4 26 March 2010 11:02PM

Abstract: If we assume that any universe can be modeled as a computer program which has been running for finitely many steps, then we can assign a multiverse-address to every event by combining its world-program with the number of steps into the world-program where it occurs. We define a probability distribution over multiverse-addresses called a Finite Occamian Multiverse (FOM). FOMs assign negligible probability mass to being a Boltzmann brain or to being in a universes that implements the Many Worlds Interpretation of quantum mechanics.

One explanation of existence is the Tegmark level 4 multiverse, the idea that all coherent mathematical structures exist, and our universe is one of them. To make this meaningful, we must add a probability distribution over mathematical structures, effectively assigning each a degree of existence. Assume that the universe we live in can be fully modeled as a computer program, and that that program, and the number of steps it's been running for, are both finite. (Note that it's not clear whether our universe is finite or infinite; our universe is either spatially infinite, or expanding outwards at a rate greater than or equal to the speed of light, but there's no observation we could make inside the universe that would distinguish these two possibilities.) Call the program that implements our universe a world-program, W.  This could be implemented in any programming language - it doesn't really matter which, since we can translate between languages by prepending some stuff to translate.

Now, suppose we choose a particular event in the universe - an atom emitting a photon, say - and we want to find a corresponding operation in the world-program. We could, in principle, run W until it starts working on the part of spacetime we care about, and count the steps. Call the number of steps leading up to this event T. Taken together, the pair (W,T) uniquely identifies a place, not just in the universe, but in the space of all possible universes. Call any such pair (W,T) a multiverse-address.

Now, suppose we observe an event. What should be our prior probability distribution over multiverse-addresses for that event? That is, for a given event (W,T), what is P(W=X and T=Y)?

## The mathematical universe: the map that is the territory

68 26 March 2010 09:26AM

This post is for people who are not familiar with the Level IV Multiverse/Ultimate Ensemble/Mathematical Universe Hypothesis, people who are not convinced that there’s any reason to believe it, and people to whom it appears believable or useful but not satisfactory as an actual explanation for anything.

I’ve found that while it’s fairly easy to understand what this idea asserts, it is more difficult to get to the point where it actually seems convincing and intuitively correct, until you independently invent it for yourself. Doing so can be fun, but for those who want to skip that part, I’ve tried to write this post as a kind of intuition pump (of the variety, I hope, that deserves the non-derogatory use of that term) with the goal of leading you along the same line of thinking that I followed, but in a few minutes rather than a few years.

Once upon a time, I was reading some Wikipedia articles on physics, clicking links aimlessly, when I happened upon a page then titled “Ultimate Ensemble”. It described a multiverse of all internally-consistent mathematical structures, thereby allegedly explaining our own universe — it’s mathematically possible, so it exists along with every other possible structure.

Now, I was certainly interested in the question it was attempting to answer. It’s one that most young aspiring deep thinkers (and many very successful deep thinkers) end up at eventually: why is there a universe at all? A friend of mine calls himself an agnostic because, he says, “Who created God?” and “What caused the Big Bang?” are the same question. Of course, they’re not quite the same, but the fundamental point is valid: although nothing happened “before” the Big Bang (as a more naïve version of this query might ask), saying that it caused the universe to exist still requires us to explain what brought about the laws and circumstances allowing the Big Bang to happen. There are some hypotheses that try to explain this universe in terms of a more general multiverse, but all of them seemed to lead to another question: “Okay, fine, then what caused that to be the case?”

The Ultimate Ensemble, although interesting, looked like yet another one of those non-explanations to me. “Alright, so every mathematical structure ‘exists’. Why? Where? If there are all these mathematical structures floating around in some multiverse, what are the laws of this multiverse, and what caused those laws? What’s the evidence for it?” It seemed like every explanation would lead to an infinite regress of multiverses to explain, or a stopsign like “God did it” or “it just exists because it exists and that’s the end of it” (I’ve seen that from several atheists trying to convince themselves or others that this is a non-issue) or “science can never know what lies beyond this point” or “here be dragons”. This was deeply vexing to my 15-year-old self, and after a completely secular upbringing, I suffered a mild bout of spirituality over the following year or so. Fortunately I made a full recovery, but I gave in and decided that Stephen Hawking was right that “Why does the universe bother to exist?” would remain permanently unanswerable.

Last year, I found myself thinking about this question again — but only after unexpectedly making my way back to it while thinking about the idea of an AI being conscious. And the path I took actually suggested an answer this time.