An Idea of Infinity

Peikoff may have said that the universe is finite, but did Rand or Aristotle? It sounds out of character, but I'll stand corrected if someone shows me the citations.

Peter

Post 21

Thursday, August 17, 2006 - 2:18pm

Peter,

I can�t think of Rand citations at the moment and I haven�t read Aristotle�s argument in the original (which would be Greek, but I haven�t even read it in English translation.) However, I understand his argument from other sources to be: �look around you. Do you see anything that is infinite? No. That�s because �infinity� is a mathematical construct with no referent in reality. So, nothing is infinite, not even everything.�

I put roughly the equivalent of my post #18 to Harry Binswanger (back when I ran an ARI-affiliated campus club and was OK) and he responded that there were only two alternatives for the universe: Either it is finite or it is infinite. And, as Aristotle has proved there are no actual infinities; therefore the universe is finite. I responded that there are no causeless events even though the universe can be causeless, so why can�t the universe be infinite even though we haven�t found any particular infinities, (which we couldn�t do anyway even if they existed)? He just said, �only two alternatives.�

Post 22

Thursday, August 17, 2006 - 2:19pm

According to Objectivism (and Aristotle), nothing can be infinite, because everything must exist in some definite size and quantity. Objectivism does recognize such a thing as potential infinity, which simply means that one can always project a size or quantity beyond any actual size or quantity that one specifies. But whatever the size or quantity of actual existents, it is of some finite magnitude, however large.

As for time, again Objectivism takes its cue from Aristotle. Quoting Rand: "My view is, in effect, Aristotelian. Aristotle's position is (in my words) that there is no such thing as independent time or space. The universe is finite, and the concept of time applies to the relationship between entities. Specifically, time is a measurement of motion, which is a change of relationship between entities within the universe. Time cannot exist by itself. It exists only within the universe; it does not apply to the universe as a whole. By 'universe' I mean the total of what exists. The universe could have no relationship to anything outside itself: no motion, no change, and therefore, no time." (Ford Hall Forum, Boston, 1968) - Quoted in Ayn Rand Answers, p. 151)

Or, in Peikoff's words: "Time is a measurement of motion; as such, it is a type of relationship. Time applies only within the universe, when you define a standard -- such as the motion of the earth around the sun. If you take that as a unit, you can say: 'This person has a certain relationship to that motion; he has existed for three revolutions; he is three years old." But when you get to the universe as a whole, obviously no standard is applicable. You cannot get outside the universe. The universe is eternal in the literal sense: non-temporal, out of time." ("The Philosophy of Objectivism," lecture series (1976), question period, Lecture 2. - Quoted in Harry Binswanger's, The Ayn Rand Lexicon, p. 503.)

Observe that by "eternal," Peikoff does not mean infinite in time. He means without time. Since the concept of time doesn't apply to the universe as a whole, it makes no sense to say that the universe exists in time or that it has a duration, either finite or infinite. Even to say that the universe will exist "forever" is incorrect, because "forever" means an infinite extension in time and there is no infinity. Typically, what people mean when they say that the universe will exist forever is that it won't end at some point in time, which is true, but not in the sense that people typically mean it. The concept of "some point in time" can apply only to events and entities within the universe, not to the universe as a whole.

The same reasoning applies to any alleged origin of the universe. (Again, by "universe" in this context is meant the sum total of that which exists, not simply the present state of the cosmos whose origin is the "Big Bang." The term "universe" gets its meaning from the term "universal" which refers to all units of a particular kind -- in this case, all the fundamental constituents of existence.) As such, the universe did not begin, let alone begin at some point in time (since for time to exist, the universe would already have to exist). Time is in the universe; the universe is not in time. The concept of a "beginning" also presupposes prior existence, because in order for something to begin, it has to begin as the result of the action of already existing entities. Nihilo ex nihilo; from nothing comes nothing.

- Bill
(Edited by William Dwyer
on 8/17, 2:26pm)

Post 23

Thursday, August 17, 2006 - 2:34pm

Thanks, Bill. That was very enlightening.

Post 24

Thursday, August 17, 2006 - 2:36pm

Bill,

If I misunderstand you, please correct me. You seem to pin your (Aristotle�s, Objectivism�s) argument on the fact that every individual entity in existence must be of finite size, position, etc. (I agree) and therefore the most we get is lots of finitude. You attempt to scale that up to an assertion that there must also be a limit to the number of individual entities in existence.

Why must there be a limit to the number of entities in existence?

Post 25

Thursday, August 17, 2006 - 2:49pm

The universe is finite; it has to be or it would lack identity, and, Rand has said, "Existence is Identity..."

Time is just a measure of motion. It is not a physical existent.

Post 26

Thursday, August 17, 2006 - 3:01pm

Bill,

You beat my short and concise post with your longer and more detailed post.:)

Oh, well, consider mine the condensed version. Excellent presentation, Bill.

Post 27

Thursday, August 17, 2006 - 3:24pm

Bob,

Don�t you commit the fallacy I wrote of earlier? That every event has a cause does not imply that the universe must have a cause.

Just as that every entity is finite does not imply that the universe must be finite�it could be that there is no end to the number of entities that exist. The universe still has its identity: All that which exists.

(Edited by Jon Letendre
on 8/17, 3:27pm)

Post 28

Thursday, August 17, 2006 - 3:44pm

Bill -- Thanks for your thoughts.

Jon, Laure -- You raise good points. My point with the Stephen Hawking example -- that asking what happened before our universe (or the Big Bang) is like asking what's north of the North Pole -- suggests that we're still grasping to understand the nature of space and time. In the usual way we speak of these matters, time and space are measures of entities in the universe. We can't speak of the universe in time.

But as we explore the implications of quantum theory, string theory, the unity of the four forces, the possibility of a fifth force, black holes, parallel universes, branes, etc., we find our vocabulary and ability to grasp these matters wanting.

I say forge ahead and don't get too hung up on making reality to conform to our understanding of certain concepts. I worry that an Objectivist who took too narrow a view of concepts 100 years ago would rejected out of hand the discoveries of quantum mechanics. It's okay to reserve one's interpretations of the implications of quantum mechanics for causality, etc. But we need to be careful about having too conventional an understanding of a concept. We might be seeing only a limited aspect of reality.

After all, Newton's description of motion in the universe was correct in context. When the context changed and we looked at a different aspect of reality -- matter accelerating to near the speed of light -- you must add Einstein's insights.

Post 29

Thursday, August 17, 2006 - 4:09pm

Hi Ed,

I have no issue with Rand�s or Bill�s or your statements regarding time.

What are your thoughts about the positive assertion that the universe is finite in quantity of entities?

Post 30

Thursday, August 17, 2006 - 9:44pm

Jon,

The number of entities in the universe must be finite, because there is no actual quantitative infinity, only potential. To better understand this, consider the process of enumeration -- of counting. No matter how long one counts, one will always be at some finite point in the process, because any actual number must be finite; it must be a particular number, however large. Now, one can certainly continue indefinitely to add to this number, but however much one adds to it, the addition will always be of a finite magnitude.

So, to say that the number of things is infinite is to say that there is no actual number. But if there is no actual number, then what is it that's infinite? It can't be the number, because there is none. Thus, the concept of a number that is actually infinite entails a contradiction. If the number is infinite, then it doesn't exist; if it exists, then it can't be infinite.

- Bill

Post 31

Friday, August 18, 2006 - 12:18am

Bill,

All you have shown is that if one sets out on the quest to count up all the entities in existence, that one will at all times necessarily have a finite running count. I agree.

However, the question is: Can one know that one will eventually find the LAST entity and be finished counting (Finitude, your assertion,) or might it be that one will never find the last entity because there will always be another to count because there is no limit to the universe (Infinitude, my posited possibility)?

The contradiction you identify does not present a problem because, while the count at any point in the quest will be finite, the point is that one would never run out of entities to add to the count.

I can already smell your demand for a final count. The point is there isn�t one. Just as the universe is eternal, that is �without time,� it may also be inextensive, or �without definite extensiveness.� Or: Just as time is in the universe, the universe is not in time�so extensiveness and enumeration are in the universe, not the other way around.

Post 32

Friday, August 18, 2006 - 6:21am

Jon,

The total mass-energy, total electric charge, and total angular momentum of the universe are each finite and constant. This claim to knowledge is a claim of contemporary general relativity. That is, it is a claim of physics. The physics uses mathematics, but unlike pure mathematics, knowledge in physics is not obtainable simply "from the armchair."

You are methodologically correct, Jon.

Here are some references for my physical claims above:
Gravitation by Misner, Thorne, and Wheeler
General Relativity by Robert Wald

Post 33

Friday, August 18, 2006 - 10:37am

Laure, you write (#19), concerning questions of cosmic infinity:

But ultimately, does it matter?

You have a kindred spirit in Ayn Rand. In her Anthem, she gives her protagonist the following deliverance:

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. I know not and I care not. For I know what happiness is possible to me on earth. (1946, p. 87)

Contrary to popular saying, the Copernican revolution has no valid implication that diminishes the importance of human life. The discovery that the sun does not orbit the earth does not change human importance. Physical size, mass, or centrality are not the scales of value. The phenomenon of life�life right here on earth�is the source and perpetual support for all value, meaning, and significance.

Like the minds posting here, and the human mind for thousands of years past, Rand wondered what is out there in the cosmos. In that sense of care, she cared. In her developed philosophy, she took her most basic axiom, existence exists, to imply �that nature, i.e., the universe as a whole, cannot be created or annihilated, that it cannot come into or go out of existence� (MvMM 25)

It is often said that the universe came into existence at the big bang (the initial singularity). That is inaccurate. The theory and observations that tell us there was a big bang also tell us that the mass-energy of the universe has been constant all along the way, even back at the beginning of time (its boundary in the past). Extended space and time may have come into existence at the big bang, but a nonzero finite amount of mass-energy was never absent.

I think that Rand was correct in saying that we know the universe did not come into existence and that we know this prior to our discoveries in contemporary cosmology grounded in physics. But it is a delicate matter, for I would not want to rule out a beginning of time or space simply from the armchair.

General relativity led us to knowledge of the big bang. The big bang does not represent an explosion of matter into preexisting space. There was no preexisting nonsingular space; there was no spatial distance, area, or volume; there was only that single point of spacetime. In addition,

since spacetime structure is itself singular at the big bang, it does not make sense, either physically or mathematically, to ask about the state of the universe �before� the big bang; there is no natural way to extend the spacetime manifold and metric [past-ward] beyond the big bang singularity. (Wald 1984, 99)

We should not, however, expect classical general relativity (which is what Professor Wald is writing about here) to apply all the way back to an absolute singularity. To fathom the very beginning, we need a quantum theory of gravity.

On the question of the beginning of time or of space, I will wait for physics.

(Edited by Stephen Boydstun on 8/18, 10:40am)

Post 34

Friday, August 18, 2006 - 10:57am

On the question of the beginning of time or of space, I will wait for physics.

I agree, Stephen. I didn't mean to imply that I'm not interested in the issue. Really what I mean by "does it matter" is, would it make me change my mind about anything else? If the universe is finite, or if it's infinite, do I have to say "oh well, then, there goes my atheism out the window." I don't think so.

Post 35

Friday, August 18, 2006 - 11:30am

Hi Stephen,

Thanks for the comments and references. My local used bookstore says they move a few copies of Wald�s book and they will be calling me when they get the next copy. (�Gravitation� seems a little harder to find, I�m still making calls.) Looking forward to reading the arguments. Of course, the follow-on question is: Might there exist expanded singularities in addition to this one. There is probably no evidence for such, but at the same time I find odd positive assertions that there cannot be, likewise for assertions that there can exist only a finite number of them.

I agree with you that these are physics/scientific questions that cannot be answered rationalistically or by philosophical deduction, or as you say, �from the armchair.�

Funny that you mention this:

�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. I know not and I care not. For I know what happiness is possible to me on earth.�

Back in high school my girlfriend and future wife made a project for an art class. The assignment was art with a calligraphic writing. She knew I was crazy for Anthem but she didn�t know any verses I liked in particular and didn�t want to spoil the surprise, so didn�t ask me. She made a watercolor of a sunset over mountains and sea, and wrote that verse. It was and is indeed one of my favorite verses from the book and has hung on my walls ever since then.

Post 36

Friday, August 18, 2006 - 2:49pm

Stephen -- Thanks for your post #33.

Especially to the point is that the Copernican revolution did not diminish man, nor for that matter did Hubble's discoveries concerning other galaxies and the expansion of the universe. In fact, they all give us a greater appreciation for the power of human reason, that we can understand the universe in which we live.

Also you write "On the question of the beginning of time or of space, I will wait for physics." I agree, as I stated a few posts ago, that we need to take care not to rule out certain discoveries, e.g., quantum theory, before the fact because they "seem" not to be in accordance with our philosophy. I do think, of course, that they will be in accordance and that we might need to refine our definition of certain concepts, such as space and time.

Post 37

Friday, August 18, 2006 - 3:22pm

Jon wrote,

All you have shown is that if one sets out on the quest to count up all the entities in existence, that one will at all times necessarily have a finite running count. I agree.

However, the question is: Can one know that one will eventually find the LAST entity and be finished counting (Finitude, your assertion,) or might it be that one will never find the last entity because there will always be another to count because there is no limit to the universe (Infinitude, my posited possibility)?

Well, there can't always be another one to count, because that would imply that there is no last entity to count, not that there is no last entity that one will succeed in counting (because one doesn't have enough time to count it). To put it another way, what I am saying is that the denial of a "last entity" entails a contradiction. As I noted earlier, the concept of "universal" refers to all units of a certain kind. Accordingly, the concept of "the universe" refers to all of the entities in existence. But if there is no last entity, then there is no such thing as "all of the entities in existence," since in referring to all of something, one is saying that there is no more, whereas to say that the universe is infinite is to say that there is always more.

The contradiction you identify does not present a problem because, while the count at any point in the quest will be finite, the point is that one would never run out of entities to add to the count.

Right, I see your point, but my point is that just as there is no actual infinity of numbers but only the potential to extend the process of counting, so there is no actual infinity of entities but only the potential of finding another one, since we are not omniscient. As I said earlier, to say that the number of things is infinite is to say that there is no actual number of things. But if there is no actual number of things, then what is it that's infinite? It can't be the number of things, because there is none. Thus, the concept of an actual number of things that is nevertheless infinite entails a contradiction. If the number of things is infinite, then there is no actual number; conversely, if there is an actual number, then that number can't be infinite. Jon, this is a real contradiction that you need to resolve if you're going to defend your statement that the number of things in the universe could be infinite. So far, I don't see that you've done that.

I can already smell your demand for a final count. The point is there isn�t one. Just as the universe is eternal, that is �without time,� it may also be inextensive, or �without definite extensiveness.� Or: Just as time is in the universe, the universe is not in time�so extensiveness and enumeration are in the universe, not the other way around.

I don't think that analogy will work. Time is in the universe, because time is relational, and there is nothing outside the universe to which the universe can be related. But if one views the universe as a collection, then one can say that the collection comprises an actual number of units and no more.

- Bill
(Edited by William Dwyer
on 8/18, 3:25pm)

Post 38

Friday, August 18, 2006 - 4:41pm

Hi Bill,

Sorry, I am not seeing any problem. As I see it, you are begging the question of infinitude vs. finitude and deciding it a priori by declaring that there must be an �actual number of things.� You show no hesitancy to say that there needn�t be an actual age of the universe�in favor of eternality (I agree,) so why the hesitancy in accepting that it needn�t be of determinate extension�in favor of inextensiveness?

Your objection to that analogy amounts to insistence that �collection� or �all that exists� must mean, �that there is no more,� but�why? Why does it have to mean that? Whatever you find I will say that it is part of �all that exists.� You will never run out of things to find and I will never stop saying, �yes, that too.� No contradiction.

Post 39

Friday, August 18, 2006 - 7:06pm