I have a question on Boltzmann brains which I'd like to hear your opinions on - mostly because I don't know the answer to this....
First of all - a Boltzmann brain is the idea that - in a sufficiently large and long-lived universe, a brain just like mine or yours will occasionally pop into existence by sheer fluke. It doesn't happen very often - in fact it happens very, very infrequently. The brain will go on to have an experience or two before ceasing to exist on the grounds that it's surrounded by vacuum and cold - which is not a very good place for a disembodied brain to be.
Such a brain would have a very short life. Well, by a greater fluke, some of them might last for a longer time, but the balance of probabilities is that most boltzmann brains that think they had a long life merely had a lot of false memories of this life, planted by the fluke of their sudden pop into existence. And in their few seconds of life, they never got to realise that they didn't actually live that life, and their memories make no sense.
Well, boltzmann brains don't pop into existence by fluke very often - in the whole life of the observable universe it's overwhelmingly likely that it's never happened.
What might be more likely to happen? Well, you could have half a Boltzmann brain instead, and by sheer fluke, have the nerves leading from that half-brain stimulated as if the other half was there during the few seconds of the half-brain's life. This is still extremely unlikely, but tremendously more likely than having a whole Boltzmann brain appear. And the half-brain still thinks it has the same experience as before.
There is of course nothing to stop us from continuing this. Suppose we have a one-quarter brain? Much more probable. One millionth? Even more probable. Maybe even single elements of a nerve cell? More probable still. The smaller the piece is, the less of a fluke you need for it to come into existence, and the less of a fluke you need to continue to supply all the same inputs that it would have had in the event that the whole brain appeared.
So we keep dividing down until we hit the opposite problem. As we divide and divide the Boltzmann brain, all the time ensuring that the part we had left had the same experience as before, we eventually get down to really simple processes that we can easily simulate, and which happen in the real universe all the time. And inputs which also happen all the time. This type of 'boltzmann brain' is extremely likely to happen.
I can sort of answer part of it. Let's forget about brains for a moment and assume it's a PC that just bubbled out of the vacuum, along with something that can power it for a bit. We can do the same thing - dividing it down into smaller parts until we get just a transistor - then below that maybe just a minimal switch made out of maybe some atomic collisions. The complete PC that bubbled up can run my web browser or whatever. The subdivided parts of it continue to act in the same way as before, because by fluke we keep providing their inputs. Right at the bottom of the stack we have a simple switch, or perhaps memory that can either be 1 or 0 - it's still doing the same things that allow the complete PC to run a complex program, but the complexity of that higher level is no longer there.
The problem is partially a sensory one. With the brain case, we accept that we have consciousness when the whole brain is there. It's still there when we separate the parts and wire them together. It's still there when we fluke out the inputs on one side?? It's still there when all the parts are not actually connected any more?
Over to you....