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thinkonaut

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Firstly, I'm explicit in my essay that I'm neither claiming that the universe is infinite nor "extending."

Alex,

You say that (what I quoted above), yet in an earlier post you said:

'I have no problem comprehending the idea that the unbounded universe is "Euclidean," and nor have I been offered any reason why it is incoherent. It doesn't steal any concepts; it just asserts that the universe is an endless realm of three-dimensional, "Euclidean" space.'

I don't see how this can be anything other than you contradicting yourself.

Please don't respond by saying "read my essay." Please address the issue point by point right here. If your time is limited, then please cut and past the relevant sentences from your essay that explain why this is not a contradiction.

Adrian

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What is truly "odd" is your reliance on a term that has virtually no meaning.

You're going to have to elaborate. Certainly the term "hypersphere" is non-controversial and well understood. A "hyperspheroid" is just a more general term that also includes shapes that are not perfectly hypersherical (which could refer to a three-dimensional surface that has elliptical qualities, for example).

If you check your database again, you will find a multitude of physics articles that include discussions of the possibility of the universe being hyperspherical. I'm assuming that those authors are just being a little loose with language in using the term "hypersphere" when they mean to also include 3D surfaces that are not perfectly hyperspherical, but are deformed in some way.

I don't see anything controversial about what I say here.

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I don't see anything controversial about what I say here.

I'm sorry to say that you just have no idea what you are talking about. You are appropriating words that you do not understand and using them to add some legitimacy to an otherwise vacuous notion. If you want to approach this mathematically, then learn topology; in particular learn about the simplicial homology groups of homology theory, essentially an algebraic study of the connectivity properties of a space.

You reject the notion of an Euclidean universe, yet you advance a mathematical notion which is based on one. The hypersphere, or n-sphere, is a compact manifold defined as the set of all points x = (x_1, x_2, ..., x_n+1) existing in E^n+1, satisfying (x_1)^2 + ... + (x_n+1)^2 = 1. E^n+1 is a Euclidean space. For the surface of the 3-sphere on the 4-dimensional hypersphere, the curved 3-dimensional space is a subspace embedded in a flat 4-dimensional Euclidean space. Your use of a "hyperspheroid" while simultaneously rejecting Euclidean space, is an attempt to have your geometry, and eat it too.

If you do not want to learn topology, then at least learn the basics of differential geometry, and then study the standard Robertson-Walker geometry of cosmology in general relativity. The geometry of 3-spaces of constant curvature have been studied in depth for three-fourths of a century, and the mathematics of, and implications for, embedding 3-surfaces in a 4-dimensional Euclidean space are well-understood.

If you want to promote the ideas you have, then either leave out your pseudo-use of mathematical terms that you do not understand, or learn the mathematics and do it right. You cannot buoy up bad philosophy with pseudo-mathematics.

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Concepts such as "curve" and "shape" are wholly formed with reference to boundaries, and simply have no meaning apart from them.

I disagree with this Alex, and we even exchanged a couple of e-mails on this point about two years ago.

I'm using the terms "curve" and "shape" here to refer to an internal metric. It does make sense to consider the idea of the universe having an internal metric. That's not controversial. One does not have to step outside of it and look at it (the universe) as being a part of something in order to do this. No reference to boundaries is called for or necessary.

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I disagree with this Alex, and we even exchanged a couple of e-mails on this point about two years ago.

I'm using the terms "curve" and "shape" here to refer to an internal metric.

Alex was not there talking about abstract mathematical spaces; he was talking about physical reality. You seem to have some difficulty separating the two. As he pointed out to you before, your assertions about the geometric structure of the universe are completely arbitrary. And, I might add, patently absurd.

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I'm sorry to say that you just have no idea what you are talking about.

No, we are just not communicating effectively with each other.

As I said before, it is not necessary that a 3-D surface be embedded in a higher-dimensional space in the metaphysical sense. If our methodology incorporates a 4-dimensional "space," that's not the same as the "space" of the universe which would be the 3-D surface of the sphere. That would be an equivocation.

All one has to do is look at a lower-dimensional example. Take the case of a sphere with a 2-D surface. If we were two-dimensional beings in a 2-D universe that wrapped onto itself, in other words, if we inhabited the surface of a 2-D sphere, it would not be necessary, in the metaphysical sense, that the sphere be embedded in three dimensions. I.e., there would simply be no "inside" and "outside" of the sphere. All that would exist would be the surface of the sphere that we inhabited. (Remember, a "sphere" is not the same thing as a "ball," which is a common confusion in popular terminology. A "sphere," as a concept of method, is the surface of a "ball.")

The same goes for the case in higher dimensions. All that exists is the 3-D surface that we inhabit. If the universe has an internal metric that causes it to join up with itself, then there does not have to be a metaphysical "inside" and "outside" of the surface, although we could imagine one for methodological purposes.

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Alex was not there talking about abstract mathematical spaces; he was talking about physical reality.

And the purpose of a concept of method is to have an application. I'm not trying to reify any concept of method and say that the universe is that thing. I'm using the concept of method to describe reality.

As I pointed out before, to say that a planet, for example, is "spherical" is not to reify the methodological concept of "sphere." Or also, to say that a tricycle has three wheels is not to reify the concept of three. The "three," as a concept, is a mental device that we use to describe it, but the three-ness, that quantity, is something on its own without our help.

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[...] your assertions about the geometric structure of the universe are completely arbitrary. And, I might add, patently absurd.

Also, you yourself criticized the idea of positing actually existing infinities, which is what one is doing when one says that the universe is Euclidean. Alex can't have it both ways. If he wants the Euclidean part, then he has to admit that he is positing the existence of infinities.

I'm trying to avoid absurdity by *not* positing the existence of infinities.

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You say that (what I quoted above), yet in an earlier post you said:

'I have no problem comprehending the idea that the unbounded universe is "Euclidean," and nor have I been offered any reason why it is incoherent. It doesn't steal any concepts; it just asserts that the universe is an endless realm of three-dimensional, "Euclidean" space.'

I don't see how this can be anything other than you contradicting yourself.

Please don't respond by saying "read my essay." Please address the issue point by point right here. If your time is limited, then please cut and past the relevant sentences from your essay that explain why this is not a contradiction.

In other words, Adrian is asking: "How can the universe be endlessly, 'Euclideanly' filled with three-dimensional objects, but not necessitate a metaphysical infinity -- whether an infinite extension (size) or an infinite number of entities?"

This, of course, is the whole question of my essay, in answer to which I gave detailed arguments. Adrian's post amounts to, then: "I can't see how every important argument of your essay isn't contradictory -- make your arguments again, please."

Adrian, in the name of posting etiquette, I cannot in good conscience response to any of your posts until you give a specific objection to a specific part of my argument, rather than simply 1) asserting ideas contrary to my essay over and over again, without support, or 2) essentially and unspecifically asserting that my whole essay is bunk.

If you think this is an unreasonable request -- and one with which you do not want to comply -- then we must, in silence, disagree.

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All one has to do is look at a lower-dimensional example.

Yes, all one has to do is look at reality. And, if one does, one then does not arbitrarily attribute to an actual existent that which belongs to the mathematical abstraction of a manifold. This is the sort of approach -- albeit not on this naive a level -- that I have come to expect from many well-educated cosmologists. I forgive some of them, in part, for their lack of philosophical expertise, because they have achieved a serious technical expertise. But you bandy about some mathematical terms with no real grasp of their meaning, and in the philosophical realm, with your exposure to Objectivism, you should know better. I cannot so forgive you.

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And the purpose of a concept of method is to have an application. I'm not trying to reify any concept of method and say that the universe is that thing. I'm using the concept of method to describe reality.

No. You are attempting to create a reality all of your own. You are (naively) using a concept of method which serves a purpose for an abstract mathematical space, and arbitrarily asserting that it applies to the real world. To do so is utterly absurd.

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Also, you yourself criticized the idea of positing actually existing infinities,

Yes, of course I have objected to such an absurdity.

which is what one is doing when one says that the universe is Euclidean.
No. That is what you are doing, apparently because you do not understand the issue and are unable to conceive of any other alternative than the ludicrous scenario you have become attached to.

Alex can't have it both ways.

Apparently Alex can't have any way with you. You do not understand his arguments (just as you have not understood my own) and you then blame him for something that he does not advocate.

This "discussion" has become utterly ridiculous. Unless you have something new of substance to offer, I really have nothing more to say.

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Ok I've read your essay again, this time a bit more closely, and yes, I think I misinterpreted you the first time. I'm still not 100% sure that I understand exactly what you are trying to say, so if I've made any further mistakes please correct me.

I appreciate your taking the time to go over my essay again, and to ask some specific questions.

As an aside, since it hasn't been linked to for a little while, here is the link to my essay again:

http://www.geocities.com/rationalphysics/U...nded_Finite.htm

...let us assume that you are correct about the universe being unbounded - assume that you can travel as far as you want in any given direction without reaching a boundary. As far as I can tell, you want to maintain this while also claiming that the spatial extension of the universe isn't infinite. But let us consult the dictionary...

While the dictionary is often a helpful tool, it cannot be referenced as if it contains all the right definitions. See the definition of "selfishness" for a good example as to why.

But, on a deeper level, allow me to give you a better idea of why I think the defintions you took the time to cite are largely definitions by nonessentials, since they are all great and explicit examples of what I'm arguing against. At the risk of not addressing every point in your post, I'll write a longer explanation, using the single example of the concept of "size." While my explanation will in principle be the answer to why the universe does not possess an infinite number of entities, don't hesitate to revisit points that I end up not addressing directly with this single, longer explanation.

In my essay, I reject the premise that metaphysical "endlessness" is to be ipso facto equated with infinity (which is what one of your definitions claimed). To take the example: a differentiation needs to be made between 1) a size which is endless, and 2) a universe which is endless. The former is contradictory and an instance of infinity; the latter is neither. As such, to say that both are "endless" and therefore infinite is to unite by nonessentials.

Why? Well, assume an advocate of an infinite size was asked: "Does the universe have a size?" He would of course answer "Yes." But then, if he was asked, "What is its size?," he would have to answer, "...it doesn't have one." He's caught positing a size, without positing a specific size. But to posit a size must be to posit a specific (i.e., finite) size, lest one be caught in this contradiction -- the contradiction which is the essence of infinity.

But there is no such contradiction when this standard is applied to the universe. "Does the universe have a size?" No. "What is the universe?" All that exists. One does not sunder the identity of the universe from its existence, because there is nothing inherent in the universe which says that its identity must lie in possessing a size.

And if one wants to still assert at this point that the universe is infinite because it is endless, my question to them would be: "What is infinite about the universe?" The universe can't be infinite unless it possesses an infinite "something" -- in this case, a size. But where is the argument that the universe possesses a size in the first place? Why must it possess a size at all? Why can't the universe be "asizal" (outside the conceptual scope of size) in the same exact way that the universe is eternal (outside the conceptual scope of time)?

The fact is, the concept of "size" is formed in reference to objects with spatial boundaries. But, the universe has no spatial boundaries. Given this fact, the universe simply can have no specific size, as there would be no boundaries to delimit the extent of such a size. Thus, not possessing a finite size, the universe must have no size (i.e., be outside the scope of "size"). Such is the conclusion that the Law of Identity mandates.

So, far from advocating an infinity, I advocate my position precisely because it avoids an infinity (i.e., a contradiction). To those who disagree, I can again only ask: what is it about the my physical picture of the universe that is infinite -- and how do you justify the claim that the universe has such a characteristic which can be infinite in the first place?

For the record, I believe it is only a rationalistic concept of "infinity" which allows one to rule out my physical picture of the universe. However, if anyone thinks that they can rule out my position via a Law-of-Identity-grounded accusation of infinity, I would be honestly interested in hearing it.

--Alex

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Alex I'm going to bed soon and will respond fully to your post tomorrow when I've read it through several times, but first could you clarify one issue for me?

Assume that you are debating someone who asserts that the universe has infinite size. You counter that this claim is false (for the reasons mentioned above), and say that actually size does not apply to the universe. Is there any actual physical difference between the two scenarios that are being asserted here? In terms of what actually exists in space-time, and what one can actually 'do' within the universe (such as travel arbitrarily far in a given direction etc etc), does your model/conception of the universe actually vary from his? I'm just trying to clarify whether you are primarilly objecting to the terminology that is being used to describe reality (because you think that 'the universe is infinite' is a misguided statement due to the non-existence of actual infinities), or whether you are actually asserting different facts about reality.

edit: Please don't think that I am attempting to reduce this to the level of 'mere' semantics; I do realise the importance of using correct terminology and concepts in areas such as this in order to avoid the false beliefs that can easily spring from misguided premises . I'm mainly interested in finding out your precise point of objection.

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Assume that you are debating someone who asserts that the universe has infinite size. You counter that this claim is false (for the reasons mentioned above), and say that actually size does not apply to the universe. Is there any actual physical difference between the two scenarios that are being asserted here?

Probably not, or at least not necessarily.

Please don't think that I am attempting to reduce this to the level of 'mere' semantics; I do realise the importance of using correct terminology and concepts in areas such as this in order to avoid the false beliefs that can easily spring from misguided premises.

I appreciate the clarification (although, just so you know, I had not interpreted your post as implying otherwise). It is indeed true that if one conceded that the universe possessed an infinite size, this would leave one in a dreadful position for ruling out other absurd positings of infinity (e.g., physical singularities). Moreover, those who were opposed to infinities to begin with would search for a different physical picture of the universe than the one I have put forth -- which leads into additional unnecessary and absurd scenarios.

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Hi Alex,

I read your essay some time ago and was very impressed with the sharp, clean epistemological style. Is it hard work to write like that or does it just come with practice?

Are there any other of Ayn Rand's statements that stand out to you as needing further analysis that haven't been analysed yet? (EDIT: I say this because I consider your essay to be an analysis of the statement "But we can't ascribe space or time or a lot of other things to the universe as a whole. " END EDIT)

Also are there any other contexts where you have smashed the traditional arguments by applying this pattern of saying the concept doesn't apply.

Thanks

Ian

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In other words, Adrian is asking: "How can the universe be endlessly, 'Euclideanly' filled with three-dimensional objects, but not necessitate a metaphysical infinity -- whether an infinite extension (size) or an infinite number of entities?"

This, of course, is the whole question of my essay, in answer to which I gave detailed arguments.  Adrian's post amounts to, then: "I can't see how every important argument of your essay isn't contradictory -- make your arguments again, please."

Adrian, in the name of posting etiquette, I cannot in good conscience response to any of your posts until you give a specific objection to a specific part of my argument, rather than simply 1) asserting ideas contrary to my essay over and over again, without support, or 2) essentially and unspecifically asserting that my whole essay is bunk.

If you think this is an unreasonable request -- and one with which you do not want to comply -- then we must, in silence, disagree.

Alex,

You did not mention the term "Euclidean" in your essay, so my question to you was entirely reasonable. Reasonable people could interpret your mentioning of Euclidean space to you making a more specific claim than what was in your essay. If you had interpreted my question in the spirit it was intended, you would not have been offended.

After all, the whole premise of your essay is that you are trying to get us to understand something new. It should be no surprise to you then if someone interprets your remarks in ways that you didn't intend. That's what forums such as this are for, and I assume that is why you are participating in this forum.

If you see the universe the way I see it, you would understand why I read the section in your essay about traveling in a spaceship and not see anything that is necessarily "Euclidean-like" about your conception of the universe. It *could* be, based on what you said, but I didn't see your language as entailing Euclidean-ness necessarily.

We are all wise people, so we ought to know that one of the pitfalls of communicating on the Internet is that it is too easy to misunderstand people. Patience and benefit of the doubt then, is order of the day.

Obviously you are a very bright person, and I believe that to be true even if I disagree with your thesis. Even so, if I thought someone was putting forth their best effort, I would never characterize their work as "bunk." Go back and read how I carefully said how I agreed with some of your essay, but not all of it.

If I have made a mistake here somehow in my attempt to communicate, I apologize. I can assure you that my motives were pure and that I *do* respect you as a thinker.

Probably one of my biggest failings in life is ineptness at predicting how people will perceive what I am saying and doing. This is not a moral failing on my part, but an error of knowledge or skill. I will try to do better.

Stephen, these remarks are intended for you, too.

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Ian, thanks for your compliments. It takes both hard work and a lot of practice in order to become a good writer, although I certainly don't want to imply that I don't have a lot more practice and hard work ahead of me, here. I'm only 21, for Galt's sake! :confused:

As to your other questions: I do have more than an inkling about other areas within philosophy to which the principles I advocated in my essay could be freshly applied. Although, I don't have any particular statements from AR in mind as much as I do certain philosophical issues. It's all relatively tentative, though; so far, I only claim certainty on what I advocated in my essay.

...I consider your essay to be an analysis of the statement "But we can't ascribe space or time or a lot of other things to the universe as a whole."

My essay could be thought of that way (at least on a broad level), although only retroactively. My thinking on the issue of the unbounded universe was independent of that statement from AR; I only came upon it after the fact.

As an aside, and for clarity's sake, I honestly don't know whether Miss Rand would have agreed with the position I've put forth in my essay, so I'm not taking a stand on that.

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Ian, thanks for your compliments. It takes both hard work and a lot of practice in order to become a good writer
Thank *you* - it's refreshing to read complex ideas presented so concisely.

My essay could be thought of that way [ian: as an analysis of one of ARs statements] (at least on a broad level), although only retroactively. My thinking on the issue of the unbounded universe was independent of that statement from AR; I only came upon it after the fact.

In that case it's even more impressive - sorry I shouldn't have assumed.

As Dr. Binswanger would say "good premises" for your future endeavors! :P

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Dear readers,

I was surprised to see that this thread has been viewed 1843 times so far, so it seems as though there are a lot of looky loos who are not chiming in. That's OK.

The topic that Alex writes about is actually quite fascinating, and I consider his paper to be a valiant effort to tackle the subject, and I applaud him for the attempt. I'm going to be responding to some of the issues he raises, though I am taking my time so as to be very careful and deliberate.

In preparation for my upcoming critique, the readers may want to reacquaint themselves with the fallacy that Ayn Rand calls the "fallacy of the stolen concept." It refers to the process of using a particular concept(s) in attempting to deny the validity of the genetic root(s) of that same concept(s). It's easy to find explanations of the fallacy by searching under the keywords "stolen concept" and "Ayn Rand."

In this connection, we should ask ourselves what "infinity" means and whether someone's use of it amounts to stealing any concepts. We need to keep in mind that it has two uses, the first being an attempted concept of entities and the second being a concept of method. Ayn Rand refers to "infinity" as a concept of method in her statement about the number series increasing toward infinity, without such increase implying that infinity exists.

Now suppose someone says the following:

"It's not logical to base an argument on the assumption that gremlins exist, even if you deny that what you are doing amounts to assuming their existence."

Is the person who uttered that statement stealing the concept "gremlin"? I'll leave that question for now as food for thought.

Recently it occurred to me that there might be a related fallacy that we could call "stealing the referent." Can any reader think of an example of trying to claim the metaphysical existence of something while denying the validity of forming a concept to refer to that thing? If so, would "stealing the referent" be a good name for attempting to do such? As a variation on the theme, how would the impossibility of the referent actually existing affect this scenario?

More food for thought: is it meaningful to speak about integrating negatives with positives? Just to take an example, does it make sense to speak about integrating atheism (a negative) with Objectivism (a positive)?

One last question: Can anyone show me an example of Ayn Rand criticizing Immanuel Kant by quoting his writings? If it's true that she criticized him without making reference to passages in his works, would her criticism then be improper? If such a form of criticism *is* proper, then why?

I'm very gratified that this thread has stirred up as much interest that it has. If anyone wants to correspond with me personally, my e-mail address is the same as the screen name I use here, plus "@yahoo.com".

Thanks.

--Adrian

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One last question:  Can anyone show me an example of Ayn Rand criticizing Immanuel Kant by quoting his writings?  If it's true that she criticized him without making reference to passages in his works ...

It is not true. For instance, there are extensive quotations from Kant -- entire paragraphs -- in Ayn Rand's Causality Versus Duty, in the July 1970 issue of The Objectivist.

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...there are extensive quotations from Kant -- entire paragraphs -- in Ayn Rand's Causality Versus Duty, in the July 1970 issue of The Objectivist.

Point well taken, thank you.

I need to get my "Objectivism Research CD ROM" re-installed in my computer so that I can look up such things.

My (modified) question is still pertinent, though: Is it proper to criticize only the unstated premises of a philosophical paper without making exact quotations of passages therein?

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Can any reader think of an example of trying to claim the metaphysical existence of something while denying the validity of forming a concept to refer to that thing? 

Yes. Your claim that the universe is a "hyperspheroid." The hypersphere is a mathematical abstraction, not a physical existent, and it serves an epistemological purpose, not a physical one. And you have already noted such a distinction in regard to the concept of infinity. In your May 18 2004, 10:03 AM post in this thread, you stated:

"Yes, but that is epistemological, not metaphysical. In other words, in mathematics, "infinity" should be properly seen to be a concept of method, not a concept of entities."

So, whether you admit or not, when you claim "I think that it makes perfect sense to identify the universe as being a hyperspheroid," that is "an example of trying to claim the metaphysical existence of something while denying the validity of forming a concept to refer to that thing."

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My (modified) question is still pertinent, though:  Is it proper to criticize only the unstated premises of a philosophical paper without making exact quotations of passages therein?

Your single-sentence question is too broad; I could write pages with a general answer. But, let's stop pussyfooting around here. You couched your first post this evening in a veil that hides it true meaning, hinting at all that you are going to do and gingerly asking questions cloaked in a false sense of mystery. I think that, in fact, you are really referring to a critique by you of Silverman's essay. Therefore, I will answer your question directly, assuming this to be the case.

In your case such a criticism without exact quotations would not be proper because you have quite adequately demonstrated that you do not understand Silverman's ideas, and that you even have difficulty grasping the actual meaning and significance of your own. Requiring you to point directly to quotations, so that third-party readers can judge your "critique" by reference to facts rather than by reference to your interpretation, is a must under these conditions.

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You couched your first post this evening in a veil that hides it true meaning, hinting at all that you are going to do and gingerly asking questions cloaked in a false sense of mystery.

What's so mysterious about saying that I am preparing a response and that here's some relevant issues to consider ahead of time? I see nothing mysterious about it. It's like a professor saying that before we talk about such-and-such paper, you should read up on these certain issues and think about how they might apply, and in particular, think about how such-and-so might apply in this way, etc. (And I'm not saying that I deserve the status of philosophy professor, only that I am doing something akin to what a professor might do, and that there's nothing wrong with doing so.)

I do think it would be helpful to debate these issues in the abstract before I apply them to his paper, if that's possible at this point. Perhaps not.

I will win you over one of these days! I know you to be an intelligent and thoughtful person, and I am perplexed really, as to the nature of your reaction to some of my posts here. Your main complaint seems to be that you say it is absurd that I attempt to apply certain mathematical concepts to the universe--but that's what valid concepts are for! That's what makes them valid, the fact that we can apply them. If it is wrong in principle wrong to hypothesize that the universe is hyperspherical (or hyperspheroidal), due to hyperspheres being concepts of method, then it is also wrong in principle to say that a planet, for example, is spherical, because spheres are also concepts of method. You've never really responded to me pointing this out.

I must be a failure when it comes to communicating. If so, that wouldn't imply that my ideas are wrong, but just that I have trouble understanding my audience and communicating my ideas to them.

I once applied for a teaching position at a community college, and the dean confided in me afterwards that the panel thought that I had real difficulties in communicating. I suppose they were right. I do have trouble seeing things from my audience's perspective, and I can see that I am just not reaching you. I'll accept the blame for that.

But may I politely request that we stop talking about talking and focus more on the issues at hand? I don't want to bore people with tangential issues.

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