Boltzmann conjectured that the low entropy universe we live in could be the result of a very big fluctuation in entropy in a universe that stays most of the time in thermodynamic equilibrium. This scenario is called "Boltzmann universe". Some have argued that if this were the case then, with a high probability, we should not be really living in the universe that we "see", we are more likely to be just a single isolated brain experiencing an illusionary ordered universe that is not real. This is because this last event seems less improbable than the formation of the entire low entropic world that we think are experiencing. This lonely brain scenario is called "Boltmann brain".

To summarize we are comparing these two situations:

  1. Boltzmann hypothesis: a very improbable random fluctuation generates a low entropic localized state that is the low-entropy world/universe we experience;
  2. Boltzmann Brain scenario: a very improbable random fluctuation generates a low entropic localized state that has the shape of a small brain that accidentally experiences the illusion of a low-entropy world that doesn't exist.

The problem is to determine which is more likely to have happened between 1 and 2 (we are not considering other possibilities). The common argument about Boltzmann Brain is that (2) has much more probability than (1), I am going to show that it may be not the case. I will quote the known argument from Wikipedia and then I will develop my objection.

The Boltzmann brain thought experiment suggests that it might be more likely for a single brain to spontaneously form in a void, complete with a memory of having existed in our universe, rather than for the entire universe to come about in the manner cosmologists think it actually did.


In 1931, astronomer Arthur Eddington pointed out that, because a large fluctuation is exponentially less probable than a small fluctuation, observers in Boltzmann universes will be vastly outnumbered by observers in smaller fluctuations. Physicist Richard Feynman published a similar counterargument within his widely-read 1964 Feynman Lectures on Physics. By 2004, physicists had pushed Eddington's observation to its logical conclusion: the most numerous observers in an eternity of thermal fluctuations would be minimal "Boltzmann brains" popping up in an otherwise featureless universe. 

I argue that this line of reasoning is missing an important point having to do with conditional probability and Bayes law. 

In order to have a Boltzmann brain you need to have a very special fluctuation: the casual brain experience needs to be coherently "compatible" with the existence of a complex low entropy external world, and it woulnd't be the result of a deterministic evolution from simple law of physics: it would be stably coherent just as a result of a continuous sequence of completely casual improbable coincidences. This would make the single Boltzmann brain scenario far less probable then a generic structured brain popping up from a fluctuation. I mean: you don't have just to compare the unlikeliness of the single brain and the full observable universe, you also have to consider the unlikeliness of the coherence of the sequence of experiences of such a brain compared with the scenario of of the external world as a source of the coherence of the experieces. In the case of the low-entropy universe (Boltzmann Hypothesis) you have complexity emerging deterministically from fixed laws of nature, in the case of the single random brain you have a complexity that is just casually resemblig that of deterministic low-entropy world, in each single moment, and this is actually really unlikely.

To be more clear I will make an analogy: you are deriving a physical law with some experiment and you find a constant  involved in the law. Suppose that you are able to check the first decimal digits of  and and you can verify that they are remarkably the same digits of . What would you say it is the scenario with higher probability between the following:

  2. The first  digits are the same by chance and it is unlikely that the other digits will be the same.

You could be tempted to say that having  identical digits with  by random chance is far more unlikely than having only  and deduce that the first option is almost impossible. But suppose that there is an a priori probability that there exists a hidden geometric reason that make , then the a posteriori probability would be hugely increased when it is conditioned by the fact that the first  digitis were those of .

To be more precise let  be the a priori probability that  for a hidden geometrical reason. Now if we find that the first  digits are the same as  we have new a posteriori probabilities that we can compute with standard Bayes formulas: 

because for a digit to be equal to the same digit of  by random chance we can assume to have probability 1/10; on the other hand we have

and so by Bayes formula we have

and for any a priori value of  the fraction goes to 0 as . That means that the a posteriori probability of the sequence being similar to  by accident becomes insignificant when we find a lot of digits of  to be digits of .

As an example of a situation where the sequence of digits of  misteriously pops out from a physical experiment because of a hidden geometric reason you can check this link.

To go back from our analogy to the Boltzmann Brain scenario: collecting an increasingly large number of experiences that are coherent with the existence of an external low-entropy big world makes the probability of the actual existence of this world more and more close to 1 and the probability that this would be the result of a radcom sequence of coincidences more and more close to 0.

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I understand your post as trying to argue that Boltzmann hypothesis is more probable than the Boltzmann brain because our subjective experiences are consistent with physical laws.

If that’s the case, your conclusions are wrong. Boltzmann brain is still more probable in this scenario. The mistake you make is excluding the 3rd option that the reality we experience is objective for some other reason (a reason we might not understand yet).

Now, your post is a good argument in favor of 3rd option and against both Boltzmann brain and Boltzmann hypothesis.

Can we estimate the probability of this 3rd hypothesis or even compare it with the probability of the other two?

Surely. You just need to make some prior on the 3rd. Note, that the 3rd hypothesis is something along the lines of “there is some other model besides Boltzmann hypothesis or Boltzmann brain”, not a specific hypothesis. It can be big bang, god created the universe, or we live inside simulation. Whatever.

Now, you would have to assign unreasonably small prior to this for it not to end up many orders of magnitude more probable than either of Boltzmann Hypothesis or Brain.

The bits of a (Boltzmann or not) brain's beliefs is limited by the bits of the brain itself. So I don't think this really works.

OTOH in my view it doesn't make sense to have a policy to believe you are a Boltzmann brain, even if such brains are numerous enough to make ones with your particular beliefs outweigh non-Boltzmann brains with your beliefs, because:

  • such a policy will result in you incorrectly believing you are a Boltzmann brain if you aren't a Boltzmann brain, but if you are a Boltzmann brain you either:
    • have such a policy by sheer coincidence, or
    • adopted such a policy based on meta-level reasons that you have no reason to believe are reliable since they came from randomness, and
  • even if a Boltzmann brain did obtain correct knowledge, this would not pay off in terms of practical benefits 

In other words, you escape the standard argument by adding an observation, e.g. the observation that random fluctuations should almost never make our universe looks obeying physical laws.

One alternative way to see this point is the following: if (2) our brains are random fluctuations, then they are exponentially unlikely to have been created long ago, whereas if (1) it is our observable universe itself that comes from random fluctuations, it could equally have been created 10 billions years or 10 seconds ago. Then counting makes (1) much more likely than (2).

A true Boltzmann brain may have an illusion of the order in completely random observations. So the fact that my observations look ordered is not evidence that they are really ordered for me as BB. In short, we should nor believe BB's thoughts. And thus I can't disprove that I am BB just looking on my observation. 

But your argument may still be valid. This is because evolving fluctuations may be more probable than momentary fluctuations. For example, imagine infinite universe filled with low concentration of gas. This gas can form a brain directly for a second, it will be BB. But this gas can also form large but fuzzy blob, which will then gravitationally collapse into a group of stars, some of them will have planets with life and such planets will produce many brains. 

While mass of initial gas fluctuation is many orders of magnitude larger than one of the brain, it is less ordered and thus more probable. Thus normal worlds are more probable than BBs.


A true Boltzmann brain may have an illusion of the order in completely random observations.

Sure, like a random screen may happen to look like a natural picture. It’s just exponentially unlikely with picture size, whereas the scenario you suggest is indeed generic in producing brains that look like they evolved from simpler brains.

I meant not that 'random screen may happen to look like a natural picture", but that BB will perceive random screen as if it has order, because BBs are more likely to make logical mistakes.


That’s an interesting loophole in my reasoning, thanks! But isn’t that in tension with the observation that we can perceive noise as noise?

(yes humans can find spurious patterns in noise, but they never go as far as mistaking white noise for natural pictures)

Inability to distinguish noice and patters is true only for BBs. If we are real humans, we can percieve noice as noice with high probability. But we don't know if we are BB or real humans, and can't use our observations about the randomness to solve this.


Yes, that’s the crux. In my view, we can reverse…

Inability to distinguish noice and patters is true only for BBs. If we are real humans, we can percieve noice as noice with high probability.

… as « Ability to perceive noise means we’re not BB (high probability). »

Can you tell more about why we can’t use our observation to solve this?

Momentary BB (the ones which exist just one observer-moment) has random thought structure, so it has no causal connection between its observations and thoughts. So even if it percieve noice and think noice, it is just a random coincidence. 

However, there is a dust theory. It claims that random BBs can form chains in logical space. In that case, what is noice for one BB, can be "explained" in the next observer moment - for example random perception can be explained as static on my home TV.  There is an article about about it 


Thanks 👍

(noice seems to mean « nice », I assume you meant « noise »)

Your point is that in the case of the low entropy universe you have much possibilities for the time to consider for its random formation compared to the single brain?


Yes, although I see that more as an alternative intuition pump rather than a different point.

I think there is a misunderstanding as what counts as an observation. I think your point of view is something like  "Yesterday I experienced a universe consistent with the laws of physics, and the day before that I experienced a universe consistent with the laws of physics and so on. I'm about 30 years old so that's 10,950 observations" to which I ask "If you have all that knowledge, what did you have for dinner 78 days ago? Was it consistent with the laws of physics?" to make the rhetorical point that a boltsman brain does not need to fake any history, it only needs to fake your current memories of history, a much easier task. Boltzman brains are very similar to Last Thursdayism.