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The Infinite and the Finite

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Grames and I were arguing this topic in another thread.

Every unit of length, no matter how small, has some specific extension; every unit of time, no matter how small, has some specific duration. The idea of an infinitely small amount of length or temporal duration has validity only as a mathematical device useful for making certain calculations, not as a description of components of reality. Reality does not contain either points or instants

Either space and time are quantized, or this is wrong. Any unit of distance (1 foot, 1 mile) or any unit of time (1 second, 1 year) is an infinite set that is bounded. If you want to assert the opposite you need to make an argument for why space is quantized.

Note that this doesn't necesarily matter, because when you are talking about "physical infinities", that doesn't include relational attributes like position and time. So while these infinities exist, they only exist as a relational attribute.

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The following quote from "The Unbounded, Finite Universe" sums up this discussion nicely.

"However, to declare existence as not possessing spatial boundaries is not to reify infinity. It is not suggesting that a size, extension or number of entities does in fact exist; to the contrary, it is suggesting that it does not. Such a claim upholds the fact that the universe possesses no size, which is fundamentally different than an infinite size – just as the universe not being in time (i.e., being eternal) is fundamentally different than the universe being infinite in time."

Size is not an attribute of the universe therefore the universe can NOT be infinite in size. The universe is not "in size".

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Indeed, it is an arbitrary assertion which cannot be proved. There isn't any logical contradiction in the hypothesis that there isn't an upper bound to the number of particles in the universe.

This from Alex's essay answers the above:

"This is my essential argument for why to posit a finite number of entities "in the universe" is contradictory.

Are all the entities in the universe a quantity? This is the criterion that would have to be filled for one to validly state that there are a finite number of entities in the universe. But, what meaning does a quantity have, if one does not provide any boundaries? After all, the universe must be unbounded.

All quantities must be able to be described by a (finite) number, because all quantities are finite in scope (i.e., extension), whether temporal or spatial. But, the universe as a whole does not encompass a (finite) spatial scope. Does this alleged quantity span out 15 billion light-years? 100 octillion light-years? This amount squared? There can be no answer to this question, because the universe has no bounds, and therefore no extension. To extend is to extend finitely, i.e., in a bounded fashion.

Just as in order to talk about an amount of time one has to say, "between now, and some other point," so with an amount of entities one has to say, "between here, and some other point." If one doesn't specify, (i.e., quantify) what one is talking about, "amount," and therefore "number" has no meaning.

So, literally speaking, there is no "in the universe" in this context, because it smuggles in "within the spatial boundaries of the universe," just as "in the entire history of the universe" smuggles in "within the temporal boundaries of the universe."

This is why the universe cannot have a finite number of entities. [5] There are no boundaries to speak of to bound this all-encompassing quantity; hence, it is not a quantity, so "number" is therefore inapplicable to it. But, at this point, there should be no temptation to jump over to the demon of infinity, which in this context means: infinite number.

The dichotomy is not: finite or infinite. But, rather: existence or nonexistence. If it has been shown that the concept of number is inapplicable to describe all the entities "in the universe," then one has blanked out the possibility of any number (including an infinite number, which is a contradiction of the Law of Identity anyway). Either a number exists in this context, or it does not. It is my belief that it does not."

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Either space and time are quantized, or this is wrong.

I would argue (and there is increasing proof that it is true) that this is exactly the case. I believe that the smallest unit of space and time respectively is the Plank length and the smallest unit of time is the Plank time. Notice how if you divide these two fundamental units by one another you get the speed of light. So light travels one Plank length in one Plank time. This is more than a mathematical trick, I think it tells about the basic structure and composition of the universe.

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Space may be quantized, and there are arguments for it out there (loop quantum gravity for instance), but the argument you have presented is not sufficient. For those of us that are not theorhetical physicists at the top of our field (ie none of us), this means we have to wait until physicists figure this one out.

Until then, you have to operate on the understanding that space is potentially a bounded infinite set of positions.

Another problem with that quote, which is the same problem with the arrow paradox, is that there is a difference between a point and an infinitessimal.

Something can have an infinitessimally small volume, but not be a dimensionless point. Two different things.

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Oh I didn't mean for that to be my argument for quantized space, that was just one piece of evidence that popped into my head. I will start a new thread listing the reasons why space has to be quantized and I will include a bunch of links to theoretical research that supports this. For what its worth loop quantum gravity is not a good candidate for this.

I absolutely do not have "operate on the understanding that space is potentially a bounded infinite set of positions", since such an idea is irrational and doesn't reflect reality.

Nothing in reality can have an infinitessimally small volume in reality, that's pure rationalism. Nor do actual points or point-like particles exist in reality.

This stuff is slightly off topic so I will explain these ideas in a new thread.

Edited by EC
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Doesn't seem off topic to me. The OP asked how the universe can be finite while being endless. The answer is that the universe is physically finite. Time and space are relational attributes, they don't actually exist.

Grames did a really good job of explaining this to me. I still find it to be a bit of a weasel definition (by restricting to non-relational attributes), but it works.

Anyways:

I absolutely do not have "operate on the understanding that space is potentially a bounded infinite set of positions", since such an idea is irrational and doesn't reflect reality.

Nothing in reality can have an infinitessimally small volume in reality, that's pure rationalism. Nor do actual points or point-like particles exist in reality.

These are two different statements. For the first statement. If you assume that space is not quantized then it IS a bounded infinite set of positions. There's nothing irrational about that.

This is different from the second statement, which I agree with. Nothing can have an infinitesimal small volume.

But, in a continuous space, it can move in infinitesimal increments.

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But, in a continuous space, it can move in infinitesimal increments.

If you cannot distinguish between where it was and where it is, did it move? Even developing sophisticated equipment that can generate a resolution greater than your perceptual abilities, if the equipment cannot distinguish between where the object was and where it currently is, did it move?

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Yea, I can see that this recent subject matter is on topic now but the direction I want to direct it would be, and will require a loooong post encompassing wide range of abstract concepts that must be true based on the law of identity. It's one of those posts you want to make but the exact way to write it out in a logical order combined with the length needed to make it understandable is making me put it off. Probably just described laziness. lol Hope calling myself lazy motivates me.

Edited by EC
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I would argue (and there is increasing proof that it is true) that this is exactly the case. I believe that the smallest unit of space and time respectively is the Plank length and the smallest unit of time is the Plank time. Notice how if you divide these two fundamental units by one another you get the speed of light. So light travels one Plank length in one Plank time. This is more than a mathematical trick, I think it tells about the basic structure and composition of the universe.

But this is just a mathematical "trick."  Planck units are defined by normalizing certain physical constants, including the speed of light, to 1.  The Planck length and Planck time are defined in terms of the gravitational constant (G), the reduced Planck's constant (h-bar), and the speed of light in a vaccuum (c).

Planck length = (h-bar*G/c^3)^1/2

Planck time = (h-bar*G/c^5)^1/2

When you divide Planck length by Planck time, the h-bar and G factors cancel, leaving c.

The Planck unit approach starts with the idea that these physical constants, including c, are fundamental.  You shouldn't be surprised when the math reflects this.

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But this is just a mathematical "trick."...

Okay let me explain why I think this is relevant, quickly. First, I was an undergrad studying physics about ten years ago, but never completed my "official" studies because of money problems at the time. That be true there is a slight chance, I already had learned this and have since forgotten the fact that what I said was true. But if I had forgotten it I should have remembered learning it when I double-checked that it was true. Why did I double check that it was true? Because I was thinking the day before I responded to this thread about the quatization and what it implies. One of the implications that I thought must be true is that the smallest possible length divided by the smallest possible time should equal the c. I then double checked the math and my idea was right. The next day I saw emorris1000's post and this immediatly came to mind.

Like I stated earlier, this is not my only argument for the quantization of space and time, but it is something that must be true if as I think such quantization reflects reality. And I will explain all my reasoning in a new thread as soon as I can organize my thoughts coherently on the subject.

Cliff Notes: I personally predicted that this had to be true if space and time are quantized. After making making my own prediction based on facts that I'm not getting into right now, I did some research and math and my prediction was proven true.

Edited by EC
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  • 10 months later...

Just wanted to drop off my recent thoughts on infinity:

Infinity is a number that is greater than any number. (according to wikipedia)

(You can't say that infinity is a concept, because only concepts that are also numbers can be greater than other numbers)

so infinity is a number therefore...

infinity is greater than infinity which means that...

infinity does not equal infinity and that...

infinity is not infinity and so ...

infinity would break the law of identity

infinity is a bound to boundless concepts, which alone is a contradiction in terms.

Infinity is a useful term, but it seems to me that it has been used the wrong way.

Like in the multiverse theory or in Religion...

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(let X=1)

(while X>0

(let X=(X/2))

)

The processing of this computer code is dependent on an allotment of physical space, i.e. a finite number of bytes available to process. The program crashes due to physical reality (hardware); not conceptual error (software). I suspect that causality is similarly constrained. Infinity has meaning, but no practical application.

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  • 4 months later...

If the Universe is finite in both time and space and if causality works as we think it does than there had to be a first cause, right?

A cause not caused by anything, an existence that is necessary for things to exist.

Or am I wrong?

Ether that or existence is a closed cycle.

ruveyn1

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When you use a word like "you", conventionally it's helpful to be clear what the referent is. In fact, etymologically it means "unboundedness" or "not-ending-ness". Typical contemporary views of "infinity" derive from a strange mathematical mis-speaking, where people talk of an "infinite number of...". I have had more than one mathematician tell me that I really ought to say something like "the cardinality of the set... is infinite". We know that there can be a billion grains of sand, or a quadrillion electrons. So whever you use the "Number+Noun" structure, that implies at least the possibility of such a thing existing. The problem is that "infinity" is not a number, yet people confuse it with a number.

Some infinite sets are bigger than others. That is, there is not a 1-to-1 mapping between them. The smaller set can be mapped bijectively into a proper subset of the larger one; with the complement of the smaller sets image being the same cardinality as the larger set. I certainly have heard people say that infinity is not a number, it is a concept. Well not quite true mathematically, it is somewhat accurate physically.

Now it turns out that recent discoveries in Astronomy suggest that the Universe may very well be Infinite. And by that I mean that it if you travel in 1 direction you will never return to where you started from nor run into any sort of spacetime boundary. This is due to the fact that based on data collected from the WMAP satellite, the Cosmos has as flat(Euclidean)geometry and with no signs of any actual boundaries, the appropriate conclusion is that it is Infinite spatially and has no end and no beginning.

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Just wanted to drop off my recent thoughts on infinity:

Infinity is a number that is greater than any number. (according to wikipedia)

(You can't say that infinity is a concept, because only concepts that are also numbers can be greater than other numbers)

so infinity is a number therefore...

infinity is greater than infinity which means that...

infinity does not equal infinity and that...

infinity is not infinity and so ...

infinity would break the law of identity

infinity is a bound to boundless concepts, which alone is a contradiction in terms.

Infinity is a useful term, but it seems to me that it has been used the wrong way.

Like in the multiverse theory or in Religion...

One infinity, two infinity, three infinity, four.....

five infinity, six infinite, seven infinity, more!

:fool:

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Now it turns out that recent discoveries in Astronomy suggest that the Universe may very well be Infinite. And by that I mean that it if you travel in 1 direction you will never return to where you started from nor run into any sort of spacetime boundary. This is due to the fact that based on data collected from the WMAP satellite, the Cosmos has as flat(Euclidean)geometry and with no signs of any actual boundaries, the appropriate conclusion is that it is Infinite spatially and has no end and no beginning.

That's far from being conclusive. No matter how long you have travelled in a infinite Universe you could never prove that it is infinite since you can only travel a finite distance. Same is true for light and anything else we can measure.

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That's far from being conclusive. No matter how long you have travelled in a infinite Universe you could never prove that it is infinite since you can only travel a finite distance. Same is true for light and anything else we can measure.

That's true. However, given what we know about astrophysics and cosmology if the Universe has a Euclidean geometry(and topology)then an infinite Universe is the most reasonable conclusion even though one cannot prove it.

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Guest Math Bot

"I certainly have heard people say that infinity is not a number, it is a concept. Well not quite true mathematically, it is somewhat accurate physically."

That is quite definitely false.  Infinity has no valid mathematical use except for as a concept, and not a number.   Mathematics treats infinity as concept that indicates that there is always the potential to progress in a series of steps, but in actuality you will have to stop eventually.  Mathematics in no way treats infinity as anything else, it certainly does not pretend it is is a number. 

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Mathematics treats infinity as concept that indicates that there is always the potential to progress in a series of steps, but in actuality you will have to stop eventually.  Mathematics in no way treats infinity as anything else

 

Mathematics as practiced by mathematicians (who are not usually Objectivists) treats infinity in a variety of aspects, not merely the particular concept you mention.

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Guest Math Bot

Mathematics as practiced by mathematicians (who are not usually Objectivists) treats infinity in a variety of aspects, not merely the particular concept you mention.

Yes, some mathematicians have the wrong idea about infinity.  However, that does not change the point that when properly understood, the mathematical concept of infinity is exactly as stated.

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