Infitine Space

[steevey]

I've said this before in other threads, but I challenge anyone here to come up with an "end of space." So, what kind of boundary can anyone imagine such a limit to be? (Hint: It's all in your head.)

But say you have a firm idea of a limit to space... so what lies beyond that limit/wall/boundary that someone might have imagined? More space? Of course. There can be no limit to space.

 

I just read the thread and copied a bunch of quotes. I'll make it simple and just reply in bold within the quote box of selected quotes:

 

Look at the meaning of the words. To "de-fine" is to make finite, at least in the collective mind, and in the dictionary. Srpace and all that it contains (visible and not) is infinite, by the above argument using logic alone. I welcome any argument with this post. Just be clear on you premise as to what space (and "the universe') is, please.

 

Actually in Einsteinian physics, a boundary could exist where the fabric of space folds in on itself, and that point is just so far away we can't see it in any way. So basically, you travel to one end of the universe and because of the folding, you end up at the other end, an inescapable box.

Edited April 23, 2011 by steevey

[owl]

Actually in Einsteinian physics, a boundary could exist where the fabric of space folds in on itself, and that point is just so far away we can't see it in any way. So basically, you travel to one end of the universe and because of the folding, you end up at the other end, an inescapable box.

I don't 'believe in' "the fabric of space" as an entity that can fold, expand, curve, etc. I consider "it" science's version of The Emperor's New Clothes. (See my thread in Philosophy, "Spacetime Ontology: the Scholarly Debate.")

So, anyway, what is outside the "inescapable box?"

[DrRocket]

I don't 'believe in' "the fabric of space" as an entity that can fold, expand, curve, etc. I consider "it" science's version of The Emperor's New Clothes. (See my thread in Philosophy, "Spacetime Ontology: the Scholarly Debate.")

So, anyway, what is outside the "inescapable box?"

 

What you don't believe in is entirely irrelevant. What is relevant is the accepted theory on which cosmology is based -- general relativity.

 

In GR there is no "fabric of space". What there is is a 4-dimensional Lorentzian manifold without boundary -- spacetime. "Without boundary" means that all points have a neighborhood that is diffeomorphic to Minkowski 4-space; i.e. there is no edge. That does not mean that spacetime is an open (i.e. non-compact, aka infinite) manifold. It may well be compact (aka finite), but still without boundary.

 

Philosophy is a good place for your ideas. They certainly do not meet the standards required for science.

[zapatos]

So, what kind of boundary can anyone imagine such a limit to be? (Hint: It's all in your head.)

But say you have a firm idea of a limit to space... so what lies beyond that limit/wall/boundary that someone might have imagined? More space? Of course. There can be no limit to space.

I find it interesting that you are so confident there is no limit to space, simply because you cannot imagine how it could be otherwise. Is your imagination really that powerful? (Hint: It's all in your head.)

[DrRocket]

I find it interesting that you are so confident there is no limit to space, simply because you cannot imagine how it could be otherwise. Is your imagination really that powerful? (Hint: It's all in your head.)

 

 

Often wrong. Never uncertain.

[IM Egdall]

DrRocket says it best. Who cares what you or I or anyone else believes. General relativity makes numerous specific detailed predictions which are verified by measurements and astronomical observations. That is why we know it is deeply connected to reality. That's why when we read that "there is is a 4-dimensional Lorentzian manifold without boundary -- spacetime . . i.e. there is no edge (to the universe)," we have to take it seriously.

 

I think it was Richard Feynam who saids something like "Nature doesn't care whether you believe her or not, that's the way She is."

[owl]

I find it interesting that you are so confident there is no limit to space, simply because you cannot imagine how it could be otherwise. Is your imagination really that powerful? (Hint: It's all in your head.)

Logically, if space is finite it has an end or boundary or wall... a limit of some kind. I simply asked two questions about that, neither of which you have answered. What is the nature of such a limit/end/boundary?

What is beyond said limit? If nothing... fine, more empty space, infinite space.

 

What you don't believe in is entirely irrelevant. What is relevant is the accepted theory on which cosmology is based -- general relativity.

 

In GR there is no "fabric of space". What there is is a 4-dimensional Lorentzian manifold without boundary -- spacetime. "Without boundary" means that all points have a neighborhood that is diffeomorphic to Minkowski 4-space; i.e. there is no edge. That does not mean that spacetime is an open (i.e. non-compact, aka infinite) manifold. It may well be compact (aka finite), but still without boundary.

 

Philosophy is a good place for your ideas. They certainly do not meet the standards required for science.

The relevance of the ontology of spacetime (see my final comments in that thread) is clear and evident from a multitude of scholarly papers presented at several conferences by members of the International Society for the Advanced Study of Spacetime. The philosophers of science, presenting at these conferences delve deeply into what it is we are talking about when we say spacetime curves, or space expands, etc. You can say 'who cares, as long as the equations work?,' but the math, by itself does not confer understanding. Does "it" have grooves/ruts to guide the observed trajectories of stuff said to be guided by spacetime? Or does gravity pull on objects without such a medium as spacetime, be it relational/dependent on matter or substantive and existing as an entity in and of itself?

Maybe if you would read a few of these papers you would gain a new appreciation for the kinds of questions philosophers (ontologists) of science are asking about spacetime... not so cut and dried as you would have it, making such inquiry irrelevant.

 

Likewise 4-D space.

PS;

Maybe you are taking my "I don't believe in spacetime" a little too seriously as a 'personal disbelief.' All ontologists who disbelieve in its existence as something with certain properties are much like the disbelievers in the "luminiferous aether" not so long ago. And I just get a kick out of stuffed shirt relativity theorists who glibly and matter of factly speak of spacetime as an established entity (ignoring the serious scholarly ontological debate about it.) The Emporer's New Clothes is a very "fitting" metaphor for this kind of quasi-scientific snobbery. You gotta love the kid who blurts out, "but he is not wearing any clothes!" Many well credentialed scientists also "believe" that 'the fabric of spacetime is equally bogus and pretentious.

Pps: I have no problem with three dimensional space and time is a factor in that everything moves around in space, requiring an 'elapsed time' for movement from point a to point B. But Minkowski reified spacetime into some malleable thing, which is how "it" has been treated ever since.

Edited April 26, 2011 by owl

[DrRocket]

Logically, if space is finite it has an end or boundary or wall... a limit of some kind. I simply asked two questions about that, neither of which you have answered. What is the nature of such a limit/end/boundary?

What is beyond said limit? If nothing... fine, more empty space, infinite space.

 

This is simply false. A finite space is simply a compact manifold, a manifold of finite volume. The surface of a sphere is such a 2-dimensional manifold. It has no boundary.

 

The question in cosmology is whether space-like slices of spacetime are compact or non-compact. In either case they are 3-manifolds without boundary, submanifolds of 4-dimensional spacetime, which also in GR has no boundary.

 

 

The relevance of the ontology of spacetime (see my final comments in that thread) is clear and evident from a multitude of scholarly papers presented at several conferences by members of the International Society for the Advanced Study of Spacetime. The philosophers of science, presenting at these conferences delve deeply into what it is we are talking about when we say spacetime curves, or space expands, etc. You can say 'who cares, as long as the equations work?,' but the math, by itself does not confer understanding.

 

It certainly aids the understanding of those with sufficient mathematical competency to comprehend the formulation of general relativity in terms of differential geometry. I can see why that is of little comfort to the attendees of your conference.

 

 

Does "it" have grooves/ruts to guide the observed trajectories of stuff said to be guided by spacetime? Or does gravity pull on objects without such a medium as spacetime, be it relational/dependent on matter or substantive and existing as an entity in and of itself?

Maybe if you would read a few of these papers you would gain a new appreciation for the kinds of questions philosophers (ontologists) of science are asking about spacetime... not so cut and dried as you would have it, making such inquiry irrelevant.

 

This is just bizarre.

 

Maybe before ontologists ask questions about spacetime they should acquire some knowledge of the actual content of the theory.

 

Spacetime is equipped with a metric that determines the geodesics. An object under no influence other than gravity follows a geodesic trajectory in spacetime -- so in that sense geodesics are "ruts". The whole content of the theory lies in the relationship between the distribution of mass/eneregy (the stress-energy tensor) and geodesics. And THAT is why you need to understand the mathematics. There is no substitute for understanding the language of the subject.

 

 

Likewise 4-D space.

 

The manifold in general relativity is not 4-D space. It is a 4-dimensional Lorentzian (not Riemannian) manifold -- spacetime.

 

It is quite simple to construct manifolds and vector spaces of any dimension, but the spacetime of physics is 4-dimensional with a metric of signature (+, -,-,-) or equivalently (-,+,+,+) -- and this conveys a great deal of understanding to those who understand mathematics.

 

 

PS;

Maybe you are taking my "I don't believe in spacetime" a little too seriously as a 'personal disbelief.' All ontologists who disbelieve in its existence as something with certain properties are much like the disbelievers in the "luminiferous aether" not so long ago. And I just get a kick out of stuffed shirt relativity theorists who glibly and matter of factly speak of spacetime as an established entity (ignoring the serious scholarly ontological debate about it.) The Emporer's New Clothes is a very "fitting" metaphor for this kind of quasi-scientific snobbery. You gotta love the kid who blurts out, "but he is not wearing any clothes!" Many well credentialed scientists also "believe" that 'the fabric of spacetime is equally bogus and pretentious.

Pps: I have no problem with three dimensional space and time is a factor in that everything moves around in space, requiring an 'elapsed time' for movement from point a to point B. But Minkowski reified spacetime into some malleable thing, which is how "it" has been treated ever since.

 

 

I find your reference to "stuffed shirt relativists" more than a bit ironic -- downright hysterical is a better description.

 

"Philosophy of science is about as useful to scientists as ornithology is to birds." -- Richard P. Feynman

[zapatos]

Logically, if space is finite it has an end or boundary or wall... a limit of some kind. I simply asked two questions about that, neither of which you have answered. What is the nature of such a limit/end/boundary?

What is beyond said limit? If nothing... fine, more empty space, infinite space.

I haven't answered because I don't know. I recognize that the depth of my ignorance is substantial.

 

The difference between us is that after I surpass the limits of my understanding, I stop making definitive statements because I realize there are things beyond my understanding. You on the other hand continue to make definitive statements. It's like you believe that if your brain is not capable of imagining anything else, then there must not be anything else. "I cannot imagine how 'x' can be true, therefore it must be false."

 

In my experience that type of attitude comes from someone who does not know enough to understand the depth of his ignorance.

[owl]

I haven't answered because I don't know. I recognize that the depth of my ignorance is substantial.

 

The difference between us is that after I surpass the limits of my understanding, I stop making definitive statements because I realize there are things beyond my understanding. You on the other hand continue to make definitive statements. It's like you believe that if your brain is not capable of imagining anything else, then there must not be anything else. "I cannot imagine how 'x' can be true, therefore it must be false."

 

In my experience that type of attitude comes from someone who does not know enough to understand the depth of his ignorance.

How about addressing the logic of what I said? There can be no end of space because whatever limit is proposed must be explained ontologically (What is it?), and then one must contemplate what lies beyond such a proposed boundary. You have not thought this through. It it is obvious to anyone who has given it serious thought.

 

I do not know what lies beyond our cosmic horizon (as far as we can see.) I do know that there can be no "end of space" (as above) no matter what exists in that space, if anything. This is a reasonable statement of my ignorance.

Btw, I am also completely ignorant of the Greek language and a multitude of other fields of knowledge in science, literature, etc. But I have contemplated the question "How big is the universe?" since I was old enough to wonder about it... and have studied cosmology all my adult life, so the above is not just spontaneously spouting off or "shooting from the hip."

 

So where is the fault in the logic as stated again above?

 

DrRocket:

 

It certainly aids the understanding of those with sufficient mathematical competency to comprehend the formulation of general relativity in terms of differential geometry. I can see why that is of little comfort to the attendees of your conference.

 

You are obviously unfamiliar with the credentials of the above attendees and the content of their papers. The conferences have always addressed the ontology of spacetime in the context of general relativity, the latter being the specialty of the attending scientists in respectful dialogue* with the attending philosophers of science, specializing in ontology. (*Perhaps a foreign concept to you.)

 

This is just bizarre.

Maybe before ontologists ask questions about spacetime they should acquire some knowledge of the actual content of the theory.

 

Maybe you should read a few of the papers presented by the participating well credentialed scientists before assuming that no presenters have any "knowledge of the actual content of the theory." I recommend Dennis Deiks' two volumes of compiled papers in "The Ontology of Spacetime." (No, I have not read both volumes in entirety, but have read a fair sampling. One such paper, linked by Spyman in the Ontology of Spacetime thread would be a good place to start. (Minkowski's space-time: a glorious non-entity.)

 

Spacetime is equipped with a metric that determines the geodesics....There is no substitute for understanding the language of the subject.

 

Case in point of your last statement above: Your first statement assumes spacetime as an established fact precluding the whole ontological inquiry as to what, exactly "it" is. The "language of the subject" (what "it" is equipped with, etc.) has the cart before the horse... what it is as a prerequisite to understanding "its" specific properties and 'equipment.' What "has grooves or ruts?" Don't you think this is a relevant question to ask, even if it is mere ontology?

 

This is simply false. A finite space is simply a compact manifold, a manifold of finite volume. The surface of a sphere is such a 2-dimensional manifold. It has no boundary.

 

The question in cosmology is whether space-like slices of spacetime are compact or non-compact. In either case they are 3-manifolds without boundary, submanifolds of 4-dimensional spacetime, which also in GR has no boundary.

 

You didn't even address what I said above your matter-of-fact judgment of falsehood. How false? Of course specific volumes of space can be designated, but there is always more space "outside the box" so defined.

Beyond any sphere of designated size/diameter is the space outside the sphere. You can keep expanding the size of the finite sphere all you want, and there will still be more space, infinite space beyond your sphere of finite size.

And you continue to assume "space-like slices of spacetime " as entities, parts of the entity spacetime, as if the debate is over and spacetime is now established as such an entity. This is what happens when scientists are weak in (or have no understanding at all of) ontology. The consensus about spacetime you keep hammering on does not exist.

 

"Philosophy of science is about as useful to scientists as ornithology is to birds." -- Richard P. Feynman

This is "hysterical,"... to dismiss out of hand all discussion of what spacetime actually is, if anything, and go on assuming "its" existence as an entity and simply ignore the ontological debate.

 

An afterthought on the relevance of philosophy to science... Here is Wikipedia's definition/intro to epistemology:

Epistemology... meaning "knowledge, science", and λόγος (logos), meaning "study of") is the branch of philosophy concerned with the nature and scope (limitations) of knowledge.[1] It addresses the questions:

 

* What is knowledge?

* How is knowledge acquired?

* How do we know what we know?

 

Much of the debate in this field has focused on analyzing the nature of knowledge and how it relates to connected notions such as truth, belief, and justification. It also deals with the means of production of knowledge, as well as skepticism about different knowledge claims.

 

The claim that the study of how we know what we know is irrelevant to science is not only laughable, it is a pathetic case of narrow mindedness.

And the Feynman quote does not compliment the intelligence of scientists. I would never call scientists bird brains, which is what the quote implies.

Edited April 26, 2011 by owl

[Iggy]

How about addressing the logic of what I said? There can be no end of space because whatever limit is proposed must be explained ontologically (What is it?), and then one must contemplate what lies beyond such a proposed boundary. You have not thought this through. It it is obvious to anyone who has given it serious thought.

 

I do not know what lies beyond our cosmic horizon (as far as we can see.) I do know that there can be no "end of space" (as above) no matter what exists in that space, if anything. This is a reasonable statement of my ignorance.

Btw, I am also completely ignorant of the Greek language and a multitude of other fields of knowledge in science, literature, etc. But I have contemplated the question "How big is the universe?" since I was old enough to wonder about it... and have studied cosmology all my adult life, so the above is not just spontaneously spouting off or "shooting from the hip."

 

So where is the fault in the logic as stated again above?

 

Imagine a child saying "logically, either the surface of the earth is infinite or you will eventually hit a boundary while walking its surface".

 

That isn't true, but the child may not be able to understand why. There is nothing illogical about finite and unbounded space, even if one finds the concept confusing.

[zapatos]

How about addressing the logic of what I said? There can be no end of space because whatever limit is proposed must be explained ontologically (What is it?), and then one must contemplate what lies beyond such a proposed boundary. You have not thought this through. It it is obvious to anyone who has given it serious thought.

 

I do not know what lies beyond our cosmic horizon (as far as we can see.) I do know that there can be no "end of space" (as above) no matter what exists in that space, if anything. This is a reasonable statement of my ignorance.

Btw, I am also completely ignorant of the Greek language and a multitude of other fields of knowledge in science, literature, etc. But I have contemplated the question "How big is the universe?" since I was old enough to wonder about it... and have studied cosmology all my adult life, so the above is not just spontaneously spouting off or "shooting from the hip."

 

So where is the fault in the logic as stated again above?

My problem is that you seem to have skipped a couple of steps in your logic. If I read the above correctly you are saying that:

 

1. Any given limit of the end of space must be explained/described/etc.

2. What lies beyond must be contemplated.

3. Therefore there is no end of space.

 

Did I get that right? If not please fill in what I overlooked. If so, then I suggest that steps 1 and 2 are insufficient argument to lead to step 3.

[owl]

Zapatos:

My problem is that you seem to have skipped a couple of steps in your logic. If I read the above correctly you are saying that:

 

1. Any given limit of the end of space must be explained/described/etc.

2. What lies beyond must be contemplated.

3. Therefore there is no end of space.

 

Did I get that right? If not please fill in what I overlooked. If so, then I suggest that steps 1 and 2 are insufficient argument to lead to step 3.

 

 

 

No. Empirical science is about observable phenomena. Only the “stuff” in space (including the effects of invisible forces) is observable, and we know that there is way more space in between things than the volume of observable things. This applies to subatomic “particles” and the space between them and to cosmic scale space between observable bodies.

Space is emptiness on all scales including infinite.There is no “wall out there,” and if there were there would only be more empty space beyond it.

 

The infinity of space needs no explanation. What “end,” as above, requires the explanation.

 

The question, “What lies beyond?” is an inevitable question unless one lives contentedly in a box (of whatever "size") that is his/her “world” and doesn’t care to contemplate “the beyond.”

 

There is no logic in a “therefore” concluded from false premises. That there can be be no end of space is a-priori knowledge, not in need of empirical data for verification. (Look into the a-priori category of epistemology.) Self evident and logically irrefutable. Will someone please explain the boundary and then what is beyond it, please.

[zapatos]

Space is emptiness on all scales including infinite.There is no “wall out there,” and if there were there would only be more empty space beyond it.

 

The infinity of space needs no explanation. What “end,” as above, requires the explanation.

 

The question, “What lies beyond?” is an inevitable question unless one lives contentedly in a box (of whatever "size") that is his/her “world” and doesn’t care to contemplate “the beyond.”

 

There is no logic in a “therefore” concluded from false premises. That there can be be no end of space is a-priori knowledge, not in need of empirical data for verification. (Look into the a-priori category of epistemology.) Self evident and logically irrefutable. Will someone please explain the boundary and then what is beyond it, please.

If it is logically irrefutable that there is no wall out there and that space is infinite, why do you keep asking someone to explain the boundary and what is beyond it? You already know the answer and have indicated that your mind cannot be changed.

[keelanz]

I think that to the ordinary layman anything that has no end point can be defined as infinite. This definition is not quite exact, though. For instance, the surface of a sphere has no end point and yet is a finite (closed) surface.

 

Chris

 

no i understand the definition of infinite, thanks, what i actually meant was obviously anything that exists doesnt exist anymore than it already existing, so our planet doesnt perceptually inflate to become infinite which means no matter how you look at it our physical existence is limited regardless if we cant express certain shapes or mathematical equations finitely. My question was actually asking if nothing could go for ever or would something have to fill the nothing for the nothing to be determined as something?

 

because if you dont need something for nothing to be determined as infinite then space is certainly infinite from our own perspective?

Edited April 27, 2011 by keelanz

[owl]

Imagine a child saying "logically, either the surface of the earth is infinite or you will eventually hit a boundary while walking its surface".

 

That isn't true, but the child may not be able to understand why. There is nothing illogical about finite and unbounded space, even if one finds the concept confusing.

The topic is Infinite Space, not "Is the surface of the earth infinite?" The metaphor does not address the topic. Of course one will not run into an insurmountable wall while circumnavigating the globe. But beyond any sphere is space, ultimately infinite space. A "wall out there" as an end of space is of course, absurd.

I know science has a special meaning for the phrase "finite and unbounded space" using a spherical surface as a metaphor. But unbounded means no boundary, and that means infinite in my vocabulary, the metaphor notwithstanding.

 

For example, DrRocket wrote:

The surface of a sphere is such a 2-dimensional manifold. It has no boundary.

...Meaning if you stay on the surface, you will not run into a surface boundary.

In the first place, a sphere occupies 3-D space,* not 2-D. (A plane is 2-D, and a line is 1-D... basic geometry.)

* Any curved surface is a 3-D object.

And, again, beyond the surface of any sphere is more space.

 

If it is logically irrefutable that there is no wall out there and that space is infinite, why do you keep asking someone to explain the boundary and what is beyond it? You already know the answer and have indicated that your mind cannot be changed.

The challenge is addressed to those who believe that space is finite.

Edited April 27, 2011 by owl

[DrRocket]

The topic is Infinite Space, not "Is the surface of the earth infinite?" The metaphor does not address the topic. Of course one will not run into an insurmountable wall while circumnavigating the globe. But beyond any sphere is space, ultimately infinite space. A "wall out there" as an end of space is of course, absurd.

I know science has a special meaning for the phrase "finite and unbounded space" using a spherical surface as a metaphor. But unbounded means no boundary, and that means infinite in my vocabulary, the metaphor notwithstanding.

 

For example, DrRocket wrote:

 

...Meaning if you stay on the surface, you will not run into a surface boundary.

In the first place, a sphere occupies 3-D space,* not 2-D. (A plane is 2-D, and a line is 1-D... basic geometry.)

* Any curved surface is a 3-D object.

And, again, beyond the surface of any sphere is more space.

 

This is vsimply wrong.

 

The cosmological model of spacetime is an intrinsic Lorentzian 4-manifold. It is not embeddd in anything. What is called "space" is any of the leaves of a timelike one-parameter foliation into 3-dimensional spacelike hypersurfaces. These hypersurfaces may be compact (aka finite) or non-compact (aka infinite) but in either vcase they have no boundary.

 

 

The challenge is addressed to those who believe that space is finite.

 

Belief has nothing to do with it. With what is known today space may be either finite or infinite. No one knows. The question of a boundary is irrelevant to the issue.

 

You seem to be arguing against the cosmological model of general relativity, while sinultaneously ignoring the meaning of the mathematics in which that theory is formulated. There is no "elsewhere" and there is no boundary to the spacetime manifold, no matter whether space is finite or infinite. If you want to argue otherwise you must of necessity put your arguement into some context other than general relativity.

 

So, what is your alternative to general relativity ?

Edited April 27, 2011 by DrRocket

[Iggy]

But unbounded means no boundary, and that means infinite in my vocabulary

 

That is wrong.

 

In the first place, a sphere occupies 3-D space,* not 2-D.

 

a sphere is a two dimensional manifold (or "space").

 

(A plane is 2-D, and a line is 1-D... basic geometry.)

* Any curved surface is a 3-D object.

 

A plane and a sphere are both 2-dimensional manifolds just like a line and a circle both 1-dimensional manifolds.

 

 

For example, DrRocket wrote:

The surface of a sphere is such a 2-dimensional manifold. It has no boundary.

...Meaning if you stay on the surface, you will not run into a surface boundary.

 

A two dimensional creature living in a curved, and closed, two dimensional space will have finite space and not run into a boundary. A three dimensional creature living in a curved, and closed, three dimensional space will have finite space and not run into a boundary.

 

This is pure logic. Like Dr. Rocket said, belief has nothing to do with it.

Edited April 27, 2011 by Iggy

[DrRocket]

 

The claim that the study of how we know what we know is irrelevant to science is not only laughable, it is a pathetic case of narrow mindedness.

And the Feynman quote does not compliment the intelligence of scientists. I would never call scientists bird brains, which is what the quote implies.

 

I don't think that any such criticism was intended by Feynman in that particular quote.

 

What you imply is more closely reflected in this quote:

 

The theoretical broadening which comes from having many humanities subjects on the campus is offset by the general dopiness of the people who study these things...” – Richard Feynman

 

 

[owl]

I think my comments here belong in the philosophy section rather than in the relativity section of Physics. Everyone by now knows that I have an ontological beef with "spacetime" as an assumed entity, and with the concepts central to non-Euclidean geometry as per, for instance, intrinsic vs extrinsic curvature as pertaining to what conceptual manifold (Euclidean space vs the Lorentzian 4-D manifold.)

 

I have studied the different manifold "dimensions." (See the Wikipedia intro: http://en.wikipedia.org/wiki/Manifold )

But I am also well aware of the ontological study of these "dimensions," and the assumptions inherent in the transition from the Euclidean paradigm to non-Euclidean. I have before cited an in-depth paper on this by Kelley Ross on the above. Here are two links to his papers, the second being very clear and specific on the varieties of manifolds and their assumed "dimensions"... for anyone not yet convinced that non-Euclidean is the absolute truth on the subject:

 

http://philpapers.org/rec/ROSTOA

 

http://www.friesian.com/curved-1.htm

 

I think it is a mistake to dismiss all philosophy as irrelevant to science. How can it be irrelevant to know exactly how we know what we think we know, i.e., epistemology? Once one abandons simple Euclidean geometry, what do these higher dimensional manifolds actually refer to in the "real world?" (See the Ross links above.) How is it that a curved surface is two dimensional since it is not a flat, 2-D plane? What happened to the line, plane, and volume geometry of the real world, being one, two, and three dimensional respectively? (Please read Ross before lecturing me yet again on higher dimensional manifolds.)

 

Anyway, I jumped into this thread to agree that space is infinite on the logical grounds that there can be no "end of space," therefore it must be infinite by definition.* In a nutshell, as I said, "A 'wall out there' as an end of space is of course, absurd."

*Here is the relevant Merriam-Webster online definition of infinite:

1

: extending indefinitely : endless <infinite space>

2

: immeasurably or inconceivably great or extensive : inexhaustible <infinite patience>

3

: subject to no limitation or external determination

 

I'll go back to the philosopher's corner now unless someone here wants to give me a reasonable account of an end of space or discuss Ross's Ontology and Cosmology of Non-Euclidean Geometry.

 

The theoretical broadening which comes from having many humanities subjects on the campus is offset by the general dopiness of the people who study these things...” – Richard Feynman

This is bigotry and arrogance at its worst, unworthy of science, which must ask not only what we know but how we know it, the field of epistemology... a branch of philosophy direcly relevant to science. The ontologists I've studied who are specialists in spacetime, for instance, are as far from "dopey" as it gets, brilliant and well credentialed philosophers of science and scientists workinf respectfully together to answer the basic "What is it?" questions.

Edited April 28, 2011 by owl

[Iggy]

I have studied the different manifold "dimensions." (See the Wikipedia intro: http://en.wikipedia.org/wiki/Manifold )

But I am also well aware of the ontological study of these "dimensions," and the assumptions inherent in the transition from the Euclidean paradigm to non-Euclidean.

 

There is only one assumption (i.e. postulate or axiom) that is different between Euclidean and non-Euclidean geometry. What is it?

 

I have before cited an in-depth paper on this by Kelley Ross on the above. Here are two links to his papers, the second being very clear and specific on the varieties of manifolds and their assumed "dimensions"... for anyone not yet convinced that non-Euclidean is the absolute truth on the subject:

 

http://philpapers.org/rec/ROSTOA

 

http://www.friesian.com/curved-1.htm

 

If you did read it, you didn't understand it. It says what everyone has been telling you and disagrees with your "objection" to space-time...

 

Einstein's general theory of relativity proposes that the "force" of gravity actually results from an intrinsic curvature of spacetime, not from Newtonian action-at-a-distance or from a quantum mechanical exchange of virtual particles. If we view Einstein's philosophical project as an answer to Kant's Antinomy of Space--to explain how straight lines in space can be finite but unbounded--the introduction of time reckoned as the fourth dimension suggests that we may separate the intrinsic curvature of spacetime into curvature based on the relationship between space and time: we can think of Einstein's theory as one that satisfies the axiom of open ortho-curvature, with the peculiarity that it is indeed time, rather than a higher dimension of space, that is posited beyond our familiar three spatial dimensions. This is a metaphysically elegant theory, since is gives us the mathematical use of a higher dimension without the need to postulate a real spatial dimension beyond our experience or our existence. Time is a dimension that is present to us only one spatial slice at a time, just as the third dimension is only intersected at one (radial) point by the curved surface of a sphere in our previous model of a positively curved space.

 

http://www.friesian.com/curved-1.htm

 

Anyway, I jumped into this thread to agree that space is infinite on the logical grounds that there can be no "end of space," therefore it must be infinite by definition.

 

That is not logical. Space can be finite and unbound so it is a false dichotomy between infinite space and a boundary.

 

Assuming that space is Euclidean is also illogical. It is an empirical matter.

[owl]

This is the conclusion of Ross' paper, not a reply to Iggy, which I may or may not get around to.

It does address the basic issue of the relevance of philosophy and the extrinsic vs intrinsic point of departure from Euclidean to non-Euclidean ontology and cosmology.

I§4. Conclusion

 

Just because the math works doesn't mean that we understand what is happening in nature. Every physical theory has a mathematical component and a conceptual component, but these two are often confused. Many speak as though the mathematical component confers understanding, this even after decades of the beautiful mathematics of quantum mechanics obviously conferring little understanding. The mathematics of Newton's theory of gravity were beautiful and successful for two centuries, but it conferred no understanding about what gravity was. Now we actually have two competing ways of understanding gravity, either through Einstein's geometrical method or through the interaction of virtual particles in quantum mechanics.

 

Nevertheless, there is often still a kind of deliberate know-nothing-ism that the mathematics is the explanation. It isn't. Instead, each theory contains a conceptual interpretation that assigns meaning to its mathematical expressions. In non-Euclidean geometry and its application by Einstein, the most important conceptual question is over the meaning of "curvature" and the ontological status of the dimensions of space, time, or whatever. The most important point is that the ontological status of the dimensions involved with the distinction between intrinsic and extrinsic curvature is a question entirely separate from the mathematics. It is also, to an extent, a question that is separate from science--since a scientific theory may work quite well without out needing to decide what all is going on ontologically. (My bold) Some realization of this, unfortunately, leads people more easily to the conclusion that science is conventionalistic or a social construction than to the more difficult truth that much remains to be understood about reality and that philosophical questions and perspectives are not always useless or without meaning. Philosophy usually does a poor job of preparing the way for science, but it never hurts to ask questions. The worst thing that can ever happen for philosophy, and for science, is that people are so overawed by the conventional wisdom in areas where they feel inadequate (like math) that they are actually afraid to ask questions that may imply criticism, skepticism, or, heaven help them, ignorance.( My bold)

 

These observations about Einstein's Relativity are not definitive answers to any questions; they are just an attempt to ask the questions which have not been asked. Those questions become possible with a clearer understanding of the separate logical, mathematical, psychological, and ontological components of the theory of non-Euclidean geometry. The purpose, then, is to break ground, to open up the issues, and to stir up the complacency that is all too easy for philosophers when they think that somebody else is the expert and understands things quite adequately. It is the philosopher's job to question and inquire, not to accept somebody else's word for somebody else's understanding.

 

BTW, I have made notes and commentaries throughout the paper if anyone wants to get into specifics... including the departure from Euclid's fifth postulate.

 

Here are quite a few quotes up front... inviting comments:

 

If we just don't think--ostrich-like--about facing an infinite universe again, then it won't happen. This is not intellectually or philosophically honest. But it is of a piece with much of the way non-Euclidean geometry and its related cosmological issues have been dealt with for a long time.

 

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. Now "intrinsic" curvature has nothing to do with any higher dimension. But how did this happen? Why did "curvature" come to have this unusual meaning? Why should we confuse ourselves by saying that "intrinsic" straight lines, geodesics, in non-Euclidean spaces have curvature?

 

...non-Euclidean planes can be modeled as extrinsically curved surfaces within Euclidean space. Thus the surface of a sphere is the classic model of a two-dimensional, positively curved Riemannian space; but while great circles are the straight lines (geodesics) according to the intrinsic properties of that surface, we see the surface as itself curved into the third dimension of Euclidean space.

 

Three dimensional non-Euclidean spaces of course cannot be modeled at all using Euclidean space.

This raises two questions: 1) what can we spatially visualize? (a question of psychology) And 2) what can exist in reality? (a question of ontology).

(Ed: But...)

....The intelligibility of non-Euclidean geometry for Kantian theory is neither a psychological nor an ontological question, but simply a logical one--using Hume's criterion of possibility as logically consistent conceivability.

 

It could well be the case that Kant is right and that we will never be able to imagine the appearance of Lobachevskian or multi-dimensional non-Euclidean spaces, or to model them without extrinsic curvature, however well we understand the analytic equations. This is purely a question of psychology and not at all one of logic, mathematics, physics, or ontology.

 

Non-Euclidean geometry was no more than a mathematical curiosity until Einstein applied it to physics. Now the whole issue seems much deeper and complex than it did in Kant's day, or Riemann's. If our imagination is necessarily Euclidean, hard-wired into the brain as we might now think by analogy with computers, but Einstein found a way to apply non-Euclidean geometry to the world, then we might think that space does have a reality and a genuine structure in the world however we are able to visually imagine it.

 

(Ed: Here the ontological water gets deeper.)

 

In light of the distinction between intrinsic and extrinsic curvature, we must consider all the kinds of ontological axioms that will cover all the possible spaces that Euclidean and non-Euclidean geometries can describe. If the limitations imposed by our imaginations present us with features of real space, we would have to say that intrinsic curvature, despite being analytically independent of extrinsic curvature, can only exist in conjunction with extrinsic curvature and so with an embedding in higher dimensions. This could be called the axiom of ortho-curvature, according to which there would actually be no true non-Euclidean geometry, for non-Euclidean geodesics would necessarily have extrinsic curvature and so would never be the actual straight lines that we need ex hypothese to contradict Euclid. The geometry of the surface of a sphere would thus involve ortho-curvature because its intrinsic straight lines, the great circles, must be simultaneously visualized and understood to be curved lines in three dimensional Euclidean space. On the other hand, it may be that intrinsically curved spaces can exist in reality without extrinsic curvature and so without being embedded in a higher dimension. This could be called the axiom of hetero-curvature, and it would make true non-Euclidean geometry possible, since lines with non-Euclidean relations to each other would be straight in the common meaning of the term understood by Euclid or Kant.

 

A further ontological distinction can be made. Even if the ortho-curvature axiom is true, a functionally non-Euclidean geometry would be possible if a higher dimension that allows for extrinsic curvature exists but is hidden from us. We must consider whether only the three dimensions of space exist or whether there may be additional dimensions which somehow we do not experience but which can produce an intrinsic curvature whose extrinsic properties cannot be visualized or imaginatively inspected by us. Thus we should distinguish between an axiom of closed ortho-curvature, which says that three dimensional space is all there is, and an axiom of open ortho-curvature, which says that higher dimensions can exist. This gives us three possibilities:

 

1. That, with the axiom of closed ortho-curvature, there are no true non-Euclidean geometries (and no spatial dimensions beyond three), but only pseudo-geometries consisting of curves in Euclidean space;

 

2. That, with the axiom of open ortho-curvature, there are no true non-Euclidean geometries but we may be faced with a functional non-Euclidean geometry in Euclidean space whose external curvature is concealed from us in dimensions (more than the three familiar spatial dimensions) not available to our inspection--this is an apparent hetero-curvature;

 

3. And that, with the axiom of hetero-curvature, there are real non-Euclidean geometries whose intrinsic properties do not ontologically presuppose higher dimensions (whether or not there are more than three spatial dimensions).

 

(Rather than quoting this whole section, please just read it from the link.)

 

§3. Geometry in Einstein's Theory of Relativity

(Continuing...)

Intrinsic curvature, which was introduced by Riemann to explain how straight lines could have the properties associated with curvature without being curved in the ordinary sense, is now used to explain how something which is obviously curved, e.g. the orbit of a planet, is really straight. Something has gotten turned around.

 

(Ed:The following is a reification of time... a departure from clear thinking in this paper, in my opinion....this stated to avoid defending the following as if I buy the whole thing.):

 

If there are no "forces" acting on the body, as Einstein says, then the only change that takes place is the body's movement along the temporal axis; and if the body is thereby displaced in space, it must be displaced by its movement along that axis. The temporal axis can displace the object if the axis is itself curved; so the curvature of spacetime in a gravitational field must result from the curvature of time, not of space. The extrinsic dimension of ortho-curvature, into which the straight lines curve, is a dimension of ordinary Euclidean space.

Edited April 29, 2011 by owl

[csmyth3025]

I have gotten into an argument with one of my friends with the size of the universe. Pretty silly argument if you ask me, but nevertheless, i am interested in what your opinions/knowledge on the subject is. Here we are:

 

The universe is huge. Can it be infinite? Where are the boundaries?

 

So, the ultimate question is, can the distance between two peices of matter farthest apart in the universe be infinite or not?

I'm amazed at how this thread got so distorted away from the OP (sumarized above). My answer would be: Yes, the universe is huge. Yes it can be infinite (but it could also be finite). The boundary of the observable universe is currently about 46 billion light years away from us in all directions according to current estimates. A boundary beyond this distance - if it exists or if it doesn't exist - is presently unobservable and, therefore, unknowable. Any speculation about whether a boundary exists outside our observable universe is not science - it's speculation.

 

As 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.

 

Chris

 

I've said this before in other threads, but I challenge anyone here to come up with an "end of space." So, what kind of boundary can anyone imagine such a limit to be? (Hint: It's all in your head.)

But say you have a firm idea of a limit to space... so what lies beyond that limit/wall/boundary...

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

[MigL]

Owl, 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.

Let me see if I can help clear up some confusion. 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. Now don't confuse this fourth dimension with time, GR says 3-d space may curve back on itself, but not time. or as DrR like to say the foliation may fold back on itself to 'unbound' the universe. You prefer infinite space with no bounds, ie flat euclidian space.

You are giving a significance to this higher dimension through which the 3-d space is curving where there is none. Assume the two dimensional, flat ( euclidian ) space with no bounds analogy that you seem to prefer to the curved sphere, in higher dimensions there is also an 'above' and 'below' the euclidian plane. You don't seem to have an objection to this, why do you have one with the outside and inside of a sphere. Again there is no significance to either.

 

I don't know if I've been very clear and that's why DrR likes to use the appropriate language, mathematics. 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.

[csmyth3025]

Nicely put MigL.

 

Chris