Monday, July 30, 2012

Confessions of a Boltzmann Brain







[slightly revised July 31]


Anonymous said...

Be sure to also give consideration to the Boltzmann-you that is in a universe where you'll gain awesome super-powers if you choose to walk around quacking like a duck this very minute.

Jamie said...


I believe I can think of an exception: if time (or space) repeats itself than having an infinite amount of it doesn't guaranteeing infinite possibilities. It would just be an infinitely repeating finite sequence of events.

If the universe follows causal rules then all that has to happen is for the universe to reach any of its previous states and it would form a loop from there/then on.

As you go through more and more states on your way to the infinite possible states you postulate (that make Boltzman brains highly likely), it seems to me like your chances of hitting a previous state would increase. I guess the question is whether or not you create a Boltzman brain before you hit a previous state (although you'd want to hit at least two identical to yourself to believe that you're more likely to be a Boltzman brain than a conventional person).

There's also the possibility of infinite amounts of time in conditions that wouldn't allow for the formation of a Boltzman brain - like the heat death of the universe.

I'm not a cosmologist so I may be way off!

Anonymous said...

reasonable reasoning about the infinte would require some nasty definitions!

Eric Schwitzgebel said...

Thanks for the comments, folks!

Anon Mon: Unlike the BB hypothesis, the contending cosmological theories in my subjectively available current database don't imply that that scenario has a high relative likelihood -- though I suppose that on the infinite cosmos + random fluctuations hypothesis, there will an in infinite number of cases like that!

Jamie: That's right. As I originally drew up the post, I had a phrase about an assumption of cosmic diversity in there, which I omitted at the last minute on the thought that I could get away with a frequentist view of probability, in which case anything not in that loop would have an infinitesmal probability. But now I think it's probably better and clearer to just insert the qualification explicitly. I have now revised the post to do that.

Anon Tues: The infinite is always tricky, but I think the finite will do. Consider the number googolplex raised to the googolplex power. Call that number G. Now consider a universe that is the following size: G raised to the G raised to the G ... iterate a G times ... raised to the G Hubble lengths in spatial extent and that exists for 10 billion times that many years. That is a finite universe, but probably large enough to contain Boltzmann brains on contemporary understandings of the likelihood of such events. Or if not, name that number G' and iterate the procedure. And if that's not enough, name the result of that procedure G" and iterate again.

Unknown said...

Wouldn't there also be an infinite number of Boltzmann brains that happen to spontaneously congeal while incorporating a vacumm-proof casing and a convenient supply of nutrients, and with all the synapses preconfigured to present an illusion of memories of your wife?

In fact, wouldn't there also be an infinite number of Boltzmann brains that spontaneously congealed with a somewhat larger life-support system, in the form of a life-sustaining planet? The entire evolution, birth, society thing would then just be illusory. Real enough right now, but temporally as fake as a salted mine.

Hmm, a new wrinkle on the anthropic principle. I am the Boltzmann brain. This planet exists to sustain me. Kneel, peasants!

Well, excuse me while I go prove to the captcha that I am not a Boltzmann brain.

Matthew Pianalto said...


Is that true? (I've been reading Davidson, and his holism about belief--having one belief requires having many--leads me to wonder here whether the odds of a complex brain emerging/congealing are higher if lots of other things organize around it (or before) than that it congeals, but nothing else does.)

candid_observer said...

Just a quick response.

I think that at least for our own universe as we understand it, it's likely pretty easy to prove that the probability of the formation, as yet, of even one Boltzmann brain would be infinitesimal. Even the likelihood that, say, 100 molecules in a room might move to one corner of the room by chance is, apparently, less than 1 in 10 to the 80th power over the entire history of the universe to date. It's pretty obvious, I think, that such a relatively simple event would be only far more likely than the formation of a Boltzmann brain.

There are at least billions of non-Boltzmann brains in our universe. Hence the relative likelihood of the coming to be of a Boltzmann brain to the coming-to-be of a non-Boltzmann brain, given a fixed amount of matter at a fixed density, would almost certainly be utterly minuscule.

I'd also dispute whether it's obvious that if our universe as we understand it is infinite in time, then a Boltzmann brain must come about.

If the universe expands forever, then the density of matter -- which is finite -- goes down forever. If the expansion never reaches a limit, then the density approaches zero. The probability of the formation of a Boltzmann brain is, most likely, going to be greatest at a density not unlike that of the current time. Past the point of greatest probability (or at a point not far removed from it), the probability will almost surely simply go down monotonically with time. It's simply not obvious that the integral of all the probabilities across infinite time will diverge -- which divergence would imply that a Boltzman brain (indeed an infinite number of them) must come to be. The integral may instead be finite -- indeed perhaps infinitesimally small.

candid_observer said...

In my comment above,

If the expansion never reaches a limit

Should be something like:

If the expansion is not bound under a limit

candid_observer said...

I guess as I think about it, if the integral I described above DID diverge, implying an infinite number of Boltzmann brains, then, contrary to what I had said in my first paragraph, the probability across the infinity of time that a brain was non-Boltzmann would actually be zero or infinitesimal. The only density at which the probability of non-Boltzmann brains would be higher would be, presumably, near our current density.

Eric Schwitzgebel said...

Thanks for the continuing interesting comments folks!

@ clasqm: Yes, but if you will allow me to compare probabilities of infinitely instantiated events, the nutrient-available event will be much less likely.

@ Matthew: Good point! I'm making assumptions about random broadly Boltzmannian/quantum mechanical congealment, and comparing that to the odds of a brain emerging from a full-on cosmological emergence from a Big Bang or other big cosmic origin. But there is an intermediate kind of case that you point to that I didn't assess: The odds of the brain arising neither from a full-on big cosmic origin nor from random Boltzmannian congealment in a moment but rather from some set of quirks in which a durable brain-sustaining environment is first created. It's an interesting thought! I'm not sure how to assess it, but if I'm flinging out armchair subjective credences, that one doesn't seem to me justifiably to deserve a higher credence than a Big Bang or a Boltzmann burst, since I haven't heard a plausible physical story of how it might go that isn't just a lower-probability version of the Boltzmann thing. If I go ahead and give it 1/3 subjective likelihood conditional on an infinite universe, I still get a non-trivial subjective probability that I'm a Boltzmann brain by the reasoning in the post.

Eric Schwitzgebel said...

Candid: Those are helpful comments. To be clear, I am assuming (as Jamie encouraged me to make explicit in my update of the post) a certain amount of cosmic diversity. My impression is that most infinite cosmologies will contain sufficient diversity. But some might not. I think you are probably right in your point that if there was only a single universe and it were to expand quickly enough, that might sufficiently reduce the probability of a Boltzmann Brain over time that we might not expect one even over an infinite period of time.

Eric Schwitzgebel said...

But unless we're highly that such a cosmology is the right one, there still remains the non-trivial chance that the real cosmology is Boltzmann Brain friendly and we're back in our main problem.

Eric Schwitzgebel said...


Charles T. Wolverton said...

This strikes me as essentially an attempt to "naturalize" Davidson's Swampman thought experiment, but I don't think it works. Eg, it seems to implicitly assume that "molecule-for-molecule identity" yields at worst a countably infinite sample space. But producing a particular brain's structure requires molecular position information, which means the sample space is not countable. In which case, even an infinite number of trials doesn't guartantee the occurrence of a specific structure.

In any event, the part of a swampman-type thought experiment (SM) that interests me is the memory issue: if brain/person Y is a structurally exact duplicate of brain/person X, why doesn't Y have exactly the same memories as B? And to the somewhat limited extent that I understand externaliism, I don't see that it applies here since I understand it to address the "natural" (ie, in our world as we know it) formation of mental states. But an SM assumes a decidedly unnatural formation.

Davidson apparently takes experience per se to be a key element of his version of externalism, and that's fine for natural memory formation. But the fact that Y has no experiential history just means that a version of externalism defined so as to require such history isn't applicable to an SM. Which doesn't seem to be a problem since the memory question posed by an SM isn't "is externalism true?" but rather "however Y is formed, does Y have X's memories?", presumably an empirical question.

My assumption is that whatever the precise mechanism of memory formation may be, the result is an exactly specifiable (in principle and modulo quantum effects) molecular structure for relevant parts of a brain. That structure will have been determined by both nature and nurture: genetics and experiences. If so, at the time of Y's emergence from any hypothesized duplicating process, Y will have exactly the memories that X has. In particular, while it's true that Y's having memories of X's wife won't be due to Y's having met her, Y will have those memories - including the memory of having met her.

So far, the discussion of SMs that I've encountered agree (at least implicitly) with Davidson and Eric, so I have to assume I'm missing something. But what?

Eric Schwitzgebel said...

Charles, I'm not sure I'm getting your first argument about non-countability. Shouldn't Boltzmannian chance be able to produce any combination of positions and momentums with finite probability? If we go quantum mechanical, shouldn't the wave equations be able to do effectively the same thing with quantum properties?

On your second point, I think most people would grant that Swampman would have my (or Davidson's) *apparent* memories, but *genuine* memory requires the right sort of causal connection to the remembered facts or events. However, some people like Dretske (1995) argue that Swampman will differ in phenomenology too (perhaps having none) because of this difference about causal origins and thus functions, and so if apparent memory is phenomenologically characterized Swampman won't even have that.

Zach Barnett said...

Charles, Eric's argument requires that there be a non-zero probability of a Boltzmann brain arising. It's true that genuinely possible events can have probability zero if they comprise a vanishingly small subset of an infinite set of possibilities (consider, for example, the probability of flipping a coin infinite times and getting tails every time). However, I believe that the probability of a Boltzmann brain arising is > 0. If you doubt this, I can explain why I think so. It's sort of a boring argument, though.

Eric, I've thought about this issue myself. We are Boltzmann brains, of a sort. We arose spontaneously, from (once) disorganized matter. Yes, evolutionary processes spent millenia organizing and reorganizing this matter before it gave rise to us, but why should this matter? Evolution is one prominent way for matter to arrange itself into an Eric-Schwitzgeblian Boltzmann brain. Indeed, this is overwhelmingly likelier than having it happen randomly as imagined.

So. If the universe is large enough, diverse enough, and long-lasting enough to make it probable that a *true* Eric-Schwitzgeblian Boltzmann brain will arise, then it should be probable that there are already millions upon millions of standard Eric Schwitzgeblian brains out there on different planets living relatively normal lives. And I see no reason for you to worry that you're the 'lucky' one.

Charles T. Wolverton said...

Re point 1, I assume the idea is that "congealing" (whatever that means in this context) produces a molecular structure that can be represented by something like a 4-tuple of molecule ID and 3-D position in some coordinate system (possibly other specs are needed in which case it might be an n-tuple with n>4, but four is enough for my argument - in fact, so is two). If so, the sample space of possible structures is uncountably infinite, so the probability that any given structure is produced by even an infinite number of congealing "trials" must be zero. (Actually, I think that's true even for the countably infinite - assuming an eternal universe - set of all structures that currently exist and ever have or ever will exist.)

When people use the kind of argument in question, I usually get the idea that they are implicitly assuming that to occur, an event must have non-zero probability. True for finite sample spaces, but false for infinite sample spaces. Eg, consider any real number in any interval of the real line. The probability that you "randomly" (ie, with equal probability) pick any particular number in that interval clearly must be zero notwithstanding that you will pick some number. And as above, I think nothing changes if I allow you to specify arbitrary sets of numbers for sets no larger than countably infinite - eg, all rational numbers - and allow you to continue picking through eternity. I may not have it quite right, but the formal statement is roughly that the measure (ie, probability fin the case of measures with appropriate characteristics) of any countably infinite subset of any interval of the real line is zero.

All of that should be qualified by "as best I can recall" - I'm digging into memory from decades ago.

As for point 2, I have no idea what an "apparent" memory might be. My understanding of memory (from reading Eric Kandell's "In Search of Memory") is as I described - a biological structure. If so, to claim he existence of an "apparent" memory would be to claim the existence of an "apparent" biological structure - which seems clearly wrong. My essential argument is that if a memory is a biological structure, that structure captures all of the causal effects leading to it.

If a memory isn't such a structure, then what and where is a memory - "genuine" or otherwise - and how is it formed? (BTW, FWIW I have speculative answers to those questions for "genuine" memories.) The problem I have with claims to the contrary (and Dretske's is one I have read) is that (again, as best I recall) they don't explicitly state what they are assuming about the structure of memory. I assume the structure is completely analogous to the structure of any complex mechanism, so that duplicating the structure duplicates the function. If memory is actually more than such a structure, exactly what is that "more"? I didn't read Kandell as claiming anything other than biological structure - although again, I may not be remembering everything in his discussion.

PS to Zach:

Obviously I do doubt, but also thrive on boring arguments - so, fire away! Altho preferably along the lines of my arguments above - I take my probability neat.

Zach Barnett said...

I'll make it quick. Suppose space is quantized. Suppose exactly one arrangement results in a Boltzmann brain, and all of the rest do not. Let's call this exact scenario "Success" and the rest of the possibilities "failures." Obviously, in this case, p(success) > 0.

Now suppose space is not quantized. The probability of *Success* is now zero, since success a lone possibility in an infinite set. But consider the set of all alternative possibilities that are closer to Success than to any case of failure (e.g. completely identical to Success, except a single particle is .00000002 angstroms to the left). On my view, these are all going to be "successful" even if they don't technically qualify as "Successful."

And if these are all successful, the probability of "success" (without a capital 'S') is going to be roughly equal to what it was in the case where space was quantized. And that was non-zero!

Charles T. Wolverton said...

By quantized, I assume you mean "conceptually divided up into small regions", and that you are taking space to be finite so that there are a finite number of such regions. I infer that these regions are roughly the size of an atom and that an "arrangement" is something like a finite set of 2-tuples each comprising an atom ID and a region ID. (See note below.) I'm not sure what you mean by "exactly one arrangement results in a Boltzman brain", but assume it's something like "for a particular organism structure, the result of a trial (ie, a congealing) is Success if that arrangement emerges".

If this is right, then I agree that the sample space of arrangements though huge is nonetheless finite, so equal non-zero probability of being the result of a "congealing" trial can be assigned each arrangement in the sample space. Then in an infinite sequence of such trials the probability that any particular arrangement does not occur will converge to zero as the number of trials increases.

So, the remaining question seems to be: how close to literally "identical" does "molecule-for-molecule identical" have to be in order to preserve whatever characteristics of the duplicated organism are of interest. Your argument for the non-quantized case seems to amount to claiming that small displacements of the atoms' locations within a sufficiently small quantizing region don't affect those characteristics. I have no idea whether that's so, but it doesn't strike me as obviously implausible. So, since I'm not really interested in that aspect of the thought experiment, I'm happy to concede that point. Congrats - your gold medal is "in the mail"!

Note: I just noticed that in my previous comment I mistakenly said that I assumed the structure of an organism to be comprised "a 4-tuple" instead of "a finite set of 4-tuples". Mea culpa.

Eric Schwitzgebel said...

Zach & Charles -- I agree with Zach's point. Another way to put it is that as long as there is some finite (although potentially very small) error region that is irrelevant to the Eric-Schwitzgeblianness of the brain, the point about the zero probability of a perfect match seems to be undercut.

Zach: Why do you think that the odds of an (Earthlike?) evolutionary process producing an Eric-Schwitzgeblian mind are much higher than the odds of it arising Boltzmannishly? One way to think about it is frequencies of human-evolution-supporting universes vs random congealments of Boltzmann brains in very large finite samples of an infinite universe. What I'm puzzled about is whether there is a reason to be confident that the first is vastly more frequent than the second.

There is still the point that there will be many evolutionarily arising duplicates of me in an infinite universe (of the right sort) too. Topic of a future post, I hope!

Anonymous said...

Hello, Charles Martelly here,

Infinity means to you apparently that it is necessarily the case
"that any event of finite probability will occur an infinite number of times". But I see no necessity in this; it is rather an assumption. It seems to me that one may coherently posit that an infinite universe might have only one such occurance, if any.
Also involved is an assumption that the same event may be repeated, whereas one may argue alternately that all events howsoever much they resemble previous events, are unique, since time does not repeat.
I find the notion of randomness odd. To generate random numbers for instance requires a highly regulated context. It seems to me one may argue that such numbers are the result of order and that randomness generally is just a particular kind of order, a particular kind of determination.
How would you know if the possibility of infinite universe

Anonymous said...

Hi Charles Martelly here, again
Sorry for the glitch, here is
my complete comment:
How would you know if the possibility of infinite universe is non-trivial? You have no experience with universes, trivial or non-trivial, to be able to calculate the probability, seems to me. If you'd had such experience perhaps you could say, "oh yes given the number of infinite versus non-infinite universes existing, I can say that the possibility of this as an infinite universe is non-trivial." Otherwise, how would you know? Probability is based upon experience.

Marshall said...

The present universe is undoubtedly finite in space and apparently so in time, according to current best thinking. Your Argument depends on the existence of an infinite Multiverse; which, fortunately for you, is not implausible.

But that isn't enough, not all infinities are the same. For a good time, read about Hilbert's Grand Hotel, which, although it's full, can still accommodate more guests. But that is about countable infinities, like the set of integers or the set of rational numbers. There are sets that the Hotel can't accommodate, such as the set of numbers that can be represented by infinitely long decimal fractions; Cantor showed that. In general, the set of all subsets of an (infinite) set is larger than the original set. So for your argument, you need to have the number of universes to be at least of the same order as the number of possible arrangements of whatever would be randomly arranged to create a Boltzmann Brain. Could be, I suppose.

Zach Barnett said...

Consider a region of space far from Earth. The region, which we can call R, is large enough to contain several stars. Let's let t be the duration over which we are observe what happens in R.

Let's compare the probabilities of two states of affairs:
(A) R will contain at least one Boltzmannishly-arising intelligent life-form during t.
(B) R will contain at least one evolutionarily-arising intelligent life form during t.

For simplicity, let's assume that the conditional probability of an Eric-Schwitzgeblian Brain (ESB) arising given A is equal to the conditional probability of an ESB arising given B. (I believe it's higher given B, but this is not required for my argument.) So let's compare (A) and (B) now.

I think you will grant that P(A) is going to be Tiny. I think P(B) is much higher. We can do a Drake-like calculation to make a rough estimate. Our estimate will depend on: the number of stars that will ever exist in R, the expected number of planets per star that could possibly support life, the probability that life will at some point arise on such a planet, and the probability that life - once it arises - will become intelligent.

We multiply those together. Or something. Bold scientists have estimated that there ought to be thousands of instances of intelligent life in our galaxy alone. More conservative estimates say that we should expect intelligent life to occur ~once per billion stars. Even this pessimism still, to me, renders P(B) much likelier than P(A).

But let's say you don't like their estimates, for some reason. Maybe you think, for some reason, that P(A) = P(B). This would astonish me, but let's suppose it's accurate. I still think it's overwhelmingly more likely that an ESB would arise evolutionarily than Boltzmannishly.

The reason is that once intelligent life arises evolutionarily, the seeds are sowed for much more of it. Intelligent life arose on Earth, and it is estimated that over 100 billion people have lived up to now. There could be billions more that live before we die out. I don't think that this is unusual for a species that reaches this point. A planet on which intelligent life evolves will get billions and billions of attempts to make an ESB. It's still going to be unlikely to occur, but billions and billions of attempts is better than just one.

Anonymous said...

I don't think you need either an infinite multiverse or a single universe of infinite spatio-temporal extent. You could make do with infinitely many undetectable fields existing on our space-time in which structures like Boltzmann brains can arise. You could even make do with there being just one extra non-quantised field, which can support brain-like structures on infinitely many scales.

Eric Schwitzgebel said...

@ Zach: Very nicely put. I think that if one assumes that the arbitrarily selected region will be populated with stars of roughly the familiar sort, then probably your reasoning works. So one issue is whether we can assume that. And that issue in turn might depend on the frequency Big Bangs (or similar events that lead to the formation of billion-year stars) in multiverse theory compared to the frequency of Boltzmannian fluctuations from whatever medium the Big Bang arose from. I find myself warming to your reasoning, though.

Eric Schwitzgebel said...

@ Anon Aug 4: Yes, that sounds right!

Eric Schwitzgebel said...

@ Marshall: Yes. Also probably there is some tolerance for irrelevant differences between the Boltzmann Brain and my own. If we allow every particle to be in the right position within 1 picometer, that probably would suffice for identical or nearly identical subjective experience.

Eric Schwitzgebel said...

@ Charles: I mean subjective probability -- my most rational best guess, which has to be grounded in whatever limited experience I happen to have -- and I mean type-identical not token-identical. Yes?

Charles T. Wolverton said...

Well, I don't know about "subjective" probability nor how type-token applies (and I mean "don't know" literally, not suggesting disagreement). But I will note that infinity is a limit concept, and as one lets the "finite" sample space size and the "finite" time available both increase without bounds you get closer and closer to the conceptual scenario of an infinite number of possibilities each with zero probability of occurrence. In which case it may not be true that every possible event has to occur.

As implied by Anonymous in comment 3 above, when dealing formally with infinities (and sometimes even non-infinities: see "Monte Hall problem" at wiki) you really need to move from verbal "analysis" to equations.

re memories:

Although I think what I said earlier was right, it was incomplete. In my hypothesized model, a memory is a brain structure that implements a context dependent disposition to respond to stimuli, external or internal. Thus, ES-BB might not experience certain memories of ES's life because the context would never occur. Other memories might require only internal context (ie, physiological state), but if ES-BB is essentially a brain-in-a-vat, perhaps even those would never occur. However, all memories would nonetheless be latent.

That still leaves me in conclict with thoase who - like Davidson - claim that SM-DD can't experience DD's memories. Ditto for ES-BB.

Zach Barnett said...

Hm. I guess my argument is a bit this-universo-centric. But I'm glad you find it compelling nonetheless.

Now if only I could persuade you in the USA consciousness case... Or maybe you will persuade me! :)

Anonymous said...

This is the sort of thing I think about too. But I'm with Zach - while the formation of my brain as per standard theory via evolution is unlikely, it isn't nearly as unlikely as it appearing out of nowhere...

What I sometimes wonder, however, is if my consciousness actually spans all of the identical scenarios until the point at which they diverge.


Charles T. Wolverton said...

Eric -

re type- vs token-identity:

Since my last response, I've gotten through another chapter of Hylton's book on Quine, in particular one that addresses that distinction in the context of DD's anolmalous monism. So, I can now better respond to your query.

I assume a memory - and in general, any "mental state", eg a PA - is a context-dependent disposition to respond to a specific sensory stimulation in a specific way, and I assume that each such disposition is implemented as a specific neurological network. Therefore, as I understand the term, I am assuming "token-identity" between a mental state and some physical state. However, given that model I don't even see what "type-identity" would mean since I have no concept of a, say, memory "type". Eg, to use a common example, although I have some fairly precise memory tokens from a visit to Vienna (eg, a humorous comment by a table mate in Grinzing circa fall 1969), I don't know what a memory type "of Vienna" - or a corresponding physical type - would even mean. The former would each be implemented in a neurological network, but it's not clear to me that those networks would belong to a physical type that would be distinguishable from a physical type to which memories of a more recent trip to the Amalfi coast would belong.

Thus, in assuming that "molecule-by-molecule identical" duplication would preserve memory I was (implicitly) assuming token identity - although at the time it didn't occur to me that I was or that doing otherwise was an option!

Anonymous said...

The idea of a Boltzmann brain seems to rely on the assumption that having a specific kind of developmental history is not necessary for a thing of a certain kind to come into existence. This seems false to me, but it's at least a pretty substantial assumption, isn't it? To motivate the falseness: it seems impossible to me for a given sort of tree to come about without undergoing some process of biological development. Removing any historically extended developmental process that leads to its coming into existence seems to remove an essential feature of its being the kind of thing it is, and it seems to make nonsense of the idea of its continuing the mature processes that make it the kind of thing that it is after it emerges. There's nothing to continue...

Eric Schwitzgebel said...

Anon Aug 9: So the question would then be, do I know what kind of thing I am? Suppose, for example, real memories are distinguished from merely apparent memories by their causal history. Sure! But then the worry is how I know I have real memories as opposed to merely apparent ones.

Anonymous said...

I see, I guess to make a point that would be more threatening to the example I'd have to say something a little stronger than just saying that there is a substantial assumption about possible development in the universe (as long as it's conceivable as potentially being really possible, the skeptical doubt seems to at least not be irrational) and instead say something like "a Boltzmann brain, although conceivable or imaginable in some sense, could not possibly actually exist" which would make it irrational to wonder whether I know I really am of that kind.

Anonymous said...