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bgg1996
BeeGees are awesome!
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 Posted: Sat Sep 03, 2011 3:57 pm Post subject: 121 |
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Doesn't light dissipate over distance?
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raekuul
Lives under a bridge & tells stories.
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| Posted: Sat Sep 03, 2011 6:08 pm Post subject: 122 |
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Yes, it does actually - unless it's laser-focused (and even then, enough of it spreads out.)
I just thought of this, but Olber's Paradox is a good argument in favor of the existence of a beginning to the universe - we (seem to) know the speed that light travels through a vacuum, and we know that it is a constant speed. Since there are dark patches in the night sky, combining that with the assumption that there is always a source of light in each direction tells us that either something is blocking the light or the light has not yet reached us - if we accept the latter to be more true than the former, then logically only a finite amount of time has passed since the light has started travelling - even if the light was from something at the beginning of the universe.
If the universe has a beginning, can it be infinite in extent? Alternately, if the universe is expanding at a speed greater than that which we can detect, wouldn't that be functionally the same as being infinite as far as we are concerned? |
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bgg1996
BeeGees are awesome!
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| Posted: Sun Sep 04, 2011 2:33 am Post subject: 123 |
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What if there was a finite, but very large amount of (not empty necessarily) space, let's say in a cube, with a single star in the center. The cube then repeats in every direction. Every cube is identical.
Skip forward infinte years. Would all space be lit up?
If you said no, then you have disproven the basic idea of that paradox.
If you said yes, recall that it is basically a cube that has the top cycle through to the bottom, etc. Now recall that this is any finite distance, no matter how large, even if filled. That means that one star could, over time, light up an infinite filled area.
Thus is my paradox.
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raekuul
Lives under a bridge & tells stories.
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| Posted: Sun Sep 04, 2011 2:47 am Post subject: 124 |
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Whether the entire thing is lit up or not depends on whether or not you have (A) actually reached an infinite amount of time passed, and (B) have a rate of expansion less than that of the speed of light. Assuming that the rate of expansion is less than that of the speed of light, as the time approaches infinity, the percentage of space that is lit approaches 100%. This is what applies to your cubes, incidentally.
The model appears to predict that the percentage approaches 0% as the time approaches infinity if the rate of expansion is greater than the speed of light, but that induces a whole new set of paradoxes. This is where the universe gets hung up - we don't know whether the rate of expansion exceeds the speed of light or not, and you'll be hard-pressed to prove it one way or another.
Last edited by raekuul on Sun Sep 04, 2011 2:56 am; edited 1 time in total |
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bgg1996
BeeGees are awesome!
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| Posted: Sun Sep 04, 2011 2:49 am Post subject: 125 |
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Expansion?
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raekuul
Lives under a bridge & tells stories.
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| Posted: Sun Sep 04, 2011 3:00 am Post subject: 126 |
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| One of the quirks about the big-bang theory is that the universe had to have become significantly larger than one light-second across within the first second of its existence in order to have not become a black hole within that selfsame second. If we start from the assumption "The Big Bang Happened", then the universe has to have been expanding at some speed ever since. |
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Thok*
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| Posted: Sun Sep 04, 2011 3:44 am Post subject: 127 |
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| raekuul wrote: |
| One of the quirks about the big-bang theory |
General Relativity doesn't work at the scale of quantum mechanics: trying to apply it to the first light second of the universe mostly gives you nonsense. Basically you get all sorts of infinities simultaneously blowing up. That's the "quirk" (and really, it's a feature, which implies that we don't understand gravity as well as we would like to; that's the major obstacle to a Grand Unified Theory.)
@extro: yes, I would consider it reasonable to say that 10% of the irrationals from [0,1] lie in [0,0.1] under the standard uniform probability distribution. And using percentage that way is fairly standard.
@Bravehat: yes, I'm saying that I believe that our understanding of the universe fundamentally comes from measurements. Or at the least, our understanding of the universe is only useful if it matches those measurements and lets us predict them in the future (within a reasonable but describable margin of error.) |
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raekuul
Lives under a bridge & tells stories.
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| Posted: Sun Sep 04, 2011 4:11 am Post subject: 128 |
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If General Relativity doesn't work at the scale of Quantum Mechanics, then our understanding of either one or the other (or both) is flawed.
And is it really better to say "Gravity didn't being until after X seconds because of Y and Z", as I've seen many timelines of the big bang say? |
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Thok*
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| Posted: Sun Sep 04, 2011 4:33 am Post subject: 129 |
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| raekuul wrote: |
| If General Relativity doesn't work at the scale of Quantum Mechanics, then our understanding of either one or the other (or both) is flawed. |
Yes. This is a known problem (and it's generally assumed that gravity is the problem given that the other forces interact nicely.)
Here's a wiki article of Quantum gravity, which describes various attempts to patch this hole (for example, this is exactly why string theory was popular.) |
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raekuul
Lives under a bridge & tells stories.
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| Posted: Sun Sep 04, 2011 4:39 am Post subject: 130 |
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I see... It still gives us the oddity of having to say "Because of Y and Z, Gravity as we understand it didn't come into being until X seconds after the Big Bang," which comes across as really strange. Plus the whole "should have collapsed in on itself" thing.
Perhaps our understanding of Gravity is an extension of the Heap Paradox - there needs to be at least so much mass before Gravity can take hold, but we haven't been able to pinpoint the exact number yet.
EDIT: It appears to me to be the case that Gravity is an extension of the Weak and Strong Whatever Forces as opposed to its own force. It would fit in nicely with my preconceived notion that there needs to be so much mass before Regular Gravity takes precedence over Quantum Forces.
Last edited by raekuul on Sun Sep 04, 2011 4:44 am; edited 1 time in total |
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Thok*
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| Posted: Sun Sep 04, 2011 4:42 am Post subject: 131 |
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| raekuul wrote: |
| And is it really better to say "Gravity didn't being until after X seconds because of Y and Z", as I've seen many timelines of the big bang say? |
I would argue that those timelines are simplifications (and bad simplifications at that.) General relatively breaks down whenever there's a singularity: far enough inside the event horizon of certain types of black holes (non-rotating ones where rotation is measure in an non-accelerating frame of reference) is just as bad as the early stages of the Big Bang.
Outside of black holes+the very beginning of the Big Bang, GR+Quantum stuff works fine, just as Newtonian mechanics works well as long as you aren't near the speed of light and aren't looking at quantum effects. There's more to be done, but what has been done is a good approximation. |
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BraveHat
Last of the Daedalians
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| Posted: Sun Sep 04, 2011 4:43 am Post subject: 132 |
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| Thok wrote: |
| @Bravehat: yes, I'm saying that I believe that our understanding of the universe fundamentally comes from measurements. Or at the least, our understanding of the universe is only useful if it matches those measurements and lets us predict them in the future (within a reasonable but describable margin of error.) |
I see, so when you say, in humor:
| Thok wrote: |
| Reality is surprisingly norealistic |
You mean, in all seriousness, that "Reality" is how the Universe actually works, and "realistic" is how we understand it to work. So we have this dichotomy, between our understanding of the Universe and the actual nature of the Universe. And the primary thing that suggests we have the ability to match the two is that we can use our understanding to predict certain things that the Universe will do.
Now I've been thinking about this. The models we use to understand the Universe, at least the ones we use that involve more than just sensory observation, and which prove to have predictive powers, are based primarily on abstract (usually mathematical) concepts. However if an abstract concept doesn't prove useful in predicting behavior in the Universe, obviously we wouldn't toss it out if it's still consistent with other abstract concepts that do have predictive powers.
The main thing I'm wondering is, is infinity one of those abstract concepts that doesn't really prove useful in predicting how the Universe behaves? Have we ever successfully predicted behavior in the Universe by relying on the concept of infinity? The reason the question seems important to me is because it occurred to me that the essential meaning of infinity may not be "no finitudes" but "no abstract finitudes". In other words, there are practical finitudes in the human endeavor to work with concepts. Even the most expansive human mind will, in the end, have conceived of only a finite number of events or objects. But since there is nothing to suggest what that finite number would have to be, we came up with the concept of infinity. I'm probably not explaining this right. :-/
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Thok*
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| Posted: Sun Sep 04, 2011 5:00 am Post subject: 133 |
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@Bravehat: You've understood my "Reality is surprisingly unrealistic" as I intended it.
| BraveHat wrote: |
| Now I've been thinking about this. The models we use to understand the Universe, at least the ones we use that involve more than just sensory observation, and which prove to have predictive powers, are based primarily on abstract (usually mathematical) concepts. However if an abstract concept doesn't prove useful in predict behavior in the Universe, obviously we wouldn't toss it out if it's still consistent with other abstract concepts that do have predict powers. |
I agree with this. For example, it looks like string theory is an interesting abstract mathematical concept, but doesn't describe our universe correctly. It's interesting math, but not obviously interesting physics. (The math obtained from string theory might be applied in a different way to get interesting physics, but not as commonly stated.)
| Quote: |
| The main thing I'm wondering is, is infinity one of those abstract concepts that doesn't really prove useful in predicting how the Universe behaves? Have we ever successfully predicted behavior in the Universe by relying on the concept of infinity? |
Depends on what you mean by infinity. I'm certain that I could find an example of a convergent infinite series that matched physical reality, or a use of an infinite power series that matched reality.
I'm equally certain that the real universe would be unaffected by any decision we make about the [url="http://en.wikipedia.org/wiki/Continuum_hypothesis"]Continuum hypothesis[/url] (this is a problem of whether there is an infinite set which is bigger than the integers and smaller than the real in the sense of cardinality. Surprisingly, the answer in the abstract sense turns out to be "what do you want the answer to be?" and then the discussion moves to trying to pick good reasons for certain answers.) |
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BraveHat
Last of the Daedalians
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| Posted: Sun Sep 04, 2011 5:14 am Post subject: 134 |
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| Thok wrote: |
| I'm certain that I could find an example of a convergent infinite series that matched physical reality, or a use of an infinite power series that matched reality. |
That would be interesting. I'd like to see if it really is the concept of infinity that is essential to the match or if finite versions of the series would have the same quality of matching reality.
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Zag
Unintentionally offensive old coot
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| Posted: Sun Sep 04, 2011 5:37 am Post subject: 135 |
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| Thok wrote: |
| I'm certain that I could find an example of a convergent infinite series that matched physical reality, or a use of an infinite power series that matched reality. |
This is easy: I'll be the tortoise and you be Achilles. We'll run a race, and Xeno will be our timekeeper.
I run at 1 mile per hour, and you run at 10 miles per hour. Knowing your greater speed, you give me a 0.9 mile head start.
In 54 minutes, you will have run to my starting point, or 0.9 miles. However, in that time, I have run 10% of the distance you ran, or 0.09 miles, so I'm at 0.99. In a little while, you will have caught up to that point, the 0.99 mile point. But in that additional time, I've made a little progress ahead of you. I'm at 0.999 mile point.
Every time you catch up to the point where I was, I have moved ahead by 10% of the distance you traveled to get there. You reach 0.999, I'll be at 0.9999. When you reach that, I'll be at 0.99999, etc. You could say that your position until you catch me is always:
Sum
(n=1 -> infinity)
(9 / 10 n )
or
0.99999... miles
You won't catch me until this infinite series is complete. Fortunately, each additional step takes less time (1/10 the time of the previous step), and you do actually catch me at exactly the 1 mile marker. You actually complete the infinite series in a finite amount of time, and we learn that the sum above actually equals 1. Note, that this is NOT that it gets very very close but never quite reaches it, because you do catch me, after all. They are actually equal, because it IS possible to complete an infinite series in finite time as long as the time it takes to add each term is diminishing geometrically. |
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BraveHat
Last of the Daedalians
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| Posted: Sun Sep 04, 2011 5:53 am Post subject: 136 |
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| Zag wrote: |
| This is easy: I'll be the tortoise and you be Achilles. We'll run a race, and Xeno will be our timekeeper. |
But that example evades what I'm looking for. You don't need to rely on the concept of infinity to predict that Achilles will catch up to you at the 1 mile mark. Infinity is not an essential ingredient in that prediction. Alls you need is the s=d/t formula to predict that.
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extropalopakettle
No offense, but....
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| Posted: Sun Sep 04, 2011 11:57 am Post subject: 137 |
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| Thok* wrote: |
| @extro: yes, I would consider it reasonable to say that 10% of the irrationals from [0,1] lie in [0,0.1] under the standard uniform probability distribution. |
If
1) 10% of the irrationals from [0,1] lie in [0,0.1]
and
2) 10% of the reals from [0,1] lie in [0,0.1]
then wouldn't it follow that
3) 10% of the rationals (reals minus irrationals) from [0,1] lie in [0,0.1] |
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Thok
Oh, foe, the cursed teeth!
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| Posted: Sun Sep 04, 2011 12:06 pm Post subject: 138 |
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| extropalopakettle wrote: |
| Thok* wrote: |
| @extro: yes, I would consider it reasonable to say that 10% of the irrationals from [0,1] lie in [0,0.1] under the standard uniform probability distribution. |
If
1) 10% of the irrationals from [0,1] lie in [0,0.1]
and
2) 10% of the reals from [0,1] lie in [0,0.1]
then wouldn't it follow that
3) 10% of the rationals (reals minus irrationals) from [0,1] lie in [0,0.1] |
No. Why should it? Without more precise conditions, a similar argument would imply that any countable set contained in [0,1] has 10% of it's elements in [0, .1]. I'm sure you can come up with "counterexamples" to that statement. (Counterexample in quotes, because we generally don't define a uniform probability on countable sets.) |
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extropalopakettle
No offense, but....
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| Posted: Sun Sep 04, 2011 12:58 pm Post subject: 139 |
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| Thok wrote: |
| extropalopakettle wrote: |
| Thok* wrote: |
| @extro: yes, I would consider it reasonable to say that 10% of the irrationals from [0,1] lie in [0,0.1] under the standard uniform probability distribution. |
If
1) 10% of the irrationals from [0,1] lie in [0,0.1]
and
2) 10% of the reals from [0,1] lie in [0,0.1]
then wouldn't it follow that
3) 10% of the rationals (reals minus irrationals) from [0,1] lie in [0,0.1] |
No. Why should it? |
Ordinary arithmetic with percentages. Hence my question of whether it is a standard practice to define 'percentage' this way. That, and that I can find no online treatment of it defined that way.
How about this:
| Code: |
1) Let X = 0
Repeat N times:
2) Select R by throwing dart at real number line [0,1]
3) If R is irrational, goto 2, else:
4) If R is in [0,0.1] then increment X |
Wouldn't X approach 10% of N for larger and larger N? ... Indicating 10% of the rationals from [0,1] lie in [0,0.1]
If you're going to argue the loop in steps 2 & 3 will never terminate, I think the argument can be adjusted using parallelism. |
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Thok
Oh, foe, the cursed teeth!
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| Posted: Sun Sep 04, 2011 2:41 pm Post subject: 140 |
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| extropalopakettle wrote: |
| Ordinary arithmetic with percentages. |
You need to actually write down the relevant equation. Because the probability of picking a rational number in [0,1] is 0, I suspect your ordinary arithmetic will give you that the probability of picking a rational number is 0/0, aka undefined.
| extropalopakettle wrote: |
| If you're going to argue the loop in steps 2 & 3 will never terminate, I think the argument can be adjusted using parallelism. |
Even if you use a countable number of parallels, the process will never end with probability 1. Using an uncountable number of parallels runs into the issue of not modeling your problem. |
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extropalopakettle
No offense, but....
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| Posted: Sun Sep 04, 2011 4:36 pm Post subject: 141 |
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| So if I pick out a random real in [0,1], and it turns out to be rational (probability 0, but that's the case for any particular real too, and we will pick a real), what is the probability it is in [0,0.1]? |
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Thok*
Guest
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| Posted: Sun Sep 04, 2011 4:53 pm Post subject: 142 |
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| extropalopakettle wrote: |
| So if I pick out a random real in [0,1], and it turns out to be rational (probability 0, but that's the case for any particular real too, and we will pick a real), what is the probability it is in [0,0.1]? |
I believe it's not defined. I could be wrong. (There's a 0/0 floating around in your question.)
I also suspect it may be influenced by how you actually pick a random real in [0,1]. |
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BraveHat
Last of the Daedalians
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| Posted: Mon Sep 05, 2011 8:08 pm Post subject: 143 |
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| zag wrote: |
| 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? |
This thread branched off from a discussion of Dr. William Lane Craig's take on the Kalam Cosmological argument for the Existence of God. My understanding of Dr. Craig's definition of the Universe is "the sum total of all matter, space, and time".
So, I suppose by this definition, there can only be one Universe, since the term itself represents the sum total of a certain kind of thing. There could not be another sum total of the same kinds of things (matter, space, and time). There could be sub totals, but those would just be elements of the set of all, which is what the term Universe is being defined as.
So, in light of this, then, the question arises is it possible to show that this sum total is not infinite in size, i.e. that there cannot be an infinite amount of any of the three things: space, matter and time? Intuitively, at least to me anyway, it seems that it is. At the very least, it seems that one of the three ingredients can be shown to not be infinite in size.
By contrast, what seems impossible to prove is that infinity is anything but an abstract concept and actually operates in the observable world. For example, π, being an irrational number, has a decimal value reaching into infinity, but we never actually use π to predict observable results. We use instead a rational number that approximates π to some finite number of decimal places. Same with e.
So, because intuitively it seems impossible to show the operation of infinity in the observable universe, but possible to show the impossibility of it operating in the observable universe, my original question remains, has anyone ever come up with such a proof?
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raekuul
Lives under a bridge & tells stories.
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| Posted: Mon Sep 05, 2011 10:50 pm Post subject: 144 |
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The limit of y = x, as x approaches infinity, is infinity.
Why do I bring this up? You can't really *prove* how things act when x equals infinity because x can't actually equal infinity (due to the logical improbability of representing an infinite value with a finite placeholder). We can model how things should act as x becomes arbitrarily close to infinity, but that's as close to a proof that you can get. |
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BraveHat
Last of the Daedalians
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| Posted: Mon Sep 05, 2011 11:55 pm Post subject: 145 |
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Right, and one wonders whether "close to infinity" can even mean anything. I mean, you have infinity, and then you have some vast finite number. But the two can't really be considered close to each other at all.
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Thok*
Guest
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| Posted: Tue Sep 06, 2011 1:19 am Post subject: 146 |
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| BraveHat wrote: |
| Right, and one wonders whether "close to infinity" can even mean anything. I mean, you have infinity, and then you have some vast finite number. But the two can't really be considered close to each other at all. |
And you're confusing two similar but distinct concepts. The idea is really "does the function approach something as the input values get larger?", where approach means that the distance from the proposed target value gets smaller over time. That doesn't require mentioning sets with infinite cardinality. |
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BraveHat
Last of the Daedalians
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| Posted: Tue Sep 06, 2011 5:04 am Post subject: 147 |
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| Thok wrote: |
| I wrote: |
Right, and one wonders whether "close to infinity" can even mean anything. I mean, you have infinity, and then you have some vast finite number. But the two can't really be considered close to each other at all. |
And you're confusing two similar but distinct concepts. The idea is really "does the function approach something as the input values get larger?", where approach means that the distance from the proposed target value gets smaller over time. That doesn't require mentioning sets with infinite cardinality. |
Are you talking to me or raekuul? For the record I know what "approaches infinity" means, I was just pointing out that the phrase "close to infinity" really has no meaning.
Thok, you say that it's "weird" for rationals to take up 0 percent of [0,1], but it seems to me it's not just weird but contradictory. It contradicts the plain fact of, like you said, our constant usage of rationals in the interval. Does not that contradiction alone compel our reasonability to conclude it is not the case that rationals take up 0 percent of the interval? Why should we trust our conclusion that rationals take up 0 percent when it's obvious they exist within that interval? This isn't merely weird or counter-intuitive, it's contradictory. I would think at the very least, it would be indeterminate what the percentage is.
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Thok*
Guest
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| Posted: Tue Sep 06, 2011 11:17 am Post subject: 148 |
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| BraveHat wrote: |
| Thok, you say that it's "weird" for rationals to take up 0 percent of [0,1], but it seems to me it's not just weird but contradictory. |
I believe I've been careful to say "under a uniform probability distribution on [0,1], the rationals make up 0 percent of the [0,1]".
The bolded part is the important part: we don't actually use a uniform probability distribution in speech, but a distribution that's weighted inversely against complexity. Where by complexity I mean how difficult it actually is to express numbers in terms of standard operations. 0 and 1 are the most commonly used numbers in the interval [0,1], and that's partially because they are the easiest to discuss/least complex. Something like e/pi is used much less frequently, and most numbers in [0,1] are too complex to ever be spoken. |
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Nsof
Daedalian Member
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| Posted: Tue Sep 06, 2011 8:54 pm Post subject: 149 |
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| BraveHat wrote: |
| is infinity one of those abstract concepts that doesn't really prove useful in predicting how the Universe behaves? |
I think Infinity was/is used to find other useful concepts. it is also useful in telling us our model of the universe is wrong.
I don’t think we can correlate infinity with physical quantities like we do with the abstract concept called ‘1’. As the same time I don’t think we can correlate irrational numbers with physical quantities.
Let us paint all rational numbers in [0,1] in black and all irrational numbers in [0,1] in white. Looking at the interval [0,1], what color would it have? Potential answers would be white, grey or black.
The answer is white and for me this is a way to understand what “rationales take up 0 percent of [0,1]” means.
There is no contradiction if you remain in the realm of math. You cannot apply it correctly to the universe so again no contradiction.
Where did this discussion branch from the Cosmological argument for the Existence of God? How is infinity related?
Interesting discussion btw
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raekuul
Lives under a bridge & tells stories.
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| Posted: Wed Sep 07, 2011 2:29 am Post subject: 150 |
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| The discussion on God was tangental; we're actually back on the original topic. As for how infinity is related, the hypothesis is that God is infinite and can act on an infinite scale without paradox or singularity. |
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BraveHat
Last of the Daedalians
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| Posted: Wed Sep 07, 2011 2:46 am Post subject: 151 |
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| Nsof wrote: |
Where did this discussion branch from the Cosmological argument for the Existence of God? How is infinity related?
Interesting discussion btw |
I had come across William Lane Craig's take on the Kalam Cosmological Argument, wondered how an atheist might refute it and posted my understanding of it in this thread. Over the course of discussing it, it became clear that (at least) three of the main premises were highly debatable 1. That the Universe must have had a beginning. 2. That anything which begins must have a cause. and 3. Any cause that is both timeless and spaceless must be an unembodied mind. Craig's support of 1. is the belief that an actual infinite number of things cannot exist, and therefore, an infinite number of past events cannot exist, and thus the Universe must have began at some point. (I attempted to exrpress it in valid syllogistic form in the actual thread). The discussion started turning to whether it was possible for the past to be infinite, and then I started thinking about infinite space as well. Apparently I'm somewhat of an oddball in that's easier for me to think there's a finite Universe than an infinite Universe, when for most people it's the opposite. So I tried to discuss this in that thread, but I suppose a good rule of thumb is that if your tangental discussion lasts more than two pages, start a new thread. So I started a thread about Infinity in OT, but I changed the discussion slightly so that others could contribute, rather than starting a thread with a direct continuation of a previous debate. And there you have it.
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BraveHat
Last of the Daedalians
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| Posted: Wed Sep 07, 2011 4:04 am Post subject: 152 |
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| Thok wrote: |
I believe I've been careful to say "under a uniform probability distribution on [0,1], the rationals make up 0 percent of the [0,1]".
The bolded part is the important part: we don't actually use a uniform probability distribution in speech, but a distribution that's weighted inversely against complexity. Where by complexity I mean how difficult it actually is to express numbers in terms of standard operations. 0 and 1 are the most commonly used numbers in the interval [0,1], and that's partially because they are the easiest to discuss/least complex. Something like e/pi is used much less frequently, and most numbers in [0,1] are too complex to ever be spoken. |
But is this really relevant? The fact that at least one rational number exists on [0,1] means that it has some chance of being picked, whether by a uniform distribution agent or a biased one, like a human. Obviously, yes, any particular rational number has less of a chance to be picked by a uniform distribution agent than by one biased towards simplicity, but the fact that it can be picked contradicts the conclusion that is has a 0 chance of being picked. The two cannot both be true at the same time. Regardless of the nature of the distribution. There seems to be only two ways to reconcile this contradiction: 1) To call indeterminate the chance of any particular element from an infinite set being picked under uniform distribution. 2)To say that there is no such thing as uniform distribution of probability on an infinite set. I tend to conclude the latter.
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extropalopakettle
No offense, but....
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| Posted: Wed Sep 07, 2011 9:24 am Post subject: 153 |
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| BraveHat wrote: |
| ... the fact that it can be picked contradicts the conclusion that is has a 0 chance of being picked. |
Not so. See: http://en.wikipedia.org/wiki/Almost_surely
| BraveHat wrote: |
| There seems to be only two ways to reconcile this contradiction: 1) ... 2)To say that there is no such thing as uniform distribution of probability on an infinite set. I tend to conclude the latter. |
The uniform probability is not the problem. You still get zero probabilities with most natural distributions. See article above - it talks about throwing a dart at a square. Even with a non-uniform but continuous probability distribution, any (every) particular point on the square has a zero probability of being hit. |
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Thok*
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| Posted: Wed Sep 07, 2011 11:17 am Post subject: 154 |
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| BraveHat wrote: |
| 1) To call indeterminate the chance of any particular element from an infinite set being picked under uniform distribution. 2)To say that there is no such thing as uniform distribution of probability on an infinite set. I tend to conclude the latter. |
You've missed 3: reject the idea that we can directly add up an uncountable number of numbers in a reasonable way. Which makes sense, since the most we ever add together is a countable number (in infinite series).
And once you reject that, there's no reason to define probability on an uncountable set point by point: you can't add up the numbers anyways to see that the entire space has probability 1. You need a generalization to do anything, and calculus is the obvious solution. |
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Thok*
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| Posted: Wed Sep 07, 2011 11:19 am Post subject: 155 |
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| extropalopakettle wrote: |
| Even with a non-uniform but continuous probability distribution, any (every) particular point on the square has a zero probability of being hit. |
For example the normal curve has this property, and it's merely the most used probability distribution in statistics. |
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extropalopakettle
No offense, but....
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| Posted: Wed Sep 07, 2011 12:40 pm Post subject: 156 |
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| Thok* wrote: |
| BraveHat wrote: |
| 1) To call indeterminate the chance of any particular element from an infinite set being picked under uniform distribution. 2)To say that there is no such thing as uniform distribution of probability on an infinite set. I tend to conclude the latter. |
You've missed 3: reject the idea that we can directly add up an uncountable number of numbers in a reasonable way. Which makes sense, since the most we ever add together is a countable number (in infinite series).
And once you reject that, there's no reason to define probability on an uncountable set point by point: you can't add up the numbers anyways to see that the entire space has probability 1. You need a generalization to do anything, and calculus is the obvious solution. |
Is countable versus uncountable really what matters here?
Are you saying there is no uniform probability distribution over the rationals in [0,1]? Nevermind, there can't be ... they can be mapped one-to-one with the integers. This is ridiculously counter-intuitive. |
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BraveHat
Last of the Daedalians
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| Posted: Wed Sep 07, 2011 1:35 pm Post subject: 157 |
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Wow, that is a new concept for me. Utterly fascinating. But I'm very curious as to why mathematicians don't give some different or added symbol to zero when meaning "almost never". Otherwise, when talking about [0,1], "probability 0" equally applies to 1/2 being hit as 5 being hit, but clearly 1/2 being hit is "almost never" whereas 5 being hit would be "absolutely never". I'm very surprised that this difference isn't distinguished symbolically....or is it?
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extropalopakettle
No offense, but....
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| Posted: Wed Sep 07, 2011 1:39 pm Post subject: 158 |
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... though the notion of a randomly selected real (in [0,1], under any continuous distribution) still bugs me. Here's why:
1) I associate probability with estimates of likelihood of real world events occurring. It's the most real-world applied/motivated math, beyond simple counting and arithmetic, that there is. In the real world we measure things to compute probabilities, and the measurements, if not rational, are at least countable.
2) The "throw a dart at the real number line" technique, as I said earlier, doesn't work. Darts and the real number line exist in different universes. One can't hit the other. |
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BraveHat
Last of the Daedalians
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| Posted: Wed Sep 07, 2011 1:45 pm Post subject: 159 |
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| extro wrote: |
| The "throw a dart at the real number line" technique, as I said earlier, doesn't work. Darts and the real number line exist in different universes. One can't hit the other. |
The unstated implication is that the dart being thrown is an abstract dart, not a real one. Both the abstract dart and the real number line co-exist in the abstract universe.
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BraveHat
Last of the Daedalians
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| Posted: Wed Sep 07, 2011 3:50 pm Post subject: 160 |
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| Thok wrote: |
| You've missed 3: reject the idea that we can directly add up an uncountable number of numbers in a reasonable way. Which makes sense, since the most we ever add together is a countable number (in infinite series). |
That might work for my specific example, but I could easily make the same case for a countable infinite set. To wit: if an abstract dart is thrown at the natural number line with uniform probability, then the probability of 5 being hit being exactly 0 contradicts that fact that 5 has a chance to be hit. As extro points, countable versus uncountable doesn't really matter to the particular type of contradiction.
No, it turns out, according to extro's link, that the real third alternative is that "probability 0" has two different meanings :"almost never" and "absolutely never". The seeming contradiction only points to the failure of mathematicians to notate the difference.
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