Not sure I get what you’re getting at. Yes, the library has less information than most of the books contained within it—that’s the point. It illustrates that collections of things with a very high information content/complexity can have a very low information content/complexity. The BBrain paradox states that since a single brain fluctuating into existence is more likely than the whole universe existing and producing that same brain, we should take ourselves to be, with overwhelming likelihood, BBrains having completely false, accidental beliefs regarding being observers in an evolved universe. The fact that collections can be less complex than any proper part of them means that the universe, contrary to the assumption of the BBrain scenario, can actually be less complex than any given brain. Thus, it may be more likely that we are brains in a universe after all.
Basically, the BBrain paradox only gets off the ground if it’s true that if a brain has complexity K, then anything containing that brain has complexity K+k, making it less likely to pop into existence due to being more complex; but this is just wrong. The library simply illustrates that, but it can be more formally proven in algorithmic information theory.
Additionally, we have good reason to suppose that the universe is actually fairly simply describable: after all, the laws we know fit on the proverbial t-shirt, and it’s conceivable that the initial state isn’t itself very complex, too. Doing something similar for a human brain is ludicrously impossible.
It’s possible to milk this idea far more than I’m doing here, by the way. Jürgen Schmidhuber, for example, constructs whole theories of everything that way.
I totally agree that the universe is less complex in its initial state. If brains within that universe were constrained to pop into existence, then I’d agree. However it appears that brains come into existence through nonrandom processes, so it is hard to compare that probability of existence with that of a BB.
I’d suspect a genetic algorithm with a filter for Shakespearean text would be a lot faster than monkeys on typewriters.
(I remember a bunch of creationists blowing their lids and accusing him of circular reasoning, because the filter was designed to promote Shakespearean text. They would have been right if the model was intended to prove the random origin of Shakespearean text, but that wasn’t what the model was intended to demonstrate.)
But the point is exactly that brains in the universe do not pop into existence. Models susceptiple to the BBrain objection are typically those in which the universe is created by a random fluctuation. The argument is that the random fluctuation that leads to a single brain is much more likely than that leading to a whole universe, thus we should believe ourselves to be BBrains, rather than actual beings in an actual universe. But if the universe is acutally less complex in its initial state than a single brain, then that argument doesn’t get off the ground. You have to compare the likelihood of a single brain popping into existence with the likelihood of the universe that eventually contains that brain doing so, not the likelihood of a brain popping into existence within that universe—the latter comparison would not tell us anything about the likelihood that what we experience is veridical, vs. it all being just a random effect of the state in which the BBrain happened to pop up.
The brain is quite complex. A BBrain which also contains lots of state information is even more complex. What’s the probability of one popping up in a finite universe? How about the many that would be need to simulate continuity of experience? Somewhat less than my computer deciding to jump 10 feet to the left I’d think.
The chaotic initial universe, on the other hand, was not very complex. It would take a lot of information to fully define its structure, but the structure it took on (in terms of placement of atoms) is only one of many possible structures, each valid, while the BBrain is a very specific configuration out of many which are not valid.
Given that the universe was formed (and we can get anthropic here) we know that brains do eventually evolve.
To go back to the library, the full library is like the universe, which is not complex, and a single book is the BBrain. If you want to generate a specific book by selecting a subset from the library, it will take longer than generating one from scratch. However if you used certain rules to generate the book, say a fitness function which drives a random beginning pattern to the book, the book will be generated much faster. That’s a genetic algorithm, clearly appropriate in this case.
I don’t think Boltzmann meant a finite universe, or that it had to happen at the beginning. I presume he was thinking of an unending universe in which after our brief time would eventually devolve to an eternity of heat death. I believe he was saying that given the enormity of that time, the random emergence of even unlikely things would outnumber anything that happened in our vanishingly tiny epoch
What kinds of complex things can be expected to pop up during the era of proton decay, which will be most of time. What is the probability of a structure as complex as a BBrain popping up? What is the probability of one with the state we see? We’ve got to compare this to the probability of an OO.
In infinite time anything can happen, but for any t less then infinity the probability of a BBrain like us emerging is going to be very small.
To be honest, I’m not sure I can tell what you’re arguing for or against (you keep saying things that I’ve said myself in a manner as if you’re disagreeing with me by doing so), so I’d be grateful if you could clarify… In the meantime, I’ll try to clarify my own views a little further.
Well, the whole problem is that if the BBrain argument goes through, then we don’t know that the universe was formed—that’s where the problem lies. You only know that you have a certain experience, which could be explained by the universe having formed. But with the hypothesis of a spontaneous fluctuation, so the story goes, it would be more likely by far for a single brain with this experience—or even a brain with a chain of states corresponding to your continuing experience—to pop into existence, than it would be for a whole universe to do so. So our experience of the universe and all the stuff in it would typically be grossly mistaken.
To resist this conclusion, I’m offering a scenario in which an actual universe is ultimately less complex than a single human brain, so it’s no longer the case that the measure of intelligent observers over the (potentially infinite) extent (spatial and/or temporal) of the universe is no longer dominated by BBrains, but by proper observers in actual universes having veridical experiences of their being proper observers in actual universes. For this, it’s besides the point whether you intend to find a specific book in the library, so to speak, or what processes can be used to do so (btw, finding a book is exactly equivalent to constructing it: for consider the catalogue listing the contents of the library, ordered, say, alphabetically; to have enough entries to differentiate between all books, every entry must be as long as the book itself; so finding a book in the catalogue is the same as writing it). All you need to know is that the book exists if the library does, i.e. that a brain in the universe exists if the universe does.