The Logical Leap: Induction in Physics

[Ellen Stuttle]

Yes, I noticed DF's retreat under fire.

There wasn't any "retreat under fire." You'd disregarded his previous comments.

The subject matter Df and and the GWers (just like theologians) study may differ in its underlying validity but the elite authoritarian attitude is exactly the same.

Well, if DF has an "elite authoritarian attitude," then so do I, since I agree with his views on criteria of competence.

Ellen

[kiaer.ts]

Can you be concretely specific as to what you are calling her a priori positions? A quote and a citation would help, if possible.

Ted,

I'm surprised that you aren't familiar with the standard Objectivist teachings about the nature of space and time. Leonard Peikoff's history of philosophy lectures from the early 1970s would do as a source; so would his 1976 lectures on the philosophy of Objectivism. If you want references to OPAR, I'll dig them up for you.

One instance, Rand's Ford Hall Forum answer to John Enright's question about the nature of time, is sitting right here on OL:

http://www.objectivistliving.com/forums/index.php?showtopic=7801&view=findpost&p=85652

Oh, ahh, then, in that sense, our position would, in effect, be Aristotelian. And Aristotle's position is that — ehh, this is not his statement, it's our statement, but the issue is the same — that there is no such thing as independent time or space. The universe is finite, and the concept of time applies to a relationship between entities, specifically a measurement of motion, which is a change of relationship, a relationship between entities within the universe, not outside. Time cannot exist by itself; it exists only in, inside the universe, but it does not apply to the universe as a whole, because time is merely a measurement of motion, a change of relationship between entities within the universe. Now if by "universe" we mean the total of that which exists, there's no relationship to anything outside itself that it could have—no motion, no change, and, therefore, no time—if you think of it as the totality.

On the nature of space, the standard Objectivist claims are that it must be finite, not infinite; that it must be relative, not absolute; and that space cannot exist without entities. "There is no nothing," Nathaniel Branden and Leonard Peikoff have both announced in lecture. Some lines of inquiry in 20th century physics are therefore rejected a priori because they allegedly reify empty space.

Like the claims about time (time is in the universe; the universe isn't in time) the arguments derive from Ancient Greek philosophy, specifically from Aristotle (and with specific regard to the denial of empty space) Parmenides.

Yes, I am familiar with those claims. I thought you were referring to something more sinister. Calling tham a priori is a bit misleading. It's not like she declared that the universe was created on a Fall afternoon in 4,004 BC.

To claim that the universe is finite is simply to assert the incoherence of saying that however big existence is, it is even bigger than that. I do think Rand should have said that existence is finite, rather than saying that the universe is finite (even if it too, by implication, must be) if only because historically what is meant by the universe has changed over time, from the solar system, to the galaxy, to now, a vast expanse whose visible portion is a sphere a dozen billion light years in radius. If we were to discover evidence that, say, the big bang was a small local event in a much wider object we were to come to name the "omniplex," the claim that the totality of existence is finite - i.e., that is it what it is, and no more - would still hold.

The same follows for absolute space and time. If absolute space means space where nothing exists, not even a measurer, then it cannot be measured. It's as meaningless a phrase as time during which nothing happens. That these philosophical conclusions, which are really just denial of absurdities, the finity of existence and the fact that time and space are relations, not entities, predate the modern scientific observations and theories that are consonant with them, relativity and the big bang, is not a proof of their invalidity.

Let me emphasize. The philosophical assertion of the finity* of existence and the relativity of time and space are not content-rich claims. They are merely the denial of absurdities. Theories have been held, such as the steady-state universe, and absolute Newtonian space, which have contradicted them. But they have never, in the past or the present, beencontradicted by observation.

I do, however, balk at accepting a priori condemnations of discussions of the nature of the vacuum. The vacuum is not conceived of as being empty nothingness, but rather a sea of virtual particles. If that is what is being condemned by the "reification of empty space," it is based on ignorance of what the scientists are talikng about.

---

* Yes, I know the preferred word is finitude, but I hold that finity is a perfectly cromulent alternative.

Edited September 1, 2010 by Ted Keer

[Xray]

But by all means, why don't you list some of your examples of what "we" "knew" about reality at one time, that was contradicted at a later time.

You really need examples? Just think of the "heliocentric" universe, to name but one.

I recount the incident because it's indicative that Rand, then, back in '69, was already of the opinion that physics had been corrupted.

Corrupted by what and by whom?

Maybe Shayne got the "corrupted physics" thing from Rand? He even called Niels Bohr "corrupt", but did not reply when he was asked to explain why.

Shayne is also of the opinion that:

What QM is actually saying is that there are no particles.

Can't wait for Shayne to explain why quantum physisicsts do mention particles all the time:

[by the famous quantum physicist Anton Zeilinger]: "The amazing thing is that there can be no exchange of information between the two particles. They react absolutely in synch, although they could could never know anything of each other's existence.

http://www.signandsight.com/features/614.html

Yes, I noticed DF's retreat under fire.

There is no retreat under any fire on DF's part. Desperate attempts by some laypersons to stone an expert with popcorn have no power at all.

Edited September 1, 2010 by Xray

[Xray]

In her 1971 piece on "psychologizing," she tried and failed to demarcate between philosophy and psychology in a way that she wouldn't have a few years earlier. She now insisted on the absolute priority of philosophy over all of the sciences and claimed that the traffic between them could go only one way (from philosophy to science, never in the opposite direction).

"Philosophy is not dependent on science; the reverse is true." (Rand, ITOE, p. 289).

Edited September 1, 2010 by Xray

[kiaer.ts]

Yes, I noticed DF's retreat under fire.

There wasn't any "retreat under fire." You'd disregarded his previous comments.

The subject matter Df and and the GWers (just like theologians) study may differ in its underlying validity but the elite authoritarian attitude is exactly the same.

Well, if DF has an "elite authoritarian attitude," then so do I, since I agree with his views on criteria of competence.

Ellen

If you are implying that I don't think a philosopher should know what he is talking about when he makes statements about the philosophical validity of scientific theories then you have obviously not paid attention to my prior comments about Harriman's and Peikoff's gross ignorance of the relevant concepts and you are attacking a straw man.

If Leonard Peikoff were able to explain to me the concept of a finite yet unbounded universe, and then tell me why he still rejected the Big Bang, I might listen to him. (He approaches the notion in one of his podcasts and then abandons it as irrelevant without quite understanding it.) Rather his objections are based on his failure to understand the geometry and an ad hominem slur. The theory is not even about a physicaal law, just an assertion of historical fact; that space is expanding (red shift); that it was once of a (not perfectly) uniform temperature (COBE map); that the Big Bang is consistent with models of hydrogen and helium frequency; and that the further galaxies are away, the more primitive they are in structure. That last fact is the clincher, unless you want to advocaate for a heliocentric universe model. Their objections are so puerile they are risible.

Questions of higher dimensions, bent space, and finite yet unbounded space time are diificult ones to discuss in an internet forum, without pictures or the ability to talk in real time

The book Geometry, Relativity, and the Fourth Dimension by Rudolph Rucker is one I read in childhood which explains these concepts which it is necessary to hold before one can understand or criticize such theories. If you wish to understand these ideas, you can acquire this small, fluent, and well illustrated book used for less than $5 at abebooks.com or elsewhere.

Does addressing my actual views deserve less diligence than addressing DF's?

My criticisms of DF's original statements stand. You are quite free to have a better attitude of his retreat position. That doesn't revise history. He didn't accidentally say that philosophers cannot understand the issues and he did not accidentally say that books for laymen cannot convey the relevant issues because of a loss in translation.

But most important, I am much more interested in the issues. If DF makes a statement I find objectionable, I will address him. I am really not interested in a derivatively derivative discussion of your opinion of my opinion of his changing opinions.

[sjw]

But by all means, why don't you list some of your examples of what "we" "knew" about reality at one time, that was contradicted at a later time.

You really need examples? Just think of the "heliocentric" universe, to name but one.

How did "we" know this? Following any rational procedure "we" would have to have said "we don't know, and don't have the means to know, yet." And there were probably people who were rational and did this. But they aren't in the history books. Only the guys who jumped to a conclusion get listed.

I recount the incident because it's indicative that Rand, then, back in '69, was already of the opinion that physics had been corrupted.

Corrupted by what and by whom?

Maybe Shayne got the "corrupted physics" thing from Rand? He even called Niels Bohr "corrupt", but did not reply when he was asked to explain why.

No I didn't get it from Rand, I got it from reading Einstein trying to reason with Bohr. If you're interested go read their actual exchanges and make your own decision about Bohr. I'm sure you'll come down on Bohr's side not Einstein's.

Shayne is also of the opinion that:

What QM is actually saying is that there are no particles.

Can't wait for Shayne to explain why quantum physisicsts do mention particles all the time:

My, you are groping to find something to argue about, aren't you? The word "particle" is rationally used to refer to something that is not actually a particle in the usual sense of the word. It's just meant to conceptually contain what is being talked about, it doesn't mean there's a literal particle there. So you're just being silly.

Shayne

[Dragonfly]

To claim that the universe is finite is simply to assert the incoherence of saying that however big existence is, it is even bigger than that.

.

There is nothing a priori incoherent in the notion of an infinitely big universe (or "omniplex" if you like), only in the way you express it: would you say that however big the set of natural numbers is, it is even bigger than that?

[sjw]

To claim that the universe is finite is simply to assert the incoherence of saying that however big existence is, it is even bigger than that.

.

There is nothing a priori incoherent in the notion of an infinitely big universe (or "omniplex" if you like), only in the way you express it: would you say that however big the set of natural numbers is, it is even bigger than that?

I wonder at the utility or meaning of answering questions like this. We know that however far we've been able to look, there's still more. Scientists talk about the known universe as if it were the whole universe, when in principle the extent we can see might be one trillionth of what is actually there -- there could be a trillion big bangs separated by a trillion times the distance of the known universe. Even if we were to observe a boundary, we would never know that it still wasn't some tiny part of something vastly more large.

Those of us who yearn to know the answers to questions like this feel stuck in their little part of time and space. Some of us learn to live with this constraint, others make things up.

Shayne

[kiaer.ts]

To claim that the universe is finite is simply to assert the incoherence of saying that however big existence is, it is even bigger than that.

.

There is nothing a priori incoherent in the notion of an infinitely big universe (or "omniplex" if you like), only in the way you express it: would you say that however big the set of natural numbers is, it is even bigger than that?

You would need to clarify to me what you mean by infinitely big universe. I would say that an infinitely big universe would be one in which no ratio between it as a whole and its parts exists. I do not see any coherent meaning assignable to the notion.

The set of natural numbers is not a thing, it is a idea left incomplete. It is defined negatively. There could be no actual enumeration of it. One would be forced to say that however large an actual enumeration of the set of whole numbers is, it is even bigger than that. The universe, unlike the set of natural numbers, is actual.

I do understand the idea of a finite yet unbounded universe. In order for me to understand an infinite universe, you'll need to unpack for me coherently what would be meant by infinite in more concrete terms.

Edited September 1, 2010 by Ted Keer

[sjw]

http://www.ronpisaturo.com/Universe.htm

[kiaer.ts]

To claim that the universe is finite is simply to assert the incoherence of saying that however big existence is, it is even bigger than that.

.

There is nothing a priori incoherent in the notion of an infinitely big universe (or "omniplex" if you like), only in the way you express it: would you say that however big the set of natural numbers is, it is even bigger than that?

I wonder at the utility or meaning of answering questions like this. We know that however far we've been able to look, there's still more. Scientists talk about the known universe as if it were the whole universe, when in principle the extent we can see might be one trillionth of what is actually there -- there could be a trillion big bangs separated by a trillion times the distance of the known universe. Even if we were to observe a boundary, we would never know that it still wasn't some tiny part of something vastly more large.

Those of us who yearn to know the answers to questions like this feel stuck in their little part of time and space. Some of us learn to live with this constraint, others make things up.

Shayne

Thanks, Shayne. That's a perfect proof to me of the "sophistication" of your knowledge in this discussion.

That "We know that however far we've been able to look, there's still more" is laughable. "However far we are able to look" is a finite distance approaching a definite limit on the close order of 13 billion light years. The older the galaxies are approaching that distance the younger they look due to the age of their light reaching us. We cannot see further than the cosmic microwave background radiation. Only someone ignorant of the science could find such an implied argument reasonable.

[sjw]

That "We know that however far we've been able to look, there's still more" is laughable. "However far we are able to look" is a finite distance approaching a definite limit on the close order of 13 billion light years. The older the galaxies are approaching that distance the younger they look due to the age of their light reaching us. We cannot see further than the cosmic microwave background radiation. Only someone ignorant of the science could find such an implied argument reasonable.

Says Ted, seething with rage and incoherence... You've really blown a fuse this time Ted, nothing I said is at all controversial. I mean, the statement that sets you off is an uncontroversial tautology pointing to the irrefutable fact that we can only see as far as we can see. It's a philosophical statement, not an expression of my scientific knowledge.

But it is good to see you finally revealing your stark naked rage at whatever it is you think I represent, it throws a nice crisp light on the BS you've been spewing at me all along without me having to do anything more than point it out.

Shayne

[kiaer.ts]

That "We know that however far we've been able to look, there's still more" is laughable. "However far we are able to look" is a finite distance approaching a definite limit on the close order of 13 billion light years. The older the galaxies are approaching that distance the younger they look due to the age of their light reaching us. We cannot see further than the cosmic microwave background radiation. Only someone ignorant of the science could find such an implied argument reasonable.

Says Ted, seething with rage and incoherence... You've really blown a fuse this time Ted, nothing I said is at all controversial. I mean, the statement that sets you off is an uncontroversial tautology pointing to the irrefutable fact that we can only see as far as we can see. It's a philosophical statement, not an expression of my scientific knowledge.

But it is good to see you finally revealing your stark naked rage at whatever it is you think I represent, it throws a nice crisp light on the BS you've been spewing at me all along without me having to do anything more than point it out.

Shayne

Uh, yeah. Narcissism, foolishness, moronicity and rage. Have you heard of projection?

That we can see 13 billion light years and that galaxies get more primitive as they reach that distance is not a tautology.

And to this you respond with neither disproof nor agreement, but with the ad hominem taunts of a junior highschooler.

I think you are one of those people Dragonfly has warned us about.

[sjw]

I would also add that no matter how small we manage to be able to look, we can never conclude that we've reached the "bottom" of reality. We can build bigger and bigger particle accelerators and potentially break the allegedly "fundamental" particles into smaller and smaller pieces, but never will we have the liberty of concluding that we have a final model of reality. We may arrive at a "complete" picture in the sense that every particle we've seen has a place in a theory that perfectly describes what we know, but we cannot know that these particles are the "bottom."

[sjw]

Ted, the more you hiss and foam at the mouth the less sense you make. Here's the deal: I know exactly what I meant by what I said. And I know what you said in response. So I know with certainty that you've popped a gasket. You've totally lost it.

[BaalChatzaf]

I would also add that no matter how small we manage to be able to look, we can never conclude that we've reached the "bottom" of reality. We can build bigger and bigger particle accelerators and potentially break the allegedly "fundamental" particles into smaller and smaller pieces, but never will we have the liberty of concluding that we have a final model of reality. We may arrive at a "complete" picture in the sense that every particle we've seen has a place in a theory that perfectly describes what we know, but we cannot know that these particles are the "bottom."

With respect to the natural Planck Units (such as Planck Length, etc.) we are 15 or 16 orders of magnitude from "rock bottom" (assuming there is a rock bottom) with the best technology currently at hand. It is highly unlikely that we will ever hit rock bottom (each advance toward smallness requires an order of magnitude more energy). The LHC is as good as we have at the moment. It generates energy of the order of magnitude of a tera-electron volt (somewhat less, actually).

Ba'al Chatzaf

[Martin Radwin]

To claim that the universe is finite is simply to assert the incoherence of saying that however big existence is, it is even bigger than that.

.

There is nothing a priori incoherent in the notion of an infinitely big universe (or "omniplex" if you like), only in the way you express it: would you say that however big the set of natural numbers is, it is even bigger than that?

You would need to clarify to me what you mean by infinitely big universe. I would say that an infinitely big universe would be one in which no ratio between it as a whole and its parts exists. I do not see any coherent meaning assignable to the notion.

The set of natural numbers is not a thing, it is a idea left incomplete. It is defined negatively. There could be no actual enumeration of it. One would be forced to say that however large an actual enumeration of the set of whole numbers is, it is even bigger than that. The universe, unlike the set of natural numbers, is actual.

I do understand the idea of a finite yet unbounded universe. In order for me to understand an infinite universe, you'll need to unpack for me coherently what would be meant by infinite in more concrete terms.

I agree with Dragonfly about this. I don't see anything at all incoherent or contradictory about an infinitely large universe. You're correct that the analogy between the universe and the set of all natural numbers is not very good, since the latter is an abstraction, while the universe actually has a physical existance. What is meant by an infinitely large universe seems self-evident. It is a universe that goes on forever, without limit. It has infinite spatial dimensions of length, volume, etc. Essentially, if you were to travel in a straight line from any point in the universe, you could go on forever and keep entering new regions of space. Whether the universe is or is not infinite is a matter of scientific study; it cannot be determined apriori. There is no philosophical contradiction in positing an infinite universe.

Martin

[sjw]

With respect to the natural Planck Units (such as Planck Length, etc.) we are 15 or 16 orders of magnitude from "rock bottom" (assuming there is a rock bottom) with the best technology currently at hand. It is highly unlikely that we will ever hit rock bottom (each advance toward smallness requires an order of magnitude more energy). The LHC is as good as we have at the moment. It generates energy of the order of magnitude of a tera-electron volt (somewhat less, actually).

Ba'al Chatzaf

There's no reason to suppose that the Plank Length is "rock bottom".

And all this raises the question of why, even if one approves of the methods of theft in order to further the "greater good", governments should be stealing money from the peasants to fund things like the LHC. Because no matter how much money they spend, it will never be enough.

Shayne

[Dragonfly]

You would need to clarify to me what you mean by infinitely big universe. I would say that an infinitely big universe would be one in which no ratio between it as a whole and its parts exists.

Why should such a ratio exist? If the universe would be finite, there would be a number M for which the distance between any two objects in the universe is always < M. But there is no a priori reason that there couldn't be an object with a distance to a given object that is > M and that argument can be repeated ad infinitum, meaning that there doesn't have to be a natural number M with that property. In such a universe there would be a bijection between the number of objects in the universe and the natural numbers. It may be a frustrating idea that you'll never finish counting them, but it is not contradictory.

[9thdoctor]

If this thread hits 800 posts, I'm just going to shoot myself... I'm warning you guys....

You have just encouraged a lot more people to contribute to this thread.

Has anyone heard from Phil lately? We're pushing 1,200 now, I hope he's not unidentified in a cooler with a tag on his toe.

We may arrive at a "complete" picture in the sense that every particle we've seen has a place in a theory that perfectly describes what we know, but we cannot know that these particles are the "bottom."

This strikes me as an extraordinary claim about the future of science. Naturally it’s unfalsifiable, and even rather vague.

[sjw]

We may arrive at a "complete" picture in the sense that every particle we've seen has a place in a theory that perfectly describes what we know, but we cannot know that these particles are the "bottom."

This strikes me as an extraordinary claim about the future of science. Naturally it’s unfalsifiable, and even rather vague.

Popper doesn't set the standards here.

It's a simple point actually. First, it's what we've observed -- no matter how small we've broken up matter, we've always managed to go smaller, up to the limits of government funding. Further, if you suppose a smallest element of reality, then how could one conceivably know that it is the smallest? You couldn't, unless God whispered it into your ear. Been hearing voices lately?

Shayne

[BaalChatzaf]

With respect to the natural Planck Units (such as Planck Length, etc.) we are 15 or 16 orders of magnitude from "rock bottom" (assuming there is a rock bottom) with the best technology currently at hand. It is highly unlikely that we will ever hit rock bottom (each advance toward smallness requires an order of magnitude more energy). The LHC is as good as we have at the moment. It generates energy of the order of magnitude of a tera-electron volt (somewhat less, actually).

Ba'al Chatzaf

There's no reason to suppose that the Plank Length is "rock bottom".

And all this raises the question of why, even if one approves of the methods of theft in order to further the "greater good", governments should be stealing money from the peasants to fund things like the LHC. Because no matter how much money they spend, it will never be enough.

Shayne

It might be enough to find the Higgs Boson (maybe). But that is no excuse for stealing from the people.

Ba'al Chatzaf

[Dragonfly]

(An aside to DF: You won't like the next part. It involves violence to spiders. Sorry. Both L and I are allergic to spider bites -- he's potentially life-threateningly allergic -- and we swat to kill the beasties when we see them in the house.)

On attempting to kill a spider, I have many times had the experience, if I swat quickly with a dry tissue so as not to let the thing get away before I can go wet a tissue, the spider will withdraw its extended legs close to its central body producing a tight ball which doesn't easily crush. The result might be that I think I've done for the creature, but upon my lifting the tissue, the spider re-extends its legs and scurries off.

Chasing a spider is not without risks, as you can read here.

[kiaer.ts]

To claim that the universe is finite is simply to assert the incoherence of saying that however big existence is, it is even bigger than that.

.

There is nothing a priori incoherent in the notion of an infinitely big universe (or "omniplex" if you like), only in the way you express it: would you say that however big the set of natural numbers is, it is even bigger than that?

You would need to clarify to me what you mean by infinitely big universe. I would say that an infinitely big universe would be one in which no ratio between it as a whole and its parts exists. I do not see any coherent meaning assignable to the notion.

The set of natural numbers is not a thing, it is a idea left incomplete. It is defined negatively. There could be no actual enumeration of it. One would be forced to say that however large an actual enumeration of the set of whole numbers is, it is even bigger than that. The universe, unlike the set of natural numbers, is actual.

I do understand the idea of a finite yet unbounded universe. In order for me to understand an infinite universe, you'll need to unpack for me coherently what would be meant by infinite in more concrete terms.

I agree with Dragonfly about this. I don't see anything at all incoherent or contradictory about an infinitely large universe. You're correct that the analogy between the universe and the set of all natural numbers is not very good, since the latter is an abstraction, while the universe actually has a physical existance. What is meant by an infinitely large universe seems self-evident. It is a universe that goes on forever, without limit. It has infinite spatial dimensions of length, volume, etc. Essentially, if you were to travel in a straight line from any point in the universe, you could go on forever and keep entering new regions of space. Whether the universe is or is not infinite is a matter of scientific study; it cannot be determined apriori. There is no philosophical contradiction in positing an infinite universe.

Martin

Can you explain, Martin, how the universe might be finite, yet unbounded?

[Martin Radwin]

To claim that the universe is finite is simply to assert the incoherence of saying that however big existence is, it is even bigger than that.

.

There is nothing a priori incoherent in the notion of an infinitely big universe (or "omniplex" if you like), only in the way you express it: would you say that however big the set of natural numbers is, it is even bigger than that?

You would need to clarify to me what you mean by infinitely big universe. I would say that an infinitely big universe would be one in which no ratio between it as a whole and its parts exists. I do not see any coherent meaning assignable to the notion.

The set of natural numbers is not a thing, it is a idea left incomplete. It is defined negatively. There could be no actual enumeration of it. One would be forced to say that however large an actual enumeration of the set of whole numbers is, it is even bigger than that. The universe, unlike the set of natural numbers, is actual.

I do understand the idea of a finite yet unbounded universe. In order for me to understand an infinite universe, you'll need to unpack for me coherently what would be meant by infinite in more concrete terms.

I agree with Dragonfly about this. I don't see anything at all incoherent or contradictory about an infinitely large universe. You're correct that the analogy between the universe and the set of all natural numbers is not very good, since the latter is an abstraction, while the universe actually has a physical existance. What is meant by an infinitely large universe seems self-evident. It is a universe that goes on forever, without limit. It has infinite spatial dimensions of length, volume, etc. Essentially, if you were to travel in a straight line from any point in the universe, you could go on forever and keep entering new regions of space. Whether the universe is or is not infinite is a matter of scientific study; it cannot be determined apriori. There is no philosophical contradiction in positing an infinite universe.

Martin

Can you explain, Martin, how the universe might be finite, yet unbounded?

No, I can't. I can neither understand nor visualize such a universe. Here's what little I do know about this subject.

Cosmologists have defined a parameter called omega, which is the average density of matter in the universe. The idea behind this is that, if omega equals one, there is just enough matter in the universe for gravity to stop its expansion. If omega is greater than one, then the force of gravity was proposed to be sufficient to stop the expansion of the universe and begin a contraction phase, or a "big crunch". If omega is less than one, then there is insufficient matter in the universe to stop its expansion, and the universe was proposed to go on expanding forever. As I recall, the most recent estimate of omega is about 0.03, or about 1/30 the density required to stop the expansion. But this is based on counting only the visible matter in the universe. Astonomers have now proposed the existence of "dark matter", which if it exists as they believe, would increase the value of omega by quite a bit. When recent astronomical observations led to the conclusion that the expansion of the universe is in fact accelerating rather than slowing, they proposed the existence of a strange, unseen entity which they now call "dark energy", which basically has the same function as Einstein's cosmological constant.

I've also read that, according to the Einstein field equations, a universe with omega less than one would be an "open", infinite universe, whereas a universe with omega greater than one would be a "closed", finite universe. This closed, finite universe would be finite, yet unbounded.

All of this, which is basically the existing standard big bang theory, is highly speculative. Noone has ever identified and proven the existence of either the dark matter or dark energy. It may even turn out that the universe is not actually expanding at all. The expansion theory is based on an interpretation of the cosmological red shift, which may turn out to have a different, not presently identified mechanism.

Anyway, getting back to my previous point, I don't see how the finitude or infinitude of the universe can be established from apriori philosophical premises. If the answer to this daunting question is ever learned, it will be by means of future scientific investigation. Philosophy can only discount the possibility of things that are self-contradictory. And there's nothing self-contradictory about either a finite or an infinite universe.

Martin