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It follows, alright, if you accept the concept of a set of infinite elements (i.e., an ACTUAL infinity).

It's not so much something you 'accept' as something that's demonstratably true. The set of integers has infinite elements. There's no way you could even possibly begin to argue that it's finite unless youre postulating some kind of intuitionism and the reconstruction of set theory.

edit: actually on second thoughts, intuitionism could possibly be the conventional philosophy of mathematics closest to what I think Rand would believe, based on IOE. I remember reading an article by David Ross where he went near this without stating it outright. Hmmmm.

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... actually on second thoughts, intuitionism could possibly be the conventional philosophy of mathematics closest to what I think Rand would believe, based on IOE. I remember reading an article by David Ross where he went near this without stating it outright. Hmmmm.

Hardly. It is true that Ross atempts to connect intuitionism ala the likes of Brouwer to Objectivism via infinity and logic, but the connection between that and the actual ideas expressed by Ayn Rand are coincidental, not fundamental, and mostly irrelevant. Rejection of Cantor is not sufficient, and the intuitionist's disregard for the law of excluded middle is not exactly a plus.

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I do not know why you say that, Tom. Why would a mathematical infinity require an actual infinity to be sensible?

Well, for starters I will admit that I am not that well-versed in set theory (I only know the basic notation and concepts). However, I have had a year of (introductory) calculus, and have used the concept of infinity only with respect to limits. And in that case, I was taught to regard infinity only as an idefinite process, never as a cardinal number.

I'll read some more into this; though I must admit I can only stand reading a book on axiomatic set theory for a few minutes before I have to put it down because my head just hurts. It's so formal and abstract!

What do you think about axiomatic set theory are as a foundation for mathematics? It seems too formal and rationalistic to me, but...

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Well, for starters I will admit that I am not that well-versed in set theory (I only know the basic notation and concepts). However, I have had a year of (introductory) calculus, and have used the concept of infinity only with respect to limits. And in that case, I was taught to regard infinity only as an idefinite process, never as a cardinal number.

But it is still a mathematical abstraction, not a physical existent.

I'll read some more into this; though I must admit I can only stand reading a book on axiomatic set theory for a few minutes before I have to put it down because my head just hurts.  It's so formal and abstract!
Well, like most learning you have to be motivated, and without a specific purpose in mind, complex mathematical theories are too abstract for casual reading.

What do you think about axiomatic set theory are as a foundation for mathematics?  It seems too formal and rationalistic to me, but...

I am not a big fan of set theory, but it is a perfectly valid and useful aspect of mathematics. However, I think that proper foundational issues in mathematics require a different basis than standard set theory. (Note also that there are several different versions of set theory.)

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Although I am also not an expert, I share Tom's reservations about set theory. The problem is that the concept "set" connotes a definite, finite group of objects, and even if some of the better mathematicians explicitly state that they are using it to denote something different (and as far as I know they don't), it's confusing to use it the way they do. I think "series" or something along those lines would be a much better term to describe open-ended "groupings" such as integers.

In my experience (in other words, at least in college classrooms), the bad terminology does in fact lead to significant philosophical errors on the part of many or most logicians and mathematicians, who often do at least implictly use the concept of "infinity" as though it is a quantity rather than a concept of method.

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Although I am also not an expert, I share Tom's reservations about set theory.  The problem is that the concept "set" connotes a definite, finite group of objects, and even if some of the better mathematicians explicitly state that they are using it to denote something different (and as far as I know they don't), it's confusing to use it the way they do.  I think "series" or something along those lines would be a much better term to describe open-ended "groupings" such as integers.

In order to properly criticize the use of terminology in a field you first need to understand the precise meaning of those terms in that field. But, even more importantly, you cannot condemn a theory because of the words used to describe the theory. One may argue that the use of the word "charm" to denote a particular property of a quark is an unfortunate choice of words, but to condemn the quantum theory of quarks based on the use of the word "charm," would be improper.

Likewise for set theory. If you object to the process of adding unlimited elements to a set, then you must also object to the process of adding numbers in an unlimited fashion. Sets are mathematical abstractions, not physical entities, and they certainly are not necessarily defined in the field of mathematics as "connot[ing] a definite, finite group of objects."

As Ash says in his post, I agree that bad terminology can lead to philosophical errors. But when concepts are clearly defined, that act, in and of itself, delimits the possibility of such an error. Mathematics is a very precise science, and its defintions are exemplary. Those not familiar with the field may not know those mathematical definitions, but the common sense meaning of a term is hardly reason to give one cause to reject an entire theory.

Note that in Objectivist epistemology a word is a symbol that denotes a concept, and that concept itself stands for an unlimited number of concretes, without implying that such an unlimited number of concretes necessarily exist in physical reality. Mathematical abstractions can be rather far removed from the chain that eventually leads to perceptual concretes, but we should applaud that level of abstraction, and not condemn it because it does not relate to some common sensical use of a term.

With that said, as I mentioned previously, I am not a fan of set theory per se, but I do defend it as a perfectly valid mathematical abstraction.

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Those not familiar with the field may not know those mathematical definitions, but the common sense meaning of a term is hardly reason to give one cause to reject an entire theory.

I don't recall saying that I rejected the entire theory on the basis of the one small criticism I made. I'm just curious why they would choose to use one word in a sense opposed to its common usage (possibly leading to confusion) when there are plenty of other words they could have easily used instead that would have been much closer in their common sense to the mathematical usage. It seems counterintuitive and counterproductive, and it makes me wonder if the abstraction was properly conceptualized by those who coined the term for it. Of course, it's also possible that there are historical or etymological reasons for the choice of that term, I don't know. But I really don't care enough about this topic to say anything more.

I guess it may not be the theory I have a problem with so much as the way that it is taught (or at least the way it was taught to me).

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There was a famous mathematician who wondered whether better sense could be made with set theory if there were limitations placed upon the "X is an element of Y such that ...." formulations in defining sets (I'm not at home now, but it was in the beginning of A. P. Morse's "A Theory of Sets.")

I've come to the tentative conclusion that that those "limitations" for defining sets should be exactly what Ayn Rand did for regular concept formation, incorporating the ideas of Conceptual Common Denominator, distinguishing characteristic and measurement-omission. In other words, we need to create a foundation for sets which is grounded in perception, via the method of integration-differentiation.

As it stands, any type of willy-nilly mathematical expression can be used to form sets, and this leads to arbitrariness in the extreme, to the point that math becomes a "science of the arbitrary," so to speak (which of course is an enormous contradiction.)

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What do you think about axiomatic set theory are as a foundation for mathematics?  It seems too formal and rationalistic to me, but...

The problem is that the primitive terms are left "undefined." In other words, such attempts are not grounded in perceptions, and are thus not proper methods of validating mathematics.

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There was a famous mathematician who wondered whether better sense could be made with set theory if there were limitations placed upon the "X is an element of Y such that ...." formulations in defining sets (I'm not at home now, but it was in the beginning of A. P. Morse's "A Theory of Sets.")

You are probably thinking of John Kelley. In the literature this is referred to as the MK set theory, for Morse-Kelley. But, I fail to see the relevance of even mentioning this.

I've come to the tentative conclusion that that those "limitations" for defining sets should be exactly what Ayn Rand did for regular concept formation, incorporating  the ideas of Conceptual Common Denominator, distinguishing characteristic and measurement-omission.  In other words, we need to create a foundation for sets which is grounded in perception, via the method of integration-differentiation.
I understand all of these words, individually, but I cannot make sense of what you are proposing, taking the words as a whole. Incidentally, for whatever it is worth, I personally think it more important to establish a foundation for mathematics independent of set theory, rather than "create a foundation for sets." But, take that as an opinion since I am not in a position to establish that here.

As it stands, any type of willy-nilly mathematical expression can be used to form sets, and this leads to arbitrariness in the extreme, to the point that math becomes a "science of the arbitrary," so to speak (which of course is an enormous contradiction.)

Sorry, but current mathematics is anything but a "science of the arbitrary," as witnessed by the endless series of incredible mathematical accomplishments in the past century. Nevertheless, as I have always said, foundational mathematics is in its infancy, and needs a separate course from set theory.

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I personally think it more important to establish a foundation for mathematics independent of set theory, rather than "create a foundation for sets."

Yes, I agree. What I am saying about sets is that if we want to use the language of sets to serve as a foundation for math, then we need to establish the proper rules for forming sets. If you look at the implicit or "set builder" notation method of forming sets, you will see that a process of identifying a superset followed by a process of selecting a subset from within that superset is involved. This is suggestive of AR's integration-differentiation method. My hypothesis is that we can use AR's exact same method in building sets.

If we can, this then represents the death of set theory and the birth of a new mathematical epistemology. Concepts of mathematics will be shown to be formed by the same method of forming regular, nonnumerical concepts. That is my goal.

Sorry, but current mathematics is anything but a "science of the arbitrary," as witnessed by the endless series of incredible mathematical accomplishments in the past century. Nevertheless, as I have always said, foundational mathematics is in its infancy, and needs a separate course from set theory. 

I wasn't saying that all of math is a science of the arbitrary, since it is not monolithic. Some mathematicians see it that way, philosophers of math primarily. They say that math is "dense" and that humans can only get glimpses of mathematical truth or snatches of insights here and there. They confuse symbolism with meaning and think it makes sense to construct systems comprised of denotationless symbols. They say that all mathematical concepts are sets, and that "any mathematical expression" can be used in the process of forming a set. This is circular, since mathematical expressions were what they were supposed to be defining in the first place (i.e., they need to define what "set" means without using the concept of "set" in the definition.)

Certainly I agree that a lot of math is wonderful and we have done wonderful things with it, however, I do not believe that concepts of mathematics have been philosophically validated, meaning, no one has shown how all mathematical concepts properly reduce back to perceptions and reality. The situation is analogous to the state of philosophy before AR. Human concept formation was not philosophically validated, though humans had used concepts to do great things before she validated it.

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... They say that math is "dense" and that humans can only get glimpses of mathematical truth or snatches of insights here and there. They confuse symbolism with meaning and think it makes sense to construct systems comprised of denotationless symbols.

I do not know you -- neither your background nor your knowledge -- so this is not addressed directly to you. Nevertheless, I have found, over the years, those most vocally antagonistic to current mathematics, and, physics too for that matter, are typically those who do not understand the math, or the physics. I do not defend the platonism in mathematical theory, and I do not defend the rationalism in physics, but both of these structures are incredibly marvelous achievements that cannot really be appreciated by those who are ignorant of the language.

It seems easy for some to sit back and berate the mathematicians for using symbols and complex structures that requires a great deal of knowledge to understand. The more I continue to study and learn the more I appreciate the brilliant minds that have touched both of these fields. There is much to fault in currrent mathematics and physics, but few of the critics that I hear -- and, I am sad to say, this is particularly true among Objectivist critics -- know the subject that they criticize. Philosophy is a powerful tool, and Objectivist philosophy is the most powerful of them all, but philosophical principles can only guide your knowledge when it comes to science. Criticism based on ignorance is worse than no criticism at all.

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I'm very sympathetic to what you are saying. I too admire what I consider virtuoso performances on the part of certain mathematicians (for example, Galois having created group theory.) Such is possible even without mathematics having been explicitly validated. I would compare it to a gifted writer composing an epic novel, while not explicitly understanding all the grammatical-epistemological nuances of his language as a grammarian would.

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Regarding the essay by Alex S., linked to by Stephen Speicher, above, I had a minor problem/question with:

"there is nothing in the Law of Identity that mandates every existent possess a (finite) size, anymore than the Law of Identity mandates every existent be (finite) in time. The concepts of size and time apply to certain existents, which have specific natures that allow for such applicability."

Is it strictly a scientific issue to insist that every existent possess a finite size and time? Or is that near the edges of what philosophy can say on the subject? It seems rationalistic to claim, on one hand, that existence necessarily implies identity, but that the identity itself need need not consist of anything we really know for sure -- maybe size, maybe time.... but maybe something else entirely.

Can't one say at least that size and time are necessary qualities of existents? Even if future knowledge should amend our contextual understanding of these concepts, the evidence for the existence of these qualities, at least, seems universal.

Another question I have with the above quote is that it is used specifically in the essay to refer to the universe as a whole. As such, it is a confusing illustration, because the object of its point, is the only exception to the illustration; it says, in effect: "that the universe has no size or time does not violate the law of identity, since the law of identity does not stipulate time or size as a necessary attribute of reality, even though every existent *does* have time and size... except the universe".

Edited by softwareNerd

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Do you have any specific objections to Cantor's work, in the sense of flaws that you can find in it? The idea of 'different levels' of infinity follows logically from the definition of sets, functions and cardinality; its not really something that you can argue with, unless you have a problem with those definitons.

Don't make the mistake of buying into Rationalist premises. Take for example, the linguist who subscribes to the premises of "generative grammar." He would think it makes sense to talk about "blackbirds" if one only has the concepts "black" and "bird" already formed and then "puts them together."

But it doesn't make sense. In order for the term "blackbird" to have real meaning, one would have to know that there is such a thing. You can't just take concepts and combine them in the absense of knowledge about the real world and claim that you have knowledge.

The same thing goes with Cantor and set theory. You can't take sets or properties of sets and then make identifications, which, in order to be true, would need to be shown to be so using induction, but then try to stick to using only deductive means. This is primacy of deduction and is Rationalistic.

In an inductive process (not to be confused with the misnamed process of "mathematical induction," which is not inductive at all, but is deductive)--in a truly inductive process, one needs to make reference to reality every step of the way. This is exactly what Cantor does *not* do when he Rationalistically makes reference to power sets without assigning them any real-world referents. It's like the "generative grammarian" who thinks he can crank up his linguistic machine (his generative "grammar") and produce meaningful sentences.

That being said, if there is any sense to made out of transfinite numbers, such truths are only implied by Cantor's theory. One would have to take his theory apart and reconstruct it in a proper manner (which would be analagous to the generative grammarian who deals with real linguistic phenomena, but doesn't know how to represent such truths in a proper theory. He's only able to toss things around as floating abstractions.)

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I'm not sure what you mean by 'refer to reality' in this context; the whole point of pure mathematics is that it _doesnt_ refer to reality directly - it primarily studies abstractions and generalisations that are divorced from any particular existents in the world. The entirity of abstract algebra for instance has no 'referent in reality', since the objects of study are purely abstract entities such as rings and groups (which are themselves defined in terms of sets). Things like power sets are perfectly well defined in mathematics, and although they dont directly refer to any real world existent, neither do many other important mathematical concepts which I assume you accept. What do imaginary numbers 'refer to in reality'? What about constuctions in non-euclidlean geometry (before Einstein)?

I'm also not sure precisely what role you want induction to play in mathematics. Induction obviously cant be used for things such as proofs, so I assume you mean it that it should have a role when it comes to forming mathematical abstractions?

They say that all mathematical concepts are sets, and that "any mathematical expression" can be used in the process of forming a set. This is circular, since mathematical expressions were what they were supposed to be defining in the first place (i.e., they need to define what "set" means without using the concept of "set" in the definition.))
No, sets are undefined. Every formal system needs to be built upon undefined terms, simpy because if you weren't prepared to have undefined terms, then everything would be defined in terms of everything else and the system as a whole would be illogically circular. You've got to stop your definitions somewhere and take some terms to be primary and undefined, and in modern mathematics sets (rather than numbers) have been chosen to play this part. This isn't a new thing; in classical euclidean geometry the basic elements such as 'points' and 'lines' were never defined.

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I'm not sure what you mean by 'refer to reality' in this context; the whole point of pure mathematics is that it _doesnt_ refer to reality directly - it primarily studies abstractions and generalisations that are divorced from any particular existents in the world. The entirity of abstract algebra for instance has no 'referent in reality', since the objects of study are purely abstract entities such as rings and groups (which are themselves defined in terms of sets).

We have some very different premises here. According to Objectivism, an abstraction must be reducible to existents in order to be an abstraction. For example, the concept "furniture" is an abstraction. It is formed by noting similarities and differences amongst real-world objects and thereby does refer to reality. It is not divorced from reality.

Things like power sets are perfectly well defined in mathematics

The problem is that the field of mathematics has not been philosophically validated, just as the field of general epistemology was not validated until Ayn Rand developed Objectivist epistemology. Therefore, there are no concepts that are well defined in mathematics. "Well defined" has to mean "refer to reality." No other meaning is sensible. Think of the "identification" part of "non-contradictory identification." Non-contradictoryness, so to speak, is not enough. You have to have identification. That means you need to attach real-world meanings to all of your concepts, including mathematical ones.

No, sets are undefined. Every formal system needs to be built upon undefined terms, simpy because if you weren't prepared to have undefined terms, then everything would be defined in terms of everything else and the system as a whole would be illogically circular.

I really recommend that you study Dr. Peikoff's book, "Objectivism--The Philosophy of Ayn Rand." You don't need undefined terms in order to avoid circularity. What you need is a base of ostensively defined terms.

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Is it strictly a scientific issue to insist that every existent possess a finite size and time? Or is that near the edges of what philosophy can say on the subject?

I don't see why one should assume that either philosophy or science would come to that conclusion. (For the record, I'm the author of the essay being discussed here.) Via philosophy, we know that all existents of consciousness (e.g., emotions) do not possess any size at all. With regard to time, the ultimate constituents of the universe would have to be eternal, and thus would not be temporal or possess an age. The universe, being all that exists, similarly must be eternal: the universe (i.e., existence) has always existed, since nothing can exist before existence.

It seems rationalistic to claim, on one hand, that existence necessarily implies identity, but that the identity itself need need not consist of anything we really know for sure -- maybe size, maybe time.... but maybe something else entirely.
I don't see why the universe as a whole has to be able to be ascribed measurable, quantifiable attributes for it to have identity. It's not rationalistic to say that the universe has identity, since every piece of knowledge consists in knowing the identity of the universe (i.e., the universe), and I don't see how such knowledge ever assumes that the universe as a whole has a measurable, quantifiable attribute. I'm not saying that the universe is this thing with an identity that we are ignorant of; I'm saying that the universe has identity, the objects of our knowledge consist in it, and the universe as a whole does not have size or age (and probably lacks every other measurable, quantifiable attribute as well, although this is more debatable and not explicitly argued for in my essay).

Can't one say at least that size and time are necessary qualities of existents? Even if future knowledge should amend our contextual understanding of these concepts, the evidence for the existence of these qualities, at least, seems universal.

Again, many things possess a size and an age, but it's simply not the case that everything does. So, my question is: why assume that the universe has either attribute? What would it mean to say that, for example, the universe has only existed for a specific time, and thus came into existence at one point? The universe can't "come into existence"; it is existence.

The main question of my essay centers around the attributes of size and age, and one must have a specific argument to show that the universe possesses these attributes. The claim that such attributes are universal simply isn't true, and does not address my positive arguments that operate in the context of knowledge that size and age are not universal attributes. One must examine the nature of the existent in question to see if it has a size or an age, rather than assume that such attributes must be present. With regard to the universe, I'm questioning a widespread (and very understandable) assumption that it does in fact have a size, so the mere fact that it is an assumption is not a sufficient objection.

Another question I have with the above quote is that it is used specifically in the essay to refer to the universe as a whole. As such, it is a confusing illustration, because the object of its point, is the only exception to the illustration...

It really isn't. But even if it was, I did try to draw out my argument (in my essay) for why it is the case that the universe cannot possess a size or an age. I did not simply make a question-begging exception to a universal law of reality; I made an argument, based on the nature of size, time, and the universe, that the latter is incompatible with the other two. If it is objected that size and time are universal and thus must apply to the universe, then -- if anything -- it is this objection (rather than my argument) which begs the question.

One could probably well say that the universe is a sui generis "exception" in the following sense: as far as I can tell, it is only in the context of the universe as a whole that one has something which lacks a size, while still having parts that do possess a size. I think (although am not entirely sure) that this is a phenomenon unique to the universe. But, either way, uniqueness is not proof of falsehood.

If anyone has any further questions on and/or objections to my essay, I'd be more than happy to respond.

-- Alex

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With regard to time, the ultimate constituents of the universe would have to be eternal, and thus would not be temporal or possess an age.

Hi Alex.

How do you reach this conclusion?

This would rule out an alternative such as the idea that these little things may interact with each other in such a way as to create new ones or destroy old ones.

The universe, as a whole, may be eternal, but I don't see why that must necessitate immortality for its ultimate constituents.

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Hi Alex.

How do you reach this conclusion?

This would rule out an alternative such as the idea that these little things may interact with each other in such a way as to create new ones or destroy old ones. 

The universe, as a whole, may be eternal, but I don't see why that must necessitate immortality for its ultimate constituents.

Create new ones? Out of what? It cannot be out of nothing. And neither can the constituents be "destroyed" (i.e., become nothing). Creation implies creation with something. If they are created and destroyed they are not fundamental.

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It's not rationalistic to say that the universe has identity, since every piece of knowledge consists in knowing the identity of the universe (i.e., the universe), and I don't see how such knowledge ever assumes that the universe as a whole has a measurable, quantifiable attribute.

Don't forget that the Axiom of Existence and the Axiom/Law of Identity both refer to the same basic fact of reality, but from a different perspective or philosophical context.

This would rule out an alternative such as the idea that these little things may interact with each other in such a way as to create new ones or destroy old ones.

One of the implications of the fact that existence exists is the fact that existence is eternal. Another implication/corollary is the fact that there has to be eternal ultimate constituents. If there was some small constituent of the type that could be created or destroyed, then it would have to be the manifestation of a composition of even smaller constituents, since there can be no such thing as spontaneous generation or spontaneous disintegration. There has to be a "bottom" or "ultimate" layer of constituents which are eternal, in order for existence to be eternal, i.e., in order for it to be true that "existence exists."

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I don't see why the universe as a whole has to be able to be ascribed measurable, quantifiable attributes for it to have identity. It's not rationalistic to say that the universe has identity, since every piece of knowledge consists in knowing the identity of the universe (i.e., the universe), and I don't see how such knowledge ever assumes that the universe as a whole has a measurable, quantifiable attribute.

I agree with that. Assuming by "universe" we mean the sum total of all that exists," I am not even sure what it would mean for philosophy to attribute characteristics to the universe. Attributes are inherent characteristics of entities, and the universe is not an entity in the primary sense of that term. But, even if we extend the notion of "entity" to include the universe -- to be a "thing" -- we cannot point to it and abstract away its attributes by observation.

In ITOE (p. 273) Ayn Rand mentions that we can only ascibe to the universe those "fundamentals that we can grasp about existence." She ascribes to the universe only "identity" and being "finite," not as characterisitics of the universe but as being inherent in existence, per se. Identity is not an attribute, and, as far as being finite, I think that Alex's essay clarifies the use of "finite" in a precise manner.

However, with that said, there is nothing that I know of within philosophy which would mandate against science identifying what could be considered to be attributes of the universe. What we can say philosophically cannot be contradicted scientifically, but that which philosophy remains silent on, becomes a strictly scientific question.

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Time being circular would require that the universe eventually collapse and then start back out the same way it did the last time. All evidence points to the universe not ever collapsing.

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Time being circular would require that the universe eventually collapse and then start back out the same way it did the last time.  All evidence points to the universe not ever collapsing.

I disagree. I don't see how one implies the other at all. If you want me to criticize your argument, you'll have to elaborate and let me know what your argument is.

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