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thinkonaut

Circular Time

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(Removed at author's request - David)

Confusing concepts of method with concepts of entities seems to be common. To wit: Just because the decimal system allows one to add zeros in an open-ended fashion to make progressively smaller and smaller decimal numerals does not mean that a "contacting/collapsing universe" can contract indefinitely and thereby become an "infinitesimally small point."

The stillborn idea of a Big-Bang creation ex nihilo is a time-reversal temporal mirror image of the above erroneous scenario. As the old Billy Preston song goes: "Nothin' from nothin' leaves nothin'..."

Similarly, time is not a concept of method, it is a concept of aspects entities. Specifically, "time" denotes a certain aspect of entities that involves the nature of the way they change their positions in relation to one another.

Finiteness is a corollary to the Axiom of Existence. To exist is to be some thing--some limitied thing. Since there are only a finite number of entities possible in the universe, it follows that there can only be a finite number of ways that these entities can change position in relation to one another. Therefore, there can only be a finite number of moments in time in the whole past, present, and future of the universe.

The way we reconcile the idea of a no-beginning/no-ending aspect of time with the idea that there can be only a limited number of moments in time is by seeing that an analogy can be made to a sphere having a limited amount of surface area without there being edges or boundaries involved. Time can be seen as being finite, but without boundaries (without beginning or end).

A good name for this is "circular time," instead of "cyclical time," since "cycle" implies repetition, whereas a temporal circle does not. If one were to orbit the earth, one would not say that there are an indefinite number of continents because they are continually coming back into view. Similarly, if one were to imagine traversing the entire temporal circle of the universe in one's mind's eye and then mentally re-traverse the entire circle a number of times, one would not then say that those mentally re-traversed moments are additional moments in time and that time is "infinite." No matter how many times one imagines a particular moment in time, it is still just one moment in time.

It is no more proper to say that the entire history of the universe "repeats" itself than it is to say that the equator of the earth is infinitely long. Both of these errors confuse imagination with reality. When one imagines the universe cycling through every moment in time and then "repeating" the cycle, one is imagining oneself to be outside of the universe as a part of a background to it. But no such background is possible. All that exists are all of finite number of moments in time, with each moment existing within a wider time-context on a local basis. "Repetition" can only be a sensible concept at a local level of time, not at the level of the entire universe. Analogously, just because one can repeatedly circle the earth at the equator, this doesn't mean that the equator is infinitely long.

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...Finiteness is a corollary to the Axiom of Existence.  To exist is to be some thing--some limitied thing. Since there are only a finite number of entities possible in the universe, it follows that there can only be a finite number of ways that these entities can change position in relation to one another. Therefore, there can only be a finite number of moments in time in the whole past, present, and future of the universe...

Huh? :confused:

Smells like rationalism...

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"Infinity" is a contradictory pseudo-concept. Whoever uses it is simultaneously claiming to be identifying and not identifying something.

"Is it this?"

"No. It's bigger than that."

"This?"

"No."

"This then?"

"No."

"Well, what is it then?"

"Can't show you. You just have to mentally extrapolate until you get the idea of it."

This shows it is "infinity" that is the Rationalistic concept. When we use the term "finite," we are conveying the idea that we are limiting ourselves to those things that have definite natures and therefore are of limited extent.

Saying that finiteness is a corollary to the Axiom of Existence is saying that we can understand the implications of it (the axiom) from this certain angle.

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I don't think I agree with you. "Infinite" is a perfectly valid concept when used in an epistemological (rather than metaphysical) context. (Otherwise, there would be no need for a distinction between it and another valid concept, "finite.") It refers to a quantity that is indefinitely open-ended. Obviously it makes no sense to say that there is an infinity of time in the sense that there literally exists an infinite quantity of time, meaning some really big numberless amount. But it is just fine to say, as a matter of method, that time will continue to extend indefinitely into the future, and in that sense is infinite. However much time has actually existed at any given point is a definite quantity--i.e., finite--but in referring to the future, we don't know how long it will continue to go on and there is no reason to think that it will ever stop--so in that sense, "infinite" is a useful and legitimate concept, I think.

I'm not sure how this would apply to the past, though. Until we know how the universe began, I don't think that we can know this. And I think it's a scientific question, not one that can be rationalistically deduced from philosophical principles (although obviously, whatever answer science comes up with can't contradict basic philosophical principles). But I'm not even sure that it really makes sense to think of time as existing in a quantity in a literal sense, but I haven't really thought about it that much.

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However much time has actually existed at any given point is a definite quantity--i.e., finite--but in referring to the future, we don't know how long it will continue to go on and there is no reason to think that it will ever stop--so in that sense, "infinite" is a useful and legitimate concept, I think.

I'm not sure how this would apply to the past, though.  Until we know how the universe began, I don't think that we can know this.

The universe didn't begin. It always was. The universe will never end. It always will be.

That's why "when" only applies to things in the universe and not to the universe as a whole. That's why time -- finite or infinite -- doesn't apply to the universe as a whole.

The proper term for referring to the duration of the universe is eternal -- always existing ... period.

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One is not being a Rationalist when one attempts to avoid contradictory concepts. "Infinity" in the metaphysical sense is a contradictory one. There can be no such thing. It's not even a possibility.

So it is not being a Rationalist to say that everything metaphysical is finite. It's just making a basic observation about existence from a different angle, i.e., from the context of saying if something exists, then that entails the idea of limitedness.

The same goes for the idea that the universe had a beginning. That's a stolen concept really. The concept "beginning" already presupposes a permanently existing universe. How could something "begin" if it didn't originate in a realm of some kind? If there was a realm from which the universe came, then that realm is a part of existence, too, and we should include that realm when we talk about the "universe."

This all has to do with understanding the implications of "Existence exists."

Of course we can use the word "infinity" in an epistemological sense, or in an "open file folder" way. I think though that we'd be better off not using the term "infinity" in this way, as it tends to obscure what we mean. We can't continually say "I mean this in the epistemological sense" every time we use the word.

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The universe didn't begin.  It always was.  The universe will never end.  It always will be.

That's why "when" only applies to things in the universe and not to the universe as a whole.  That's why time -- finite or infinite -- doesn't apply to the universe as a whole.

The proper term for referring to the duration of the universe is eternal -- always existing ... period.

Right. I didn't mean "began" in a literal sense, which is why I said "how" (hoping it would be taken more metaphorically), not "when." To be honest, this is a topic that I still find somewhat confusing. I think my confusion is what led me to say this:

"But I'm not even sure that it really makes sense to think of time as existing in a quantity in a literal sense..."

I see now that it is just in regard to the universe as a whole that that doesn't make sense, and thinking about it in that context is what led me to say that. But I'm still kind of confused.

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Think of it this way: time is an *aspect* of the universe. It's a part, so to speak, of the whole thing. It just doesn't make sense to suppose that time can apply to the whole thing, because then there would be a bigger whole, something outside of what you said was the whole thing in the first place.

If you were supposing that you can measure the whole universe on a scale of time, then what would the scale be? What would the standard be? If there were some kind of transcendent scale and standard, then they would be part of a bigger whole.

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I see now that it is just in regard to the universe as a whole that [time] doesn't make sense, and thinking about it in that context is what led me to say that.  But I'm still kind of confused.

Maybe this will help.

Time is directly perceivable, even by young children, in terms of "before" and "after." We measure the how-long-it-took-to-change, from what is was before to what it is after, in units of "time."

Since the universe is everything that exists, there is no "before" everything existed and no "after" everything existed.

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But I'm still kind of confused.

Have you read this, by a young Objectivist student of philosophy?

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

I think of this essay as the best single piece of metaphysics to come out in many decades. It may seem deceptively simple, but the essay should probably be read several times. This might help you clear up your confusion.

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Betsy--

I get (and agree with) that much. What I'm confused about are some of the further conclusions that thinkonaut tries to draw.

Stephen--

Thanks for the link. I will definitely read that and give it some thought.

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"Infinity" is a contradictory pseudo-concept.  Whoever uses it is simultaneously claiming to be identifying and not identifying something.

What about mathematics then? It uses the concept of infinity in many aspects. In fact, higher mathematics even breaks the curve into infinite number of infinitely small parts in order to calculate its length. (see integrals)

Not to say that

lim (1/x) = 0 when x -> infinity.

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What about mathematics then? It uses the concept of infinity in many aspects.

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.

What is the method? "As x approaches infinity" is a poorly phrased, but the idea is sound. It means you have a process where you consider what happens when x gets larger and larger. Eventually the process has to stop, since in reality we cannot subdivide or multiply things indefinitely.

I prefer to think of such processes as being "open ended." This is similar to the "open file folder" idea where you know that the folder is open, you can put more in there when appropriate, but the contents are limited and the potential contents are limited, too.

So to say that lim (1/x) = 0 when x approaches infinity simply means that when x gets bigger and bigger, 1/x gets smaller and smaller. Implied is the idea that the process has to stop eventually when applied to some particular scenario. One never "reaches" infinity.

There was confusion about this metaphysical/epistemological distinction in mathematics, especially after Newton used "infinitesimals" in his calculus, but by around 1842, Karl Weierstrass had solved the problem along the lines of what I say here. It's the famous "epsilon-delta" argumentation used in the concept of "limit" that all calculus students become acquainted with.

Most mathematicians do not grasp the significance of what Weierstrass did, including Weierstrass himself evidently, as his student Georg Cantor went on to develop his theory of "transfinite" numbers, or different so-called levels of infinity.

Keep in mind that an objective philosophy or theory of mathematics has not yet been developed. Ayn Rand has shown us where the door is with her theory of concept-formation and her basic identification of how certain concepts of numbers are formed (in "Introduction to Objectivist Epistemology"), but that's only the very beginning of what needs to be done.

See: http://math.nist.gov/opsf/personal/weierstrass.html

See also: http://encyclopedia.thefreedictionary.com/Infinitesimal

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thinkonaut,

So in fact when x grows, 1/x approaches zero asimpthotically. (did I spell this right?) I uderstand what you're saying. To be approaching infinity for x only means "as x is growing." Or something similar, right?

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"Asymptotically" is correct. Yes, you seem to understand. "As x grows without bound" is a commonly used phrase, but of course there has to eventually be a bound, since nothing is infinite in nature.

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"As x grows without bound" is a commonly used phrase, but of course there has to eventually be a bound ...

Are you familiar with the mathematical work of Abraham Robinson? Some forty years ago Robinson developed what is now known as nonstandard analysis, in which he established a rigorous foundation to the vague notion of infinitesimals devised by Newton centuries ago.

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Are you familiar with the mathematical work of Abraham Robinson? Some forty years ago Robinson developed what is now known as nonstandard analysis

Based on what little of it I can understand at this point, it looks like a great work of methodological improvement--an improvement over Weierstrass in methods in the field of analysis. But since we are still in the age of pre-reason, the work hasn't been validated.

I think one of the biggest challenges facing the philosopher of math who comes up with an objective theory of mathematics is to untangle all of the blending of concepts of entities and concepts of method. Surely that confusion exists in non-standard analysis, too.

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But since we are still in the age of pre-reason ...

You think that mathematics is in the age of "pre-reason?" Are you not aware of the brilliant mathematics which lies at the heart of all the physics which makes all of our modern technology possible? Do you really think that particle accelerators, computers, PET scans, GPS satellites, 2-photon laser-scanning microscopes, etc. are the result of "pre-reason?"

Mathematics is one of the most brilliant creations of the human mind, a tribute to the reasoning powers of men.

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I was just using the term "pre-reason" in the spirit of the way it was used in the appendix of IOE, using reason in a meta or philosophical sense. In other words, yes, ever since the first human conceived his first concepts and made sense of the world with them, we can say that reason was born. But it took until Ayn Rand's theory of concept formation for the birth of reason in the meta sense--the sense of "reasoning about reason."

So I agree with what you said in your post. Scientists are using reason when they use math. But I think you'll agree that what we need to do now is not just use mathematics, but validate it philosophically, i.e., do for mathematics what Ayn Rand did for the field of concept formation.

This calls first for the development of a theory of number-formation and a unified theory of concept-formation/number-formation. After that will come validating arithmetic, then algebra, etc. Isn't this what AR was working on before she died?

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I was just using the term "pre-reason" in the spirit of the way it was used in the appendix of IOE

Where exactly in the appendix to Introduction to Objectivist Epistemology is the term "pre-reason" used? The only time I recall Ayn Rand using that term was in reference to a comment by Peikoff which used that term, and that was not in ITOE.

But I think you'll agree that what we need to do now is not just use mathematics, but validate it philosophically ...
Mathematicians, and physicists too, do not just "use" math, they also create and develop new mathematical approaches and disciplines all the time. This is theory, as well as application.

But, what I do agree with -- and what I think you mean -- is that much work remains to be done in the foundational aspects of mathematics. That is true, and it is an important part of mathematics, but keep in mind that mathematics is not philosophy, it is a science.

This calls first for the development of a theory of number-formation and a unified theory of concept-formation/number-formation.  After that will come validating arithmetic, then algebra, etc.  Isn't this what AR was working on before she died?

No, I do not think so. I think she was learning some elements of mathematics primarily to connect to the problem of induction.

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Most mathematicians do not grasp the significance of what Weierstrass did, including Weierstrass himself evidently, as his student Georg Cantor went on to develop his theory of "transfinite" numbers, or different so-called levels of infinity. 

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. Personally I think that the mathematical notion of 'cardinality' is translated somewhat imprecisely into the English word 'size', and this is where a lot of the problems stem from. I agree that it doesnt make much sense to talk about "different sizes of infinity" in the standard English sense of the word, but talking about 'greater cardinality' in the sense of the non-existence of an injective function from one infinite set to another seems sound.

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Where exactly in the appendix to Introduction to Objectivist Epistemology is the term "pre-reason" used? The only time I recall Ayn Rand using that term was in reference to a comment by Peikoff which used that term, and that was not in ITOE.

I haven't read IOE for some time, but I'm sure I can remember a comment in the appendix along the lines of "I am talking about _rational_ mathematics, ie that which existed before Russell", or something similar. Perhaps he is referring to that?

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I haven't read IOE for some time, but I'm sure I can remember a comment in the appendix along the lines of "I am talking about _rational_ mathematics, ie that which existed before Russell", or something similar. Perhaps he is referring to that?

Your rememberance was close:

"When I say 'mathematics,' I really don't mean the modern status of the science, but proper mathematics, rational mathematics."

In that context I think she is referring to foundational issues.

But, anyway, when thinkonaut says

I was just using the term "pre-reason" in the spirit of the way it was used in the appendix of IOE

I take the part in quotes literally.

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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. Personally I think that the mathematical notion of 'cardinality' is translated somewhat imprecisely into the English word 'size', and this is where a lot of the problems stem from. I agree that it doesnt make much sense to talk about "different sizes of infinity" in the standard English sense of the word, but talking about 'greater cardinality' in the sense of the non-existence of an injective function from one infinite set to another seems sound.

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

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

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

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