A ramble about omniscience


I was talking a couple of weeks ago to a few friends from the glowfic community about whether it was logically consistent for a power to completely erase a concept from existence to also be able to guarantee that no one would die or cease to exist due to this change. I argued that no, this is impossible, because there exist concepts such that, if you erase them, you will change some (or even most) people enough that it will be equivalent to killing them and replacing them with someone else. I pointed to a parallel between this situation and the definition of “omniscience” and then the discussion went to that but it was past 10PM and I’m no longer a spry young person who can stay up that late and stay functional, so I couldn’t explain exactly what I meant very well. Then I kinda forgot to return to this post and it sat in my drafts for weeks.

I’m gonna explain what I meant here, and I’m gonna start by talking about determinism.

A friend said that, from a philosophical point of view, determinism is the position that “everything follows from prior events in a definite, non-random way.” She furthermore said that you could then say that this could include “thirty percent of universes do this, seventy percent of universes do that,” but not “one universe has a chance of doing one thing, or a chance of doing the other thing.” I sort of disagree that that is a sensible definition of determinism.

Let’s go with the first definition. Its strongest form is that there are no “probabilities” – not in the sense of “descriptions of uncertainty,” but rather in the sense of “the future of the world is itself intrinsically uncertain” – involved anywhere: there is one and exactly one possible configuration of matter following any given present configuration of matter, no two ways about it. Equivalently, the relation that takes as input a “world state” and has as output the world state immediately following it is a proper function. Let’s call this strongest form determinism-1.

We can be a bit less strict about what a “world state” is, though. Suppose there’s an infinity of universes, in the same sense there’s an infinity of numbers between 0 and 1. In this case, you could say that if you have this infinity of universes in a certain state and then in the next time step 30% of them are in a state and 70% of them are in another state, that’s a form of determinism, too. From the inside, someone who was in this universe ensemble would experience a form of nondeterminism, if they had no way of knowing in what fraction of universes they were, but since the state evolution of the entire universe ensemble is uniquely determined by its current state, it fits; a bird’s-eye description of all of it is also a proper function. This would, then, include the Everett or Many Worlds interpretation of Quantum Mechanics; we’ll call this determinism-2.

Now suppose I have a black box that takes certain inputs – like, say, a world state – and can spit out a large number of possible outputs. The process used to select a given output is unobservable and completely independent of the input, which means that just using that box I can’t really know in advance what its output is going to be. Suppose I have a second box, though, that takes as input the same input as the first box, and has as output a probability distribution over all possible outputs of the first box. This second box is, additionally, deterministic-1: given a specific input there is one and only one possible output (i.e. it’s a proper function). Further suppose that a bird outside the world knows that the first box doesn’t do the world ensemble thing (it’s not deterministic-2), it merely chooses amongst its possible outputs in a way that follows the probability distribution of the second box exactly but is independent of the rest of the world (i.e. it’s “nondeterministic”).

So the first box is neither deterministic-1 nor deterministic-2, but the probability distribution over its outputs is deterministic-1. I’ll call this first box deterministic-3, and I will call determinism-3 the state of affairs describing a world that evolves according to a rule like the first box’s: “nondeterministic”, but following a specific probability distribution.

(There is no determinism-4. I’ll explain why in a bit.)

Now let’s say I made observations that led me to believe that I’m either in a deterministic-2 or a deterministic-3 universe – that is, I have observed states that, from within the universe, seem to evolve in a completely but predictably random way. Assuming I can’t talk to the birds (and if I could they can hardly be said to be outside the universe so the premise is false), there’s no possible observation I can make that will distinguish definitively between those two hypotheses.

So if the two descriptions are in principle indistinguishable, what’s the point of calling one of them “determinism” but not the other?

(I should make an aside here that while you can’t disprove any of those you can in fact be more certain of one or the other using Occam’s Razor and other experiments. You just will never make an observation that is definitely ruled out by one but isn’t ruled out by the other.)

So I’m gonna call determinism-2 and determinism-3 “probabilistic determinism” or “p-determinism.” It’s easy to see that the determinism-1 is just p-determinism with p = 1 always, but that’s a special enough case I’m okay with separating them; it’s just determinism-2 and determinism-3 that I have trouble separating.

Regardless of whether you agree with my reasoning so far, at least in terms of “experiences an observer can anticipate in their universe” this division seems natural, and that’s all I need for the rest of my argument.


Why is there no determinism-4? Well, suppose the second box also only has a probabilistic output (over the probabilities of the outcomes of the first box) but there is a third box that predicts the probabilities of the outputs of the second box. But then, if you create a box that concatenates these two boxes you’ll get the same effect the second box had in determinism-3. This argument can be extended for any finite sequence of boxes, so that determinism-3 includes all of these cases.

Okay, so what if there is in principle a second box to predict the probabilities of outputs of the first box, but these probabilities change over time? But time is part of the world state, it’s part of the input of the box, there is no meta-time over which the first box’s rule changes. And if there is, well, then you just have to find the box that describes the evolution of the first box in meta-time and then you have a new timeless box. This always bottoms out.

So if I’m calling all these things “determinism,” what aren’t I calling determinism?

Well, nothing really.

The only case that’s not (straightforwardly) covered by those descriptions is the case where the probabilities are impossible to determine; that is, there is no possible second box, or no finite sequence of boxes that can be concatenated into a second box, or no box that describes how the first box evolves in meta-time. But suppose that’s the case. If there is no way even in principle to predict the probabilities of your next observation, if the present moment contains no information whatsoever about the next moment, then what’s left is white noise: every possible observation is as likely as any other.

But wait, isn’t that exactly a statement about the probabilities of the possible outputs? If the next moment is impossible to predict even in principle, even in rough more-likely-than-not terms, based on the present moment, that’s just saying that there is a second box and it says that all possible outcomes of the first box are equiprobable.

So chaos is also a special case of p-determinism. It too is special enough that I’m comfortable separating it into a different class just like regular determinism, but there doesn’t seem to be a way in principle for it to be impossible to describe any world with the two boxes.

That does seem to make the definition useless, though, doesn’t it? Maybe. So I guess I don’t have many objections to whatever definitions of nondeterminism. But it does seem to me that experience in any possible universe can be described by at most two boxes.


I have Microsoft Word installed on my computer. While my computer is turned off, can Microsoft Word be said to exist in it? Or does it only exist while my computer is turned on and I’m running the program? What if I write out a sequence of zeroes and ones such that, if I switch the contents of my computer’s HD to these zeroes and ones, I will have Microsoft Word installed in it? What if I write out a Turing Machine (on paper, using a pencil) that simulates my computer running Microsoft Word?

I don’t know. I genuinely do not know. I think it’s very obvious that Microsoft Word “exists” while my computer is turned on running it. It’s still a bit obvious that Microsoft Word “exists” while my computer’s turned off but I could at any moment turn it back on and run Word. But I don’t know at which step in this chain of logic I can stop saying that Microsoft Word exists, if I ever can.

And this is somewhat scary! If we take it to its extreme conclusion I can’t see anything stopping us from believing in Tegmark’s Level IV multiverse: every describable mathematical computation is “real” in exactly the same sense you and I are “real” because that’s exactly what we are. That’s scary.

I don’t need to bite this bullet for my argument, though. I just need the two first questions: if Microsoft Word is running I’m pretty cool with saying it exists, and I’m only slightly less cool with saying it exists if my computer is turned off. I’m even cooler about it if my computer is actually turned on and all I need to do is find Word’s icon and double-click it to run.


What’s omniscience?

I think its strongest form is “can answer every question that can be asked in principle.” If God is omniscient, they should have this property: any question you ask them, they should be able to answer. “What’s a full description of world X at time Y?” is a class of questions that can be asked, for all worlds X and all times Y in these worlds.

Now God can answer “Z.” If they’re right, if they’re really omniscient, then we can say that X is a deterministic-1 world, or perhaps a deterministic-3 world but God somehow has access to the information the first box uses to determine its output (and if God steps into this world then it automatically becomes a deterministic-1 world since now we have a deterministic-1 box, in the form of God themself).

If God answers “Z1 happens with probability p1, Z2 happens with probability p2, …” then X is a p-deterministic world. It can be a deterministic-2 world or a deterministic-3 world, depending on God’s temperament.

If God answers “this question doesn’t have a well-defined answer” then either God is lying to you or the world is chaotic. Or God is a very annoying prick who doesn’t want to tell you the probability distributions or doesn’t consider them a valid answer but they must exist.

And regardless of whether God will answer you truthfully, if they are omniscient then they must know the answer to those questions. They’re in there, in God’s mind, the full descriptions of the state of every particle in the universe at any given time.

If Z1 happens with probability p1, Z2 happens with probability p2, and so on… then what happens in world X at time Y+1 if Z1 happened?

What about at time Y+2?

And Y+3?

God must know the answers to these questions. And I can keep asking them, forever. Each time I ask them, God will output (or at least compute, in their head) X’s world state. I can figure out every single world state of every single world.

Does Microsoft Word exist in God’s mind?

God’s not running Microsoft Word. But they could, if they wanted to. They can answer every question, they can output every output of every computation performed by Microsoft Word. When my computer runs Microsoft Word, it’s displaying colourful bits on my screen but what Microsoft Word really is is the bits running there. If, for every key I press on my keyboard and every mouse stroke, God can tell me what Microsoft Word would show me and what computations it would perform, is that different than my computer doing it instead?

Do you exist in God’s mind? Does the world?

God’s not running the world. But they could, if they wanted to. They can answer every question, they can output every output of every computation performed by the laws of physics. If a given world is p-deterministic, then God’s answer may be in the form of the probabilities of each possible configuration of all the particles in the universe, but all of those configurations are in fact computed there.

I’m comfortable saying that Microsoft Word exists in my computer while it’s on but not running it. Does this mean I must be confortable with saying that I exist in God’s mind?

Does omniscience imply that every world exists?

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