The mathematical universe: the map that is the territory


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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. As I worked on writing it up, I noticed that it sounded familiar. After I remembered what that Wikipedia article was called, and after actually looking up Max Tegmark’s papers on it this time, I confirmed that it was indeed the same essential idea. (Don’t you hate/love it when you find out that your big amazing groundbreaking idea has already been advocated by someone smarter and more important than you? It’s so disappointing/validating.) One of the papers briefly explores reasoning similar to that which I had accidentally used to convince myself of it, but it’s an argument that I haven’t seen emphasized in any discussions of it hereabouts, and it’s one which seems inescapable with no assumptions outside of ordinary materialism and reductionism.

I shall now get to the point.


Suppose this universe is a computer simulation.

It isn’t, but we’ll imagine for the next few paragraphs that it is.

Suppose everything we see — and all of the Many Worlds that we don’t see, and everything in this World that is too distant for us to ever see — is the product of a precise simulation being performed by some amazing supercomputer. Let’s call it the Grand Order Deducer, or G.O.D. for short.

Actually, let’s say that G.O.D. is not an amazing supercomputer, but a 386 with an insanely large hard drive. Obviously, we wouldn’t notice the slowness from the inside, any more than the characters in a movie would notice that your DVD player is being choppy.

Clearly, then, if G.O.D. were turned off for a billion years, and then reactivated at the point where it left off, we wouldn’t notice anything either. How about if the state of the simulation were copied to a very different kind of computer (say, a prototypical tape-based universal Turing machine, or an immortal person doing lambda calculus operations by hand) and continued? If our universe’s physics turns out to be fundamentally time-symmetrical, then if G.O.D. started from the end of the universe and simulated backwards, would we experience our lives backwards? If it saved a copy of the universe at the beginning of your life and repeatedly ran the simulation from there until your death (if any), would it mean anything to say that you are experiencing your life multiple times? If the state of the simulation were copied onto a million identical computers, and continued thence on all of them, would we feel a million times as real (or would there be a million “more” of each of us in any meaningful sense), and would the implausibly humanlike agent who hypothetically created this simulation feel a million times more culpable for any suffering taking place within it? It would be hard to argue that any of this should be the case without resorting to some truly ridiculous metaphysics. Every computer is calculating the same thing, even the ones that don’t seem plausible as universe-containers under our intuitions about what a simulation would look like.

But what, then, makes us feel real? What if, after G.O.D. has been turned off for a billion years… it stays off? If we can feel real while being simulated by a hundred computers, and no less real while being simulated by one computer, how about if we’re being simulated by zero computers? More concretely, and perhaps more disturbingly, if torturing a million identical simulations is the same thing as torturing one (I’d argue that it is), is torturing one the same as torturing zero?

2 + 2 will always be 4 whether somebody is computing it or not. (No Platonism is necessary here; only the Simple Truth that taking the string “2 + 2” and applying certain rules of inference to it always results in the string “4”.) Similarly, even if this universe is nothing but a hypothetical, not being computed by anyone, not existing in anything larger, there are certain things that are necessarily true about the hypothetical, including facts about the subjective mental states of us self-aware substructures. Nothing magical happens when a simulation runs. Most of us agree that consciousness is probably purely mechanistic, and that we could therefore create a conscious AI or emulate an uploaded brain, and that it would be just as conscious as we are; that if we could simulate Descartes, we’d hear him make the usual arguments about the duality of the material body and the extraphysical mind, and if we could simulate Chalmers, he’d come to the same familiar nonsensical conclusions about qualia and zombies. But the fact remains that it’s just a computer doing what computers always do, with no special EXIST or FEEL opcodes added to its instruction set. If a mind, from the outside, can be a self-contained and timeless structure, and the full structure can be calculated (within given finite limits) from some initial state by a normal computer, then its consciousness is a property of the structure itself, not of the computer or the program — the program is not causing it, it’s just letting someone notice it. So deep runs the dualist intuition that even when we have reduced spirits and consciousness and free will to normal physical causality, there’s still sometimes a tendency to think as though turning on a sufficiently advanced calculator causes something to mysteriously blink into existence or awareness, when all it is doing is reporting facts about some very large numbers that would be true one way or the other.

G.O.D. is doing the very same thing, just with numbers that are even more unimaginably huge: a universe instead of an individual mind. The distilled and generalized argument is thus: If we can feel real inside a non-magical computer simulation, then our feeling of reality must be due to necessary properties of the information being computed, because such properties do not exist in the abstract process of computing, and those properties will not cease to be true about the underlying information if the simulation is stopped or is never created in the first place. This is identically true about every other possible reality.

By Occam’s Razor, I conclude that if a universe can exist in this way — as one giant subjunctive — then we must accept that that is how and why our universe does exist; even if we are being simulated on a computer in some outer universe, or if we were created by an actual deity (which, from a non-intervening deity’s perspective, would probably look about the same as running a simulation anyway), or if there is some other explanation for this particular universe, we now see that this would not actually be the cause of our existence. Existence is what mathematical possibility feels like from the inside. Turn off G.O.D., and we’ll go on with our lives, not noticing that anything has changed. Because the only thing that has changed is that the people who were running the simulation won’t get to find out what happens next.


Tegmark has described this as a “theory of everything”. I’d discourage that use, merely as a matter of consistency with common usage; conventionally, “theory of everything” refers to the underlying laws that define the regularities of this universe, and whatever heroic physicists eventually discover those laws should retain the honour of having their theory known as such. As a metaphysical theory (less arbitrary than conventional metaphysics, but metaphysical nonetheless), this does not fit that description; it gives us almost no useful information about our own universe. It is a theory of more than everything, and a theory of nothing (in the same way that a program that prints out every possible bit string will eventually print out any given piece of information, while its actual information content is near zero).

That said, this theory and the argument I presented are not entirely free of implications about and practical applications within this particular universe. Here are some of them.

  • The simulation argument is dissolved. At this point, the idea of “living in a computer simulation” is meaningless. Simulating a universe should properly be viewed as comparable more to looking in a window than building the house. (Most of Robin Hanson’s thoughts about metaethics and self-preservation within a simulation are similarly dissolved, since a reality doesn’t pop out of existence when people stop simulating it; the only relevant part is the section about “If our descendants sometimes play parts in their simulations”, and this doesn’t seem to be the case anyway.)

  • As I mentioned, this significantly changes the dynamics of thought experiments like The AI In A Box Boxes You. Torturing a thousand identical simulations is the same as torturing one, and torturing one is the same as torturing zero — if and only if the structure within the simulation(s) is not being causally influenced by any ongoing circumstances in this universe. If it is, then the two realities are entangled to the point where they are essentially different parts of the same structure, and it is worth thinking about how much we should care about each one.

  • That leads me to a more general point about metaethics: although there are other realities out there where there are very sentient and very intelligent beings experiencing suffering literally 3^^^3 times greater than anything we can imagine, and others where there are beings experiencing bliss in the same proportions, we must resist the urge to feel (respectively) sorry for them or jealous of them. Your intuitive sense of what “really exists” should remain limited to this universe.

    Perhaps this caution only applies to me in the first place. I am, admittedly, the only person I know who has to leave the room when people are playing The Sims because I can’t stand to watch those little nowhere-near-sentient structures being put in torturous or even merely uncomfortable situations, so maybe it’s only my own empathy that’s a bit overactive. However, when we’re talking about sentient, sapient structures, we really do need to think about where to draw the line. I’d draw it at the point where a simulation starts to interact with this universe, in both directions — of course it will affect our universe if we are observing the simulation and reacting based on it, but we should only start caring about its feelings if we have designed the software such that it is affected by our actions beyond our choices for its initial conditions. That’s what I referred to as entanglement earlier. Once there’s that bilateral feedback, it’s no longer one structure observing another; they are both part of the same reality. (Take that as a practicality, not as a statement of an alleged metaphysical law. We’re trying to eliminate the need for metaphysical laws here.)

  • This theory results in a variation on the Boltzmann brain scenario: regardless of this universe’s ability to create Boltzmann brains, there’s also the possibility (and, therefore, necessity) of disembodied mind-structures hallucinating their own realities. My best guess as to the solution to this problem (if we’re to take it as a problem) is that any mind-structure that contains enough information to reliably hallucinate an orderly, mechanistic reality must be isomorphic to that reality.

  • It raises other strange anthropic questions too. The one that comes most immediately to my mind is this: If every possible mathematical structure is real in the same way that this universe is, then isn’t there only an infinitesimal probability that this universe will turn out to be ruled entirely by simple regularities? Given a universe governed by a small set of uniformly applied laws, there will be an infinity of universes governed by the same laws plus arbitrary variations, possibly affecting the internally observable structure only at very specific points in space and time. This results in a sort of anti–Occam’s Razor (Macco’s Rozar? Occam’s Beard Tonic?), where the larger the irregularity, the more likely it becomes over the space of all possible universes, because there are that many more ways for it to happen. (For example, there is a universe — actually, a huge number (possibly infinity) of barely different universes — identical to this one, except that, for no reason explainable by the usual laws of quantum mechanics, but not ruled out as a logically possible law unto itself, your head will explode as soon as you finish reading this post. I hope that possibility does not dissuade you from doing so, but I accept no responsibility if this does turn out to be one of those universes.)

    From the outside, this would appear to be a non-issue. Consider people in some other reality simulating this one (assuming that this one really is as simple and consistent as it appears). By some extraordinary luck, they’ve zoomed in on this exact planet in this exact Everett branch, and from there, they’ve even zoomed in on me writing this. “What does this guy mean,” they ask themselves, “wondering what the probability is that this particular reality will have the laws that it does? It’s not like anyone had any choice in the matter.” Yes, there will be versions of the universe that really are that orderly, and if this is one of them, than that would be why this universe’s version of me is wondering about the apparent astronomical unlikelihood of being in this universe. But from the inside, this seems terribly unsatisfying — if these slightly-irregular universes are possible, then we don’t know for sure what kind we’re in, so should we expect to find such irregularities? Perhaps such exceptions would constitute such a departure from quantum mechanics that they couldn’t be made consistent with it even as a special case. (Tegmark makes a related point in one paper: the hypothesis “does certainly not imply that all imaginable universes exist. We humans can imagine many things that are mathematically undefined and hence do not correspond to mathematical structures.”) Or perhaps the infinity of universes where such irregularities exist in places we’ll never observe (outside our light cone, in vast areas of empty space, etc.) is a much larger infinity (in probability density, not cardinality) than that of those universes where any of those irregularities will actually affect us. I’m leaning toward that explanation, but maybe a simpler one is that I’m reasoning about this incorrectly — after a conversation about this with Justin Shovelain, I’m reconsidering whether it’s actually correct to use probabilities to reason about an infinite space of apparently equally-likely items — or maybe this reasoning is correct and it observationally refutes the hypothesis. We’ll see.


One last comment: some people I’ve discussed this with have actually taken it as a reductio ad absurdum against the idea that a being within a simulation could feel real. As we say, one person’s modus ponens is another person’s modus tollens. Since the conclusion I’m arguing for is merely unusual, not inconsistent (as far as I can tell), that takes out the absurdum; therefore, in the apparent absence of any specific alternatives at all, you can weigh the probability of this hypothesis against the stand-in alternatives that there is something extraphysical about our own existence, something noncomputable about consciousness, or something metaphysically significant about processes equivalent to universal computation (or any other alternatives that I’ve neglected to think of).

Finally, as I mentioned, the main goal of this post was to serve as an intuition pump for the Level IV Multiverse idea (and to point out some of the rationality-related questions it raises, so we’ll have something apropos to discuss here), not to explore it in depth. So if this was your first exposure to it, you should probably read Max Tegmark’s The Mathematical Universe now.