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The Finite Universe and the Fallacy of Composition


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#41 Dennis Hardin

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Posted 21 September 2010 - 02:56 AM



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


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

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

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

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

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

Ghs


George,

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

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

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

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

Repeating the last paragraph of my response to Ted:

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

#42 Stephen Boydstun

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Posted 21 September 2010 - 03:11 PM

Endless Existence

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

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

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

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

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

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

From Between Metaphysics and Science:

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

Dennis opened the present thread this way:

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

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

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

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

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

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

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

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

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

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

References

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

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

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

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


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

#43 Ted Keer

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Posted 21 September 2010 - 05:56 PM

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


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

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

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

Posted Image



Edited by Ted Keer, 21 September 2010 - 05:58 PM.




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#44 BaalChatzaf

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Posted 21 September 2010 - 07:05 PM


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


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

Posted Image




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

Ba'al Chatzaf



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#45 Ted Keer

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Posted 21 September 2010 - 07:37 PM



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


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

Posted Image




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

Ba'al Chatzaf


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

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

What do you say, Bob?



Edited by Ted Keer, 21 September 2010 - 07:39 PM.




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#46 Dennis Hardin

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Posted 22 September 2010 - 02:26 AM

Endless Existence

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


Stephen,

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

#47 BaalChatzaf

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Posted 22 September 2010 - 04:36 AM

What do you say, Bob?




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

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#48 Stephen Boydstun

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Posted 22 September 2010 - 05:19 AM


Endless Existence

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


Stephen,

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

Dennis,

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

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

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

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

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

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

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

#49 Dennis Hardin

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Posted 23 September 2010 - 02:51 AM

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


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

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

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

#50 BaalChatzaf

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Posted 23 September 2010 - 03:24 AM

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


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

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

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

Ba'al Chatzaf

Edited by BaalChatzaf, 23 September 2010 - 03:26 AM.

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#51 Brant Gaede

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Posted 23 September 2010 - 09:09 AM


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


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

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

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

Ba'al Chatzaf

Insofar as "space" is empty it does not exist, but it is not empty; it is suffused with radiation. Even then what exists is something else and space is just an idea connoting place and distance. Time is also only an idea--a measurement of motion. Space-time is an epistemological construct whose metaphysical referents are other things.

--Brant

Rational Individualist, Rational self-interest, Individual Rights--limited government libertarian heavily influenced by Objectivism


#52 Dennis Hardin

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Posted 23 September 2010 - 11:01 AM



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


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

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

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

Ba'al Chatzaf

Insofar as "space" is empty it does not exist, but it is not empty; it is suffused with radiation. Even then what exists is something else and space is just an idea connoting place and distance. Time is also only an idea--a measurement of motion. Space-time is an epistemological construct whose metaphysical referents are other things.

--Brant


Brant’s analysis is excellent.

Bob--The fact that the theoretical notion of ether has been superseded by energy fields and radiation is totally irrelevant to the validity of the concept of "empty space."

#53 BaalChatzaf

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Posted 23 September 2010 - 01:01 PM

Brant's analysis is excellent.

Bob--The fact that the theoretical notion of ether has been superseded by energy fields and radiation is totally irrelevant to the validity of the concept of "empty space."


radiation is the motion of particles. One does not need a visco-elastic medium for particles to move through empty space.

Ba'al Chatzaf



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#54 Dennis Hardin

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Posted 23 September 2010 - 02:44 PM



Brant's analysis is excellent.

Bob--The fact that the theoretical notion of ether has been superseded by energy fields and radiation is totally irrelevant to the validity of the concept of "empty space."


radiation is the motion of particles. One does not need a visco-elastic medium for particles to move through empty space.

Ba'al Chatzaf


Nobel Prize laureate and MIT Physics Professor Frank Wilczek talks about the Long Hadron Collider project and the search for the actual physical nature of so-called ‘empty space.’

The Long Hadron Collider Project

Firstly, equations tell us that what appears to us in ordinary life as empty space is actually a material which affects the properties of matter in ways that our very successful equations tell us, but we've never yet really broken down this material to see what it's made out of. So it's as if we have been fish in an ocean, surrounded by water, and for a long time we've taken the water for granted because it's the only thing we know, we couldn't imagine a world without it.

But then if we were smart fish we would eventually realize that we are surrounded by something that is slowing us down and changing the way we move and function, and then the next step would be to figure out what this founding medium is made out of.

That's what the LHC is going to be able to do for the world we actually live in and the complex medium that we call empty space. For the first time we're going to be able to tell what kinds of things it is made out of. This is usually called looking for the “Higgs particle,” but it might not be one particle, it might be a whole bunch of things.



#55 dlewis

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Posted 08 September 2013 - 01:25 PM

I find this topic of spatial infinity and (or versus) finitude pretty interesting.  I think, Dennis, you are right--this topic hasn't been addressed as much as it should be by philosophers.  However, although I think agnosticism is an honest answer to the question, I don't think agnosticism in general is where any philosophy should end--shouldn't we always probe the mysteries of things to better understand them, and use the best logic or sustem we can attain at the time, even if the questions can't ever be fully centain to us?   
 

I generally agree with Ted that there can be a finite matter-energy in the universe yet an unbounded construct to space-time, making sense with what scientists have observed and Big Bang theorists have posited.  But I may even go further than this.  For example, in comment to the post by George H. Smith above:

 

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

Peikoff, OPAR, p,. 31-32."

 

This to me is too limited an observation.  Isn't Rand's concept of "existence" as anything and everything, really not limited by anything as well, yet still definable?  If Peikoff means that metaphysically, in the context of space-time, that infinity cannot be defined, what about the idea that space could nave no external spatial boundary, yet have internal ones, as may have been already suggested when Ted explained the universe as "self-bounded."  Couldn't some infinite, most basic, spatial substance isotropically be moving inward at the same rate and "inward" direction (like gravity) from all points in space to manifest or evolve into finite forms? Furthermore, I don't think infinity has to be caught up in the idea of curvature to be classified only that way as "unbounded."  For example, what exists on each side of that curve?  If "something" permeates all space, then there shouldn't be any gaps in the space-time continuum, right?






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