In my previous post, I introduced the idea of an "l-zombie", or logical philosophical zombie: A Turing machine that would simulate a conscious human being if it were run, but that is never run in the real, physical world, so that the experiences that this human would have had, if the Turing machine were run, aren't actually consciously experienced.
One common reply to this is to deny the possibility of logical philosophical zombies just like the possibility of physical philosophical zombies: to say that every mathematically possible conscious experience is in fact consciously experienced, and that there is no kind of "magical reality fluid" that makes some of these be experienced "more" than others. In other words, we live in the Tegmark Level IV universe, except that unlike Tegmark argues in his paper, there's no objective measure on the collection of all mathematical structures, according to which some mathematical structures somehow "exist more" than others (and, although IIRC that's not part of Tegmark's argument, according to which the conscious experiences in some mathematical structures could be "experienced more" than those in other structures). All mathematically possible experiences are experienced, and to the same "degree".
So why is our world so orderly? There's a mathematically possible continuation of the world that you seem to be living in, where purple pumpkins are about to start falling from the sky. Or the light we observe coming in from outside our galaxy is suddenly replaced by white noise. Why don't you remember ever seeing anything as obviously disorderly as that?
And the answer to that, of course, is that among all the possible experiences that get experienced in this multiverse, there are orderly ones as well as non-orderly ones, so the fact that you happen to have orderly experiences isn't in conflict with the hypothesis; after all, the orderly experiences have to be experienced as well.
One might be tempted to argue that it's somehow more likely that you will observe an orderly world if everybody who has conscious experiences at all, or if at least most conscious observers, see an orderly world. (The "most observers" version of the argument assumes that there is a measure on the conscious observers, a.k.a. some kind of magical reality fluid.) But this requires the use of anthropic probabilities, and there is simply no (known) system of anthropic probabilities that gives reasonable answers in general. Fortunately, we have an alternative: Wei Dai's updateless decision theory (which was motivated in part exactly by the problem of how to act in this kind of multiverse). The basic idea is simple (though the details do contain devils): We have a prior over what the world looks like; we have some preferences about what we would like the world to look like; and we come up with a plan for what we should do in any circumstance we might find ourselves in that maximizes our expected utility, given our prior.
In this framework, Coscott and Paul suggest, everything adds up to normality if, instead of saying that some experiences objectively exist more, we happen to care more about some experiences than about others. (That's not a new idea, of course, or the first time this has appeared on LW -- for example, Wei Dai's What are probabilities, anyway? comes to mind.) In particular, suppose we just care more about experiences in mathematically really simple worlds -- or more precisely, places in mathematically simple worlds that are mathematically simple to describe (since there's a simple program that runs all Turing machines, and therefore all mathematically possible human experiences, always assuming that human brains are computable). Then, even though there's a version of you that's about to see purple pumpkins rain from the sky, you act in a way that's best in the world where that doesn't happen, because that world has so much lower K-complexity, and because you therefore care so much more about what happens in that world.
There's something unsettling about that, which I think deserves to be mentioned, even though I do not think it's a good counterargument to this view. This unsettling thing is that on priors, it's very unlikely that the world you experience arises from a really simple mathematical description. (This is a version of a point I also made in my previous post.) Even if the physicists had already figured out the simple Theory of Everything, which is a super-simple cellular automaton that accords really well with experiments, you don't know that this simple cellular automaton, if you ran it, would really produce you. After all, imagine that somebody intervened in Earth's history so that orchids never evolved, but otherwise left the laws of physics the same; there might still be humans, or something like humans, and they would still run experiments and find that they match the predictions of the simple cellular automaton, so they would assume that if you ran that cellular automaton, it would compute them -- except it wouldn't, it would compute us, with orchids and all. Unless, of course, it does compute them, and a special intervention is required to get the orchids.
So you don't know that you live in a simple world. But, goes the obvious reply, you care much more about what happens if you do happen to live in the simple world. On priors, it's probably not true; but it's best, according to your values, if all people like you act as if they live in the simple world (unless they're in a counterfactual mugging type of situation, where they can influence what happens in the simple world even if they're not in the simple world themselves), because if the actual people in the simple world act like that, that gives the highest utility.
You can adapt an argument that I was making in my l-zombies post to this setting: Given these preferences, it's fine for everybody to believe that they're in a simple world, because this will increase the correspondence between map and territory for the people that do live in simple worlds, and that's who you care most about.
I mostly agree with this reasoning. I agree that Tegmark IV without a measure seems like the most obvious and reasonable hypothesis about what the world looks like. I agree that there seems no reason for there to be a "magical reality fluid". I agree, therefore, that on the priors that I'd put into my UDT calculation for how I should act, it's much more likely that true reality is a measureless Tegmark IV than that it has some objective measure according to which some experiences are "experienced less" than others, or not experienced at all. I don't think I understand things well enough to be extremely confident in this, but my odds would certainly be in favor of it.
Moreover, I agree that if this is the case, then my preferences are to care more about the simpler worlds, making things add up to normality; I'd want to act as if purple pumpkins are not about to start falling from the sky, precisely because I care more about the consequences my actions have in more orderly worlds.
Imagine this: Once you finish reading this article, you hear a bell ringing, and then a sonorous voice announces: "You do indeed live in a Tegmark IV multiverse without a measure. You had better deal with it." And then it turns out that it's not just you who's heard that voice: Every single human being on the planet (who didn't sleep through it, isn't deaf etc.) has heard those same words.
On the hypothesis, this is of course about to happen to you, though only in one of those worlds with high K-complexity that you don't care about very much.
So let's consider the following possible plan of action: You could act as if there is some difference between "existence" and "non-existence", or perhaps some graded degree of existence, until you hear those words and confirm that everybody else has heard them as well, or until you've experienced one similarly obviously "disorderly" event. So until that happens, you do things like invest time and energy into trying to figure out what the best way to act is if it turns out that there is some magical reality fluid, and into trying to figure out what a non-confused version of something like a measure on conscious experience could look like, and you act in ways that don't kill you if we happen to not live in a measureless Tegmark IV. But once you've had a disorderly experience, just a single one, you switch over to optimizing for the measureless mathematical multiverse.
If the degree to which you care about worlds is really proportional to their K-complexity, with respect to what you and I would consider a "simple" universal Turing machine, then this would be a silly plan; there is very little to be gained from being right in worlds that have that much higher K-complexity. But when I query my intuitions, it seems like a rather good plan:
- Yes, I care less about those disorderly worlds. But not as much less as if I valued them by their K-complexity. I seem to be willing to tap into my complex human intuitions to refer to the notion of "single obviously disorderly event", and assign the worlds with a single such event, and otherwise low K-complexity, not that much lower importance than the worlds with actual low K-complexity.
- And if I imagine that the confused-seeming notions of "really physically exists" and "actually experienced" do have some objective meaning independent of my preferences, then I care much more about the difference between "I get to 'actually experience' a tomorrow" and "I 'really physically' get hit by a car today" than I care about the difference between the world with true low K-complexity and the worlds with a single disorderly event.
In other words, I agree that on the priors I put into my UDT calculation, it's much more likely that we live in measureless Tegmark IV; but my confidence in this isn't extreme, and if we don't, then the difference between "exists" and "doesn't exist" (or "is experienced a lot" and "is experienced only infinitesimally") is very important; much more important than the difference between "simple world" and "simple world plus one disorderly event" according to my preferences if we do live in a Tegmark IV universe. If I act optimally according to the Tegmark IV hypothesis in the latter worlds, that still gives me most of the utility that acting optimally in the truly simple worlds would give me -- or, more precisely, the utility differential isn't nearly as large as if there is something else going on, and I should be doing something about it, and I'm not.
This is the reason why I'm trying to think seriously about things like l-zombies and magical reality fluid. I mean, I don't even think that these are particularly likely to be exactly right even if the measureless Tegmark IV hypothesis is wrong; I expect that there would be some new insight that makes even more sense than Tegmark IV, and makes all the confusion go away. But trying to grapple with the confused intuitions we currently have seems at least a possible way to make progress on this, if it should be the case that there is in fact progress to be made.
Here's one avenue of investigation that seems worthwhile to me, and wouldn't without the above argument. One thing I could imagine finding, that could make the confusion go away, would be that the intuitive notion of "all possible Turing machines" is just wrong, and leads to outright contradictions (e.g., to inconsistencies in Peano Arithmetic, or something similarly convincing). Lots of people have entertained the idea that concepts like the real numbers don't "really" exist, and only the behavior of computable functions is "real"; perhaps not even that is real, and true reality is more restricted? (You can reinterpret many results about real numbers as results about computable functions, so maybe you could reinterpret results about computable functions as results about these hypothetical weaker objects that would actually make mathematical sense.) So it wouldn't be the case after all that there is some Turing machine that computes the conscious experiences you would have if pumpkins started falling from the sky.
Does the above make sense? Probably not. But I'd say that there's a small chance that maybe yes, and that if we understood the right kind of math, it would seem very obvious that not all intuitively possible human experiences are actually mathematically possible (just as obvious as it is today, with hindsight, that there is no Turing machine which takes a program as input and outputs whether this program halts). Moreover, it seems plausible that this could have consequences for how we should act. This, together with my argument above, make me think that this sort of thing is worth investigating -- even if my priors are heavily on the side of expecting that all experiences exist to the same degree, and ordinarily this difference in probabilities would make me think that our time would be better spent on investigating other, more likely hypotheses.
Leaving aside the question of how I should act, though, does all of this mean that I should believe that I live in a universe with l-zombies and magical reality fluid, until such time as I hear that voice speaking to me?
I do feel tempted to try to invoke my argument from the l-zombies post that I prefer the map-territory correspondences of actually existing humans to be correct, and don't care about whether l-zombies have their map match up with the territory. But I'm not sure that I care much more about actually existing humans being correct, if the measureless mathematical multiverse hypothesis is wrong, than I care about humans in simple worlds being correct, if that hypothesis is right. So I think that the right thing to do may be to have a subjective belief that I most likely do live in the measureless Tegmark IV, as long as that's the view that seems by far the least confused -- but continue to spend resources on investigating alternatives, because on priors they don't seem sufficiently unlikely to make up for the potential great importance of getting this right.