. Then you'll ask questions like "Even with a supposed 'finite universe,' what lies beyond an 'enclosed, finite universe?'" because you've falsely concluded, from your example, that there has to be something beyond. Posit any form with boundaries/edges any size and shape you want. Beyond that, there is no limit, no "brick wall", and if there were (duh!), ... beyond that, either more stuff scattered about in space or infinite empty space. No, I have not concluded "that there has to be something beyond." Nothing beyond is still empty space, whether or not there is more "stuff" out there or just endless (what end!?) no-thing-ness, empty space.
md65536 Posted May 6, 2011 Posted May 6, 2011 (edited) In the context of a space-like slice of spacetime, aka "space", a manifold is "finite" if it is compact as a topological space -- or equivalently has finite volume. It is "infinite, or "open" otherwise. In either case there is no boundary (aka edge). Posit any form with boundaries/edges any size and shape you want. Beyond that, there is no limit, no "brick wall", and if there were (duh!), ... beyond that, either more stuff scattered about in space or infinite empty space. No, I have not concluded "that there has to be something beyond." Nothing beyond is still empty space, whether or not there is more "stuff" out there or just endless (what end!?) no-thing-ness, empty space. If there's no boundary either way (whether or not space is finite) I don't see what it would prove to imagine such a space. I won't posit it (as in, "assume the existence of") because that goes against the assumptions that DrRocket laid out early in this thread: "Under the assumptions of homogeneity and isotropy it can be decomposed as a one-parameter foliation of space-like 3-dimensional hypersurfaces (aka "slices"), without boundary." I have no special understanding that supersedes all of known science, that would even suggest that space (whether finite or not) has a boundary. But for the sake of imagination, I'll try anyway. Here's how to imagine it. Imagine first a flat and infinite homogeneous space, which is what I think you are claiming is the only possibility. Now curve or stretch that space so that you map any point at a distance of d from some arbitrary point, to a distance of 1/(d+1). This effectively turns space inside out, which I know is weird. All of space is now contained in a sphere with a radius of 1 unit. This is an infinite bounded space; if we remove the singularity at the center, it becomes a finite bounded space. Imagine existing within this sphere. There is NO SPACETIME beyond the sphere. There is no emptiness, there is no volume to the emptiness, there is no measurement of the emptiness. All of space, all emptiness and all "stuff" (energy and matter) within that space, are contained within the sphere. Now, I will admit this: I don't know if this even counts as a bounded space, because from within the universe, you can still measure space with your original mapping in which the space was flat and infinite and unbounded. It requires defining distance from some external perspective, which has its own measure of distance, which we might again be compelled to imagine as another space in which the first is embedded. We must resist that compulsion, but in doing so I may be forced to concede that this is an abstract concept only, which may not have any possible real existence in any way. But that doesn't even matter, because this space requires severe curvature. It is not homogeneous and isotropic in terms of curvature or distance. This whole discussion of bounded space may be a complete waste of time discussing, because even if it does make sense (I'm not sure it does) and even if it made sense and you understood it (doubtful), it's still not the finite space that we should be talking about, which is closed, unbounded space of a constant curvature. I know that you and michel123456 don't get it that space can have properties such as curvature without making it an entity (which by the way I'm taking to mean "The existence of something considered apart from its properties"), and I don't know if anyone gets anything out of what I'm saying, so it probably indicates "pure imbecility" on my part to bother writing at all. Yes, it does seem to indicate moderate or severe mental retardation. Thank you. Everybody here, except Owls, feels O.K. to discuss about "nothing" as it was 'something' with properties. But IF "nothing" has properties, then "nothing" is not nothing anymore, it becomes "something" mysterious. The ontologic problem that Owls tries to discuss is very well existing. I don't understand what you are saying here. Ontology is a branch of philosophy dealing with existence. An entity is something that has a physical existence. What is the ontology of a non-entity? I think there is none. There's only a problem if spacetime is an entity. I also don't get how defining something's properties makes it mysterious. "Nothing" was a bad word for me to use, because it is too ambiguous. It can be used as a name for empty space, or it has even been used to describe vacuum energy (which I'd agree could be an entity with an "ontological problem"), or something that has no properties, but I meant only that it has no physical existence apart from its properties. It is nothing besides its properties. It is an evidence, better say an axiom: if something (some-thing) has properties, it is an entity. It is Not a false assumption. I don't think that's an axiom at all. Edited May 6, 2011 by md65536
owl Posted May 6, 2011 Posted May 6, 2011 (edited) I spent much of this morning on the net researching this topic. One thing which particularly struck me is how all the technical lingo of math, geometry and various concepts associated with complex "manifolds" seldom explicitly refers to observable phenomena in "the real world." I'll start with the special use of "compact" gleaned from Wikipedia: ...a compact space is an abstract mathematical space (my bold) whose topology has the compactness property...compact manifold ...it should be noted that the term "compact manifold" often implies "manifold without boundary," ... When there is need for a separate term, a compact boundaryless manifold is called a closed manifold. ... closed manifold: A compact manifold means a "manifold" that is compact as a topological space, but possibly has boundary. More precisely, it is a compact manifold with boundary (the boundary may be empty...my italics??). By contrast, a closed manifold is compact without boundary. In contrast (but still "without boundary") is Mathworld's definition of an open manifold as a " noncompact manifold without boundary." So apparently both an open and a closed manifold can be "without boundary." But an open manifold can be a manifold with boundaries. Wikipedia also says: In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold. So "closed" in this context has a special, counter- intuitive meaning. One usually thinks of closed as having an inside and an outside, and in the context of this thread, the space outside of a closed cosmos (I will not say "universe" in this context) has no end or boundary, so is therefore infinite. While I was cruising Wiki's links on the size and shape of the universe, I came across the following: The shape of the universe is a matter of debate within physical cosmology over the geometry of the universe including both local geometry and global geometry. It is loosely divided into curvature and topology, even though strictly speaking, it goes beyond both. More formally, the subject in practice investigates which 3-manifold corresponds to the spatial section in comoving coordinates of the 4-dimensional space-time of the Universe. Yes, in the real world, the debate does indeed "go beyond both." And of course, the " 4-dimensional space-time of the Universe" is used without a second thought to the ontology of its referents in the real world, i.e., beyond the obvious, that space/volume has three dimensions/axes and that everything moves "in time." I'll finish my wiki note presentation with it's comments on infinity, global shape, and the observable universe: The word (infinity) comes from the Latin infinitas or "unboundedness" Compactness of the global shape Formally, the question of whether the universe is infinite or finite is whether it is an unbounded or bounded metric space. An infinite universe (unbounded metric space) means that there are points arbitrarily far apart: for any distance d, there are points that are of a distance at least d apart. A finite universe is a bounded metric space, where there is some distance d such that all points are within distance d of each other. The smallest such d is called the diameter of the universe, in which case the universe has a well-defined "volume" or "scale." Observable Universe: No evidence exists to suggest that the boundary of the observable universe constitutes a boundary on the universe as a whole, nor do any of the mainstream cosmological models propose that the universe has any physical boundary in the first place. In other words, in lieu evidence to the contrary, (no wall out there by any stretch of common sense or intelligent thinking) and to the point of this thread, space (the universe) is infinite. Edited May 6, 2011 by owl
Iggy Posted May 7, 2011 Posted May 7, 2011 Iggy: Iggy:Space-time has intrinsic curvature and there is no 4th spatial dimension. Similarly, we can describe a closed infinite space without the need for a 4th dimension. Space-time is now an established entity with intrinsic curvature? md65536 said "Similarly, we can describe a closed infinite space without the need for a 4th dimension", not me. The curvature is intrinsic. Inform the International Society for the Advanced Study of Spacetime! You can reference chapter 7.4 of Einstein's general theory of relativity: with modern applications in cosmology Thus the specification of the metric does not presuppose any embedding of the curved space in a higher-dimensional flat space. The Riemann curvature tensor represents intrinsic geometric properties of space, which may be measured by the inhabitants of that space. (Such inhabitants are always assumed to be creatures with the same number of dimensions as that of the space they inhabit.) Therefore one says that the Riemann tensor is a measure of the intrinsic curvature of space. Write a paper on it and maybe you can deliver it at their next conference. I don't believe that would be helpful. It would be helpful if you acknowledged that this is not logical, Logically, if space is finite it has an end or boundary or wall... a limit of some kind. Finite space does not necessitate or imply a boundary. Define "closed." (No, don't. Just explain what lies beyond the "closure" or "form" on whatever scale.) Closed means compact without boundary. You could imagine a creature living in a small closed three dimensional space throwing a ball in any direction. The ball would eventually hit the creature in the back. Or imagine the creature looking through a powerful telescope and seeing the back of its head.
owl Posted May 7, 2011 Posted May 7, 2011 (edited) Iggy: Finite space does not necessitate or imply a boundary. You don't seem to know what the words finite and infinite mean. You can make up your own definition, of course, but here are the definitions I have already quoted earlier. Wikipedia: Infinity (symbol: ∞) is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity. The word comes from the Latin infinitas or "unboundedness". Merriam-Webster: Definition of INFINITE1 : extending indefinitely : endless <infinite space> 2 : immeasurably or inconceivably great or extensive : inexhaustible <infinite patience> 3 : subject to no limitation or external determination 4 a : extending beyond, lying beyond, or being greater than any preassigned finite value however large Finally, a repeat of Wikipedia on the topic question (from my post yesterday which you apparently didn't read): A finite universe is a bounded metric space, where there is some distance d such that all points are within distance d of each other. The smallest such d is called the diameter of the universe, in which case the universe has a well-defined "volume" or "scale." Logically, what confines the above arbitrary "diameter of the universe" to "d?" (What prevents "d+1" of whatever units of distance?) You seem incapable of thinking about what would lie beyond such a well defined (finite) volume. If "something," what?... if nothing... more empty space, ad infinitum. Observable Universe:No evidence exists to suggest that the boundary of the observable universe constitutes a boundary on the universe as a whole, nor do any of the mainstream cosmological models propose that the universe has any physical boundary in the first place. (my bold.) Edited May 7, 2011 by owl
Iggy Posted May 7, 2011 Posted May 7, 2011 (edited) Iggy: Finite space does not necessitate or imply a boundary. You don't seem to know what the words finite and infinite mean. You can make up your own definition, of course, but here are the definitions I have already quoted earlier. Wikipedia: Infinity (symbol: ∞) is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity. The word comes from the Latin infinitas or "unboundedness". Your logic may be failing you. You think that because "infinite space" implies "unbounded" then "finite space" must imply "bounded"? That would be a logical fallacy called "denying the antecedent". If "unripe banana" implies "not delicious" then you should not think that "ripe banana" must imply "delicious". If "infinite space" implies "not bounded" then you should not think that "finite space" must imply "bounded". Logically, what confines the above arbitrary "diameter of the universe" to "d?" (What prevents "d+1" of whatever units of distance?) You seem incapable of thinking about what would lie beyond such a well defined (finite) volume. If "something," what?... if nothing... more empty space, ad infinitum. This would help, if you read it, The Possibility of a "Finite" and yet "Unbounded" Universe Edited May 7, 2011 by Iggy
md65536 Posted May 7, 2011 Posted May 7, 2011 The bounded infinite spaces I proposed are metric spaces that are only bounded according to a different metric. I don't think they can be called bounded metric spaces. Pure imbecility!, sorry about that. I think we should stop talking about boundaries, because that only answers the question about whether or not space is infinite, if there IS a boundary. Since no one is arguing for the existence of a boundary, we may as well assume there is no boundary. For a metric space to be finite and have no boundary, does this mean that no spatial dimension has a bound, yet the distance metric has an upper bound? Is it a confusion of spatial dimensions and the distance metric that is causing all the trouble?
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