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What is the nature of "time"?

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W A Dunkley,

Here and there do not exist, just as past and future do not exist. But they are all often reified.

Also, rational != rationalist. Reason includes identification of fact. Rationalism includes making things up as you go along.

"Here" and "there" are relative terms and so to are "past" and "future." but they have reference to reality. If I am sitting in Atlanta, not doing the boot-scoot in Ft. Worth, then I refer to Atlanta as "here" and Ft Worth as "there," but Atlanta and Ft. Worth both exist. Four-dimensionalism maintains that "past," "present" and "future." are also relative. (what does this "make up)" The part of me sitting here calls the part of me boot-scooting in Texas last summer "past", but both exist. Three-demensionalism makes "present" absolute and existence relative.

With all due respect, it seem that reason becomes rationalism when it fails to support ones position.

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Another powerful argument for four-dimensionalism might be referred to as a "cosmological argument."

Stating it briefly:

Three-dimensionalism leaves one with the unattractive choice between the absurdity of a universe that pop out of nothing and returns to nothing or the contradictions of infinity. Four-dimensionalism liberates us from such conundrums by providing us with a view of the universe that is finite, but yet eternal.

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Dunkley:

It is a view that holds that the passage of time is psychological (i.e., pertaining to the way one perceives the dimension of time, not an actual component of the physical universe) and existence is an unchanging four-dimensional totality.[emphasis added]

Does this mean "we're all in this together" because "we're all parts of a whole?"

Three-dimensionalism leaves one with the unattractive choice between the absurdity of a universe that pop out of nothing and returns to nothing

Why?

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What do "coupled" and "uncoupled" mean, in general and as applied to the contexts of the GRT field equations and the field equations of electromagnetism? Why is (D'Alembertian A = 4 pi J) uncoupled and (G = 8 pi T) coupled? What do "linear" and "nonlinear" mean?

Let me sketch out a few basics and establish the context while answering your questions.

A differential equation involves one or more unknown functions along with one or more of their derivatives. This unknown function is called the dependent variable, since it depends on one or more independent variables. When there exists only a single independent variable then only ordinary derivatives are required and we refer to the equation as an ordinary differential equation. When there exists two or more independent variables then partial derivatives are required and we refer to the equation as a partial differential equation. The Maxwell electromagnetic and Einstein gravitational field equations both involve partial differential equations.

When the unknown functions and their derivatives appear in the equation only to the first degree, and when there are no mixed products of these functions, then we refer to this as a linear differential equation. If there exist mixed products of the unknown functions, or higher than first degree derivatives, then we refer to this as a nonlinear differential equation. So, for instance, if the differential equation contains x + y, or derivates such as x' or x''', it is linear. If the derivates are of the form like (x')^2 or (x''')^3, or a product such as xy, then the differential equation is nonlinear. Finding exact solutions to nonlinear differential equations is, in general, much more difficult than for linear differential equations. Maxwell's electromagnetic equations are linear, whereas the Einstein field equations are nonlinear.

Physical problems often have multiple unknown functions which lead to a system of differential equations. When each of these differential equations in the set can be solved independently, then the system is known as uncoupled. When the unknown functions appear together in each of the differential equations in the set, each equation cannot be solved independently and the system of equations is said to be coupled. Special techniques are required to solve these coupled system of differential equations. Both Maxwell's electormagnetic and Einstein's gravitational equations are a set of coupled partial differential equations, but the gravitational field equations are much more complex and much, much more difficult to solve.

The electromagnetic equation you wrote for the vector potential A is one portion of the result of introducing the potential to reduce the number of, and uncouple, Maxwell's electromagnetic equations. It represents half of two equations that are equivalent to the full set of Maxwell's equations.

Can you give me any more examples of finding the solutions (ie a metric) for some given kind of stress tensor?
If you send me your email address I'll email you a couple of papers.

Also, if you could clarify exactly what these two metrics are, that would be great.

I am not sure how this will be helpful to you, but the Schwarzschild metric is

ds^2 = (1-2m/r)dt^2 - (1-2m/r)^-1 dr^2 - r^2(d{theta}^2 + sin^2{theta} d{phi}^2)

There are several Kerr forms that are generally used, but the original metric that Kerr discovered was in Cartesian-type coordinates, and it took this form,

ds^2 = dt^2 - dx^2 - dy^2 - dz^2 - (2mr^3/{r^4 + a^2 z^2})[dt + (r/{a^2 + r^2}) (x dx + y dy) + (a/{a^2 + r^2}) (y dx - x dy) + z/r dz]^2,

where,

t = v - r

x = r sin{theta} cos{phi} + a sin{theta} sin{phi}

y = r sin{theta} sin{phi} - a sin{theta} cos{phi}

z = r cos{theta}

Also, though you didn't bring it up, how are point charges and singularities (point masses?, eg within a black hole's horizon) different?

A point-charge is just the classical model of an electron, whereas the central singularity of a black hole derives from several solutions to the field equations. (And, as I have mentioned several times, not all solutions of mathematical equations reflect what exists in physical reality.)

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Dunkley,

Ah, you see I stopped keeping up with philosophy (historically at least) after Sarte. So I had to look it up.

This Four-Dimensionalism is as old as Plato in essential form. First it attempts to describe human consciousness, and from this basis to define what reality must be like. This is the primacy of consciousness in its original formula, and the most common form of rationalist metaphysics. It also violates, I believe, Occam's razor, by giving existence to time as if it is something rather than a relation. And, this violation is the result of the primary fallacy of the primacy of consciousness.

That is all that I've gleemed from it thus far, and am not going to argue it any further since it violates one of the cardinal rules of a rational metaphysics. Besides, I'd have to update my philosophy texts!

PS. Try to be a little more intelligible to the audience you are writing for. Objectivists are the last people to think a river is deep because the water is murky, whether by intention or not.

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Dunkley:

Does this mean "we're all in this together" because "we're all parts of a whole?"

Why?

I am not sure exactly what this first question is asking in context to the quote. My philosophy maintains that everything that exists shares the same primary part and the same first cause, which is self-sameness. This affirms that the universe is in fact a universe (i.e., an integrated totality.)

In response to the second question, I will address the problem of assuming that there is no beginning in time. Objectivism acknowledges that an actual infinity is incompatible with the axiom. (Why this does not constitute "rationalism," but what I do with the axiom does, has not been explained to my satisfaction.) The Aristotelean approach to the problem is to maintain that an infinite past does not constitute an actual infinity. But, surely, when Ayn Rand was born, this was an actual event, an actual moment. When the last Neanderthal died this was actual. If there is no beginning, then you are stuck with an infinite regression of actual moments. The other three-dimensional alternative seems to leave us with an actual nothing!

I am reminded of the saying. "Once you have eliminated the impossible, what remains, however unlikely, most be true." What remains is four-dimensionalism!

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Thoyd Loki

You seem ready to discuss everybody's philosophy but mine. The primacy of identity and four-dimensionalism do not affirm the primacy of consciouseness.

"A river is not deep because the water is murky." This sounds like a wise aphorism, but it should also be remembered that when we learn new ideas they often seem murky in the beginning. Four-dimensionalism is a simple idea, but hard to get a handle on, because it is so divergent.

I address this "audience" because, while some of my metaphysical views diverge from Objectivism, I have great respect for Ayn Rand and her philosophy. My philosophy and objectivism maintain the same conviction that A is A and that this truth is profoundly important. Like myself, objectivism also maintains that it is wrong to evade and to misrepresent. Here and elsewhere, I see much of this when confronting my ideas.

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Thoyd Loki

You seem ready to discuss everybody's philosophy but mine.

Have you considered the possibility that the problem lies, not with Thoyd Loki, but rather with the "philosophy" that you think you have presented? Frankly, I find your view of "four-dimensionalism" to be lacking clarity and intelligibility, a jumble of floating abstractions with no tie to reality. I suppose that was what Thoyd Loki had in mind when he referred to your posting as a "word salad." One cannot really discuss a "philosophy" when there is nothing meaningful there to discuss.

Now, I have come across the philosophical views of "four-dimensionalism" before, in several different guises. People like Theodore Sider have written about this in The Philosophical Review. Others have tried to tie it directly to special relativity. But what you have written here is so nebulous -- such a faint reflection of meaningful ideas -- that you really should not be surprised that no one has taken a serious interest in what you have said.

p.s. If you do not mind me asking ... I am curious. Are you perchance a university philosophy student?

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Thoyd Loki

You seem ready to discuss everybody's philosophy but mine. The primacy of identity and four-dimensionalism do not affirm the primacy of consciouseness.

"A river is not deep because the water is murky." This sounds like a wise aphorism, but it should also be remembered that when we learn new ideas they often seem murky in the beginning. Four-dimensionalism is a simple idea, but hard to get a handle on, because it is so divergent.

I address this "audience" because, while some of my metaphysical views diverge from Objectivism, I have great respect for Ayn Rand and her philosophy. My philosophy and objectivism maintain the same conviction that A is A and that this truth is profoundly important. Like myself, objectivism  also  maintains that it is wrong to evade and to misrepresent. Here and elsewhere, I see much of this when confronting my ideas.

Actually, there are several philosophies that I do not discuss. Leibniz's, Comte, Existentialism, and others. The reasons for not doing so are two: 1) they are boring and not too important (Kant bores me, but he is important) 2) they are very recent and rehash well worn fallacies.

I didn't have a problem grasping four-dimensionalism which I looked up from various sources. I got the gist of the idea in about 5 minutes, while an eternity of staring at your obscure sentences would have yielded zilch.

Am I missing something by throwing four-dimensionalism away with such a quick look?. No. It is simply another passing bundle of jargon that will pass away. It is only of interest to PhD canidates and tired old professors who have given up on the real business of philosophy.

Nobody here has evaded, nor misrepresented your ideas because nobody here knows what the hell you are talking about. Sorry, not trying to offend for the sake of offending, but your posts are reading more like comedy for Objectivists than philosophy. I am merely suggesting that you take the time to make your ideas clearer to a mind other than your own. It may make perfect sense in your own mind, and is just not making it out of you coherently.

Come to think of it, I probably will look deeper into four-dimensionalism. As a science-fiction writer I am always looking for weird ideas to serve as vehicles for my stories! Where would the whole gerne be without the screw-ups of the philosophers! B)

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This is what I have learned about time:

Time is just a form of measurment--motion of entities in relation to each other. It does not exist in and of itself. Without entities in motion, there is no time. The universe or existence itself is eternal. But I need help with this one. Why is the universe or existence eternal? Is existence itself featureless--timeless, motionless, etc. Confused.

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Why is the universe or existence eternal? Is existence itself featureless--timeless, motionless, etc. Confused.

Things IN the universe change and time is one of the ways we measure that change, but the universe AS A WHOLE always was and always will be. The concept of time is not applicable to the universe as a whole.

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Have you considered the possibility that the problem lies, not with Thoyd Loki, but rather with the "philosophy" that you think you have presented? Frankly, I find your view of "four-dimensionalism" to be lacking clarity and intelligibility, a jumble of floating abstractions with no tie to reality. I suppose that was what Thoyd Loki had in mind when he referred to your posting as a "word salad." One cannot really discuss a "philosophy" when there is nothing meaningful there to discuss.

Now, I have come across the philosophical views of "four-dimensionalism" before, in several different guises. People like Theodore Sider have written about this in The Philosophical Review. Others have tried to tie it directly to special relativity. But what you have written here is so nebulous -- such a faint reflection of meaningful ideas -- that you really should not be surprised that no one has taken a serious interest in what you have said.

p.s. If you do not mind me asking ... I am curious. Are you perchance a university philosophy student?

Well no, I majored in Football and wild Texas women. It was a long tome ago. But, perhaps it can be argued that I received to may helmet to helmet hits, I am substantial convinced, nonetheless, that I have expressed meaningful issues. These issues cannot be seeped away with lazy reason.

When Objectivism states that there can be nothing outside of existence, or as I put it, everything exists, it is not a "floating abstraction." This logical truth does not conveniently become meaningless when it is weighted against the notion of a changing world, where things come in to and go out of existence.

I am also supremely confident that the three-dimensional eternity embraced by Objectivism would indeed constitute an actual infinity, which Objectivism and my philosophy, both reject.

P.S. If this message is incomprehensible to you, I'll try to bring in a young philosophy major to explain it.

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I am also supremely confident that the three-dimensional eternity embraced by Objectivism would indeed constitute an actual infinity, which Objectivism and my philosophy, both reject.

Objectivism does hold that the universe is eternal (and there is not contradiction in that), but three-dimensionality is simply the form in which we perceive metaphysical reality. Evidently, not only do you need to strive harder, intellectually, to make your "four-dimensionalism" sensible, but you also have to learn to distinguish the form from the object.

P.S. If this message is incomprehensible to you ...

I can read the message just fine. It is a good deal of the content of your postings that is consistently incomprehensible.

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Objectivism does hold that the universe is eternal (and there is not contradiction in that), but three-dimensionality is simply the form in which we perceive metaphysical reality. Evidently, not only do you need to strive harder, intellectually, to make your "four-dimensionalism" sensible, but you also have to learn to distinguish the form from the object.

I may be just an old punch-drunk quarterback, but I cannot comprehend how this answers to the statement you quoted. Defining "form" into oblivion (which is all I can make of what you said) does not resolve the conundrum to my satisfaction.

You may argue that space/time (i.e., relative position) is something less basic than objects. (And I would disagree.) Asserting that it is not something that exists, however, is fundamentally absurd. And since it does exist, I'm afraid it brings us back to the dilemmas that I so poorly describe.

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I may be just an old punch-drunk quarterback, but I cannot comprehend how this answers to the statement you quoted.

You mistakenly claimed that "the three-dimensional eternity embraced by Objectivism would indeed constitute an actual infinity." I point out, again, that Objectivism states that the universe is eternal, but three-dimensionality is simply the form in which we perceive metaphysical reality. I can understand why you might not want to grasp this, since, if three-dimensionality is no longer a primary, but just an effect of something more fundamental, then your "four-dimensionalism," as bizarre and nebulous as it already is, is now relegated to the minors when considering the universe.

Defining "form" into oblivion (which is all I can make of what you said) does not resolve the conundrum to my satisfaction.
Sorry, but since you presumed to assert that Objectivism leads to certain contradictions, one might think that, at least, you knew what Objectivism avocated before you would claim that it contradicted itself. That you do not understand Objectivist metaphysics is your own private "conundrum," and whether or not you resolve it to your "satisfaction" is entirely up to you. Fortunately, it has nothing to do with Objectivism.

You may argue that space/time (i.e., relative position) is something less basic than objects.

Space and time are relational concepts, relations involving entities.

Asserting that it is not something that exists, however, is fundamentally absurd.
Why would I ever argue that relational concepts do not exist?

And since it does exist, I'm afraid it brings us back to the dilemmas that I so poorly describe.

Well, maybe one day you will not do so poorly. In the meantime, the "dilemmas" are of your own confused making, not a problem for Objectivism.

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You mistakenly claimed that "the three-dimensional eternity embraced by Objectivism would indeed constitute an actual infinity." I point out, again, that Objectivism states that the universe is eternal, but three-dimensionality is simply the form in which we perceive metaphysical reality. I can understand why you might not want to grasp this, since, if three-dimensionality is no longer a primary, but just an effect of something more fundamental, then your "four-dimensionalism," as bizarre and nebulous as it already is, is now relegated to the minors when considering the universe.

Sorry, but since you presumed to assert that Objectivism leads to certain contradictions, one might think that, at least, you knew what Objectivism avocated before you would claim that it contradicted itself. That you do not understand Objectivist metaphysics is your own private "conundrum," and whether or not you resolve it to your "satisfaction" is entirely up to you. Fortunately, it has nothing to do with Objectivism.

Space and time are relational concepts, relations involving entities.

Why would I ever argue that relational concepts do not exist?

Well, maybe one day you will not do so poorly. In the meantime, the "dilemmas" are of your own confused making, not a problem for Objectivism.

If things in the past and future, did not exist, then if you were to fall into a meat grinder, something more than a "relational concept" would be lost, It would be of little consolation if no matter was lost. It is significant to note that, if the fundamental constituents of matter are extentionless (which Ayn Rand conceded to at least be possible,) then these fundamental objects would also become "relational concept." and no more basic that space/time. Actually, that both of these concepts are relational only means that the things to which they refer cannot be conceived as an independent totality like a chair or cup (but the only truly independent totality is the totality of existence. All other things that exist are dependent parts of this all inclusive unchanging universe.) The intrinsic characteristics if a thing are usually used to determine the concepts it can be correctly subsumed. In the case if the physically simple object as is with space?time, this is not possible. They must be conceived of extrinsically as part of a greater whole. To deny, however, the existence of a thing because its conception defies a given epistemological methodology, is to do exactly what (some) Objectivist have tried to accuse me and impose methodology on reality.

The proponents of reason should be careful of what they throw away. It may be claimed by there adversaries resulting in the surrender of intellectual territory to the anti-rational, to the mystic. Worse than this, they may throw away the very foundation that sets them apart from there antagonists. If one clings to the notion that the usual means of formulating concepts is the only means possible or that the things that defy it are not things at all, one would falsely conclude that space/time, extension-less objects and even more fundamental, identity itself, the very foundation of a rational world, are either impossible or unknowable.

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three-dimensionality is simply the form in which we perceive metaphysical reality.

In consideration of this statement and Rand's concession that extensionless objects are possible, I may indeed not have a handle on the the Objectivist posistion. My only interpretation is that this statement implies if someone was not perceiving, then "metaphysical reality" would collapse.

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If things in the past and future, did not exist, then if you were to fall into a meat grinder, something more than a "relational concept" would be lost, It would be of little consolation if no matter was lost.
And you think Objectivism says otherwise? Ayn Rand made perfectly clear that the concept of "existence" refers to every entity, action, event, etc. that has ever or will ever exist. Also, you do not seem to understand what a relational concept is. Indeed, I wonder if you grasp the Objectivist view of concepts at all? "Time" is a relational concept, and in that sense it is epistemological. But a relationship such as time is just as metaphysical as are attributes of enities. Neither time nor attributes can exist independent of entities, but both are certainly metaphysical.

It is significant to note that, if the fundamental constituents of matter are extentionless (which Ayn Rand conceded to at least be possible,) then these fundamental objects would also become "relational concept." and no more basic that space/time.
You really do not know what you are talking about.

Actually,  that both of these concepts are relational only means that the things to which they refer cannot be conceived as an independent  totality like a chair or cup (but the only truly independent totality is the totality of existence.

I see. A chair is "independent totality" but not a "truly independent totality." Thanks for clearing that up.

All other things that exist are dependent parts of this all inclusive unchanging universe.) The intrinsic characteristics if a thing are usually used to  determine the concepts it can be correctly subsumed. In the case if the physically simple object as is with space?time, this is not possible. They must be conceived of extrinsically as part of a greater whole. To deny, however,  the existence of a thing because its conception  defies a given epistemological methodology, is to do exactly what (some) Objectivist have tried to accuse me and impose methodology on reality.

I tried translating this using Altavista's Babel Fish Translation, but neither Greek, nor Dutch, nor Portugese, nor Chinese, nor any of their other provided languages worked. I guess we are back where you started, with a word salad.

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In consideration of this statement and Rand's concession that extensionless objects are possible, I may indeed not have a handle on the the Objectivist posistion. My only interpretation is that this statement implies if someone was not perceiving, then "metaphysical reality" would collapse.

As I said previously, you have to learn to distinguish between form and object. Objects exist in metaphysical reality and they have a nature independent of us, so they certainly do not rely upon our perception for their existence. But our senses are the means by which we directly perceive those objects, and we must distinguish between the forms of our perceptions and the objects themselves. Color, for instance, is not "out there" in the object; color is the form in which we perceive certain wavelengths of light associated with an object.

Another example. We directly perceive macroscopic objects and infer their atomic structure. Suppose a being existed who could directly perceive atomic structure. Such a being would have to infer the existence of macroscopic objects. Both we and that being perceive the same reality -- there is only one metaphysical reality -- but the form in which we each perceive it is different. We can each eventually accumulate the same knowledge of reality; we just used different means.

When it comes to the ultimate constituents of reality, all we can say positively is that they have identity and act causally in accordance with their identity. We cannot philosophically prescribe the exact nature of the ultimate constituents; that is the job of science. Just as color is an effect on our senses, so may be size and shape. That does not make color, size, and shape any less real. It simply means that they may not be primary aspects of metaphysical reality, but rather effects of the ultimate constituents as perceived in the form of our senses.

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If things in the past and future, did not exist, then if you were to fall into a meat grinder, something more than a "relational concept" would be lost, It would be of little consolation if no matter was lost. It is significant to note that, if the fundamental constituents of matter are extentionless (which Ayn Rand conceded to at least be possible,) then these fundamental objects would also become "relational concept." and no more basic that space/time.  Actually,  that both of these concepts are relational only means that the things to which they refer cannot be conceived as an independent  totality like a chair or cup (but the only truly independent totality is the totality of existence. All other things that exist are dependent parts of this all inclusive unchanging universe.) The intrinsic characteristics if a thing are usually used to  determine the concepts it can be correctly subsumed. In the case if the physically simple object as is with space?time, this is not possible. They must be conceived of extrinsically as part of a greater whole. To deny, however,  the existence of a thing because its conception  defies a given epistemological methodology, is to do exactly what (some) Objectivist have tried to accuse me and impose methodology on reality.

The proponents of reason should be careful of what they throw away. It may be claimed by there adversaries resulting in the surrender of intellectual territory to the anti-rational, to the mystic. Worse than this, they may throw away the very foundation that sets them apart from there antagonists. If one clings to the notion that the usual means of formulating concepts is the only means possible or that the things that defy it are not things at all, one would falsely conclude that space/time, extension-less objects and even more fundamental, identity itself, the very foundation of a rational world, are either impossible or unknowable.

I have no idea what any of this means.

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And you think Objectivism says otherwise? Ayn Rand made perfectly clear that the concept of "existence" refers to every entity, action, event, etc. that has ever or will ever exist. Also, you do not seem to understand what a relational concept is. Indeed, I wonder if you grasp the Objectivist view of concepts at all? "Time" is a relational concept, and in that sense it is epistemological. But a relationship such as time is just as metaphysical as are attributes of enities. Neither time nor attributes can exist independent of entities, but both are certainly metaphysical.

I began thinking otherwise after listening to you try to wiggle out of the actual infinity of an infinite past. If time is metaphysical and has no begining, then infinity is what you are stuck with, and no amount of entrenched dogma can obscure it.

To say that color is perception, not the thing perceived, is one thing, (and I would agree) but, to say this about space/time, (i.e., relative position) is quite another. This is an assault on the very notion of reality. Even the one who proclaims that "the only thing I know exists is me" cannot obliterate the self-evidence of the objective existence of relative position, of extension. Even if one assumed reality to be nothing but sensations feelings and thoughts conspiring to deceive themselves into believing that there is an external world, the concept of relative position would still have an objective reference to reality. Even sensations exist in relative position. And if these sensations in relitive posistion had no beginning, your stuck with actual infinity.

Obviously we are not on the same page, but, it seems that when I am not comprehended it is "word salad" and when you are not comprehended it is ignorance.

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I see. A chair is "independent totality" but not a "truly independent totality." Thanks for clearing that up.

I hope I am wrong, but this sounds like deliberate misrepresentation! At the very least it is obscurantism not Objectivism.

We can conceive of a chair independent of other things because it has intrinsic characteristics. A characteristic is a part that can be used to distinguish one thing from another. Identity, by the way, is a part, but not a characteristic, because all A is A.

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If things in the past and future, did not exist, then if you were to fall into a meat grinder, something more than a "relational concept" would be lost, It would be of little consolation if no matter was lost.

Yes, if things that exist are obliterated, then they do cease to exist. If the fabric of spacetime falls into a meat grinder, however, it does not cease to exist: it never existed, and things that don't exist cannot be affected by meat grinders. Recognizing that existents exist is a requirement; reifying concepts is a fallacy.

It is significant to note that, if the fundamental constituents of matter are extentionless (which Ayn Rand conceded to at least be possible,) then these fundamental objects would also become "relational concept." and no more basic that space/time.

If the fundamental constituents of matter do not have the property of extension, then extension is a relationship between these fundamental constituents. Converting existents into concepts is just as bad as reifying concepts into existence.

Actually,  that both of these concepts are relational only means that the things to which they refer cannot be conceived as an independent  totality like a chair or cup (but the only truly independent totality is the totality of existence. All other things that exist are dependent parts of this all inclusive unchanging universe.)

[What Stephen said.]

Actually, that space and time are concepts only means that we cannot conceive of them as existent in their own right. But that the fundamental constituents of matter do exist only means that we cannot treat them as relational concepts.

The intrinsic characteristics if a thing are usually used to  determine the concepts it can be correctly subsumed.

Usually? Are there really no absolutes?

In the case if the physically simple object as is with space?time, this is not possible. They must be conceived of extrinsically as part of a greater whole.

Exactly as (1) they are not and (2) as the human mind does not work.

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