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BraveHat
Last of the Daedalians

Posted: Sat Aug 20, 2011 3:01 am    Post subject: 1

Inspired by, though not a direct off shoot of, a discussion in the "Ask About Atheism" thread in the "Science, Art, and Culture" forum:

 I almost wrote: If the Universe is infinite in space, does Earth take up some percentage of it's space, or no percentage of it's space? It seems that if Earth takes up some percentage of it's space, then that percentage, no matter how infinitesimally small would imply a finite amount of space in the Universe, which would be wrong, since we are assuming the Universe is infinite in space. However, if Earth takes up no percentage of space, then that would seem to contradict our plain observation, which is that Earth is taking up some of the Universe's space. Does the apparent wrongness of either answer imply that the Universe is finite? Same question with time: If time is infinite, does the present day take up some percentage of time, or nopercentage of time? Does this dilemma imply that time is finite?

And then I realized that the term "percentage" doesn't really apply. It would seem as if the answer would be Earth is taking up no percentage, but some portion of space. The portion having no mathematical ratio and just simply meaning "chunk".

I've observed that every time I try to prove my intuition that the Universe can't be infinite (in space or time), I only end up (though not intending to) using some trickery of argument. In this case, the tricky word "percentage" which automatically implies a portion of the finite. I really don't mean to, though. The argument just comes up in my head, I scream "Eureka!" and then write it down, and then once it's written down, careful scrutiny (either by myself or others) reveals that the argument is invalid.

Has anyone ever come up with any good arguments for the impossibility of infinite quantities in physical reality? Meaning an argument that comes up with legitimate contradictions, rather than mere absurdities (like the "Hilbert's Hotel" scenario)?

If not, I may have to conclude that actual infinities are possible (at least theoretically)
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax

Last edited by BraveHat on Sun Aug 21, 2011 12:32 pm; edited 1 time in total
Chuck
Daedalian Member

 Posted: Sat Aug 20, 2011 3:47 am    Post subject: 2 I thought that since infinity is not a number it can't be used as a divisor.
Zag
Tired of his old title

 Posted: Sat Aug 20, 2011 4:46 am    Post subject: 3 Although I'm inclined to trust the scientists who say that the universe is finite, I struggle to comprehend how that can be. I mean, what if you got near the end of it? Obviously, you could never say, "this is the edge" because then you'd be able to point beyond it and say, "what's over there?" They say that the big bang happened about 13.7 billion years ago. What if there was another, equivalent creation of matter and energy, say 8 billion years ago, but it's 50 billion light years away, so light from it hasn't reached us yet? Is that not inside our universe? What does "the universe" mean, really? Anyway, you're assuming that the universe is infinite, which, as I said, is not the current thinking amongst physicists. However, with that assumption, then any finite amount of space (i.e. the Earth, the Solar System, the Milky Way galaxy, etc.) takes up 0 percentage of it, in the same way that the numbers from 1 to 1 billion is zero percent of all the available integers. There are lots of counter-intuitive things when you talk about infinity. My favorite one, which I read here on GL, is this: If you took all the integers, what percentage of them have the digit 3 somewhere in their decimal representation? The somewhat counter-intuitive answer is 100%. Maybe someone more adept at math than I could answer this question: Is it appropriate for me to say that, of all the integers, half of them are even?
Scurra
Daedalian Member

Posted: Sat Aug 20, 2011 8:56 am    Post subject: 4

 Douglas Adams wrote: It is known that there is an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so the average population of all the planets in the Universe can be said to be zero. From this it follows that the population of the whole Universe is also zero, and that any people you meet from time to time are merely the products of a deranged imagination.

_________________
still Quiz Olympiad champion. Must get a life.
New definitions: COFFEE - someone who is coughed upon
Antrax
ESL Student

 Posted: Sat Aug 20, 2011 9:04 am    Post subject: 5 Percentage doesn't imply that at all. Ten percent of the real numbers from 0 to 1 lies within the region 0 to 0.1, despite there being an infinite amount of real numbers between 0 and 1. Your other argument is just as flawed. Mathematically, it's quite possible that earth takes up no percentage of the universe's space, and still exists. Let's say you pick a random integer with uniform probability. Mathematically, the odds for you to pick any specific number (1, 2, 1000000) or even any finite group of numbers (1 to 10) are exactly 0. Yet, these numbers exist, and in an intuitive sense you might pick any one of them. The lesson is "don't try to intuitively understand infinities", I believe._________________After years of disappointment with get rich quick schemes, I know I'm gonna get rich with this scheme. And quick!
extro...*
Guest

Posted: Sat Aug 20, 2011 11:15 am    Post subject: 6

 Zag wrote: Although I'm inclined to trust the scientists who say that the universe is finite, ...

You don't trust the scientists who say evidence suggests it's infinite? http://map.gsfc.nasa.gov/universe/uni_shape.html

Is there a consensus?

 Quote: ... I struggle to comprehend how that can be. I mean, what if you got near the end of it?

The standard analogy is to a 2-dimensional universe being curved, and shaped like the surface of a sphere. The 2-dimensional surface is finite, but inhabitants of that 2-dimensional universe would never encounter an edge. Of course, it's important with analogies to separate similarities from dissimilarities. We envision the 2-dimensional surface of the sphere existing in a 3-dimensional space, but it need not be so. It seems easy to envision an infinite flat 2-dimensional plane, without a higher 3-dimensional space, but when we put curves in it, we automatically picture a higher dimension.

And of course a sphere is not the only edgeless finite surface. There's the torus, and others. Sphere is the simplest.

 Quote: Anyway, you're assuming that the universe is infinite, which, as I said, is not the current thinking amongst physicists.

In the thread this spun off from, the question was more whether there could be a proof of the necessary impossibility (as opposed to actual non-existence) of an infinite temporal past. In general, it's more about logical possibilities as opposed to observable reality.
Thok*
Guest

Posted: Sat Aug 20, 2011 11:18 am    Post subject: 7

 Zag wrote: Maybe someone more adept at math than I could answer this question: Is it appropriate for me to say that, of all the integers, half of them are even?

Before you answer this question, you need to describe what you mean by "half of them are even". There's no "natural" probability distribution on the integers or any infinite set. (There's no "natural" probability distribution for any finite set with at least two elements either, but there is a seemingly preferred one for finite sets.)

As for integers, the problem is that the integers have more structure than just being points. You really shouldn't have problems with the multiples of 60 being the same amount as the multiples of 1 (do you measure time in hours or minutes?), but there's implicit extra structure in the integers as well. (Either by fixing a distance scale or by insisting on multiplication working as well: that doesn't break the time examples since for the most part 5 hours * 15 hours doesn't make sense and certainly isn't an hour.)
Antrax
ESL Student

 Posted: Sat Aug 20, 2011 11:35 am    Post subject: 8 Thok, I think you can answer "yes" to Zag's question, if you interpret it as asking "if I uniformly pick a random integer, is the probability it's even half?"._________________After years of disappointment with get rich quick schemes, I know I'm gonna get rich with this scheme. And quick!
extro...*
Guest

Posted: Sat Aug 20, 2011 11:35 am    Post subject: 9

 Antrax wrote: Let's say you pick a random integer with uniform probability.

Not possible. The probabilities have to add up to 1.
Thok*
Guest

Posted: Sat Aug 20, 2011 1:23 pm    Post subject: 10

 Antrax wrote: Thok, I think you can answer "yes" to Zag's question, if you interpret it as asking "if I uniformly pick a random integer, is the probability it's even half?".

As extro said, you can't uniformly pick a random integer.

(There's also an argument of why a uniform distribution is the right one for the integers or any set.)

The closest way of dealing with Zag's question in sort of the spirit you have in mind is "What happens to the ratio of (even integers within distance d of 0)/(all integers within distance d of 0) as the distance d goes to infinity?" The answer to that question is the ratio approaches 1/2. That sort of approach is used in number theory.
extro...*
Guest

 Posted: Sat Aug 20, 2011 2:14 pm    Post subject: 11 Two related question, though not sure how well they fit, but they also spun off the original discussion: 1) If the universe is infinite, was it ever finite? (I can't see the finite to infinite threshold being crossed) 2) My understanding is that the Big Bang was the beginning of an expansion of not only space, but space-time. Is this correct? If so, what does it mean for time to be expanding? We could conceivably measure the rate of expansion of space, but we do that against time. Would it make sense to talk of the rate of expansion of time itself?
Chuck
Daedalian Member

 Posted: Sat Aug 20, 2011 3:27 pm    Post subject: 12 The percentage of integers that are even depends on the order in which you list them. If you list them in numerical order then it's 50%. If you list the first two odd numbers and then the first even, then the third and fourth odd numbers and then the second even, then the fifth and sixth odd numbers and then the third even, etc, then only one third of the integers are even. This alternate order includes every integer once each so it's just as reasonable a way to list them as numerical order, so the percentage of integers that are even can be arbitrarily chosen. The same can be done with integers containing a 3.
extropalopakettle
No offense, but....

Posted: Sat Aug 20, 2011 6:19 pm    Post subject: 13

 BraveHat wrote: Has anyone ever come up with any good arguments for the impossibility of infinite quantities in physical reality? Meaning an argument that comes up with legitimate contradictions, rather than mere absurdities (like the "Hilbert's Hotel" scenario)?

This question is more loaded than you think, I think. What do we mean by "impossibility ... in physical reality"? The discussion branched off from arguments that the universe must have a beginning, and I think we distinguished that from the universe has a beginning. I'm trying to ask, but not doing it well, what is the difference between "impossible in physical reality" (as we're talking about it), versus "impossible as a coherent abstraction"? We can have a continuous real number line extending infinitely in both directions, positive and negative. No contradictions arise from assuming such a thing exists. It's possible. So what is it about "possible in physical reality" that it should be any different? Would we say that it's impossible for physical reality to have 2 or 4 spatial dimensions (assuming we know our physical reality has 3)? Is that we we mean by "impossible" here - just that it isn't that way in this universe? Or do we mean that it couldn't be in any universe, without contradictions arising? If the latter, then I'd say any coherent (not leading to contradictions) abstraction is possible in physical reality.
raekuul
Lives under a bridge & tells stories.

 Posted: Sat Aug 20, 2011 10:17 pm    Post subject: 14 Consider the following possibility: The universe is spherical, but we're on the inside surface as opposed to being on the outside surface (Iike how we're on the earth). If we accept that to be true, then there is no possible observation that any native inhabitant of this universe can make that could lead to a provable conclusion. We can theorize all we want, but can the theories be proven before we have observations that can back them up?
Thok*
Guest

Posted: Sat Aug 20, 2011 10:31 pm    Post subject: 15

 raekuul wrote: If we accept that to be true, then there is no possible observation that any native inhabitant of this universe can make that could lead to a provable conclusion.

I'm pretty sure you are wrong about the lack of provable conclusions. But I'd need to see an attempt at an argument before I respond. (You basically state this as a fact without any sort of reasoning and I'm not even sure what your model is in this situation.)
extropalopakettle
No offense, but....

Posted: Sat Aug 20, 2011 11:53 pm    Post subject: 16

 raekuul wrote: Consider the following possibility: The universe is spherical, but we're on the inside surface as opposed to being on the outside surface (Iike how we're on the earth).

In the 2-dimensional analogy of the surface of a sphere, it's a 2-dimensional universe, it's inhabitants are in it, not on it, and there is no inside or outside. They're 2-dimensional beings living in a 2-dimensional plane that seems flat to them (as it's a large sphere). We, having evolved in, and living in, a 3-dimensional universe, automatically think of 2-dimensional surfaces, especially curved ones, as existing in a higher (3-dimensional) space. It needn't be so. This relates to my previous point (or question) about the difference between an abstraction and the real physical world. Non-Euclidean geometry, like Euclidean geometry, is an abstraction. We tend to think of Euclidean geometry as holding in our world, but on a very large scale, if space is "curved", so to speak, a non-Euclidean geometry may hold.

If our 3-dimensional spatial universe is finite, it either has to be closed, curved in a sense (like the surface of a sphere for the 2-dimensional case), ... or ... it would have to have edges. The notion of edges seems absurd. What happens to a bullet fired at the edge? And if there are edges, it seems there'd have to be something somewhat like a center - a place far from the edges. If so, it seems we just happen to be there (what are the odds?), as the cosmic background microwave radiation seems to emanate from all directions equally.

 raekuul wrote: If we accept that to be true, then there is no possible observation that any native inhabitant of this universe can make that could lead to a provable conclusion. We can theorize all we want, but can the theories be proven before we have observations that can back them up?

Not sure I follow this, but in our universe, there are observations that can be made to determine that. Google "shape of the universe". Or check the link I included in a previous post above.
Death Mage
Raving Lunatic

 Posted: Sun Aug 21, 2011 12:54 am    Post subject: 17 The amount of space taken up by a finite object in an infinite space can be best described as the difference of 1-0.999...(repeating). Therefore, if you believe that 0.999...(repeating) = 1, you do not exist._________________* These senseless ramblings brought to you by Insanity™. If you just can't figure the dang thing out, it must be Insanity™. [YOUR AD HERE!]
MNOWAX
0.999... of a Troll

 Posted: Sun Aug 21, 2011 2:02 am    Post subject: 18 oh god, not another 0.99999....=1 thread...._________________The Man The Myth The Legend MNOWAX
BraveHat
Last of the Daedalians

Posted: Sun Aug 21, 2011 2:09 am    Post subject: 19

extro wrote:
 I wrote: Has anyone ever come up with any good arguments for the impossibility of infinite quantities in physical reality? Meaning an argument that comes up with legitimate contradictions, rather than mere absurdities (like the "Hilbert's Hotel" scenario)?

This question is more loaded than you think, I think. What do we mean by "impossibility ... in physical reality"? The discussion branched off from arguments that the universe must have a beginning, and I think we distinguished that from the universe has a beginning. I'm trying to ask, but not doing it well, what is the difference between "impossible in physical reality" (as we're talking about it), versus "impossible as a coherent abstraction"? We can have a continuous real number line extending infinitely in both directions, positive and negative. No contradictions arise from assuming such a thing exists. It's possible. So what is it about "possible in physical reality" that it should be any different? Would we say that it's impossible for physical reality to have 2 or 4 spatial dimensions (assuming we know our physical reality has 3)? Is that we we mean by "impossible" here - just that it isn't that way in this universe? Or do we mean that it couldn't be in any universe, without contradictions arising? If the latter, then I'd say any coherent (not leading to contradictions) abstraction is possible in physical reality.

Good question. And I'm not sure what the answer is, only that it seems that the only time we deal with a certain kind of infinite set is in the abstract. The certain kind of infinite set I'm talking about is an infinite set in which each individual element is of the same finite measurement. A Universe with infinite space, for example, can be thought of as that kind of infinite set. The set of all "space cubes" of 1 inch dimensions or of 5 inch dimensions, or of 100 billion mile dimensions or of square root of 3 millimeter dimensions.

Now, in the abstract, It seems that every conclusion needs to be consistent with an established system of laws built on axioms in order for us to accept it. In the physical world, we can accept something that isn't consistent with established law simply because we observe it happening physically, and we need to either adjust our system of laws or come up with an entirely new system to explain it. So, if we assume some theoretical reality, like a Universe of infinite space, and our abstract principles of infinity leads us to one conclusion about how it should be, but we observe something in the physical world that contradicts that conclusion, it would imply that the kind of infinity we're talking about is a physical impossibility. One could say, well, we would just need to change our abstract principles about infinity in that case. Well, that depends on how close to the definition of infinity the principle is. We can't change a principle if it makes something finite out of infinity. And I don't even know if we really can change abstract principles, since they so rigorously follow from other abstract principles.

I thought, for example, I had a valid contradiction when I thought that, if one assumes the Universe is infinite, the Earth could neither take up some percentage of the Universe's space because of abstract law, nor no percentage of the Universe's space because of physical observation. In this case, however, I was wrong, because the word "percentage" entails a comparison to a finite quantity to begin with, so the answer would be Earth would take up no percentage of an infinite Universe's space, though it would take up some portion of it. But in any case, the possibility of some physical observation contradicting some abstract law implies the possibility of concluding soundly that the abstract law doesn't apply to physical reality.
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax

Last edited by BraveHat on Sun Aug 21, 2011 3:31 am; edited 1 time in total
raekuul
Lives under a bridge & tells stories.

 Posted: Sun Aug 21, 2011 2:46 am    Post subject: 20 Our observations are limited by our understanding, aren't they? If whatever is past the (hypothetical?) boundary of the universe is literally incomprehensible, how would we observe it at all? Infinity is the same way. It's there, it exists (logically, at least), but we can't wrap our minds around it. It's beyond what we can understand, so we cannot actually observe infinity. If you had an infinite amount of time and an infinite amount of patience, would you be able to count to infinity?
Thok
Oh, foe, the cursed teeth!

Posted: Sun Aug 21, 2011 3:17 am    Post subject: 21

 BraveHat wrote: But in any case, the possibility of some physical observation contradicting some abstract law implies the possibility of concluding soundly that the abstract law doesn't apply to physical reality.

Yes. The point of science is to rephrase real world phenomena as a mathematical model that can be used to make predictions about the real world phenomena. Those models are only so accurate and get tweaked when we don't like what they are.

(That said, mathematics is the art of taking a small set of rules and seeing what we can say about the objects described by those rules, for a varying definition of rule. Applications of those objects to the real world is the domain of physicists, chemists, biologists, and other science types.)

 Quote: If you had an infinite amount of time and an infinite amount of patience, would you be able to count to infinity?

Define what count to infinity means and I'll answer the question. Infinity isn't an integer, and isn't part of what we normally count.

If you mean that I should say the sizes of sets in increasing order of cardinality without any missing cardinalities and try to reach the cardinality of the integers in a finite time, the answer is yes. I leave that as an exercise to the more mathematically astute, with the hint that (1+1/2+1/4+1/8+1/16+....)+1 is a finite number. Whether you can reach the cardinality of the real numbers depends on your model of set theory.
extropalopakettle
No offense, but....

Posted: Sun Aug 21, 2011 4:08 am    Post subject: 22

BraveHat wrote:
 extro wrote: Would we say that it's impossible for physical reality to have 2 or 4 spatial dimensions (assuming we know our physical reality has 3)? Is that we we mean by "impossible" here - just that it isn't that way in this universe? Or do we mean that it couldn't be in any universe, without contradictions arising?

Good question. And I'm not sure what the answer is, only that it seems that the only time we deal with a certain kind of infinite set is in the abstract.

I'm not sure my question was understood, because you say you're not sure what the answer is, but here you answer it:

 BraveHat wrote: So, if we assume some theoretical reality, like a Universe of infinite space, and our abstract principles of infinity leads us to one conclusion about how it should be, but we observe something in the physical world that contradicts that conclusion, it would imply a physical impossibility.

That answers my question. You're saying if we observe something in the physical world that is inconsistent with infinite space, then infinite space is a physical impossibility. Similarly, since we observe the physical world has 3 spatial dimensions, that makes a universe with 2 or 4 spatial dimensions a "physical impossibility". You're equating "physically impossible" with "it's not that way in this universe".

But in our other discussion, I suggested:

 Quote: How about a single prior universe that had a temporal existence extending infinitely into the past (it's own past), ending in a Big Crunch, "followed" by our Big Bang?

(the point there to posit something possible without a beginning)

You eventually responded with a lengthy argument that included:

 Quote: The following series of syllogisms does not argue that infinite space can't exist, nor does it argue that infinite time can't exist, but only that a physical object can't have existed from infinite time.

And there I'm getting the sense that you're arguing for the physical impossibility of something not just in this universe, but in any possible universe. It doesn't sound like "we observe this universe had a beginning", but anything physical, in any possible universe, must have a beginning.

Do you see the difference, and the confusion?

And there I would say "possible in some universe" is equivalent to "possible in the abstract".
extropalopakettle
No offense, but....

Posted: Sun Aug 21, 2011 4:17 am    Post subject: 23

 raekuul wrote: Our observations are limited by our understanding, aren't they?

No. We often observe what we don't understand.

 raekuul wrote: Infinity is the same way. It's there, it exists (logically, at least), but we can't wrap our minds around it. It's beyond what we can understand, so we cannot actually observe infinity.

Explain what is it you don't understand about infinity.
bgg1996
BeeGees are awesome!

 Posted: Sun Aug 21, 2011 5:20 am    Post subject: 24 If I put together two seperate units, then observed that there were three of them, I would not rewrite arithmetic, I would blink a few times, rub my eyes, and observe again._________________The one member below 18
MatthewV
Daedalian Member :_

 Posted: Sun Aug 21, 2011 6:31 am    Post subject: 25 Assume that you are really small and living on an atom. What would you think of as the universe? And what would be microscopic to you? On the atomic scale, the earth is practically infinite. But what is the actual edge of its universe? I feel the same is true with our universe. It is all a perspective of scale. The influences of far off galaxies are insignificant on our world. Why would the influence of all the junk in my room be significant on the atoms in my fingers? Even the motion of my fingers is so great the relative motion translates to nothing. So I believe in the possibility of a multiverse of many different scales.
extropalopakettle
No offense, but....

Posted: Sun Aug 21, 2011 12:05 pm    Post subject: 26

 BraveHat wrote: So, if we assume some theoretical reality, like a Universe of infinite space, and our abstract principles of infinity leads us to one conclusion about how it should be, but we observe something in the physical world that contradicts that conclusion, it would imply a physical impossibility.

I think the points I'm trying to make are that:

1) We should be careful to distinguish between the following:
a - It is not possible that it's true in the physical reality of this universe. (example: only 2 spatial dimensions - we know we have 3)
b - It is not possible that it's true in the physical reality of any possible universe. (example: space being both finite and infinite)

2) If "impossible in physical reality" means the latter of the two above, then it's no different than "impossible in the abstract". Abstractly, logically, mathematically possible means possible in physical reality.

And relating to Craig's argument from the other discussion,

3) If "impossible in physical reality" means the first of the two above, it does not apply to the possibilities of another "universe" that might in some sense have given rise to this one, and that may have had no beginning. The abstract, logical, mathematical possibility of a timeline with an infinite past and a terminating endpoint is thus a possibility in physical reality.
BraveHat
Last of the Daedalians

Posted: Sun Aug 21, 2011 12:26 pm    Post subject: 27

 extro wrote: That answers my question.

Well, actually it doesn't answer your question if your question is what's the difference between what's true in the abstract world, and what must be true in some Universe. All I seem to be doing, like you said, is talking about this Universe. I would be lying, however, if I said the reason I said "I'm not sure" was because I couldn't answer that question. But happened, pecularly, is that I forgot you had asked that question, was thinking of only this Universe, and wasn't sure if I could find some proposition to explain my reasoning. Once I did, it just didn't occur to me to delete/change "I'm not sure". lol.

In any case, to answer your large question about any Universe, we need to note that the term Universe some definition. Any properties that make something a Universe have the potential to be observed, and if any observation of those properties contradict an abstract law, that would imply that the abstract law does not apply to any Universe. So the possibility still exists to soundly conclude that certain kinds of infinity does not apply to any Universe.

extro wrote:
You eventually responded with a lengthy argument that included:

 I wrote: The following series of syllogisms does not argue that infinite space can't exist, nor does it argue that infinite time can't exist, but only that a physical object can't have existed from infinite time.

This wasn't actually responding directly to anyone's comments, it was just me trying to start afresh in expressing my general beef with the concept of actual infinity. Sorry if it seemed like a direct response to what you were saying.
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax
extropalopakettle
No offense, but....

Posted: Sun Aug 21, 2011 12:52 pm    Post subject: 28

 BraveHat wrote: Any properties that make something a Universe have the potential to be observed, and if any observation of those properties contradict an abstract law, that would imply that the abstract law does not apply to any Universe.

Either I'm reading that wrong, or it makes no sense.

First, I'm not clear on what you mean by an abstract law or how it relates to the discussion.

In the abstract, we can have a 2 dimensional surface that is an infinite flat plane. We can also have a 2 dimensional surface that is like the surface of a sphere, finite, but without borders or any point on it having properties different from another. In the abstract, we have 3 dimensional analogs of each of those 2 dimensional constructs. The are well defined by axioms and rules of inference.

Are you saying that if we observe one of these to be the case for our universe, then the other is a physical impossibility in any universe?
BraveHat
Last of the Daedalians

Posted: Sun Aug 21, 2011 1:06 pm    Post subject: 29

 extro wrote: 1) We should be careful to distinguish between the following: a - It is not possible that it's true in the physical reality of this universe. (example: only 2 spatial dimensions - we know we have 3) b - It is not possible that it's true in the physical reality of any possible universe. (example: space being both finite and infinite)

If any possible Universe needs certain properties to even be considered a Universe, than it's possible to observe the defining behaviors of those properties and find they contradiction abstract. And it's possible to repeat those observations as a means of verification to avoid the bgg1996 scenario:

 bgg1996 wrote: If I put together two seperate units, then observed that there were three of them, I would not rewrite arithmetic, I would blink a few times, rub my eyes, and observe again.

Perhaps, but if you kept repeating that action, putting together two separate units, and the same result of three kept being observed, not just by you, but by everyone else you show this to, you will have hit upon something that is a cause of scientific and mathematical concern, to say the least. Eventually, science will have to adjust it's laws or come up with a new system of laws to explain what you keep doing.

 extro wrote: 2) If "impossible in physical reality" means the latter of the two above, then it's no different than "impossible in the abstract".

False. The veracity of abstract conclusions is determined by consistency with abstract principle alone. The veracity of physical conclusions is determined by consistency with physical observation.

 extro wrote: Abstractly, logically, mathematically possible means possible in physical reality.

False. Working with the square root of a negative number, for example, is mathematically possible, but not possible in physical reality. In order for us to work with it in physical reality, it has to translate to a physical quantity, but it has no quantity at all. It is still a mathematical object with properties of it's own, but those properties and object do not exist in physical reality.
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax
BraveHat
Last of the Daedalians

 Posted: Sun Aug 21, 2011 1:11 pm    Post subject: 30 Sorry, I will have to respond to your other questions tomorrow. Busy day today_________________"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax
extropalopakettle
No offense, but....

Posted: Sun Aug 21, 2011 5:46 pm    Post subject: 31

BraveHat wrote:
 extro wrote: 1) We should be careful to distinguish between the following: a - It is not possible that it's true in the physical reality of this universe. (example: only 2 spatial dimensions - we know we have 3) b - It is not possible that it's true in the physical reality of any possible universe. (example: space being both finite and infinite)

If any possible Universe needs certain properties to even be considered a Universe, ...

What does that mean? Do you have in mind some properties you think a universe needs to have to be considered a universe? I think for me it includes no logical contradictions, like space being both finite and infinite (when each word means "not the other").

 Quote: ... than it's possible to observe the defining behaviors of those properties and find they contradiction abstract.

Meaning not clear.

In any case, I hope you don't consider either finiteness or on infiniteness of space to be a defining property. Science is still undecided on which our universe has, but we do live in a universe.

BraveHat wrote:
And it's possible to repeat those observations as a means of verification to avoid the bgg1996 scenario:

 bgg1996 wrote: If I put together two seperate units, then observed that there were three of them, I would not rewrite arithmetic, I would blink a few times, rub my eyes, and observe again.

Perhaps, but if you kept repeating that action, putting together two separate units, and the same result of three kept being observed, not just by you, but by everyone else you show this to, you will have hit upon something that is a cause of scientific and mathematical concern, to say the least. Eventually, science will have to adjust it's laws or come up with a new system of laws to explain what you keep doing.

No mathematical concern. What are the objects? I can see putting 1 rabbit in a cage, then another 1, and after a time finding 3. That may need study. It doesn't mean 1+1=3

BraveHat wrote:
 extro wrote: 2) If "impossible in physical reality" means the latter of the two above, then it's no different than "impossible in the abstract".

False. The veracity of abstract conclusions is determined by consistency with abstract principle alone. The veracity of physical conclusions is determined by consistency with physical observation.

To be physically observed in some possible universe. If it's abstractly possible, then such a universe is possible where it might be observed.

(I'd also contend it's possible for a tree to fall in a forest, and to make a sound, when there is no one there to see it or hear it. A universe without observers might exist.)

BraveHat wrote:
 extro wrote: Abstractly, logically, mathematically possible means possible in physical reality.

False. Working with the square root of a negative number, for example, is mathematically possible, but not possible in physical reality. In order for us to work with it in physical reality, it has to translate to a physical quantity, but it has no quantity at all. It is still a mathematical object with properties of it's own, but those properties and object do not exist in physical reality.

Do I need to remind that when you, here, say "do not exist in physical reality", you must mean "cannot exist in any physical reality, no matter how radically different from the physical reality of this universe"?

1) Complex numbers have an imaginary part and a real part (and don't let the names fool you, they're historical throwbacks), and are often plotted in two dimensions on a plane. Just as we have spatial and time dimensions that are different in this universe, why is it not possible to have, in some universe, at least two distinct kinds of dimensions, where "real" and "imaginary" numbers measure distances in each of those dimensions, where motion in those dimensions is possible, and where distances between locations is measured by complex numbers that obey the mathematical operations of addition, multiplication, etc?

As for our universe,

2a) Here, in our physical reality, we find that electromagnetic fields have two components, an electrical field and a magnetic field, both existing in independent measurable quantities. Electromagnetic fields can be measured with complex numbers, and multiplication and addition used to calculate physical outcomes.

2b) http://en.wikipedia.org/wiki/Imaginary_time#In_cosmology
Thok*
Guest

 Posted: Sun Aug 21, 2011 6:17 pm    Post subject: 32 Complex numbers can also be realized as a subset of 2x2 matrices, where a+bi is represented by the matrix [a b] [-b a] One can check that addition and multiplication works out correctly. If you don't believe 2x2 matrices are real, I'd like to see you try to rotate.
raekuul
Lives under a bridge & tells stories.

Posted: Sun Aug 21, 2011 10:30 pm    Post subject: 33

 Quote: Infinity isn't an integer, and isn't part of what we normally count.
That was kind of the point I was trying to get at, I think. I'm not sure.
But I get the feeling that that's exactly what we've been trying to do in this thread. We're trying to put a finite label on an afinite concept.

If the universe is infinite, then it is not something we can clearly and completely observe. We can make models of it, but we can't see the whole thing at once and call the model accurate. Likewise, if the universe is not infinite, then there has to be a point where the universe ends and something else begins. Either way, our understanding of what the universe actually is is limited by our observations. The math supports certain conclusions, but it does not definitively prove those conclusions in one direction or another. And before you argue that point, just remember that math majors are taught when 2 + 2 really does equal 5.
Thok*
Guest

Posted: Mon Aug 22, 2011 12:05 am    Post subject: 34

 raekuul wrote: I'm not sure.

This sentence describes the speakers of roughly half of the posts in this thread. People aren't sure of what they are saying, and it's because they've tried to make mathematical statements without anything resembling precision.

If you aren't sure what a mathematical statement means, you are doing it wrong and should go back and define terms.

Vagueness is useful in other subjects, but not mathematics.

(And since I'm bored, here's how to count to infinity in finite time. Say one in 1/2 seconds, two in 1/4 seconds, three in 1/8 seconds, and so on. Then after 1/2+1/4+1/8+1/16+... = 1 second, you've said every positive integer, so the next thing you can do is say infinity [really Aleph 0 , since there are different sizes of infinity. Continuing from there runs immediately into trouble with the continuum hypothesis.

Obviously, in reality, one can't speed up their speaking speed at such a rate. But you need to do something like that to "count to infinity".)
Zag
Tired of his old title

Posted: Mon Aug 22, 2011 12:36 am    Post subject: 35

 raekuul wrote: Likewise, if the universe is not infinite, then there has to be a point where the universe ends and something else begins.

This is not true. I can't explain why, because I don't really get it myself, but you have to understand that the 3-dimensional space that we understand intuitively gets warped. Consider that a black hole has a finite circumference but an infinite radius.
Chuck
Daedalian Member

Posted: Mon Aug 22, 2011 12:40 am    Post subject: 36

 Thok* wrote: Obviously, in reality, one can't speed up their speaking speed at such a rate. But you need to do something like that to "count to infinity".)
You can choose some other representation for the numbers. You can walk toward a doorway of the room you're in and decide in advance that reaching the halfway point counts 1, three fourths of the way counts 2, seven eights of the way counts 3, etc. Reaching the doorway counts infinity.
extropalopakettle
No offense, but....

Posted: Mon Aug 22, 2011 1:08 am    Post subject: 37

 raekuul wrote: If the universe is infinite, then it is not something we can clearly and completely observe. We can make models of it, but we can't see the whole thing at once and call the model accurate.

This has nothing to do with the universe, finite or infinite. It's just human limitations in knowing. We have a theory that there's a world we live in that corresponds to what our senses observe.

As for what we can't observe, Occam's razor (law of parsimony, law of succinctness) applies. The simpler explanation for what's actually observed applies. If the universe is observed to be infinite (see http://map.gsfc.nasa.gov/universe/uni_shape.html ... "We now know that the universe is flat with only a 0.5% margin of error. This suggests that the Universe is infinite in extent; however, since the Universe has a finite age, we can only observe a finite volume of the Universe."), and it's also observed to look very much the same in every direction we look for as far as we can see, it's a strange and random conclusion to suggest we might just happen to be at the very center of a region beyond which it all starts to change.

 raekuul wrote: Likewise, if the universe is not infinite, then there has to be a point where the universe ends and something else begins.

False. The 2-dimensional surface of a sphere is a finite 2-dimensional space, but it has no edges or boundaries, taken by itself. Again, the 3-dimensional space of a finite universe may be shaped like the surface of a 4-dimensional hypersphere.

 raekuul wrote: Either way, our understanding of what the universe actually is is limited by our observations. The math supports certain conclusions, but it does not definitively prove those conclusions in one direction or another.

There are a LOT of observations.

And again, everything you think you know is theory. "I think, therefore I am" is about as much as you can be certain of.

 raekuul wrote: And before you argue that point, just remember that math majors are taught when 2 + 2 really does equal 5.

I've no idea what that refers to. Please explain.
Thok*
Guest

Posted: Mon Aug 22, 2011 1:24 am    Post subject: 38

extropalopakettle wrote:
 raekuul wrote: And before you argue that point, just remember that math majors are taught when 2 + 2 really does equal 5.

I've no idea what that refers to. Please explain.

I assume he's talking about finite groups/finite fields. But 2+2 = 5 is equivalent to 0 = 1, which leads to mostly boring groups/fields. (R/Z is one of the few interesting exceptions).

2+2 = 6 is much more interesting. (Except there isn't really a 6 there, but an equivalence class of multiples of 2.)
BraveHat
Last of the Daedalians

Posted: Mon Aug 22, 2011 2:28 am    Post subject: 39

 extro wrote: First, I'm not clear on what you mean by an abstract law or how it relates to the discussion.

Is this discussion, by abstract law, I meant any inference that must be made by the positing of an abstract concept, such as, in this case, a certain type of infinite set. In my failed attempt at disproving that that type of infinite set applies to the Universe, I used the "abstract law" that "an element of an infinite set takes up no percentage of the cardinality". The Earth takes up no percentage of space within an infinite Universe. This is an abstract inference, and that was what I mean by "abstract law", but I suppose inference is a better word.

 extro wrote: In the abstract, we can have a 2 dimensional surface that is an infinite flat plane. We can also have a 2 dimensional surface that is like the surface of a sphere, finite, but without borders or any point on it having properties different from another. In the abstract, we have 3 dimensional analogs of each of those 2 dimensional constructs. The are well defined by axioms and rules of inference. Are you saying that if we observe one of these to be the case for our universe, then the other is a physical impossibility in any universe?

No, here's what I am trying to say.

It's possible that there are observable physical occurrences that are prohibited by a certain type of infinity (infinite space, infinite time). I thought that one of those observable physical occurences was "the Earth taking up some percentage of the Universe", but I was wrong. The idea that "the Earth takes up some percentage of the Universe" is prohibited by the concept of infinite space, but it is not observable. "The Earth takes some portion of the Universe" or "The Earth takes some chunk of the Universe" is observable, but is not prohibited by the concept of infinite space. Nevertheless, that does not make it impossible that some observable physical occurrence is prohibited by the concepts of infinite time or infinite space. Until that is proven impossible, it remains possible.

Now space exists within the third dimension, just as in Euclidean geometry, a plane exists within the 2nd dimension, and a line exists within the first dimension. So either there is a higher dimension that contains space or there isn't.

If there is no higher dimension which contains space, then the conclusion that our Universe is of finite space would imply that either every Universe is of finite space or that our Universe is a sub-Universe of an infinite space Universe. But if our Universe is a sub-Universe of an infinite space Universe, then that means the infinite space Universe is actually our Universe, and our assumption is contradicted. Therefore, if there isn't a higher dimension which contains space, then the conclusion that our Universe is of finite space implies that every Universe is of finite space.

Now let's suppose that there is a higher dimension which contains space. Then the objects(s) in that dimension which contain our Universe space within it also contain within it every object within our Universe. A 1-inch line segment that is contained within a line in Euclidean geometry, for example, would also be contained within any of the planes that contain that line. Likewise, if I observe hair on my knuckles, I need not only conclude that there is hair within this Universe, but also that there is hair within whatever object in a higher dimension contains this Universe, and within whatever object(s) in an even higher dimension contains object(s) in the next lowest dimension which contain our Universe, and so on and so on. Instantly, by observing my knuckle hair, I can conclude that hair is a part (if only a small part) of all those objects in all those dimensions.

Now, we already assume it's possible that some observable physical occurrence could be prohibited by the concept of infinite space. Still assuming that's possible, then, it's also possible to conclude soundly that any object which contains that observable physical occurrence is prohibited by the concept of infinite space. This means all objects in higher dimensions which contain the observable physical occurrence would be finite in space. The Universe would be finite in space, and any object containing the Universe would be finite in space because it also contains the observable physical occurrence, and any object containing that object would be finite in space, because it contains within it the observable physical occurrence. (I include as an "object" the dimension itself, if no other object directly contains the object in question). If the occurrence of my knuckle hair were somehow prohibited by the concept of infinite space then all objects in this dimension and in all higher dimensions which contained my knuckle hair would be finite in space. Now if our Universe is an element of a set of many Universes (a multiverse) then whatever object in whatever dimension contains all these Universes, also contains our Universe. And because it's possible to soundly conclude that this object is finite, it is possible to soundly conclude this finite object cannot contain Universes of infinite space, just as it's possible to sound conclude a finite square cannot contain an infinite line or a finite cube cannot contain an infinite plane. Therefore it's possible that concluding soundly that our Universe is of finite space can lead to concluding soundly that every Universe is of finite space.

I can sum up my whole feeling thusly: If my original argument had worked, i.e. if "the Earth is some percentage of the Universe" was a physically observable occurence, then I could have used the same exact argument for the object containing our "Multiverse": "the Earth is some percentage of the object containing the Multiverse". And the "proof" of the object containing the Multiverse being finite in space would be the same. I'd assume the object containing the Mulitverse is infinite in space, and point out the "contradiction" that the Earth can't be some percentage of the object containing the Multiverse, because percentage implies finitude, and it can't be no percentage of the object containing the Multiverse because it's percentage of the object containing the Multiverse is physically observed. And the object containing the Multiverse having been proved finite, would then prove that every Universe is also finite.

I need to sleep. I will attempt to explain my other answers later...
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax
extropalopakettle
No offense, but....

 Posted: Mon Aug 22, 2011 3:34 am    Post subject: 40 No idea what you're talking about. Our 3-dimensional universe being embedded in a higher dimensional one? And that implying that if our universe is finite, only finite universes are possible?
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