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I Boltzmann brains

  1. Nov 26, 2016 #1
    Hi guys

    I recently read a lot about Boltzmann brains and their possibility of existence. Are there any models in cosmology which exclude them and make them a physical impossibility? How can we safely conclude that we are not Boltzmann brains which last for a moment and then vanish? Are the Boltzmann brains seriously taken by the scientific community?

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  3. Nov 26, 2016 #2


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  4. Nov 26, 2016 #3


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    As I understand it, cosmologist Sean Carrol originally thought they were a problem, and then later convinced himself that they were not.

    Here's my take on why some people worry about Boltzmann brains. On the one hand, it's obviously a stupid thing to worry about, because the assumption that the universe is real seems necessary for the scientific method to work. If I'm just a brain in a vat being fed false sensory data by a mad scientist, what hope is there for ever figuring out what's going on? So certain assumptions seem necessary to make any progress at all in science. Everybody makes those assumptions, and it's sort of useless to speculate about the possibility that they are false.

    That being said, it's still a logical puzzle---a scientific brain teaser---and it would be nice to have a satisfying answer.

    Here's my understanding of the original (non-quantum, non-relativistic) version of the puzzle. Suppose that you put our entire galaxy into an impenetrable box that no energy or matter can flow into or out of. Now wait a billion years, a trillion years, [itex]10^{10^10}[/itex] years, whatever the number is, and the box will approach thermal equilibrium, where everything is run down and used up and at its highest entropy state. Very sad...the end of history. But no! Wait even longer... Even though the box is at thermal equilibrium, that doesn't mean absolutely nothing ever happens, because thermal equilibrium is a state in which the particles are still bouncing around and colliding and interchanging energy. By far, the most likely outcome of one of these microscopic events is to return the system to the highest entropy state again. But every once in a while, the system entropy will dip lower by a microscopic amount, for a tiny amount of time. And the size and duration of these dips is itself statistically distributed. Wait long enough (I mean, a REALLY long time), and the entropy will dip arbitrarily low. Wait long enough, and there will once again be life and humans and shining stars. So heat death is not forever...

    Now, suppose you are a "second generation" human--one that popped up after the heat death of the galaxy. By far, the most likely situation is that you are the ONLY human in an otherwise dead galaxy. If you are relying on random thermal fluctuations to lower the entropy, it's enormously more likely that the entropy is lowered only in a small region of space, than that it is lowered everywhere. But what if this second generation human has memories of a childhood, of parents, of visiting the big city teeming with hundreds of thousands of people? Well, the odds are still greater that those memories are fakes---the accidental artifact of random fluctuations--than that all those things actually happened. So if you knew you were a second-generation human, you would have no logical reason to think that you aren't a brain floating in space and all your memories are false.

    Thank God (or whoever) that we're first-generation humans, and we can believe our senses. But then...how do you know that you're a first-generation human? And why can first-generation humans believe their senses? For us to be able to believe our senses means that the rest of the galaxy is not heat-dead, which means that in the far past, the entropy was very, very low compared to the maximum possible entropy. Why did that happen? What reason do we have for believing that it happened (as opposed to our memories being false)?

    So to me, the logical puzzle, or brain-teaser, is to understand why the universe should be in a low-entropy state (compared to the highest possible entropy state). Boltzmann's brains are just one way to illustrate the puzzle. I'm not sure that there is any answer other than to shrug our shoulders and say: We're just going to assume that the universe is in the low-entropy state that it appears to be in, and stop worrying about it.
  5. Nov 26, 2016 #4
    But you could still be a BB, it really doesn't matter which generation human are you, if the BB's from the future outnumber humans through history than chances are you should be a BB. The ratio is what matters and if there are infinitely many BB's in spacetime the odds would be that you are a BB. So what is needed is a model where BB's are really an impossibility.
  6. Nov 26, 2016 #5


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    It seems to me that of course they are possible, if you say that something with a nonzero probability is possible (no matter how small that probability is).
  7. Nov 27, 2016 #6
    i dont see that you need BB to be impossible, you only need them to be less probable than normal observers. A model like eternal inflation can generate this. Alan Guth discusses this 27: 13 into this film:
  8. Nov 28, 2016 #7


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    I don't think that's accurate. They are a problem in a number of specific cosmological models. They are not a problem in others.
  9. Nov 28, 2016 #8


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    This question can only be answered through observations and any observation can be called a fake memory later. So, the way you described it, there is no way to say whether we are first or second generation humans.
    But let's say the probability for a Boltzmann brain to form is p. And the probability for it to have a memory worth of one-day is q. And of course you should consider that the probability of having d days of memory is actually less than ## q^d ## because the memories can give you an illusion of life only if they're consistent with each other. Also, the more complicated the memory, the less likely. Then as you get older, which means for someone who is there and has a lot of memory, you know that its very improbable for you to be a Boltzmann brain and its more likely that you are a complete human in a real world with other real humans. So I can say, almost with certainty, that I'm not a Boltzmann brain.
  10. Nov 28, 2016 #9
    What would be the chances of creating a Boltzmann brain which is identical with some human who lived in some time in human history (doesn't matter if it's the past, present or the future), so the person would say there's a 50-50 chance that he is a BB. Is that kind of event far unlikely to even consider, despite that there will be many, many humans and many, many brains in the history of universe?
  11. Nov 28, 2016 #10


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    The probability of a Boltzmann brain with some special type of memory to come into existence has nothing to do with the number of actual people with that type of memory hat have existed or will exist!
  12. Nov 28, 2016 #11


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    But the issue is not the absolute probability of a BB forming, it's the conditional probability. We have certain observations [itex]O[/itex]. We're trying to figure out the most likely explanation for those observations. For simplicity, let's suppose that there are only two possible explanations:
    1. Evolution: The universe started at an extremely low entropy (compared to the maximum possible entropy), and the usual story is true: hydrogen formed into stars and planets, and one of those planets evolved life.
    2. Boltzmann's Brain: The universe is heat-dead except for just me, and my memories are all false.
    So, let [itex]P(L)[/itex] be the a-priori probability of an initially low-entropy universe, and let [itex]P(H)[/itex] be the probability of a high-entropy (heat-dead) universe. Then the conditional probability that we evolved or that we are BBs, given observations [itex]O[/itex] is given by:

    [itex]P(L|O) = P(L \wedge O)/P(O) = \dfrac{P(L) P(O|L)}{P(O)}[/itex]

    [itex]P(H|O) = P(H \wedge O)/P(O) = \dfrac{P(H) P(O|H)}{P(O)}[/itex]

    where [itex]P(A|B)[/itex] means the probability of [itex]A[/itex] given [itex]B[/itex]. From probability theory, we have:
    [itex]P(A|B) = \dfrac{P(A \wedge B)}{P(B)} = \dfrac{(B|A) P(A)}{P(B)}[/itex]

    So the argument for Boltzmann's Brain is that, even though [itex]P(O|H)[/itex] is extremely low, the probability [itex]P(H)[/itex] is very high compared with [itex]P(L)[/itex]. So the product [itex]P(O|H) P(H) > P(O|L) P(L)[/itex].

    The fuzzy part of this is figuring out how to calculate [itex]P(L)[/itex] and [itex]P(H)[/itex]. Boltzmann just assumed it was a matter of the volume in phase space. There are many, many more ways to have high entropy than to have low entropy, so [itex]P(H) \gg P(L)[/itex].

    When it comes to our universe, we don't really know how to judge the probability of a low-entropy universe versus a high-entropy universe. Some people (Alan Guth, from the above-posted video) believe that eternal inflation naturally leads to lots and lots of low-entropy universes, so maybe [itex]P(L)[/itex] is greater than Boltzmann thought.
  13. Nov 28, 2016 #12


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    Let me illustrate the problem with a simpler example:

    Suppose I thoroughly shuffle an ordinary deck of 52 cards. Then I deal out the first 4 cards, face-up. They happen to turn out to be, in order,
    1. Ace of spades
    2. King of spades
    3. Queen of spades
    4. Jack of spades
    People give two explanations for this result:
    • The deck was in descending order (according to the usual ranking of card values).
    • The deck was random, but there just happened to be those 4 cards on top.
    (For the purposes of this discussion, I'm just going to equate "random" with "NOT being in descending order").

    So we can give two conditional probabilities and two a-priori probabilities:
    • [itex]P(result| ordered) = 1[/itex]
    • [itex]P(result| random) \approx 1.5 \times 10^{-7}[/itex]
    • [itex]P(ordered) \approx 6 \times 10^{-90}[/itex]
    • [itex]P(random) \approx 1[/itex]
    It is very unlikely that a random arrangement of cards just happened to have Ace, King, Queen and Jack of spades as the first four cards. On the other hand, it is very likely (probability = 1) that an ordered arrangement had those cards as the first four. But the odds are tremendously against the deck being ordered. So the random explanation is by far the most likely.

    (Unless you propose a mechanism that makes ordered arrangements more likely--that would completely change the calculations.)
  14. Dec 12, 2016 #13
    It is supposed in cosmology that for a theory to be valid it must yield more regular observers than Boltzmann brains. But is that enough?

    For instance, we know that the overwhelming majority of Boltzmann brains will have disordered consciousness and observations, vastly different than what we experience. So the minority of BB observations will be like our.

    Now suppose that the number of ordered human observers throughout the history is a extremely large number. So large that all the observations that we consider normal and possible brain/mental states that are ordered have been realized in the history of our universe, but without duplication of humans.

    So the overwhelming minority of BBs that are ordered would be let's say, 1 percent of the total number of BBs - the total number of BBs is smaller but approximately identical to the number of normal observers. Then the BB minority would have a ratio of 1/100 to normal observers. If all the possible normal observers mental states have been realized there would be a real, real chance of duplication of a brain of an ordinary observer - so even the theory where BBs are in a minority would have some ludacris consequences.

    So my question is - what is wrong with this reasoning, and what is really meant when it is said that normal observers should outnumber BBs - do they have to outnumber them by a very, very big number so that the eventual duplication should become extremely unlikely, or is it extremely unlikely that every human brain configuration (or every possible person for that matter) is realized before the thermal equilibrium.

    Thanks for the patience.
  15. Dec 12, 2016 #14
    It seems the answer to your original question "Are the Boltzmann brains seriously taken by the scientific community?" is no; like ShayanJ said earlier, any observation made can be called a fake memory later. I like what the Wikipedia article says on this:
  16. Dec 14, 2016 #15
    Do a YouTube search for the biggest number ever printed in a scientific paper. I think the channel was numberphile. It dealt with something like this.

    I don't see any way to show that it's impossible, only that it's extremely unlikely. When you deal with very complicated things that require randomness, the time frames involved make heat death seem like next week by comparison.
  17. Dec 14, 2016 #16


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    I'm sure it has nothing on calling the Ackerman Function with Graham's Number as the arguments, though (https://xkcd.com/207/).

    In all seriousness, conscious brains are extraordinarily complex, and are necessarily extremely far from maximum entropy. It is certainly possible, according to the math, to produce a human brain out of thermal quantum fluctuations, but the frequency is small enough the possibility can be safely ignored in nearly all contexts.

    The one context where it does come up is if you have a universe that expands eternally but maintains a finite, non-zero temperature. In such a universe, there's a finite number of real brains produced, because heat death stops the production of new brains after a sufficient amount of time has passed. But after that there's an infinite amount of time to wait, such that even events with absurdly tiny probability will happen an infinite number of times.

    The Boltzmann Brain paradox shows that this cannot be right. Any number of things could solve the paradox. It could be that the method of counting probabilities is way off, so that it doesn't make sense to compare a few billion brains that exist right now to the very occasional but still infinite Boltzmann brains that may occur in the far distant future. It could be that the supposition that quantum fluctuations keep happening is wrong: perhaps the fluctuations stop once the universe reaches some final ground state. It could be that Boltzmann Brains are produced, but new pocket universes are produced in greater number, so that real brains vastly outnumber the Boltzmann Brains.

    There are lots of ideas here, but the main point is that this is a bit of arcane solipsistic discussion that isn't well-grounded in reality, as it relies upon a lot of unevidenced assumptions. For any real problem that matters for our lives here on Earth, Boltzmann Brains are irrelevant.
  18. Dec 15, 2016 #17


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    Last edited: Dec 15, 2016
  19. Dec 20, 2016 #18
    On wikipedia there is an article about the timeline of the far future:


    So it's stated that the estimated time for a Boltzmann brain to appear is 101050.

    It is also stated that the estimated time for the Universe to reach its final energy state is 1010120.

    Seems like a quite great difference in time. So even before the heat death/thermal equilibrium there seems to be a risk of overproducing BB-s, because the number of the estimated time for the heat death is so much larger than the time for a BB to appear.

    What is your take on this, and why or why not is this a problem? Is it valid to compare these two numbers?
  20. Dec 20, 2016 #19
    I don't think it's valid to compare the two numbers because of entropy. Things will happen at a slower rate as the universe ages.
  21. Dec 20, 2016 #20
    So the BB production rate/probability to happen would decrease in that age relative to now?
  22. Dec 20, 2016 #21
    Wouldn't the production of the new brains stop even way, way before heat death, because - for instance - the Sun will burn out way before heat death etc. So that gives a long period between the end of life and coming to thermal equilibrium. I doubt that, like you said, natural human brains can exist and make observations of disorder in a high entropy universe between the end of the Sun and the thermal equilibrium. What's your take on this?
  23. Jan 10, 2017 #22
    I've often red that the implications of an truly infinite universe are that every event we can imagine of can happen. Now this sounds pretty speculative and it's doubtful that probability has the same meaning in an infinite universe as it does in a finite universe, but are these consequences really true?

    For instance, is it reasonable to expect a huge number of Boltzmann brains already (not in the far future) if the universe is truly infinite? Also, is it reasonable to expect many of our copies to exist in the same way?

    Both conclusions seem counter-intuitive and weird to me, but it seems that this is what you get if you combine small-probability events in combination with infinity or large numbers.

    Thanks in advance for the answers.
  24. Jan 10, 2017 #23


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    There are, IMHO, a number of problems with applying a raw probability calculation to an infinite universe. First, you have to make some assumptions, which you don't know to be true. Second, you are dealing with probabilities so low that there is no practical way to verify anything.

    For example, suppose you conclude that there is another planet somewhere that is "identical" to Earth by some description. The condition of being identical can be as strong or as weak as you like. The strongest condition would be that everything (all the people and animals that have ever lived) is exactly the same. The number of days in a year, the calendar system, in 1815 there was a battle of Waterloo, between armies of precisely the same numbers with precisely the same DNA etc.

    If you ascribe a probability, no matter how small, that the Earth's history is as it is and not something else, then you get some finite number. And, if there are infinitely many planets out there each with the same finite probability of being identical to the Earth, then, of course, you conclude that (even down to the finest details) there are infinitely many planets identical to Earth.

    But, there are a lot of assumptions that underlie this calculation. First, that the probability of an Earth-like planet in all respects is common everywhere. If the universe varies so that the probability becomes less likely as we move away from our region, then the calculation might fail.

    There's also something of a paradox that creeps in, as follows:

    Suppose you toss an infinite number of coins, then you get an infinite number of heads and an infinite number of tails, with equal probability. Right?

    But, wait, how exactly do you do that? There is no way to toss an infinite number of coins. Instead, all you can do is toss a finite number of coins in a finite time and count what happens.

    So, "an infinite universe that has produced an infinite number of planets" is already a statement outside of (practical) probability theory.
    Instead, what you can do is go through each planet one at a time and check its relation to Earth. You could imagine that a computer model gives you the specification of each planet one at a time accoding to your probabilistic rules. So, there is no need for space travel to see what happens.

    But, let's assume anyway, that even checking a planet every Planck time for the current duration of the universe, you still haven't found another battle of Waterloo in the universe. Let alone everything else you need to be identical to Earth. The probability of your finding an Earth-like planet (in the precise sense) is almost infinitesimally small in this experiment. If you tried to calculate the probability, you wouldn't even be able to write the numbers down, they would be so small.

    So, if someone says there is another planet Earth out there, what does that actually mean?
  25. Jan 10, 2017 #24
    It seems to me, that in an infinite universe there are also an infinite number of possibilities, therefore you could never satisfy them all. I don't know, is that possible to express mathematically?

    I'm struggling for just the words to try to say it...
  26. Jan 10, 2017 #25


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    Yes, that's a good way to look at it.

    Imagine that the universe has produced an infinite number of planets and that the information about all of them is known - in some sense. If you went out and tested this information, you would never find an error.

    But, the totality of that information, for an infinite number of planets, represents something that does not have a finite probability. You can illustrate this with an old example of trying to generate an infinite random sequence. Any particular sequence has probability 0 (in a test, you would always diverge from the precise sequence at some point). So, in a mathematical sense, there is a paradox that the totality of information randomly generated by an infinite universe has 0 probability of occurring.

    Quite what the resolution to this paradox is, I'm not sure. But, I think that's where the people who say there are an infinite number of identical Earths out there go wrong.
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