View Full Version : Writing a very big number

jogleby

2010-May-14, 07:41 PM

I was recently having a conversation with a friend about the Multiverse episode of Radio Lab. In that episode, Brian Greene, a physics and mathematics professor at Columbia University, stated that in an infinite universe, everything that is possible has happened an infinite number of times. That means that there is an infinite number of planets exactly like ours that are inhabited by people that look exactly like us some where out there. The average distance that you would have to travel to get to those planets, according to Greene, was 10^10^23 light years. That number is so big that I can't even comprehend it. I've read that the total number of particles in the visible universe is 10^89, which is dumbfounding huge, but if I took the time I could write it out. It is 1 followed by 89 zeros.

My question is how long would it take to write out 10^10^23 in its full form? I tried to do the math, but very quickly got confused. I was figuring an average of writing 2 digits a second. Thanks for your help.

Geo Kaplan

2010-May-14, 07:45 PM

My question is how long would it take to write out 10^10^23 in its full form? I tried to do the math, but very quickly got confused. I was figuring an average of writing 2 digits a second. Thanks for your help.

Short answer: A really, really long time.

Longer answer: The exponent of the first 10 tells you how many zeros to write. In this case, you'll be writing 10^23 zeros. At two digits each second, you're looking at 5x10^22 seconds. Given that a human lifetime is of the order of 3x10^9 seconds, you'll need to involve many generations of offspring to complete the task.

Argos

2010-May-14, 07:56 PM

Bear in mind that the universe has been around for only 432,329,886,000,000,000 seconds.

tdvance

2010-May-14, 08:28 PM

and that is the "expected" distance---the uncertainty is enough that it would take less effort to build a copy of Earth and its inhabitants, down to the molecule, than to find the nearest (or even, any) copy. (it's like what is the fastest way to get Monkeys to write Shakespeare? answer---wait a few million years for them to evolve into humans and look up the one named William Shakespeare--it's a joke and not strictly true, but illustrates the point).

Geo Kaplan

2010-May-15, 01:58 AM

Bear in mind that the universe has been around for only 432,329,886,000,000,000 seconds.

And that our own sun will become a red giant in approximately half that time, with the earth becoming uninhabitable sometime before then. The OP will have to outsource the writing of the digits.

Murphy

2010-May-15, 03:56 AM

If I'm understanding this right then 10^10^23, is the same as 1.e+230 (1x10^230), or at least that's what I get when I do it in Windows calculator (10^10 = 10,000,000,000 then 10,000,000,000^23 = 1e+230). With 2 digits each second that's 5.e+229 seconds, that's a lot bigger number than 5x10^22 seconds as you calculated Geo Kaplan. Something like 1.58e+222 years, just to write it down. Unless I'm misinterpreting the number in some way?

Geo Kaplan

2010-May-15, 04:00 AM

If I'm understanding this right then 10^10^23, is the same as 1.e+230 (1x10^230), or at least that's what I get when I do it in Windows calculator (10^10 = 10,000,000,000 then 10,000,000,000^23 = 1e+230). With 2 digits each second that's 5.e+229 seconds, that's a lot bigger number than 5x10^22 seconds as you calculated Geo Kaplan. Something like 1.58e+222 years, just to write it down. Unless I'm misinterpreting the number in some way?

Your calculation yields the time it would take to count from 1 to that final number, two at a time. The OP asked how long it would take to write down the number, which is what I calculated.

Murphy

2010-May-15, 04:25 AM

Your calculation yields the time it would take to count from 1 to that final number, two at a time. The OP asked how long it would take to write down the number, which is what I calculated.

I'm not sure I understand the difference. Maybe I'm just getting confused here, what exactly is the initial number? Is it not 1x10^230?

DrRocket

2010-May-15, 04:58 AM

I was recently having a conversation with a friend about the Multiverse episode of Radio Lab. In that episode, Brian Greene, a physics and mathematics professor at Columbia University, stated that in an infinite universe, everything that is possible has happened an infinite number of times. That means that there is an infinite number of planets exactly like ours that are inhabited by people that look exactly like us some where out there. The average distance that you would have to travel to get to those planets, according to Greene, was 10^10^23 light years. That number is so big that I can't even comprehend it. I've read that the total number of particles in the visible universe is 10^89, which is dumbfounding huge, but if I took the time I could write it out. It is 1 followed by 89 zeros.

My question is how long would it take to write out 10^10^23 in its full form? I tried to do the math, but very quickly got confused. I was figuring an average of writing 2 digits a second. Thanks for your help.

Greene's statement is flat out wrong.

It is based on the law of large numbers in probability theory, which implies that, given a probability space any event of positive probability will, with probability one, occur infinitely many times in infinitely many independent trials.

However, in order to apply this theorem you first have to have a probability space.

Greene's statement has been made many times by string theorists who advocate the "multiverse" or "cosmic landscape" hypothesis -- a main advocate of which is Leonard Susskind (author of The Cosmic Landscape), There is zero basis for this assertion.

The origin of the assertion lies in the failure of string theory to produce any model of the universe as it is observed, and in the failure of the the original hope of the theory, which was to produce one single theory will no arbitrary constants on the basis that such a theory would be the only mathematically consistent "theory of everything". Instead several things have happened. First, no one has yet produced any rigorously well-defined string theory, and several potential string theories have been partially developed. Second, there appear to be potentially something like 10^500 possible string theories that, if anyone ever manages to clearly define what a string theory is, would me mathematically consistent. These correspond to the various Calabi-Yay manifolds that are used in string theory to "compactify" hidden dimensions, and they result in different physical laws. Now, overcome by the aesthetic beauty of these as-yet-undefined theories and some string theorist have decided that ALL of the 10^500 or more theories must be true and that the universe therefore consists of a an infintite number of "pocket universes" each of whch has physical laws determined by these various string theories.

Then with a giant leap of something, maybe faith, they apply probability theory to this collection of pocket universes, which results in this statement that everything that is possible has happened an infinite number of times. However, nowhere in this menagerie of illogic has anyone identified any probability space. In fact, the more honest of them admit that there is no known way to impose the structure of a probability space on this "multiverse". This problem has even been given a name -- "the measure problem". The name comes from that the fact the critical ingredient in a probability space is a probability measure, which is what assigns probabilities to events. So the plain fact of the matter is that these guys are using a major theorem from probability theory in a situation in which there is no sensible way to talk about probability. Utter nonsense.

They then go forth and simply lie to the public in a scam to make it appear that string theory offers answers that it simply cannot offer at the current stage of maturity of the subject.

Now, having said that answer to your queston is that at 2 digits a second it woudl take 5 x ((10^23)-1) seconds to write 10^(10^23) . That is still one hell of a long time.

Geo Kaplan

2010-May-15, 05:01 AM

I'm not sure I understand the difference. Maybe I'm just getting confused here, what exactly is the initial number? Is it not 1x10^230?

Ok, let's try a simpler example: How long would it take to write the number 5842, two digits per second? There are four digits, so it would take two seconds.

Your calculation would be to write 1, then 2, then 3...all the way to 5482. That's a lot more operations. Is the difference clear now? That's why your calculation yields an exponentially greater value. The OP didn't ask to write every number from 1 to the final value, he's just asking how long it would take to write down that final number.

Shaula

2010-May-15, 08:29 AM

I'm not sure I understand the difference. Maybe I'm just getting confused here, what exactly is the initial number? Is it not 1x10^230?

No it is 10^(10^23) or 1 followed by 100000000000000000000000 zeroes. You are calculating 10^10 * 10^23 which is 10^(10*23) = 10^230. FWIW the first number, if written in full in ASCII characters (one byte per character) would fill 23 million, million DVDs with text. I think (my maths is not good on a Saturday morning...)

Geo Kaplan

2010-May-15, 08:37 AM

No it is 10^(10^23) or 1 followed by 100000000000000000000000 zeroes. You are calculating 10^10 * 10^23 which is 10^(10*23) = 10^230. FWIW the first number, if written in full in ASCII characters (one byte per character) would fill 23 million, million DVDs with text. I think (my maths is not good on a Saturday morning...)

No, sorry. Murphy's math is correct, yours is wrong (consider how one would write 10^23, and compare with what you wrote out). Where Murphy went off the rails is in his interpretation of the question.

Shaula

2010-May-15, 08:45 AM

No, sorry. Murphy's math is correct, yours is wrong (consider how one would write 10^23, and compare with what you wrote out). Where Murphy went off the rails is in his interpretation of the question.

Ouch - you are dead right. I feel rather stupid for having even posted that! I'll leave it up to remind myself that I am rather rusty at even the most basic maths....

Geo Kaplan

2010-May-15, 08:58 AM

Now, having said that answer to your queston is that at 2 digits a second it woudl take 5 x ((10^23)-1) seconds to write 10^(10^23) . That is still one hell of a long time.

If we're going to be this precise about calculating such an inaccurate estimate, then let's do it: The number of digits is (1+log(N)), or 10^23 +1 in this case. So the total time would be (5x10^22 + 0.5) seconds.

Geo Kaplan

2010-May-15, 09:02 AM

Ouch - you are dead right. I feel rather stupid for having even posted that! I'll leave it up to remind myself that I am rather rusty at even the most basic maths....

No need to beat yourself up! Chalk it up to needing that extra cup of coffee before manipulating exponents. :)

jogleby

2010-May-15, 02:11 PM

Short answer: A really, really long time.

Longer answer: The exponent of the first 10 tells you how many zeros to write. In this case, you'll be writing 10^23 zeros. At two digits each second, you're looking at 5x10^22 seconds. Given that a human lifetime is of the order of 3x10^9 seconds, you'll need to involve many generations of offspring to complete the task.

Scientific notation has alway been really hard for me. If I remember right, if 3x10^9 is one human life time, then 3x10^10 is 10 life times. So writing out 10^10^23 digits would take 3x10^14 or 300,000,000,000,000 human life times just to write the number. Is that right or did I completely screw up the math?

tdvance

2010-May-15, 05:09 PM

If I'm understanding this right then 10^10^23, is the same as 1.e+230 (1x10^230), or at least that's what I get when I do it in Windows calculator (10^10 = 10,000,000,000 then 10,000,000,000^23 = 1e+230). With 2 digits each second that's 5.e+229 seconds, that's a lot bigger number than 5x10^22 seconds as you calculated Geo Kaplan. Something like 1.58e+222 years, just to write it down. Unless I'm misinterpreting the number in some way?

nope it is 10^(10^23).

10^230 isn't all THAT big---and even a back of the envelope calculation would give the probability much higher than that.

Geo Kaplan

2010-May-15, 05:23 PM

Scientific notation has alway been really hard for me. If I remember right, if 3x10^9 is one human life time, then 3x10^10 is 10 life times. So writing out 10^10^23 digits would take 3x10^14 or 300,000,000,000,000 human life times just to write the number. Is that right or did I completely screw up the math?

Your final value is more or less right (you're off by a factor of 2 because you didn't take into account a speed of 2 digits per second, and you lost a factor 10 somewhere along the line), but the statement that immediately precedes the number isn't quite right -- you're not writing out 10^10^23 digits, but rather the digits of 10^10^23. These aren't the same thing, and I know you know that, but given Murphy's posts, perhaps it wouldn't hurt to highlight the difference again. For example, the number 10^3 is 1000, but it doesn't have 1000 digits; it has 4. The number 10^4 has 5 digits, not 10000. The general procedure for calculating the number of digits is to take the logarithm of the number itself (use base 10 here), and then add 1 to that. So, for 10^3, we find the log as 3 (just read off the exponent), and then add 1, to get 4 for the total number of digits.

In your example, the number itself is 10^10^23. The exponent is therefore 10^23, and that's the log we're looking for. Add 1 to that if you want to be precise, but adding 1 to a gigantic number like 10^23 is a bit silly, especially given that we're dealing with wild estimates anyway. So let's just leave the number of digits, D, as 10^23.

If we take a human lifetime as H seconds, and you are writing W digits per second, then the number of lifetimes L would simply be L = D/(WH).

Plugging in the numbers, we get L = 10^23/(2*3x10^9) = 1.7x10^13 lifetimes.

And yes, that's a very, very long time.

jogleby

2010-May-15, 05:44 PM

So that is something like 10,000 times the age of the universe just to write the number. I don't think I'll be seeing my doppelganger anytime soon.

Ken G

2010-May-15, 06:13 PM

In that episode, Brian Greene, a physics and mathematics professor at Columbia University, stated that in an infinite universe, everything that is possible has happened an infinite number of times.

What I don't get about this statement is the meaning of the term "possible." I agree with DrRocket that citing the frequency of occurrence of something requires a probability measure, and we don't see anything like that here (though I don't think Greene is talking about the multiverse here, he is literally talking about a single infinite universe, which I think means, if our one universe is spatially infinite-- something we also have no way of knowing, and no way to ever know). But what the heck does "everything that is possible" mean? It cannot mean that we take laws of physics and determine what is possible from them, because that is just obviously backward logic-- the job of the laws of physics is to explain what we discover is possible, not to tell us what is possible-- what is possible comes before the laws, we create laws to help us understand why the things we have found to be possible are indeed possible. So if it doesn't mean "what the laws allow" (which would be the worst possible logical error), it must mean "whatever does happen." So then the claim is, whatever does happen, happens an infinite number of times. That logic isn't much better! What about the things that happen ten times in an infinite universe-- are they not possible, and do they not happen less than an infinite number of times? The assertion really doesn't mean anything at all without significant additional assumptions.

I think what he is actually saying is that things that the laws of physics attribute a finite probability to, no matter how small that probability is, must happen an infinite number of times. That's OK to a point, but we must recognize that this is clearly a hypothesis-- it is not itself a law. The statement has all the earmarks of a hypothesis-- it starts with a big "if" (which we don't know is true), and then continues with "the laws of physics as we currently understand them would then dictate..." This is just precisely what a hypothesis is. So what is the problem with recognizing a hypothesis when we see one? Well, it's the usual purpose of a hypothesis to guide some test of that hypothesis-- that is the point of forming hypotheses, to help us see the next test we need to advance our understanding. Hypotheses were never intended to tell us something about the universe by themselves, and they were never intended to give us a "warm fuzzy feeling" that we understand something-- they were always intended to bring attention to what we aren't sure about, and help us find ways to discover greater certainty.

But where is the observation here that is guided by the hypothesis that in an infinite universe, everything that is possible happens an infinite number of times? There's no hint how to find out if the universe is infinite, and there's no hint how to test if things happen an infinite number of times. I therefore cannot see anything in the remark that reminds me of physics at all. DrRocket sees nothing in the remark that reminds him of mathematics. So what does it remind us of? Hmmm, well, the last time I heard that kind of logic, I was sitting in a pew.

Geo Kaplan

2010-May-15, 06:27 PM

What I don't get about this statement is the meaning of the term "possible." I agree with DrRocket that citing the frequency of occurrence of something requires a probability measure, and we don't see anything like that here (though I don't think Greene is talking about the multiverse here, he is literally talking about a single infinite universe, which I think means, if our one universe is spatially infinite-- something we also have no way of knowing, and no way to ever know).

Yes, Greene seems to be just making up stuff. No one should take it too seriously.

Geo Kaplan

2010-May-15, 06:31 PM

So that is something like 10,000 times the age of the universe just to write the number. I don't think I'll be seeing my doppelganger anytime soon.

Your last sentence is correct, but the first one still needs revising (those darned factors of 10 again). It'll take 1.7x10^15 years to get the job done, and the universe is about 1.4x10^10 years old. So we're looking at roughly 100,000 times the age of the universe to get the job done.

Murphy

2010-May-15, 07:10 PM

Ok, let's try a simpler example: How long would it take to write the number 5842, two digits per second? There are four digits, so it would take two seconds.

Your calculation would be to write 1, then 2, then 3...all the way to 5482. That's a lot more operations. Is the difference clear now? That's why your calculation yields an exponentially greater value. The OP didn't ask to write every number from 1 to the final value, he's just asking how long it would take to write down that final number.

Ah, Ok. I understand what you mean now. Yes that was a bit confusing, but I think I get it now.

jogleby

2010-May-15, 08:04 PM

Your last sentence is correct, but the first one still needs revising (those darned factors of 10 again). It'll take 1.7x10^15 years to get the job done, and the universe is about 1.4x10^10 years old. So we're looking at roughly 100,000 times the age of the universe to get the job done.

With my math skills, only missing one zero is awesome for me.

jogleby

2010-May-15, 08:19 PM

What I don't get about this statement is the meaning of the term "possible." I agree with DrRocket that citing the frequency of occurrence of something requires a probability measure, and we don't see anything like that here (though I don't think Greene is talking about the multiverse here, he is literally talking about a single infinite universe, which I think means, if our one universe is spatially infinite-- something we also have no way of knowing, and no way to ever know).

I think what Greene was trying to say was more of a thought experiment. In an infinite universe, if something is allowed by the laws of physics, in matter how improbable, it is bound to happen, and to happen an infinite number of times. His analogy was that atoms can only be arranged and interact with each in a finite number of ways. It is a lot of ways, but finite. In an infinite universe, those patterns of atoms will eventually repeat. It would be like flipping a coin and have it come up heads a million times in a row. That is very unlikely, but if you flip the coin an infinite number of times, the odds will go up to 100% that it will happen eventually. Of course, all of this is mute if the universe isn't infinite, but it is fun to think about.

Ken G

2010-May-15, 08:28 PM

I think what Greene was trying to say was more of a thought experiment. In an infinite universe, if something is allowed by the laws of physics, in matter how improbable, it is bound to happen, and to happen an infinite number of times. His analogy was that atoms can only be arranged and interact with each in a finite number of ways. It is a lot of ways, but finite. In an infinite universe, those patterns of atoms will eventually repeat. It would be like flipping a coin and have it come up heads a million times in a row. That is very unlikely, but if you flip the coin an infinite number of times, the odds will go up to 100% that it will happen eventually. Of course, all of this is mute if the universe isn't infinite, but it is fun to think about.Yes, I think what he is saying very much falls under the "fun to think about" category. Unfortunately, he fails to preface the remarks with that qualifier! A thought experiment is basically saying, "if we take the laws as we know them perfectly seriously, and apply them to some highly idealized situation, what happens?" So it's a way to look at your laws, not a way to look at the universe-- it is "kicking the tires" of your own laws, and should never be mistaken for a statement about the universe itself, unless it is applied to situations that are expressly like those for which the laws were determined. "If the universe is infinite", then we haven't the vaguest clue what is going on 10100 Hubble distances from us, because every philosophical claim we could make that we do (say, based on the cosmological principle), comes with an equally valid one that says we don't (say, based on the track record of sweeping principles in the history of science).

Geo Kaplan

2010-May-15, 11:07 PM

With my math skills, only missing one zero is awesome for me.

:)

And especially with these very large numbers, a factor of 10 just takes us from a "really, really long time" to a "really, really, really long time."

grapes

2010-May-16, 08:03 AM

No it is 10^(10^23) or 1 followed by 100000000000000000000000 zeroes. You are calculating 10^10 * 10^23 which is 10^(10*23) = 10^230. FWIW the first number, if written in full in ASCII characters (one byte per character) would fill 23 million, million DVDs with text. I think (my maths is not good on a Saturday morning...)

No, sorry. Murphy's math is correct, yours is wrong (consider how one would write 10^23, and compare with what you wrote out). Where Murphy went off the rails is in his interpretation of the question.

Murphy did indeed come up with 10^230 (which would have only needed 231 digits to write out, so he did misinterpret that). Shaula's "1 followed by 100000000000000000000000 zeroes" is correct, the only thing I see in his post that is an error is where he assumes what Murphy was calculating, what I highlighted in red, the rest (in blue) does seem to be what Murphy was calculating.

Schneibster

2010-May-16, 12:39 PM

Greene's statement is flat out wrong.No, it's not.

It is based on the law of large numbers in probability theory, which implies that, given a probability space any event of positive probability will, with probability one, occur infinitely many times in infinitely many independent trials.Yes, that's correct. There is a reason it's called the Law of large numbers.

However, in order to apply this theorem you first have to have a probability space.What is a "probability space?" What you have to have is a nonzero probability of an event or circumstance, in this case the existence of another Earth identical in every detail to ours, and an infinite universe.

Period.

Nothing more is required.

tdvance

2010-May-16, 02:20 PM

No, it's not.

Yes, that's correct. There is a reason it's called the Law of large numbers.

What is a "probability space?" What you have to have is a nonzero probability of an event or circumstance, in this case the existence of another Earth identical in every detail to ours, and an infinite universe.

Period.

Nothing more is required.

You need a probability space to compute an exact expected value. For this, about all one can do is guess (and the number WILL be very large). A probability space is a mathematical construct modeling the situation you wish to compute probabilities for. Most people, when they speak of probability, have an implicit probability space somewhere, but to be precise and not make mistakes, one ultimately has to deal with it explicitly.

mugaliens

2010-May-16, 06:13 PM

In an infinately large universe, all finate numbers, regardless of their "very big size," are effectively equal to zero. However, this only holds true from an infinately large perspective...

From our perspective, it matters little whether the universe is infinate or not.

Ken G

2010-May-16, 06:23 PM

What is a "probability space?" What you have to have is a nonzero probability of an event or circumstance, in this case the existence of another Earth identical in every detail to ours, and an infinite universe.

Period. Actually, you need at least one more thing-- you need assumptions about the probabilities of various things happening. Probability is a shifty business-- it is a reflection both of what you do know, and what you do not know. So a "probability space" is an explicit model that treats everything you know about what you don't know. As such, there's no such thing as "the probability" of something happening-- it all depends on what information you have, what constraints you wish to apply, and what data you are averaging over. Details about all those assumptions are what is missing from Dr. Greene's statement.

For example, one could simply relax the cosmological principle, which is by no means a law of physics (it's more a default convenience we'll use in any situation we feel justified using it-- but for the whole infinite universe?). Then one couldn't say squat about the likelihood of Earth's showing up elsewhere, even in an infinite universe.

In my view, the problem with Greene's statement is that it reverses the correct logic of science-- we make sense of what we observe, we do not make sense of what we have no chance to observe, based on what we do observe, because we have no idea if the lessons apply or not. I can't see any content in it other than "if we make the necessary assumptions to assert that an infinite universe has an infinite number of Earths, then an infinite universe will have an infinite number of Earths." If there is any content other than that in his remark, I surely don't know what it is. It's fine as a point of speculation, a "kicking the tires" of the cosmological principle if that principle is taken to its logical (and most likely absurd) extreme. But it doesn't actually tell us anything about the universe we actually live in-- it's just not that easy to make assertions about what we have no experimental access to.

Geo Kaplan

2010-May-16, 08:34 PM

Murphy did indeed come up with 10^230 (which would have only needed 231 digits to write out, so he did misinterpret that). Shaula's "1 followed by 100000000000000000000000 zeroes" is correct, the only thing I see in his post that is an error is where he assumes what Murphy was calculating, what I highlighted in red, the rest (in blue) does seem to be what Murphy was calculating.

I'm not sure what your point is. Shaula's math is not correct (the part you highlighted in red, in fact, is wrong -- look at it carefully; it is just plain wrong, and he agreed). 10^10*10^23 is not equal to 10^230. It is equal to 10^33, unless you are adopting a unique convention for operator precedence.

George

2010-May-16, 08:43 PM

Oops, I saw the thread title just after having watched our symphony do a Rodgers and Hammerstein special. Don't let me interrupt. ;)

Geo Kaplan

2010-May-16, 11:33 PM

In my view, the problem with Greene's statement is that it reverses the correct logic of science-- we make sense of what we observe, we do not make sense of what we have no chance to observe, based on what we do observe, because we have no idea if the lessons apply or not. I can't see any content in it other than "if we make the necessary assumptions to assert that an infinite universe has an infinite number of Earths, then an infinite universe will have an infinite number of Earths." If there is any content other than that in his remark, I surely don't know what it is.

I'm with you, Ken. It's all gnomes.

DrRocket

2010-May-17, 03:40 AM

No, it's not.

Yes, that's correct. There is a reason it's called the Law of large numbers.

What is a "probability space?" What you have to have is a nonzero probability of an event or circumstance, in this case the existence of another Earth identical in every detail to ours, and an infinite universe.

Period.

Nothing more is required.

Given that you don't know what a probability space your initial comments are not only wrong they are directly traceable to ignorance of the very baasics of the theory of probability.

A probability space is what is needed to even make the the term "probability" have meaning.

Specifically a probability space is a set S, a sigma-algebra of subsets of S (called events), and a positive measure of total mass 1 defined on that sigma algebra.

In layman's terms it is the structure required to define the probability of events. It is what allows one to develop the theory of probability in a rigorous fashion. Kolmogorov is generally credited with developing rigorous measure-theoretic probability theory.

DrRocket

2010-May-17, 03:49 AM

You need a probability space to compute an exact expected value. For this, about all one can do is guess (and the number WILL be very large). A probability space is a mathematical construct modeling the situation you wish to compute probabilities for. Most people, when they speak of probability, have an implicit probability space somewhere, but to be precise and not make mistakes, one ultimately has to deal with it explicitly.

And sometimes they are just using the intuitive language of probability theory without any basis and are just talking through their hat.

Probability theory is one of the most misused branches of mathematics, apparently because the language is suggestive and people who have no idea what they are talking about invoke that language in ways that are simply not valid.

Schneibster

2010-May-18, 01:16 PM

You need a probability space to compute an exact expected value. No, I don't. If you claim I do, prove it. That's the nice thing about math; you actually have to prove it if you're asked to, and if it's nothing but hand-waving, everybody can tell.

Ken G

2010-May-18, 03:02 PM

I think that's what everyone here is saying to you right now.

tdvance

2010-May-18, 04:39 PM

Hey, don't presume to teach a mathematician math, if you don't know what a probability space is!

Geo Kaplan

2010-May-18, 04:53 PM

I think that's what everyone here is saying to you right now.

Certainly Kolmogorov's ghost is waiting to see Schneibster's ATM formulation. I'm interested, too.

Ken G

2010-May-18, 05:26 PM

To be fair, we all know Schneibster is new here, and there's always a tendency for the new kid to want to make a splash. Don't worry Schneibster, we're not giving you a hard time and we can tell you will make a contribution here-- once you get the lay of the land a bit better!

DrRocket

2010-May-18, 05:27 PM

No, I don't. If you claim I do, prove it. That's the nice thing about math; you actually have to prove it if you're asked to, and if it's nothing but hand-waving, everybody can tell.

What you are doing is hand-waving of the first order. Utter ridiculous.

You don't know what you are talking about. Your simple question "what is a probability space ?" proves that you lack the background to be making such (ridiculous) assertions.

The DEFINITION of the expected value of a random variable is the integral of the variable against the probability measure over the entire probability space. So you need the space and the measure for the integral, and you need the sigma algebra to define what an integral is and also to know that the random variable is measurable.

You might want to consider the fact that tdvance and I both are PhD mathematicians before you try to instruct us, incorrectly, in basic mathematics.

Now, go read a good book on probability. Loeve's book, page 1 might be a good start.

DrRocket

2010-May-18, 05:28 PM

To be fair, we all know Schneibster is new here, and there's always a tendency for the new kid to want to make a splash. Don't worry Schneibster, we're not giving you a hard time and we can tell you will make a contribution here-- once you get the lay of the land a bit better!

That splash was the water going over his head.

Jeff Root

2010-May-18, 06:07 PM

It must be 35 years since I first got the notion that in an infinite

Universe, everything that can happen will happen in infinitely many

places, infinitely many times. I took that to be a logical necessity,

but I didn't imagine that it might be possible to calculate a mean

distance between occurences of identical things.

-- Jeff, in Minneapolis

Ken G

2010-May-18, 06:55 PM

But it is not a logical necessity. There are an infinite number of integers, are there not? How many times does the integer "42" come up in that list? The point is, additional assumptions are required, beyond just being infinite. Do we have any right to make those assumptions? We have no idea-- we can certainly choose to make them, but when do our assumptions tell us about things other than ourselves?

DrRocket

2010-May-18, 07:18 PM

But it is not a logical necessity. There are an infinite number of integers, are there not? How many times does the integer "42" come up in that list? The point is, additional assumptions are required, beyond just being infinite. Do we have any right to make those assumptions? We have no idea-- we can certainly choose to make them, but when do our assumptions tell us about things other than ourselves?

Not only is is not a logical necessity, the rationale presented makes no sense.

The Law of Large numbers simply implies that, given an event of probability greater than zero, and given infinitely many independent trials the probability of that event will converge, with probbility 1 (aka almost surely) to the ratio of the number of occurences to the number of trials as the number of trials increases without bound.

This has absolutely nothing to do with whether or not the underlying probability space is infinite or finite. It has a lot to do with the application of rigorous probability theory, and that depends on having a sensible way to impose a probability measure.

Take your example of the integers. There is no way to impose a probability measure so that each integer has the same probability of occurence. You can impose a probability measure on the integers, but it is likely to be sometihng ad hoc. For instance you could make the probability of "n" be simply

1/n^2 (divided by (pi^2)/3 or something like that to normalize) and that would give you a probability measure. But unless there is some justification for that assignment it is a purely artificial construction.

Any probability measure defined on the real numbers using the elementary technique of density functions will assign to any single real number the probabilty zero.

HenrikOlsen

2010-May-18, 08:12 PM

If I'm understanding this right then 10^10^23, is the same as 1.e+230 (1x10^230), or at least that's what I get when I do it in Windows calculator (10^10 = 10,000,000,000 then 10,000,000,000^23 = 1e+230). With 2 digits each second that's 5.e+229 seconds, that's a lot bigger number than 5x10^22 seconds as you calculated Geo Kaplan. Something like 1.58e+222 years, just to write it down. Unless I'm misinterpreting the number in some way?

You calculated (10^10)^23, where the number we're talking about is 10^(10^23).

Geo Kaplan

2010-May-18, 08:28 PM

You calculated (10^10)^23, where the number we're talking about is 10^(10^23).

Good catch, HO.

DrRocket

2010-May-18, 09:16 PM

It must be 35 years since I first got the notion that in an infinite

Universe, everything that can happen will happen in infinitely many

places, infinitely many times. I took that to be a logical necessity,

but I didn't imagine that it might be possible to calculate a mean

distance between occurences of identical things.

-- Jeff, in Minneapolis

It is not a logical necessity.

It was nonsense 35 years ago and it has not gained any validity in the intervening third century..

Schneibster

2010-May-18, 09:26 PM

I think that's what everyone here is saying to you right now.Burden of proof rests with the claimant. Good luck with that.

DrRocket

2010-May-18, 09:29 PM

Let us take a well-known result from probability theory.

If you put a monkey at a typewriter and wait long enough he sill produce the complete works of Shakespeare.

This is based on the law of the large numbers and the simple recognition that if symbols are typed out at random then one of the random sequences will indeed be the complete works of Shakespeare, with some positive probability. Therefore in and INFINITE sequence of trials, with probability one, the complete works of the bard will be produced.

Now, suppose that one assumes a reasonable typing rate. Then, given the known length (number of symbols) in the works of Shakespeare one can can calculate the probbility of that sequence and the expected time in which it would be produces by random typing. That has been done (see Innumeracy by Paulos) and the result is the expected time to produce this feat is quite a bit longer than the age of the universe.

Schneibster

2010-May-18, 09:30 PM

Hey, don't presume to teach a mathematician math, if you don't know what a probability space is!Given you haven't proven what you claimed, and seem unfamiliar with the facts of supporting claims, overall I'm not particularly impressed with your reasoning skills. Color me skeptical on the "mathematician" claim.

Once again: what is "probability space" and why is something other than ordinary probability calculations needed in this case? I see lots and lots of handwaving.

DrRocket

2010-May-18, 09:30 PM

Burden of proof rests with the claimant. Good luck with that.

The proof has been provided to you. Your inability to recognize that fact is your problem. No luck required, just a knowledge of mathematics.

DrRocket

2010-May-18, 09:32 PM

Given you haven't proven what you claimed, and seem unfamiliar with the facts of supporting claims, overall I'm not particularly impressed with your reasoning skills. Color me skeptical on the "mathematician" claim.

Once again: what is "probability space" and why is something other than ordinary probability calculations needed in this case? I see lots and lots of handwaving.

Color you ignorant. That is the basis for your failiure to be impressed. What has been described to you is absolutely standard probability theory. Go read a book.

Schneibster

2010-May-18, 09:32 PM

To be fair, we all know Schneibster is new here, and there's always a tendency for the new kid to want to make a splash. Don't worry Schneibster, we're not giving you a hard time and we can tell you will make a contribution here-- once you get the lay of the land a bit better!Whether someone is right or not has nothing to do with how many posts they've made. You can prove it or you can't, and of all places on the Internet this should be one where that is the case more than any other.

You haven't met your burden of proof. This is more handwaving, and in fact what you're doing is moving the conversation to the social arena to attempt to divert attention from that fact. Which is not exactly what I expected to see on BAUT.

Ken G

2010-May-18, 09:33 PM

Burden of proof rests with the claimant. Good luck with that.

The imperatives of logic are not so clear-cut as you imagine. You have claimed one does not need a probability space to meaningfully talk about probability. Does that not make you a "claimant" too? What proof would you offer? It seems quite natural to me that any probability would be meaningful only in the context of a probability space of some kind, so in my view, the claim that is not obvious here is yours. I see your position as logically equivalent to saying "vectors mean things even when not part of a vector space, and anyone who thinks they don't needs to prove it." It's hard to prove a logical necessity.

This is more handwaving, and in fact what you're doing is moving the conversation to the social arena to attempt to divert attention from that fact. Which is not exactly what I expected to see on BAUT.Again, you have a tendency to place on others shoes that fit more naturally your own feet-- here you object to "diverting" attention from the argument by bringing up "social" issues, and then you say you expected something different from BAUT. If the shoe fits....

Schneibster

2010-May-18, 09:34 PM

Color you ignorant. That is the basis for your failiure to be impressed. What has been described to you is absolutely standard probability theory. Go read a book.Horsepucky. Claim without proof = false claim. No tickee no laundry. Try again. Stop playing social games; it makes you look like an idiot. Stick to the science.

Schneibster

2010-May-18, 09:35 PM

The imperatives of logic are not so clear-cut as you imagine. A common claim by those who cannot meet their burden of proof. No tickee no laundry.

tdvance

2010-May-18, 09:35 PM

yeah---too bad he's the first to point it out!

Schneibster

2010-May-18, 09:36 PM

The proof has been provided to you. Your inability to recognize that fact is your problem. No luck required, just a knowledge of mathematics.No, it hasn't. I see three people doing a bunch of hand-waving and taking the argument to the social arena because they got pwnt in the scientific one.

No tickee no laundry.

DrRocket

2010-May-18, 09:39 PM

Horsepucky. Claim without proof = false claim. No tickee no laundry. Try again. Stop playing social games; it makes you look like an idiot. Stick to the science.

You are just plain wrong,

I have provided you with definitions of a probability space and of the mean value of a random variable. The fact that you do not unbderstand them is entirely your problem.

The proof is there, and it is quite clear to anyone who actually does understand the rudiments of probability.

DrRocket

2010-May-18, 09:40 PM

No, it hasn't. I see three people doing a bunch of hand-waving and taking the argument to the social arena because they got pwnt in the scientific one.

No tickee no laundry.

What you see is called a halucination.

You do realize, don't you that those three people are 2 PhDs in mathematics and one PhD in physics ?

Schneibster

2010-May-18, 09:57 PM

You are just plain wrong,No, I'm not. Here, this (http://en.wikipedia.org/wiki/Law_of_Large_Numbers) says why. I suggest you read it.

You claim there's something wrong with the law of large numbers. Wikipedia doesn't agree with you. Prove it. Burden of proof rests with the claimant, and you haven't met yours.

Schneibster

2010-May-18, 09:58 PM

What you see is called a halucination.

You do realize, don't you that those three people are 2 PhDs in mathematics and one PhD in physics ?More claims, more handwaving.

It's spelled "hallucination" and PhDs generally know how to spell.

DrRocket

2010-May-18, 10:09 PM

No, I'm not. Here, this (http://en.wikipedia.org/wiki/Law_of_Large_Numbers) says why. I suggest you read it.

You claim there's something wrong with the law of large numbers. Wikipedia doesn't agree with you. Prove it. Burden of proof rests with the claimant, and you haven't met yours.

That is ridiculous.

I most certainly did not claim tha there is anything wrong with the law of large numbers. I qoted the law of large numbers and explaiined where it applies.

I simply told you that to apply the theory of probability you must first have a probability space. That is what tdvance also told you.

You not only don't understand the mathematics, you don't apparently understand what you have been told in plain English.

DrRocket

2010-May-18, 10:10 PM

More claims, more handwaving.

It's spelled "hallucination" and PhDs generally know how to spell.

We make typos just like everybody else.

Schneibster

2010-May-18, 10:14 PM

Here is your post:

Greene's statement is flat out wrong.

It is based on the law of large numbers in probability theory, which implies that, given a probability space any event of positive probability will, with probability one, occur infinitely many times in infinitely many independent trials.

However, in order to apply this theorem you first have to have a probability space.I've helpfully bolded your claim that I dispute. Please point out where it discusses "probability space" in the Wikipedia article. Please explain how Wikipedia can explain the Law of Large Numbers without it if, as you claim, it's necessary to have one in order to apply it.

Thanks.

DrRocket

2010-May-18, 10:37 PM

Here is your post:I've helpfully bolded your claim that I dispute. Please point out where it discusses "probability space" in the Wikipedia article. Please explain how Wikipedia can explain the Law of Large Numbers without it if, as you claim, it's necessary to have one in order to apply it.

Thanks.

I don't give a damn what Wikipedia says. I also don't give a damn what you dispute.

Try reading any good book on probability. Michael Loeve's Probability will do nicely.

The point is that you cannot even state the Law of Large numbers without use of a probability space. Wikipedia does that. Their use of words like "almost surely" require the notion of a probability space ( a measure space with unit measure) in order to have meaning.

Since, as you have clearly stated, you don't even know what a probability space is, your argument is just plain ridiculous.

Schneibster

2010-May-18, 10:39 PM

I don't give a damn what Wikipedia says. I also don't give a damn what you dispute. Then we're done here; you make claims you can't support, and get huffy when someone shows it.

Good bye.

DrRocket

2010-May-18, 10:40 PM

Then we're done here; you make claims you can't support, and get huffy when someone shows it.

Good bye.

Please be as good as your word.

Jeff Root

2010-May-19, 12:45 AM

It must be 35 years since I first got the notion that in an infinite

Universe, everything that can happen will happen in infinitely many

places, infinitely many times. I took that to be a logical necessity,

but I didn't imagine that it might be possible to calculate a mean

distance between occurences of identical things.

But it is not a logical necessity. There are an infinite number of integers,

are there not? How many times does the integer "42" come up in that list?

The point is, additional assumptions are required, beyond just being infinite.

Do we have any right to make those assumptions? We have no idea-- we

can certainly choose to make them, but when do our assumptions tell us

about things other than ourselves?

I think that in an infinite Universe, everything that can happen will happen

in infinitely many places, infinitely many times. I still consider that to be a

logical necessity. The notion does not depend in any way on probability.

However, it may have been inspired in part by the comment made by some

famous physicist (I forget who) that anything that isn't prohibited by the

laws of physics seems to be required.

-- Jeff, in Minneapolis

Ken G

2010-May-19, 01:03 AM

Horsepucky. Claim without proof = false claim. No tickee no laundry. What is so odd about your stance is how completely you miss that your words are directed at yourself. I see your claim that probability does not require a probability space. I see no proof of that whatsoever. I see a reference to a Wiki article on the law of large numbers, but Wiki articles attempt to match their degree of sophistication to the level of the query, based on whether or not mathematically precise terms are used in the subject heading. For example, if you just look up the Law of Large Numbers, you get the basic idea of the law, but if you look up probability theory (http://en.wikipedia.org/wiki/Probability_theory#Measure-theoretic_probability_theory) you will get more of the details. Look up "measure theory", and you'll get even more. There's little doubt from any degree of basic research into the question that putting probability onto a formal (axiomatic) mathematical basis requires concepts from measure theory like a probability space. It is also clear that more informal uses of the term can skip that step. So really the only question here is, do you take Dr. Greene's remarks as correct in some formally logical sense, or do you just take them as informal hand-waving? Ironically, it is the latter you seem to be objecting to-- yet your stance relies on it.

DrRocket

2010-May-19, 02:08 AM

I think that in an infinite Universe, everything that can happen will happen

in infinitely many places, infinitely many times. I still consider that to be a

logical necessity. The notion does not depend in any way on probability.

However, it may have been inspired in part by the comment made by some

famous physicist (I forget who) that anything that isn't prohibited by the

laws of physics seems to be required.

-- Jeff, in Minneapolis

I am sure that you do consider that to be a logical necessity.

There is a simple explanation.

You are wrong.

Ufonaut99

2010-May-19, 02:49 AM

I haven't heard BRian Greene's claim (I'll have to listen in when I get home from work), but it seems similar to an article in Scientific American a while back.

It wasn't about the multiverse and string theory, but went something like this :

Take a Noughts-And-Crosses board (OK, Tic-Tac-Toe if you prefer ;) ), and fill it with a combination of noughts and crosses. Now imagine your board is in the middle of an infinitely large sheet. It is "certain" that your pattern of noughts and crosses will repeat infinitely many times. You can certainly calculate the average distance you would have to go to find a match (eg. probability of a match > 90%).

Turning to the universe, pick a grid size - say, the size of a neutron. Define a set of variables to describe that volume of space (eg. "Neutron here", "Proton here", etc). Now, assuming an infinite universe, you can calculate the average distance you would have to go to find a match to an Earth-sized (or solar-system sized - or observable-universe sized) region matching our space. Clearly, the finer the grid-size, and the more variables you choose to include, then the further the matching region will be - but there'll still be infinite repeats, each by definition including a RobA typing into a bautforum.

(IIRC in Sci-Am, he was claiming repeats not only in an infinite universe, but even in one having the smallest size assuming the universe is not flat within the error-limit of current measurements)

Fun to think about certainly. Also certainly totally untestable. If we leave aside the fuzziness of QM, though (admittedly a big IF, since we're leaving the real universe!), are we just playing with figures or could there be arbitrarily close repititions?

Schneibster

2010-May-19, 02:59 AM

What is so odd about your stance is how completely you miss that your words are directed at yourself. So you think that's as odd as someone using "I know you are but what am I" on a science forum? I disagree. I think using arguments worthy of a 3-year-old on a science forum is ludicrous. Almost as ludicrous as an anonymous poster on the Internet claiming to be smarter than the Felix Bloch Professor of Theoretical Physics at Stanford University, at one of the most famous centers of physics research in the world.

DrRocket

2010-May-19, 03:04 AM

I haven't heard BRian Greene's claim (I'll have to listen in when I get home from work), but it seems similar to an article in Scientific American a while back.

It wasn't about the multiverse and string theory, but went something like this :

Take a Noughts-And-Crosses board (OK, Tic-Tac-Toe if you prefer ;) ), and fill it with a combination of noughts and crosses. Now imagine your board is in the middle of an infinitely large sheet. It is "certain" that your pattern of noughts and crosses will repeat infinitely many times. You can certainly calculate the average distance you would have to go to find a match (eg. probability of a match > 90%).

This is just wrong.

Let's mark your spot with an X. Fill in all the other positions on the board with an 0. Your X is never repeated.

The argument only works if you take the pattern to be randomly generated, which requires a probability space, and then apply the Law of Large numbers to conclude that you get repeats "almost surely".

However, you don't have a probability space in evidence, and the reasoning doesn't work.

Ken G

2010-May-19, 04:20 AM

So you think that's as odd as someone using "I know you are but what am I" on a science forum? I disagree.You accuse others of handwaving, but nothing you've posted yet is anything but handwaving. That's just a fact here.

Almost as ludicrous as an anonymous poster on the Internet claiming to be smarter than the Felix Bloch Professor of Theoretical Physics at Stanford University, at one of the most famous centers of physics research in the world.Correction, the claim is that your version of what the Felix Bloch Professor is saying is all muddled. Unless you are the Felix Bloch Professor-- no, I didn't think so. Perhaps what you fail to recognize is that oftentimes, even the Felix Bloch Professor makes comments that are designed for the ears of nonspecialists, nonphysicists, and those not particularly skilled in mathematics. When you hear comments like that, and take them completely at face value as official pronouncements of truth, you are simply being naive. He did not lay out his assumptions behind that remark-- that much is perfectly obvious, even to a lowly physics Ph.D. with no big titles.

Ken G

2010-May-19, 04:29 AM

I haven't heard BRian Greene's claim (I'll have to listen in when I get home from work), but it seems similar to an article in Scientific American a while back.The point we are making is not that there is no set of assumptions that will make Brian Greene's claim true. There are obviously sets of assumptions that will make it true-- like the assumptions that go into your "noughts and crosses" example (and it certainly doesn't take an expert theoretical physicist to draw that conclusion, any high school student learning about probability knows that). What we are saying is that Brian Greene has no idea if those assumptions are appropriate to our universe-- it has nothing to do with his title or standing in the field, he just doesn't know. Why would being a great theoretical physicist tell him what assumptions to make about parts of our universe we've never seen and never will?

What is going on here is pretty clear-- he is making a comment intended for a general audience, and he is being uncareful about laying out his assumptions. I'm certain that in an audience of physicists and mathematicians, he would be the first to admit that he is making certain assumptions not included in those general remarks for laymen, and he would also be happy to lay out in detail what those assumptions were. This is really not that profound-- some posters on here are taking his remarks as some kind of gospel, and they are simply out of context.

Fun to think about certainly. Also certainly totally untestable. Which is exactly why no physicist, even Brian Greene, is in any position to make the claim on our universe. As I said, at the very least it requires taking the cosmological principle quite seriously, and the cosmological principle is not even a law of physics. I have no doubt Brian Greene would have to agree. His comments are doing nothing but "kicking the tires" of certain assumptions we might like to make, but as I also said, in physics his statement has the status of a hypothesis-- yet without any of the purpose that we actually use hypotheses for. If he could disagree with anything I've said here, I'd surely like to know which. This is all very simple-- he can lay out clearly his assumptions that make his statement true, and anyone can feel free to accept or reject those assumptions, and no experiment is ever going to tell them one way or the other. Some day the assumptions he is making might be useful for some prediction or other, and it might even become accepted as the standard model for thinking about things (it has as yet made no such useful predictions-- Weinberg's cosmological constant from the anthropic principle is not a prediction of a multiverse, it is merely a requirement for our being here). But even if it becomes the standard model (a big if), it will still be just that.

DrRocket

2010-May-19, 04:46 AM

You accuse others of handwaving, but nothing you've posted yet is anything but handwaving. That's just a fact here.

Correction, the claim is that your version of what the Felix Bloch Professor is saying is all muddled. Unless you are the Felix Bloch Professor-- no, I didn't think so. Perhaps what you fail to recognize is that oftentimes, even the Felix Bloch Professor makes comments that are designed for the ears of nonspecialists, nonphysicists, and those not particularly skilled in mathematics. When you hear comments like that, and take them completely at face value as official pronouncements of truth, you are simply being naive. He did not lay out his assumptions behind that remark-- that much is perfectly obvious, even to a lowly physics Ph.D. with no big titles.

And then there is Gerardus 'tHooft who is not the Felix Block Professor of Physics either, but who has also critized the holder of that position. Poor 'tHooft. I think I will side with him anyway. Not because of who he is, but because the Felix Bloch Professor has stepped way outside the bounds of science, and because poor old 'tHooft was right to criticize him.

Also note that the Felix Bloch Professor, in his subsequent book, The Black Hole War, goes so far as to take as fact, without even being honest to clearly state it, the AdS/CFT correspondence. But in fact the Ads/CFT correspondende is an unproved conjecture of Maldecena from about 1997. Theree is great deal of intellectual dishonesty that is apparent in that book.

Ken G

2010-May-19, 05:21 AM

Yes, Brian Greene has his detractors, and it's perfectly valid to call people when they are being imprecise, no matter what their title. I will grant him that there are assumptions which can be made to make his remarks valid-- he just needs to identify them and spell them out. Whether those assumptions are valid elements of physics, or something that has left science itself far behind, can then be debated in the open. But the assumptions, and what he sees as their justification, need to be made clear before that can even happen.

Schneibster

2010-May-19, 06:43 AM

You accuse others of handwaving, but nothing you've posted yet is anything but handwaving. That's just a fact here.No, actually it's not a fact at all. I quoted from Gravitation, linked to Wikipedia articles on the Law of Large Numbers, and made arguments based on well-known characteristics of black holes.

Have you people been getting away with sloppy stuff like this until now or something?

Schneibster

2010-May-19, 06:45 AM

Yes, Brian Greene has his detractors, and it's perfectly valid to call people when they are being imprecise, no matter what their title. I will grant him that there are assumptions which can be made to make his remarks valid-- he just needs to identify them and spell them out. Whether those assumptions are valid elements of physics, or something that has left science itself far behind, can then be debated in the open. But the assumptions, and what he sees as their justification, need to be made clear before that can even happen.I haven't seen any of his remarks quoted here so I'm sorry but I have to classify this as bloviation.

Ken G

2010-May-19, 07:13 AM

I haven't seen any of his remarks quoted here so I'm sorry but I have to classify this as bloviation.

I call it responding to the OP (original post), you know: "In that episode, Brian Greene, a physics and mathematics professor at Columbia University, stated that in an infinite universe, everything that is possible has happened an infinite number of times." Now, it is certainly true that I am taking the OPer's word for what was stated, but that's more or less obvious. All I've said is that it is quite clear to me that Brian Greene is making assumptions that cannot be known to be applicable, when he makes that remark. I view that as perfectly obvious. Whether he actually specified those assumptions on that program or not is unclear, but it's not much of a stretch-- that program is for laymen, and the assumptions would sound like jargon were they spelled out. As I said before, I don't see anything going on here that is not as plain as day.

Ken G

2010-May-19, 07:17 AM

No, actually it's not a fact at all. I quoted from Gravitation, linked to Wikipedia articles on the Law of Large Numbers, and made arguments based on well-known characteristics of black holes. For those not reading the other thread, you are talking about two separate threads there. But it's no matter-- we on this forum are all very well acquainted with the fallacy of "quote mining." When you are not understanding the basic lessons of the book Gravitation, you can draw out quotes out of context and support all kinds of false positions. That is nothing new to us here. We also know that Wiki articles do not always give the complete amount of information necessary to make an argument rigorous-- sometimes they just explain stuff, not anticipating every possible objection that would be raised. As such, absence of evidence that a particular issue is important (like probability spaces) should not be construed as evidence of the absence of that importance. Your position is a logical fallacy we are also well acquainted with here.

Have you people been getting away with sloppy stuff like this until now or something?Now you are shooting for some kind of record for making criticisms that actually apply much better to your own argument. What is "sloppy" is mining quotes out of context, and not understanding what they mean.

mugaliens

2010-May-19, 07:45 AM

But it is not a logical necessity. There are an infinite number of integers, are there not? How many times does the integer "42" come up in that list? The point is, additional assumptions are required, beyond just being infinite. Do we have any right to make those assumptions? We have no idea-- we can certainly choose to make them, but when do our assumptions tell us about things other than ourselves?

Furthermore, in a self-assembling universe, only certain things are probable, and the space of those things is miniscule compared to what's possible. For example, it's possible for an animal to exist which is mobile, able to put down a mesh of roots anywhere water exists and which is endowed with chlorophyl as a means of producing sugar for energy. The reason we don't see this possibility as a reality is because the nature of this process is way too slow to produce the magnitude of daily energy required by animals.

It's a simple matter of phsyics, chemistry, and biology. The same laws of nature vastly reduce the realm of what's possible to the realm of the improbable.

Nevertheless, we may one day discover a planet populated by intelligent phytonoids, who, after soaking up the sun for a week might take a day off to move across a clearing towards a more suitable location before plunking down roots and repeating the process.

I'm not holding my breath...

tusenfem

2010-May-19, 09:43 AM

Okay guys, that is enough bickering here. I do no longer want to see any personal attacks, discuss the topic, not the poster. If the poster makes a mistake then clearly show where (s)he is doing so (and typos don't count). And please please please, wikipedia is NOT the ultimate resource for scientific definitions.

Come with arguments and definitions and maybe even an equation or two to show what you want to express. Especially in probability that can be very usefull, e.g. in showing what a "probability space" is.

Now, please, continue your discussion.

DrRocket

2010-May-19, 03:38 PM

Given that you don't know what a probability space your initial comments are not only wrong they are directly traceable to ignorance of the very baasics of the theory of probability.

A probability space is what is needed to even make the the term "probability" have meaning.

Specifically a probability space is a set S, a sigma-algebra of subsets of S (called events), and a positive measure of total mass 1 defined on that sigma algebra.

In layman's terms it is the structure required to define the probability of events. It is what allows one to develop the theory of probability in a rigorous fashion. Kolmogorov is generally credited with developing rigorous measure-theoretic probability theory.

Okay guys, that is enough bickering here. I do no longer want to see any personal attacks, discuss the topic, not the poster. If the poster makes a mistake then clearly show where (s)he is doing so (and typos don't count). And please please please, wikipedia is NOT the ultimate resource for scientific definitions.

Come with arguments and definitions and maybe even an equation or two to show what you want to express. Especially in probability that can be very usefull, e.g. in showing what a "probability space" is.

Now, please, continue your discussion.

A probably space was defined in post 36, quoted above. The definition given is the standard mainstream definition.

One does not use equations to define a probability space. One requires a probability space in order to write equations reflecting probabilities.

DrRocket

2010-May-19, 03:46 PM

The DEFINITION of the expected value of a random variable is the integral of the variable against the probability measure over the entire probability space. So you need the space and the measure for the integral, and you need the sigma algebra to define what an integral is and also to know that the random variable is measurable.

This might be consider an equation defining the expected value of a random variable. In fact that is what it is, although this board does not support the symbology necessary to write it in the customary form.

Note that it is necessary to first have a probability space in order for the integral in question to make sense.

Ken G

2010-May-19, 06:41 PM

Yes, there is a kind of "layman's version" of probability, which has to do with a kind of emotion (I don't think that's too likely, or that seems like the probable answer to me) based on experience in a loose way, but then there is also the mathematical version of the meaning of probability. As usual, you need the mathematical version to have both rigor, and confidence that you are not fooling yourself. In particular, mathematical definitions become most important when dealing with somewhat unusual or singular kinds of situations, such as when talking about infinite universes and so forth. I think it is safe to say that Brian Greene was handwaving on that program from the OP, but it is also somewhat natural that he would be-- it was a program for laymen, not mathematicians. This is nothing new folks-- respected physicist goes out on a bit of a limb to express a personal view that he/she finds insightful, but when packaged for a popular audience, it comes out sounding like a well-known and widely accepted scientific fact.

DrRocket

2010-May-19, 07:13 PM

This is nothing new folks-- respected physicist goes out on a bit of a limb to express a personal view that he/she finds insightful, but when packaged for a popular audience, it comes out sounding like a well-known and widely accepted scientific fact.

Yes.

This seems to have become a bit too common ever since Hawking published A Brief History of Time. Hawking was rather careful in that book to separate speculation from established science, but what he demonstrated is that there is a significant commercial market for popularizations of modern physics.

Since Hawking's book there have been many popularizations of modern physics. Some of them are objective and honest. Gordon Kane, for instance, in his book on supersymmetry is very careful to clearly identify speculation. Roger Penrose is equally careful in The Road to Reality Others are not so careful and tend towards the sensational at the expense of being honest. IMO Sussking and Kaku fall into the latter camp. One might guess that hype sells more books than does solid science, and the objective of a commercial endeavor is sales and profit. In a forum designed for laymen a specialist can get away with statements that would simply not fly in front of a knowledgeable audience. Some people take advantge of that.

The problem is magnified when people without adequate background in science and mathematics, and that is by far the majority of the population, are lead to believe that speculation is established science. String theory appears to be particularly susceptible to such misinterpretation. That is understandable since no one, repeat no one, has ever rigorously even defined what string theory really -- read Witten (http://www.sns.ias.edu/~witten/) for honest assessments ([URL="http://www.sns.ias.edu/~witten/papers/Unravelling.pdf).

One might also suspect that those without direct profit motives related to books, might be lobbying for support for their area of research. This might lead to advocacy that clouds objectivity. I think Greene may fall into this genre. He is certainly a string theory advocate.

Just to be clear, I am of the opinion that string theory is a valid avenue of research. It offers promise, and already serves as a tremendous conjecture machine for mathematics. However, it is not at all clear that it has anything to do with physics. It may eventually become a great physical theory. Or it may not. But it has not yet fulfilled the obligation of a physical theory to provide new testable hypotheses, or in fact to even provide a well-defined consistent model from which to develop such hypotheses. Major gaps remain in the logical structure. Witten's 1995 conjecture of an over-arching structure, M-theory, remains unproved, as does Maldacena's 1997 conjecture of the AdS/CFT correspondence. Nevertheless, some string theorists write for the public as though these conjectures are established facts. Susskind, in The Black Hole War uses the AdS/CFT correspondence without even clearly stating that he is doing so, let alone noting that is an unproven conjecture.

Fiery Phoenix

2010-May-19, 07:31 PM

I remember reading a paper that included that number as well. I couldn't even read it. It just seems beyond normal human comprehension. I was trying to calculate the corresponding number of light-years, until I found out even my scientific calculator couldn't do it. Oh well.

tdvance

2010-May-19, 07:55 PM

For numbers that size, I doubt a mere human can get a handle on it through the difference between an angstrom and a light year, etc. which is just a small (relatively speaking) order of magnitude separation, whereas the number in question is 10^23 orders of magnitude from familiar measurements. I personally think numbers like that as "effectively infinite" --- if a computer with as many processors and memory cells as the number of particles in the observable universe, making one computation every Planck time for the current age of the universe, can't make a dent in counting to the number, then traveling that many Planck units, much less...what was the original unit, meters? Light years? Angstroms? (answer is the same regardless---see what I meant by the first sentence!) is a forgetaboutit issue. So you see, effectively infinite is an apt description!

tdvance

2010-May-19, 07:59 PM

Or to put it another way, 10^(10^23) light years, when expressed in feet, Comes out to.... 10^(10^23) to the same level of precision inherent in this way of writing the number (i.e. assuming 23 is "something between 22.5 and 23.5").

Fiery Phoenix

2010-May-19, 08:00 PM

For numbers that size, I doubt a mere human can get a handle on it through the difference between an angstrom and a light year, etc. which is just a small (relatively speaking) order of magnitude separation, whereas the number in question is 10^23 orders of magnitude from familiar measurements. I personally think numbers like that as "effectively infinite" --- if a computer with as many processors and memory cells as the number of particles in the observable universe, making one computation every Planck time for the current age of the universe, can't make a dent in counting to the number, then traveling that many Planck units, much less...what was the original unit, meters? Light years? Angstroms? (answer is the same regardless---see what I meant by the first sentence!) is a forgetaboutit issue. So you see, effectively infinite is an apt description!

More or less, seeing as how it's practically impossible to deal with these kinds of numbers.

DrRocket

2010-May-19, 08:01 PM

For numbers that size, I doubt a mere human can get a handle on it through the difference between an angstrom and a light year, etc. which is just a small (relatively speaking) order of magnitude separation, whereas the number in question is 10^23 orders of magnitude from familiar measurements. I personally think numbers like that as "effectively infinite" --- if a computer with as many processors and memory cells as the number of particles in the observable universe, making one computation every Planck time for the current age of the universe, can't make a dent in counting to the number, then traveling that many Planck units, much less...what was the original unit, meters? Light years? Angstroms? (answer is the same regardless---see what I meant by the first sentence!) is a forgetaboutit issue. So you see, effectively infinite is an apt description!

Yep. One can become too easily accustomed to large numbers expressed in "scientific notation" and fail to realize just how big they are.

The flip side is that some things that are not only possible, but probable in infinite time (like monkeys at a typewriter typing out the complete works of Shakespeare) are not likely even in gangantuan time intervals.

This number is so big that I agree that for many purposes it is indeed, "effectively infinite". But for other purposes it might be pretty small.

BTW I know that you know this. Not everyone does.

Schneibster

2010-May-20, 12:17 AM

I call it responding to the OP (original post), you know: "In that episode, Brian Greene, a physics and mathematics professor at Columbia University, stated that in an infinite universe, everything that is possible has happened an infinite number of times." Yep. That's bloviation all right.

Schneibster

2010-May-20, 12:23 AM

Well, now, I haven't seen anyone actually explain anything about why his views are wrong, and if anyone's going to do that then he needs the courtesy of being actually quoted, what he said, not what someone with potential (and, in fact, based on the quote from the OP, actual and demonstrated) bias, claims he meant. That's the way these things are done among polite people. I don't see anyone here doing that, with one or two possible exceptions. I am judging this forum accordingly. You'll note that I have been a member a very long time but had a very low post count until the recent position was taken on AGW. If you're perceptive you will be aware of the probable meaning of this in terms of what other judgments I've made so far.

Ken G

2010-May-20, 02:52 AM

Well, now, I haven't seen anyone actually explain anything about why his views are wrong, and if anyone's going to do that then he needs the courtesy of being actually quoted, what he said, not what someone with potential (and, in fact, based on the quote from the OP, actual and demonstrated) bias, claims he meant.We're confused. Above, you were arguing that the statement attributed to him was perfectly correct, so we pointed out why it was lacking. Now you seem to be expressing skepticism that he even said it. If you think it was correct, why would you doubt that he said it? If you think it isn't correct, without significant further assumptions being explained (which is our view), then why did you defend it? If you were simply defending the man, Brian Greene, and not the statement, that's another matter that you should have clarified. This thread really doesn't have enough primary sources involving him to speak authoritatively about what he did or did not say, or does or does not believe, you are right about that much. That just isn't the purpose of the thread-- the purpose is to answer the question posed in the OP.

Since Brian Greene has been somewhat impugned in this thread, you have every right to raise the separate issue you are now asking. I can agree that nothing said in this thread about the man is terribly authoritative, nor well supported by evidence, as that's just not what this thread is about. All we can really claim is that if he really said what he was purported to say, then he left out certain important assumptions. If he didn't say it, then the statement by itself leaves out those assumptions, and none of this should be taken as a reflection on Brian Greene. Personally, I think he is clearly a very bright guy who has chosen a path for himself and has made important contributions. Like any important physicist who tries to popularize his discoveries, he walks a fine line between bringing insight to laymen, versus creating a kind of gospel of speculative physics. I don't personally pass any judgement on his efforts-- I merely respond to the OP question, taken at face value.

If you're perceptive you will be aware of the probable meaning of this in terms of what other judgments I've made so far.I think the general relativity thread should tell you everything you need to know about how much this forum could actually benefit you, if you let it. That's the tricky think about judgements-- by definition, they can be wrong.

Schneibster

2010-May-20, 10:54 PM

We're confused. I can see that. I knew it as soon as I saw that Brian Greene wasn't even done the courtesy of being quoted before someone started denigrating what someone else claimed his views were.

Good bye. This isn't a particularly science-friendly place.

DrRocket

2010-May-20, 11:17 PM

I can see that. I knew it as soon as I saw that Brian Greene wasn't even done the courtesy of being quoted before someone started denigrating what someone else claimed his views were.

Good bye. This isn't a particularly science-friendly place.

Ta ta

Ken G

2010-May-21, 01:06 AM

This isn't a particularly science-friendly place.If by "science" you mean "clinging to misconceptions about coordinates in general relativity," I guess it isn't very friendly to that, no. But if you define science as a "sincere effort to replace misconceptions with a more functional understanding", then I think it actually is pretty friendly, if at times argumentative.

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