Nate Smith

An Ageless Universe

84 posts in this topic

I think there is a long philosophical tradition regarding the concept of "infinity" that many of us who lack formal academic training in philosophy are missing out on. From a strictly etymological viewpoint, "infinity" would seem to mean simply anything that is not "finite." But Alex and others repeatedly mention things that are not finite, yet not infinite, or things to which the concept of finitude do not apply yet are not infinite. It has been mentioned that Dr. Binswanger argued that infinity is impossible-- yet surely it is possible for something to exist and not be finite.

Stephen mentioned that in ITOE infinity is described as valid only as a concept of method, and I've seen it mentioned elsewhere that Dr. Binswanger has argued against it even being used as a concept of method, although I haven't been able to hear this argument (and if anyone knows the source of this, btw, I'd like to know). It's clear to me that in all of this, there is a meaning to "infinity" beyond what would be clear from its etymology.

Does anyone know the historical source and contextual meanings of these concepts of "infinity"? I have seen arguments against "actual infinites" in a collection of medieval philosophy that I own, but I don't know who were the actual first philosophers to discuss this or what they meant by "infinites."

For me, this is the source of a lot of confusion in these types of cosmological discussions. It took me a long time to figure out that these writers were using "infinite" to mean something other than simply "not finite." So you can see why "unbounded but not infinite" seemed like a direct contradiction in terms, at first. If someone can give me a little more context in this regard, I think I'll have a much better perspective for evaluating Alex's essay and other similar topics that come up.

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Thanks for your kind words, here and elsewhere.  However, I do hope you go back and read my essay, and what's more I strongly encourage you to directly address more than what I have written here.  For, I agree with Stephen that my arguments have not yet really been addressed in this thread, and in order to do so it is imperative that my essay itself be addressed.  That is, after all, where I have laid out my arguments for all to see, for which this (or any other) thread is not a substitute
.

Yes, I'll read it again, as soon as time permits, and post my thoughts.

Again, there really isn't a substitute for actually pointing out in my essay where you think a particular argument of mine goes astray, and I hope you will choose to do this as our discussion continues.

Again, I will - point for point. Judging from just your criticisms here, I may even find that I've misinterpreted your positions, in which case I apologize in advance. As you say, to be continued...

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If someone can give me a little more context in this regard ...

Below is a short post I wrote a number of years ago on a different forum. Perhaps it will be helpful to you as a starting point in seeking what you want to understand. Feel free to ask specific follow-up questions.

**************************************

XXXX XXXXX has expressed some concerns about mathematical

infinites, and if I understand him correctly he seems to hold a

mathematical infinity as some sort of anomaly, one for which,

XXXX believes, we need to revise our mathematics. There are two

levels on which I would like to respond.

The first level is to identify that the use of infinity in

mathematics is as a method of concept, and follows Aristotle's

distinction between the potential and the actual -- that the

infinite is a mathematical process, while the actual is always

finite. In this sense, as a concept of method, the infinite is

similar to 'i', the square root of minus one. Mathematics itself

does not say that 'i' exists in the world as a real physical

object, but it does have an undeniable and valid use throughout

most all fields of mathematics. The method of 'i' is not an

anomaly which requires a revision to mathematics, but rather a

powerful tool which is extremely useful to mathematical pursuits.

Likewise, infinity naturally arises in mathematics, from simple

calculus to some of the most abstract of mathematical

disciplines. There are almost countless physical problems in the

field of mathematical physics where the mathematically infinite

applies, and it is an invaluable tool in formulating and solving

the mathematical means by which a physical process can be

grasped. On the other hand, if there is an attempt to declare

that a mathematical infinity corresponds directly to a fact of

reality, then that is indeed a premise to be rightfully

condemned.

The second level has to do with an argument offered by Harry

Binswanger: namely that the infinite is an invalid concept, in

that counting numbers lose their meaning beyond a certain range;

a range beyond which we cannot physically represent a number, and

it therefore becomes meaningless, and a range by which our own

epistemologies can sensibly grasp, by whatever notation, the

limited, as opposed to the unlimited. It is by this argument

that Harry concludes the number system is not even _potentially_

infinite. (If I have done any injustice to the essence of Harry's

formulation, I apologize, and I'm sure he can correct me.)

I have found Harry's arguments to be rather convincing, and I

think a basic foundational mathematics without infinitesimals is

entirely appropriate. However, in some sense this is more a

philosophic problem than a practical one (not to mean that

philosophy should not be practical). Such a reformulation could

indeed change some of the more esoteric abstractions, such as

transfinite infinities, but the same mathematical series, and the

same mathematical integrations, will still give the same

mathematical results, because limits will still apply no matter

how they are understood to be. So, for instance, the series:

SUM{from n = 1 to infinity} of 1/(2^n)^2

will still equal the value 1/3 even if infinity is replaced with

a more rationally justified limit. Note that, as discussed by Leo

Zippin, more than 2000 years ago Archimedes encountered this

series when he applied to an infinite process the method of

exhaustions devised by Eudoxus some two hundred or so years

before him. Archimedes was able to logically prove that the

limit of that series is 1/3, because that limit itself has a

physical meaning and it represents a mathematical truth.

**********************************************

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I see that Stephen has posted a reply to Bold Standard, covering some issues with regard to the mathematical concept of infinity. So, allow me to just make clear that my discussion of infinity in this post refers to "infinity" taken metaphysically, not mathematically.

I think there is a long philosophical tradition regarding the concept of "infinity" that many of us who lack formal academic training in philosophy are missing out on.  From a strictly etymological viewpoint, "infinity" would seem to mean simply anything that is not "finite."

That may be true (I'm not at all knowledgeable with regard to etymology), although I just wanted to point out that I don't think "infinite" today literally means "not finite." For, I think everything that exists is finite; so, as soon as you talk about something that is "not finite," you've moved into a fantasy world populated by identity-dropping human creations. And, an infinity is certainly one way to violate the law of identity in this regard, but it is not the only way. (See, for example, Locke's substratum, or the nothingness of the Existentialist.) So, when you say that something is "not finite," I don't think it necessarily implies that it's infinite.

But Alex and others repeatedly mention things that are not finite, yet not infinite, or things to which the concept of finitude do not apply yet are not infinite. 

For myself, I haven't posited anything that is not finite. The most I have said is that the universe does not, e.g., have a finite or infinite size, but by this I mean that "size" is just wholly inapplicable. So, the universe has no size, no age, etc.

It has been mentioned that Dr. Binswanger argued that infinity is impossible-- yet surely it is possible for something to exist and not be finite.

Could you give an example? The only one I would entertain is the universe itself, even though I do ultimately think that it should be described as finite.

Stephen mentioned that in ITOE infinity is described as valid only as a concept of method, and I've seen it mentioned elsewhere that Dr. Binswanger has argued against it even being used as a concept of method, although I haven't been able to hear this argument (and if anyone knows the source of this, btw, I'd like to know).

He makes this argument in his course, Selected Topics in the Philosophy of Science, given in (I believe) 1987. This lecture is available through the Ayn Rand Bookstore (click here).

So you can see why "unbounded but not infinite" seemed like a direct contradiction in terms, at first.  If someone can give me a little more context in this regard, I think I'll have a much better perspective for evaluating Alex's essay and other similar topics that come up.

If infinity is going to refer to something metaphysically invalid, then part of the context that you need is the locus of infinity's contradiction of the law of identity. And, on this score, I (and Objectivism, so far as I can tell) say that the locus is this: "infinity" attempts to posit a quantity that is "too big" or "too small" to be of any particular quantity.

But, observe that this doesn't mean that anything "unbounded" must be infinite; rather, it simply means that any quantity that is unbounded (i.e., not delimited) must be infinite. So, an unbounded size, age, or quantity of entities would indeed be contradictory, since such existents inescapably refer to quantities of something ("age" refers to a quantity of temporal units, "size" to a quantity of spatial units, etc.) However, positing an unbounded universe harbors no such contradiction, because there is no reason why the universe as such as to be a quantity. And it is in this sense that one can speak of something that is unbounded, but not infinite.

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For myself, I haven't posited anything that is not finite.  The most I have said is that the universe does not, e.g., have a finite or infinite size, but by this I mean that "size" is just wholly inapplicable.  So, the universe has no size, no age, etc.

Alex, I think I have a full understanding of the points you raised in your essay, but there are a couple of integrations I want to check with you.

Would you say that the universe is "without size" in the same way that the air around us and water are both "without color," or is there some difference? They are both cases in which a certain property just doesn't apply to certain existents, but is there any difference aside from the concretes involved?

Is th asizal nature of the universe different in some way from the asizal nature of concepts of consciousness? To clarify my question further, I'm not asking about any difference between concepts of consciousness and the universe themselves, but specifically about their asizality. It seems like there may be some difference between the two, since the universe contains sizal elements, whereas consciousness does not. I'm not sure if positing a difference on these grounds is fallacious.

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Alex, I think I have a full understanding of the points you raised in your essay, but there are a couple of integrations I want to check with you.

The examples you raise are fundamentally similar to the universe, in the sense that in every case you have something that, by its very nature, lacks a certain attribute. There are non-fundamental differences that may be nagging at you, though, and here are three:

1) As you point out, there are components of the universe that possess size, whereas there are no components of consciousness that possess size or no components of air that possess color. (Incidentally, components of the universe also possess color, but the universe as such has no color.) This is something that can happen with collections -- the characteristics of its parts are not shared by the whole -- and it is why we have the fallacy of composition. (However, it is worth noting that, with regard to size, the universe may be the only thing that lacks size but nevertheless has "sizal" parts.)

2) The fundamental reason why the universe lacks such attributes is because it is the sum total of that which exists. And this is different, of course, from why consciousness lacks size, or why air lacks color.

3) To make an epistemological point, direct awareness can more or less tell you that consciousness lacks size, or that air lacks color. However, with the universe, one needs to engage in very abstract thought in order to see that it lacks such attributes.

I'm not sure if that's the kind of answer you were looking for, but that's what I came up with off the top of my head. The most important point, however, is that whatever the differences, "size" is just as inapplicable to the universe as it is to consciousness, or as "color" is to air.

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Adding up the masses of all the matter, if that could be done, would yield just that: a sum, and the total mass of things in the universe would be 50 times X, to use your example.  But that is not the same as saying that the mass of the universe would be 50 times X, or even that there is such a thing as the mass of the universe.  (I agree that there isn't.)

Why aren't they the same thing? You've asserted it, but not proved it.

Because the properties of the parts are not necessarily shared by the whole.

Another example would be color. There are entities in the universe that have color. It might even be the case that they have some average color. But the universe as a whole does not have color.

Stepping away from cosmology: a person has a height. One could sum up the heights of the people in a group, but one would then have just that: a sum of heights in the group. That might or might not be a useful number to know, but it would not be the same as "the height of the group", because the group does not have a height.

I don't know how to state my position any more clearly, so if we're still in disagreement, I'm content to leave it at that.

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I'm not sure if that's the kind of answer you were looking for, but that's what I came up with off the top of my head.

Yes, it helps. Thanks.

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Allow me to frame the issue this way: assuming you think that metaphysical infinities contradict the law of identity, try and see if you can draw out what it is about my physical picture of the universe which contradicts the law of identity.  Where is the contradiction in saying: "The universe is neither a finite quantity nor an infinite quantity, but simply outside the context of 'quantity' altogether"?

I've been thinking about this for a while, and I think you're right, it doesn't contradict the Law of Identity. The LoI says that that which exists does so in some particular way. At first I was tempted to say that it also existed in some particular amount, but I don't think this follows. Once this clicked, your article started to make a lot more sense to me. It's very persuasive.

I do have a question about the way you use the word infinite though. In the beginning of your article you state that "the infinite is the impossible." Given my new understanding of the LoI, I needed to remind myself why this is. Towards the end of your article you state:

The concept of infinity is metaphysically invalid because it attempts to describe an existent (e.g., an attribute) as existing, but as nothing in particular.

I take this to mean you define finite and infinite (in the metaphysical sense) to mean ‘with identity’ and ‘without identity’ respectively. As further evidence for this conclusion, you also said that:

What exactly do you mean when you say that the universe is not finite in size, but it is finite. In what respect is it finite?

That being said, my essential answer to your question is that there is no "respect" in which the universe is finite, at least in the sense of there being some finite attribute that the universe possesses. Rather, when one says that the universe is finite, I don't think that this need mean more than the simple fact that the universe possesses identity.

Please correct me if I am mistaken.

I mention this because the definition of infinite is 'having no boundaries or limits'. When you use it to mean, 'having idenitity', why do you use it that way?

Why not say that everything in the universe is finite (has boundaries and identity), but the universe itself is infinte (in that it has no boundaries) and finite (in that this is its identity)? In other words, in one context, it is infinite (in the mathematical sense of there always being one more) and in another context it is finite (in that this is its identity).

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Thanks for the responses.. The way I use words, I would have described anything to which "finity" does not apply as "not finite" or "infinite." Thus attributes such as "hardness" or abstractions such as "justice" or anything that doesn't exist in the metaphysical world, I would think of as not finite, but still existing.

Your description of infinity as meaning something that is a quantity but too big to be a quantity seems invalid like the typical meaning of "selfishness." Could it be "infinity" just needs a better definition?

Could you give an example?  The only one I would entertain is the universe itself, even though I do ultimately think that it should be described as finite.

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Why not say that everything in the universe is finite (has boundaries and identity), but the universe itself is infinte (in that it has no boundaries) and finite (in that this is its identity)?  In other words, in one context, it is infinite (in the mathematical sense of there always being one more) and in another context it is finite (in that this is its identity).

What do you mean by "always being one more?" If you mean in relation to size, as in "always being one more inch," then I think Alex has more than adequately demonstrated in his essay why "size" does not apply to the universe. In fact, I think he has made clear that there is no respect in which the universe is quantifiable. That means that there is no respect in which the universe can be considered either finite or infinite, other than perhaps being finite in the sense of having identity, i.e., all that exists.

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Thus attributes such as "hardness" or abstractions such as "justice" or anything that doesn't exist in the metaphysical world, I would think of as not finite, but still existing.

This would mean that concepts of consciousness would not be quantifiable, which surely is not the case.

Your description of infinity as meaning something that is a quantity but too big to be a quantity seems invalid like the typical meaning of "selfishness."

I'm not sure of your meaning in how this relates to "selfishness." But, as I understand Alex, he has made clear that, in the metaphysical sense, to claim an infinite quantity is to assert a quantity that has no identity since that quantity is either too small or too big to be any particular quantity. This is perfectly consistent with my understanding of the Objectivist view.

Could it be "infinity" just needs a better definition?

What's the problem?

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This would mean that concepts of consciousness would not be quantifiable, which surely is not the case.

Hm. That's true. I think I'm starting to understand the way you're using "finite." But I'm still a little foggy on "infinite."

It would seem to me that by Aristotle's usage as you've explained it, "infinite" could be used to describe time into the future, since it has the potential to stretch on into a limitless series of events-- Maybe this is where I keep missing the whole point of Alex's essay, which I have read several times, but each time I think I understand it, some hidden premise of mine or unclear point of his (not sure which yet) knocks me back into a conceptual limbo. I'm not sure about the past, since it's already happened and wouldn't properly be a "potential," although I guess conceptualizing any given amount of time in the past would be a potential. But in regards to the future, and in regards to space-- as Alex describes his astronaut who never reaches "an end" to the universe-- I don't see any difference between those metaphysical potentials and the epistemological potential of natural numbers.

So if it's true that there are things epistemological, metaphysical, or both, that are not conceptually quantifiable-- or that have a potential to eventually exceed their presently quantifiable limit, it would seem to me that the word "infinite" would be just as useful to describe them as "unbounded" or another word that seems at least superficially synonomous.

As to my comparison to "selfishness," (the library was closing last night as I was posting so I was admittedly too vague, sorry :) ) I meant that people traditionally used "selfishness" as a word to describe an impossible contradiction, when the word itself, from an analyses of its root words, and as it was defined in dictionaries until recently, described a perfectly valid and essential concept.

I was wondering out loud whether "infinity" might be a similar type of word. Possibly "finite" too, since "has identity" is a concept that describes everything, therefore, nothing in particular; I would think a definition more along the lines of "is conceptually quantifiable" would be more useful-- as a concept that is descriptive of the relationship between metaphysical reality which is limited to be only what it is and the open ended nature of concepts, which are all potentially unlimited, from "number of chairs possible" on to "number of galaxies possible" and beyond.

But if "infinity" has too much bad history, it might be easier to just start saying "unbounded." That's why I brought up the historical issue. But I think I might need to hear that Binswanger lecture before I can ask for, or contribute anything else that might be useful. Also, I'm interpolating that in Alex's essay, "amount of space in the universe" is an presented as an invalid question, since "in the universe" implies a limited amount, which there is not. Is that an acceptable way of stating it?

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... But in regards to the future, and in regards to space-- as Alex describes his astronaut who never reaches "an end" to the universe-- I don't see any difference between those metaphysical potentials and the epistemological potential of natural numbers.

So if it's true that there are things epistemological, metaphysical, or both, that are not conceptually quantifiable-- or that have a potential to eventually exceed their presently quantifiable limit, it would seem to me that the word "infinite" would be just as useful to describe them as "unbounded" or another word that seems at least superficially synonomous.

I think we need to re-emphasize that "infinite" refers to a process, not to a "thing." Infinity as a concept of method, not as a metaphysical existent. The notion of a metaphysical infinity is a contradiction of identity; the claim that an attribute of an entity is infinite is the claim that at an attribute exists metaphysically, but in no specific amount. It is impossible to make sense, metaphysically, of, say, the infinite length of an object. An unbounded length is no particular length at all.

But "unbounded" is not synonymous with the "infinite," as witnessed by our notion of the universe. As Alex demonstrates in his essay, the universe cannot be bounded temporally or spatially, yet it is not infinite. And, the typical two-dimensional example of the surface of a sphere is also unbounded, but not infinite. Hence there is a need to maintain some distinction between "unbounded" and "infinite."

Also, I'm interpolating that in Alex's essay, "amount of space in the universe" is an presented as an invalid question, since "in the universe" implies a limited amount, which there is not.  Is that an acceptable way of stating it?

To ask about the amount of space is to ask a metaphysical question. And, as Alex demonstrates in his essay, the question itself only makes sense when you specify bounds, i.e., between "here" and "there." But, since the universe is unbounded, the question makes no sense.

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But "unbounded" is not synonymous with the "infinite," as witnessed by our notion of the universe. As Alex demonstrates in his essay, the universe cannot be bounded temporally or spatially, yet it is not infinite. And, the typical two-dimensional example of the surface of a sphere is also unbounded, but not infinite. Hence there is a need to maintain some distinction between "unbounded" and "infinite." 

Would it be accurate to say that, if one began traveling along the surface of a sphere, one would cover a potentially infinite distance without ever reaching a bound? (If this is impossible, would it at least be a correct usage of the term "infinite" as Alex and/or Aristotle used it?)

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Would it be accurate to say that, if one began traveling along the surface of a sphere, one would cover a potentially infinite distance without ever reaching a bound?  (If this is impossible, would it at least be a correct usage of the term "infinite" as Alex and/or Aristotle used it?)

I don't care for the expression "cover a potentially infinite distance." You can never "cover" the infinite; whatever distance you actually cover, that distance is finite. You are speaking about a method, a method which extends into infinity. The method exists as a potential, not as an actual, for whatever steps you take will exist finitely.

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It's funny that someone can say the same thing a dozen different ways, then suddenly it just clicks. Thanks for your patience, that last one clarified my confusion, for now at least. I can see the distinction between "bounded" and "finite" now-- I was conflating the two concepts before. Alex, I can see how you arrived at your definitions for "infinite" and "finite" now. No lack of clarity on your part.

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Maybe one of the problems here is that we have to clearly define what we mean by quantity. A quantity is a measurement of an attribute -- i.e., a quantity of length, duration, mass, intensity, etc.

The universe contains different things with different attributes. Pencils and fingers have various finite quantities of length (extension in space) and concepts of consciousness like happiness and justice do not. Emotions have various finite quantities of intensity and pencils do not.

The reason you can't get a finite mass of the universe by summing up all the masses of the things in the universe is because the universe includes a lot of things that don't have mass at all. Ditto for any other quantifiable characteristic you choose to name.

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But "unbounded" is not synonymous with the "infinite," as witnessed by our notion of the universe. As Alex demonstrates in his essay, the universe cannot be bounded temporally or spatially, yet it is not infinite. And, the typical two-dimensional example of the surface of a sphere is also unbounded, but not infinite. Hence there is a need to maintain some distinction between "unbounded" and "infinite." 

Sorry if I'm a little slow, but I've read all the posts in this topic as well as Alex's essay, and I am still not grasping the distinction between "unbounded" and "infinite". Would you please provide an explicit, concise definition of each term (if that's possible)?

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Sorry if I'm a little slow, but I've read all the posts in this topic as well as Alex's essay, and I am still not grasping the distinction between "unbounded" and "infinite".  Would you please provide an explicit, concise definition of each term (if that's possible)?

Have you read this thread: http://forums.4aynrandfans.com/index.php?showtopic=200, where Alex first introduced his essay to THE FORUM?

Definitions are given therein, and the difference between the two is discussed at great length.

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Sorry if I'm a little slow, but I've read all the posts in this topic as well as Alex's essay, and I am still not grasping the distinction between "unbounded" and "infinite".  Would you please provide an explicit, concise definition of each term (if that's possible)?

Infinity is a concept of method in which a process can potentially be extended without bounds.

Unbounded means without limits in some respect; without endpoints.

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...The reason you can't get a finite mass of the universe by summing up all the masses of the things in the universe is because the universe includes a lot of things that don't have mass at all.  Ditto for any other quantifiable characteristic you choose to name....

Your post is helpful to me. I have a question for you.

Is there anything contradictory in discovering the sum of the mass of all objects in the universe that have mass? Would this exist as a finite quantity of mass, if we had the means to discover it?

Also, to Stephen, the surface of a sphere analogy is a bit confusing to me. Yes, the sphere is unbounded, that is, is doesn't have an endpoint. It is also finite, it is a sphere and all that goes with that classification. My problem is this. Even though I agree that the surface sphere is finite and unbounded, it does have a surface area, a radius, a volume, etc. I am also not clear how the sphere example relates to the universe. Also, couldn't you say the surface of anything is finite and unbounded, a cube for instance, the skin on my body?

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...To ask about the amount of space is to ask a metaphysical question. And, as Alex demonstrates in his essay, the question itself only makes sense when you specify bounds, i.e., between "here" and "there." But, since the universe is unbounded, the question makes no sense.

This also confuses me, I am assuming you are using "unbounded" consistently throughout your posts. If this is so, then the surface of a sphere or the surface of anything is unbounded, yet it has a finite amount of space (is this synonymous with volume?) or surface area.

The problem for me seems to be that the examples that are typically given for the unbounded are metaphysical in nature (e.g. the sphere example). Yet when speaking of the universe, its unbounded-ness is not a metaphysical. Are we speaking of metaphysical unboundedness when we speak of the universe or is it something else?

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Also, to Stephen, the surface of a sphere analogy is a bit confusing to me.  Yes, the sphere is unbounded, that is, is doesn't have an endpoint.  It is also finite, it is a sphere and all that goes with that classification.  My problem is this.  Even though I agree that the surface sphere is finite and unbounded, it does have a surface area, a radius, a volume, etc.  I am also not clear how the sphere example relates to the universe.  Also, couldn't you say the surface of anything is finite and unbounded, a cube for instance, the skin on my body?

The two-dimensional surface of a sphere is an (imperfect) analogy just meant to underscore the notion of the finite yet unbounded, in some respect. The respect which is analogized is in that no matter how far you travel along that two-dimensional surface you will never reach an edge. This is contrasted with a two-dimensional infinite plane surface where likewise no matter how far you travel you never reach an edge. The difference being, of course, the former is finite while the latter is not.

I myself am not too fond of using that analogy, but many non-technical people have difficulty in conceiving the finite yet unbounded, and some find it helpful. If the analogy confuses you rather than illuminates, then perhaps it is best to dispense with it entirely and just deal directly with the uniqueness of the finite yet unbounded universe.

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...To ask about the amount of space is to ask a metaphysical question. And, as Alex demonstrates in his essay, the question itself only makes sense when you specify bounds, i.e., between "here" and "there." But, since the universe is unbounded, the question makes no sense.

This also confuses me, I am assuming you are using "unbounded" consistently throughout your posts. If this is so, then the surface of a sphere or the surface of anything is unbounded, yet it has a finite amount of space (is this synonymous with volume?) or surface area.

The problem for me seems to be that the examples that are typically given for the unbounded are metaphysical in nature (e.g. the sphere example). Yet when speaking of the universe, its unbounded-ness is not a metaphysical. Are we speaking of metaphysical unboundedness when we speak of the universe or is it something else?

First, we can say that the two-dimensional surface of a sphere is finite in size only because we see the sphere as embedded in a three-dimensional space. That is the geometric fact that allows us to establish the boundaries to the sphere. However, when considering the universe -- the totality of existence -- we cannot embed the universe into some other dimension because the universe is all that exists. There is nothing outside of the universe by which we can establish boundaries.

Second, the sense in which the universe is unbounded is temporally and spatially, which is why we cannot speak of the age or size of the universe. Without bounds, age or size has no meaning. And, as I said in my last post, the two-dimensional surface of the sphere analogy is quite imperfect, and the respect in which we there speak of unbounded is not the same respect in which we regard the universe. Rather than using the analogy in too literal a manner, I think it might be better for you to discard the analogy altogether and just deal directly with these issues as they apply to the universe.

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