The mathematical universe: the map that is the territory

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New Comment

124 comments, sorted by Click to highlight new comments since: Today at 4:52 PM

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You take too much confidence in a confusing idea. What does it tell, exactly (in anticipated experience, moral evaluation of decisions, etc.)? What does the original question about the cause for the world mean?

When considering the simulation scenario, you are discussing the world in which a simulation is running. In that world, it does matter how many simulations are running, and what they are running: the stuff of the world is organized in certain patterns, the patterns that implement the simulations. For any non-trivial preference about states of that world, different ways of organizing its matter will be valued differently, and any (hypothetical) change in content of the simulations or their number is a change to the content of that world, change that will be either preferred or not.

Now, with the perfectly autonomous simulations, the inhabitants of those simulations have no knowledge of the content of the outside world, they don't even have any knowledge about whether their own simulation exists (they are not able to pick out that particular hypothesis). But this lack of knowledge is a separate issue from moral weight of possible states of the (outside) world, or their own decis...

414y

Ditto. I'm sure you and others realize that given its length your post is highly scrutinizable, but if you replace "must" with "may", I did like your statement in italics as food-for-thought:
... though my thoughts will have to eat it awhile longer before I digest and judge it as an actual idea :)

I'm already a bit late to be commenting here, but I would suggest that people who are interested in further thought experiments along these lines read Egan's /Permutation City/. I don't totally agree with how the author answers his own thought experiments, but the experiments themselves are very closely related to the topic of the post.

214y

While I agree this is an excellent and relevant book, I would like to warn that it's a horror novel, and you may want to take that into consideration. As a horror novel it is outstanding, I couldn't put it down. But don't expect a pleasant reading experience.

311y

Horror Novel? I thought it was a light-hearted piece of metaphysics. Surely it's more 'Star Trek' than 'Misery'.
I would have lent this to any child capable of understanding it without the slightest worry until I read this comment. Now I'd worry very slightly. Am I a bad person who shouldn't lend books to children?

411y

Yeah it's a horror novel.
Read Ultimate Meta Mega Crossover for a fix fic that tries to repair the damage.

011y

I dunno, I'm not good a modeling children who can read Egan. Peer's storyline in particular was horrifying, especially his eternal suicide at the end of Part 1.

211y

As a teenager, I don't see how it can be horror. I thought it was inspiring, honestly.

611y

One of the major messages I got from it was thus: even if you never physically die, eventually over eternity one of these two things will happen - 1) your utility function will drift enough, and your memories fade and change enough, they you will be unrecognizable as the person you were. You as you are now will effectively be dead. 2) you will successfully resist change, and will be stuck thinking and doing the same things endlessly in a loop. You might as well be dead.
Even if we defeat death, living long enough is essential death anyway. You are doomed, there is no escape.

611y

Isn't that just due to the author's inability to imagine/describe a mind capable of becoming increasingly and unboundedly complex without losing its identity? Why take it as an inevitable conclusion?

211y

Well sure, but that wouldn't make for a good horror novel. :)

611y

Teenagers are immune to cosmic horror. Well-known fact.

014y

Hmm the "map that is the territory"... The essence of ata's post is a mirror of Daniel Dennett's argument against p-zombies.

I wonder, how much people are there who 'invented' this idea independently? I came up with this idea on my own when I was 15, and I know at least one other person who did. And now you and Tegmark, so this is not unique at all. Is that sort of thing widespread in rationality/math/INTJ/technophile/whatever?

310y

Greg Egan's book "Permutation City" was published in 1994. Max Tegmark's paper "Is "the theory of everything'' merely the ultimate ensemble theory?" was published in 1998. It is ironic that the paper gets more formal credit in a lot of writing than the book, even though the book explores more implications and has priority :-)
Based on my reading of Egan's other works, he has a minor sense of rivalry with Tegmark. One possible reading of Egan's Orthogonal Series is that it was an inspired way of pointing out that Tegmark's intuitions about the kinds of math that might contain minds are wrong because Tegmark's intuitions are grounded in thinking that "local physics" is especially privileged, perhaps from too little attention to ideas like "kolmogorov complexity" that are more native to artificial intelligence. The Orthogonal series is set in one of the kinds of physics that Tegmark claimed was probably uninhabited.

Ata, there are many things wrong with your ideas. (Hopefully saying that doesn't put you off - you want to become less wrong, I assume.)

it is more difficult to get to the point where it actually seems convincing and intuitively correct, until you independently invent it for yourself

I have indeed independently invented the "all math exists" idea myself, years ago. I used to believe it was almost certainly true. I have since downgraded its likelihood of being true to more like 50% as it has intractable problems.

...If it saved a copy of the univ

814y

I'm not so sure, Mallah. Your first argument seems to say that if someone simulated universe A a thousand times and then simulated universe B once, and you knew only that you were in one of those simulations, then you'd expect to be in universe A. I think your expectation depends entirely on your prior, and it I don't see why your prior should assign equal probabilities to all instances of simulation rather than assigning equal probabilities to all computationally distinct simulations.
(I'm assuming the simulation of universe A includes every Everett branch, or else it includes only a single Everett branch and it's the same one in every instance.)
What if you run a simulation of universe A on a computer whose memory is mirrored a thousand times on back-up hard disks? What if it only has one hard disk, but it writes each bit a thousand times, just to be safe? Does this count as a thousand copies of you?
As for wavefunction amplitudes, I don't see why that should have anything to do with the number of instantiations of a simulation.

114y

That's right, Nisan (all else being equal, such as A and B having the same # of observers).
In the latter case, at least in a large enough universe (or quantum MWI, or the Everything), the prior probability of being a Boltzmann brain (not product of Darwinian evolution) would be nearly 1, since most distinct brain types are. We are not BBs (perhaps not prior info, but certainly info we have) so we must reject that method.
No. That is not a case of independent implementations, so it just has the measure of a single A.
A similar argument applies - more amplitude means more measure, or we would probably be BB's. Also, in the Turing machine version of the Tegmarkian everything, that could only be explained by more copies.
For an argument that even in the regular MWI, more amplitude means more implementations (copies), as well as discussion of what exactly counts as an implementation of a computation, see my paper
MCI of QM

29y

The choice of your turing machine doesn't much matter, since all turing machines can simulate each other. If you choose the "wrong" turing machine, your measures will be off by at most a constant factor (the complexity penalty of an interpreter for the "right" machine language).

114y

For continuous functions, we do. See "abstract stone duality".

114y

Interesting. Do you know of place on the net where I can see what other (independent, mathematically knowledgeable) people have to say about its implications? It's asking for a lot maybe, but I think that would be the most efficient way for me to gain info about it, if there is.

(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.)

Indeed! I haven't read Tegmark's paper yet (but I've resolved to do so now), and I sorta came to the same conclusion myself a while back. Here are some more thought-experiments that can be shortcuts to the idea you have described:

Zhuangzi didn't know whether he had dreamt he was a butterfly, or whether he was a butterfly dreaming of being Zhuangzi. The two point

Universe as a counterfactual or hypothetical object seems intuitive, but claiming that it doesn't make a difference if you torture two or one identical beings in identical ways seems to be a bit problematic here. I think there are two ways to understand this, given two ways you can interpret the word "existence", as in, location within a given universe, and this metaphysical "does universe exist at all" -sense, and I also think that ethical worth of happiness or sadness can add up in both ways without this argument failing.

Thanks for writing this, I enjoyed learning more about the mathematical universe theory, and it at least seems I've gained a useful intuition. I'm not sure the intuition has practical uses, and I also think I disagree on what can be drawn from this.

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.

When the simulation is stopped, later to be restarted, we experience no difference because nothing is changed in our simulation by the progression of time in ...

09y

I think the situation here can be resolved using an analogue of the limiting behavior of functions in mathematics. If, from the perspective of the beings simulated, nothing happens when the simulation turns off for an arbitrary amount of time (call this number n), then as n approaches infinity, the simulation beings' consciousness will still continue. (If a function f(x) = c for all real x, then the limit of f(x) as x approaches infinity will also be c.)

When I was a kid I was led to the Ultimate Ensemble hypothesis from thinking about time travel. I imagined a scenario where I had just witnessed a time traveler disappearing into the past to go kill my parents before I was born. I concluded that nothing would 'happen' to me. After all, my future was determined by the current state of the universe, and that state certainly existed at that moment, since I had just observed it.

From there I hit on the Everything-Is-Math theory, which I lovingly called the Eris-Polygenesis Effect.

I think Eliezer's Anthropic Trilemma is relevant to this discussion.

The fundamental problems don't seem to require an Ultimate Ensemble to appear. They already exist in our current universe, most likely.

I'm super late to the party, but... The argument that convinces me that something like the MUH must be true is:

Suppose you simulate universes until you find one that evolves intelligent life. You could then theoretically synthesize an individual of this population in your own ("real") universe. This individual would have memories of their past life-- they would act as if they really existed to experience it. So sentient, experience-having life could in theory exist in a mathematical structure. (I do not believe you created this individual in the ...

010y

I don't follow that. Being able to agree on a value of pi - epistemological objectivism -- does not imply the immaterial existence of pi, mathemtical realism. It is clear enough that MUH follows from mathemtical realism (and a few other assumptions, such as the supervenience of conscious experience on suitable mathetmatical structures), but mathematical realism is a very non-trivial claim.

110y

What would "immaterial existence" even mean?
I think my claim is that the above argument shows that whatever that might be, it's equivalent to epistemological objectivism.
Specifically, to believe that they're separate, given the scenario where you simulate universes until you find a conscious mind and then construct a replica in your own universe, you have to believe both of the following at the same time:
(1) Mind X didn't have real memories/experiences until you simulated it in the "real" world (i.e., yours), and (2) proof of mind X's running existed previously to you computing it (in the form of an execution history).
To me, accepting both points requires me to believe something like "Proofs that I don't know about aren't true", and I'll be happy if you can show me why that's not true.

-110y

I don't know exactly, but if "material existene" means something, so does "immaterial existence".
I think you argument assumes that. You say that the simulated person must have had a pre-existence (ontology) because mathematicians agree about pi (epistemolology)
Specifically, to believe that they're separate, given the scenario where you simulate universes until you find a conscious mind and then construct a replica in your own universe, you have to believe both of the following at the same time:
You seem to be assuming that if a mind has "memories", then it must have pre-existed, ie that the only way a mind can have "memories" at time T, i sby expereincing things at some previous time and recording them.
Rather than assuming that there are infinite numbers of real but immaterial people floaitng around somewhere, I prefer to assume that "memories" are just data that don't have any intrinsic connection to prior events.. Ie, a memory proper is a record of an event, but neurons can be configured as if there were a trace of an even that never happened.
I don't see your point.

010y

Hm. I don't think "material existence", if it's a thing, has a unique opposite.
I guess I'd define exists-in-the-real-world as equivalent to a theoretical predicate function that takes a model of a thing, and a digitized copy of our real world, and searches the world for patterns that are functionally isomorphic (given the physics of the world) to the model, and returns true iff it finds one or more.
This model of existence doesn't work if you don't supply the real world (or at least a world) as an argument. I'm interpreting "immaterial existance" as "exists, but not in a world" which seems like a logical impossibility to me. Of course, this is a function of how I've defined "exists", but I don't know of a better way to define it.
OK, that's a reasonable position. I'll adjust my argument. My claim now is:
(1) Given that you've found a bitstring in a simulation that represents a mind existing, taking actions, and feeling things,
(2) this bitstring is quite astonishingly long,
(3) most long bitstrings do not similarly describe minds by any reasonable mapping function,
it's therefore (4) vastly more probable that a mind actually ran to produce that bitstring, than it is that you found it randomly.
Basically, I'm treating the outputs of a mind as being something like a proof that there was a mind running. Similarly to the idea that publishing an SHA-256 hash of some data is proof that you had that data at the time that you published the hash.

010y

Failing to have a unique referent is not meaninglessness.
That is rather beside the point, since none of that is necessarily material.
Most people would interpret it as "exists, but is not made of matter". To cash that out, without contradiction, you need a notion of existence that is agnostic about materiality. You have given one above. Tegmarkians, can input a maximal mathematical structure as their world, and then say that something exists if it can be pattern-matched within the structure.
So far, none of this tells us what immateriality is. But then it isn't easy to say what matter is either. For immaterialists, anything physics says about matter boils down to structures, behaviour and laws that are all mathemaitcal, and therefore within some regions of Tegmarkia.
There are more than two options. If you had evidence of a bitstring corresponding to a billions of years of biological development involving trillions of organims -- amuch more comple bitstring than a mind, but not a mind,-- it might well be most probable to assign the production of a mind to that.
I don't know if you realise it, but your argument was Paleyian

010y

Yeah. There's supposedly two mysterious substances. My claim is that I can't see a reason to claim they're separate, and this thought experiment is (possibly) a demonstration that they're in fact same thing. Then we still have one mysterious substance, and I'm not claiming to make that any less mysterious with this argument.
Whoa, I think you understood something pretty different from what I was trying to say. I was definitely not claiming a deity of some sort must have been responsible! Let me repeat with unambiguous labels:
(1) Given that you've found a bitstring in a simulation that represents mind M existing, taking actions, and feeling things, ... it's therefore (4) vastly more probable that mind M actually performed computations in the course of producing that bitstring, than it is that you found it randomly.
Yes, of course mind M will have been the result of some evolutionary process in any universe I can imagine finding via simulation, but that dosen't make mind M less real. I suppose you could probably see this as a special case/expansion of the Anti Zombie Principal-- any system that produces the outputs of a mind (at least) contains that mind.
(If you meant something else by the Paleyian comment, you'll have to spell it out for me. I'm not the one that downvoted you and I appreciate the continued interaction.)

-210y

EDITED
No, the claim of Tegmarkian immaterialism is not that there is another substance other than matter.
You were previously saying that a log or record of mental style acrivity was probably produced by a mind. This is an explanation of an argument that you said supports "something to.like the MUH ". I still don'see how it does,. I am also puzzled that you thave been arguing against immaterialism throughout.

210y

Thanks for editing-- I'm still puzzled.
I also don't know what "Tegmarkian immaterialism" is and I'm not arguing for or against it. I do not know what "immaterialism" is and I'm also not arguing for or against that. (Meta: stop giving either sides of my arguments names without first giving the names definitions!)
If anything, let's call my position "non-distinctionalism"-- I maintain that there's no other coherent models for the word "exist" than the one I mentioned earlier, and people who use the word "exist" without that model are just talking gibberish. There's no distinction between "material existence" and "immaterial existence" in the sense that the first clause of this sentence is meaningless noise. I can be disproved by being informed of another coherent model. I maintain that my thought experiment shows that it's difficult to hold any distinction between exists-in-reality and exists-in-the-mathematical-multiverse.
(If I were king of the world, "immaterial existence" would mean "exists in an inaccessible place of the mathematical universe", but for me to use the term that way would currently be idiosyncratic and just confuse everyone further.)

010y

You have mentioned the Mathematical Universe hypothesis several times, and Tegmark's is a name very much associated with it, as WP states:
"In physics and cosmology, the mathematical universe hypothesis (MUH), also known as the Ultimate Ensemble, is a speculative "theory of everything" (TOE) proposed by the theoretical physicist, Max Tegmark.[1]"
You second sentence doens't follow from your first. Someone can define "material existence" as existence in your sense, plus some additional constraint, such as the "world" in which the pattern is found being a material world.
Standard arguments against MUH (etc) are that they predict too much weirdness. But that is an arguemnt against the truth of MUH, not for the coherence of materialism. However, you have not acutally argued against the coherence of materialism. Your definition of existene doesn't requires worlds to be material or immaterial, but it also doesn't require them to be neither.

110y

Right, I know what the MUH is, I know who Tegmark is, I just don't recognize terms that are a combination of his name and (im)materialism. Please taboo your terms! I don't know what they mean to you!
Yes, if you look at the last paragraph, I (tried to) say explicitly that is my desired state of the terms, it's just not the standard usage, afaik.
So, if I were writing the terms, "material existence" = exists in our sub-branch of the MUH, "immaterial existence" = exists in a sub-branch of the MUH inaccessible to us (except via simulation). However, I don't think people ordinarily use the word "material" to mean this.
Actually, now that I think about it, I think that's a bad way to think of things. Really, the concept those terms would express is accessibility, not existence. When I hear other people say things like "immaterial existence", my brain interprets that as "exists, but not in a branch of the MUH," which I think is gibberish.

010y

Perhaps you could interpret their remarks according to the Principle of Charity: since their remarks are nonsense under you interpretation, they probably have a different one in mind.

010y

I believe if you read my previous comments, you'll see that they all are attempts to do exactly this. I will bow out of this conversation now.
(Meta: you're tripping my troll sensers. I'm sorry if it's unintentional on your part. I'm just not getting the sense that you're trying to understand me. Or it's the case that the two of us just really cannot communicate in this forum. Either way, it's time to call it quits.)
EDIT: Your response to this has caused my P(you're trolling me) to rise from ~60% to ~95%.

-310y

You mean you had some even more nonsencial interpetations in mind, and chose the most charitable?

010y

I think we are having very significant communication difficulties. I am very puzzled that you think that I think that I'm arguing against MUH. I think something like MUH is likely true. I do not know what "Tegmarkian materialism" is and I'm not defending or attacking it. I also cannot make sense of some of your sentences, there seems to be some sort of editing problem.

010y

I think you have been arguing against immaterialism, and that Tegmarkian MUH is a form of immaterialism.
I have edited my previous comment.

The Tegmark Level IV multiverse idea is quite elegant, and this was a fairly well-thought-out post, although with some problems, most of which Mallah already identified. One more thing though:

...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 un

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.

Why is that? Your argument is that if there are a large or infinite number of parallel universes being simulated (or just existing) I have to resist the urge to care about the feelings of the creatures being simulated in those parallel universes. My question is why? You haven't developed this argument at all, you just claime...

511y

A long time ago, in a galaxy far, far away...

112y

Yes, but perhaps not very much, and definitely not because we wrote a story about them; that's pure coincidence. Any story with finite information content should be coincidentally similar to some actual real universe, but following Solomonoff induction there's a sense in which the more complex ones are not as real as, say, us.

In my opinion, the exploding head objection is fatal to this theory. I'm not morphing into a pheasant right now.

214y

I agreed intuitively, but then as I proceeded to flesh out the logic, I (remembered) that inconsistency of that kind is impossible anyway. (According to the theory, only what is possible exists.)
Suppose that a simulation is run for a certain number of flops under a certain set of rules, and then the rules are abruptly changed. Then this means the simulation wasn't closed -- there was something external to the simulation changing the rules. Thus, something simulated within a proper subset of the universe can have the property of being self-inconsistent, but it's impossible for the universe itself to not be running a single set of self-consistent rules.
A common worldview among scientists is that as we evolve a deeper and deeper understanding of the universe, we increasingly find that self-consistency narrows what is possible: the universe had to have these particular rules. So maybe we are residing in the only possible universe.
As a thought experiment, we could instead discover that exactly n universes were possible. Then I guess I agree with the intuition of this post: I would think it is unlikely that someone or something 'chose' this particular universe, I would guess that all exist and I happen to be in this one.

014y

There are quite a many coherent simple mathematical entities, so it's really unlikely that it turns out that number of those that allow subentities that are conscious is anyhow limited.

014y

I'm currently in yellow/red hat mode for the idea that the theory described above would result in a single universe.
(I thought about moving this comment to a more appropriate thread, but it was too much work. Unless someone picks up on the idea, I'll let the idea drop for a while and imagine there are multi-verses in order to be more on-page. Anyway, what follows below is my argument for why there would be a UNIverse, a continuation of a theme begun here.)
Given a set of observations, there can be a number of coherent mathematical entities that fit those observations. Also, some of those coherent mathematical entities may be independent and/or mutually exclusive, giving rise to our notion of many possible but not-simultaneous universes. However, the production of a universe from a void is a completely different context in which 'false' and 'independent' may not .. exist.
I don't know how it would go, obviously, but suppose it begins something like this: just a void, so you have the empty set. You have all sets of the empty sets, so your mathematical structure is a little more complex. Then you have more complex relationships between sets, etc.
* At what point does an inconsistent fact ever come in? (never)
* At what point is an independent fact introduced? (never)
You just have an evolution of complexity, each step derived true relationships from lower level relationships. Eventually, 'stronger'/'abundant' relationships yield the particles we know of, them and the relationships between them define our universe and the rules.
There's just one universe. A universe that is the deduction from nothing.
Contrari-wise, independent coherent mathematical entities and multiverses would be interesting too.
(We all agree this is metaphysics, right? Which is to say..interesting thought exercises only?)

014y

I'm not sure how to interpret this thought experiment, but it seems that you could still produce indistinguishable-from-independent-fact-in-the-eyes-of-in-world-observers -like things.
Like, if we take empty set, set that contains empty set, then form a set that contains all earlier sets, and this way construcy natural numbers, and then produce a function that takes one set and gives it's "successor", as in, the next natural number. F(1) = 2, F(11) = 12 and so forth.
This could be taken as "natural, simple world", where there are no surprises, like, F(15) giving 1337 as a result, or anything. That F(15) = 1337 would count as an independent fact. But obviously rule like could be formulated, and I'm really unsure about what's there to prevent this "add one expect if 15, give 1337" rule from happening, given that the original successor function did 'happen'.
But then again, maybe I just misunderstood, I got kind of an impression that you're thinking Universe as some sort of limit as length of deduction chains approach infinity, but this idea seems counter-intuitive to me.

014y

No external meddling needed, the simulation just has an extra rule that says "cousin_it morphs into a pheasant on turn N".

114y

The simulation wouldn't know what 'cousin_it' is or what a pheasant is. These are higher level things that evolve from the lower level rules. (They don't exist independently, and don't really exist except as categories in our mind.) All of the rules in the universe have to be in terms of the lowest level things because reductively, these are the things that are real. So the universe wouldn't be able to say "cousin_it morphs into a pheasant on turn N", it would only be able to say that "things like cousin_it morph into pheasants under these conditions", in which case we would discover that the rule happens reliably and predictably, not arbitrarily.

414y

Another way of expressing this is that a universe which has our laws of physics is a shorter encoding than a universe that has our laws of physics, plus a detailed exception for "cousin_it turns into a pheasant." This makes it more likely, but, the set of all universes with rules much more complicated than ours which would still allow conscious observers makes it unlikely that we made it into such a simple universe. This has been covered on lesswrong and overcomingbias before.

114y

I disagree that you're responding to my argument. I'm not making an argument about whether the universe is simple or not: I'm making the argument that if the universe has an encoding, "cousin_it turns into a pheasant", it's not going to be an exception. If the universe has that encoding, we would find that cousin_it turns into a pheasant, and upon further study, would find this was predicted all along by the lower level rules. Simply because nothing exists beyond the lower level rules. We can't expect inconsistencies at higher levels because the higher levels are just derivative of the lower ones.

714y

What do you mean? There's no law of nature saying laws of nature must be low-level in all possible worlds - that's just an observation about our world. I can write a simulator that runs the low-level rules as you'd expect, but also it's preprogrammed to search for cousin_it in the simulation at a certain moment, turn him into a pheasant, then resume business as usual. If all simulated worlds "exist", this one "exists" too.

-114y

On the one hand, it a matter of the definition of 'low-level': all laws of nature must be low-level or derived from a low-level law because if you had a law that wasn't derived from a lower-level law, that would make it low-level.
Yes, I agree and I'm generally interested in this case. But as I explained, this is only possible in a subset of a universe. The whole universe could not be such a simulation, because there would be nothing 'outside it' to swoop down and make the arbitrary change.

514y

The hypothesis says that all universes that can be simulated by computer programs exist. It doesn't restrict those computer programs by saying they must use "only local laws", or "can't swoop down", or whatever. What does this even mean? Moving the universe one step forward in time according to the Schroedinger equation qualifies as "swooping down" just as much as turning me into a pheasant, they're both things that the program just does.
I assume you're thinking of some other hypothesis, like "all universes that exist must start from a simple core and proceed logically from there", but unfortunately no one has a "logicalness predicate" that would say whether a given program simulates a "logical" universe without "swooping down". In fact, looking at a program you may not even tell if it's "simulating" any universe at all, as opposed to moving some bytes around in weird patterns.

014y

Perhaps it is a matter of relying on different analogies for our intuitions. I was thinking of each possible universe as being identified with a single self-consistent mathematical structure. In which case, I expect the universe to be organized and coherent because any set of mutually consistent initial facts would generate a universe with structure rather than one with haphazard and inexplicable events.
I hadn't thought of all possible computer programs each mapping to a universe ... what is the truth value of a random string of instructions? But since I don't know much about computer science, I don't expect that to work as an intuition pump for me.
Instead I was thinking along the lines of there being a universe -- say -- for each exotic algebra there might be and all the facts that are derived from it. An "algebra-generated-universe" wouldn't say something random or arbitrary; instead everything about the universe would be derivable from a few facts. (Note you couldn't have too many facts, or they'd self-contradict.)

314y

I can remove contradictions by making up a new rule that specifies which rules take precedence over other ones.
Let's build a simpler intuition pump: instead of universes, we'll talk about infinite sequences of integers that can be specified by finite sets of rules. For example, (1, 1, 1, ...) is such a sequence. (1, 2, 3, 4, 5, ...) is another. Those look regular all right. But this sequence: (1, 2, 3, 4, ..., 999999, 1000000, 12345, 1000001, 1000002, ...) can also be specified by a finite and self-consistent set of rules, even though something seems to have "swooped down" and changed it in one place. There's no hard difference between "regular-looking" and "irregular-looking" sequences. All finite sets of rules have equal footing.
Does this make sense to you? Now imagine those integers encoding the time evolution of your toy universe...

014y

It seems to me we're conflating 'possible as an output' and 'true' but since I don't really know what 'true' means in this context, let's conflate them.
The fact that you've written the sequence (1, 2, 3, 4, ..., 999999, 1000000, 12345, 1000001, 1000002, ...) is evidence that it's the possible output of an algorithm. (Indeed, it was the output of an algorithm you ran.) However, this means that the output was possible (and true) for a subset of the universe. How do you know this rule could be universally true?
I say that it could not be universally true, because it has this property of arbitrariness. I think to answer, 'what could possibly be universally true?', you would have to answer the question, 'what can be deduced as true from nothing?' or at best, 'what can be deduced as possibly true from nothing?' From nothing, the universe might deduce the natural numbers. By definition of what the natural numbers are they have an ordering 1, 2, 3, ..., n, n+1, ... This ordering really could not be different.
Suppose that the universe had a way of "knowing about" a single element '12345' that it places in a new position between 1000000 and 1000001. Simultaneously, it could have placed this element in any position, so universally, you would get a much, much larger structure in which (1, 2, 3, 4, ..., 999999, 1000000, 12345, 1000001, 1000002, ...) was only a tiny subset.
My point, whether I can figure out how to make it or not, is that the universe doesn't 'know about' 12345, it only knows about all the numbers and evolves this structure universally. You can look at particular components of the structure and observe that individual components seem arbitrary, but the entire structure cannot be.

014y

Imagine a "universe" that consists of all streams of natural numbers that can be specified by algorithms. Is that fundamental and non-arbitrary enough for you? This universe contains many "sub-universes" that cannot communicate, so we can call them "universes" in their own right. One of them is my 12345 sequence, and many others have me spontaneously turn into a pheasant a week from now.

014y

Exactly, yes.
How do you know they don't communicate? This would be a very non-trivial claim. I'm saying that the set of things that could be independently true (and thus universally true) might be extremely small, and certainly much smaller than the set of possibly-possible things you can think of. Most things we can think of as possible are going to be entangled in ways we aren't aware of with other truths.
Instead of being where you are thinking of things that could be ('I turn into a pheasant in 5 minutes'), you would need to turn it upside down and think if there was nothing, what would be true? Perhaps not so many things ... perhaps this experienced universe is the only one that was possible. How do we know without developing a theory about what truths self-generate from a void?

214y

Same way I know natural numbers don't communicate. The output of one algorithm can't "communicate" with the output of another algorithm, whatever that means.

014y

Each possible universe corresponds to a different set of axioms, right? (If two universes have exactly the same axioms, then they'll be the same, and the addition of any new axiom that is consistent with but not deducible from the others will make a new universe.)
I've maintained all along that arbitrary and weird sets of rules can occur in subsets of the universe, but should not be universally true (true throughout a universe) because there cannot exist a set of axioms that would result in these rules. For example, you can build possibly a machine that turns someone into a pheasant and an extra bag of sand, even in this universe, but it wouldn't ever be a universal rule that a person with cousin-it-specifying-characteristics turns abruptly into a pheasant.
Now we are considering whether two algorithms A1 and A2 that generate distinct streams of numbers "communicate" (whether they're independent). They are independent if they are generated by different axioms. We would have that there are two sets of axioms, one which possibly generates A1 but not A2, and one which possibly generates A2 but not A1. How do we know that we could find a set of axioms that results in the possibility of only A1 or A2, but not both? I think this is very unlikely, because the possibility of an algorithm already requires a lot of structure, and I doubt you could consistently add to it a set of axioms that specify that A1 is possible but not A2. In our own universe, all the computable algorithms are possibly generated, and this has the symmetry and non-arbitrariness I've come to expect from the structure of an entire set of facts deduced from a set of axioms. Sets of axioms don't result in a fact like '12345 can move to a position between 1000000 and 1000001 but no other numbers can ever be moved to any other positions'.
It's conceivable, in contrast, that you have a fact, "numbers can be listed in different orders". So that moving 12345 would be a possibility but not universally true.

014y

We agree on that. But why does it have to be a universal rule? In other words, where am I? In the single universe that is 100% lawful, or in one of the myriad chaotic sub-universes embedded within larger lawful structures? For example, the perfectly lawful "universe of all algorithms" contains a lot of entities indistinguishable from me that will horribly disappear the next instant. I'm not insisting on a pheasant - a banana will do as well. If you really believe that all axiomatic structures exist, each passing second of lawfulness should surprise you tremendously.
Why not? "Axioms" aren't syntactically distinct from "facts". You can take any fact and bless it as an axiom.

014y

It could mean something that allows them to "communicate"...

0[anonymous]14y

All that means is that 'cousin_it turns into a pheasant' has to be taken as an axiom for the algebra you're using...

314y

Higher-level behavior can be explicitly coded into lower level rules.

114y

Hi. After reading your posting on the mathematical universe, my coments are:
* The only way "2+2=4" can exist is if there are first two existent objects and then a mind to come up with the construct describing their addition. "2+2=4" doesn't exist on its own.
* My own view for why there is "something" rather than "nothing" is:
1. There are two choices for why there is "something" rather than "nothing":
A. "Something" has always been here. B. "Something" hasn't always been here.
2. Choice A is possible but doesn't offer much explanatory power so it won't be
pursued here.
3. Going with choice B, if "something" hasn't always been here, then "nothing" must have beeen here before it. By "nothing", I mean complete non-existence which would be the lack of all volume, matter, energy, ideas/concepts, etc. However, in "nothing", there is no mechanism to change this "nothing" into "something". So, if "something" is here now, the only possible way is if "nothing" and "something" are one and the same thing. I think this is logically required if we go with choice B.
* If it's logically required that "nothing" and "something" are the same thing, the next step is to try and figure out how this can be since they seem different. My view on how this can be is that they only seem different because we're looking at them from two different perspectives. In thinking about "nothingness", we use our mind, which exists. Next to something that exists, "nothing" just looks like nothing. But, in true "nothing", there would be no minds there, and only then would "nothing" be completely self-defining (it says exactly what is there) and therefore existent.
An idea that's helpful in thinking about this topic is that the mind's conception of something ("nothing" in this case) and the thing itself are different. Thanks for listening!

314y

Hi! You seem to be asserting anti-realism and trying to arrive at a correct ontology using an Aristotelian application of deductive logic. Would you like some help with that?
Max Tegmark, the physicist who proposed the mathematical multiverse theory, was aware of the anti-realist position. However, there's good evidence that minds are made out of math, instead of the contrary position. It's a fairly mature debate, and it pays to be aware of the strongest arguments both sides.
This awareness also applies to the universe's beginnings, or lack thereof. Historically, deductive logic has had some problems locating true beliefs.
Also, welcome to Lesswrong! Feel free to post on the introduction thread; and start working your way through the sequences so you understand where other people here are coming from.

010y

Is there? Evidence from simulations running on material comptuers doens't show you can make minds out of immaterial math.

014y

Kharfa,
Thanks for your comment. In response,
o You seem to be asserting anti-realism and trying to arrive at a correct ontology using an Aristotelian application of deductive logic. Would you like some help with that?
I'm not denying the reality of anything that you can show me. Please show me where "2+2=4" is or where it exists. Using that type of argument that things like this exist is like saying Santa Claus exists. That's possible, but we can't prove it or disprove it, and you can't show him to me. There's no point in discussing it. And, by the way, I don't need any help with that. Patronizing attitudes especially when not backed up by sound reasoning are of no interest to me.
o there's good evidence that minds are made out of math, instead of the contrary position.
I believe there's good evidence that minds are in heads and are made out of matter and energy, not mathematics.

514y

Not disagreeing, but fleshing out part of what it seems you're trying to say:
Numbers don't exist, that much ought to be clear. I think Eliezer says that numbers are in our minds, and our minds exist, but this is not the case: it's not numbers that are in our minds but representations of numbers.
Mathematical Platonism is, to me, religion for intellectuals. Mathematicians as esteemed at Kurt Goedel have even gone so far as to postulate that mathematics exists in an alternate universe. This is a basic error or at least wildly unparsimonious, akin to saying that modus ponens exists in an alternate universe.
To see how silly this is, it helps to realize that a sufficiently intelligent being would find all our mathematical theorems just alternative ways of stating the axioms, and all our mathematics just axioms and definitions with a bunch of obvious rephrases of the same. It would find our most advanced theorems as simple and obvious as modus ponens is to us - as just rewordings of the axioms and definitions.
From the perspective of a sufficiently intelligent being, mathematics is just a set of initial statements (axioms and definitions), along with humans' silly little demonstrations to help each other realize that a bunch rewordings of those statements (theorems) all mean the same thing.

114y

From the perspective of a sufficiently intelligent being, physics is just a set of initial statements, along with a silly demonstration that history is what you get when you apply those statements over and over. How dull!

014y

Amanojack,
Hi. I agree with you completely and like the phrase "religion for intellectuals". I just don't see the difference in saying that numbers and mathematics exist somewhere but we can never show you where and saying that other things exist somewhere but we can't show you where. But, trying to get even very intelligent people (ie, your example of Goedel) to see this or even listen to this type of reasoning seems almost impossible. Oh, well! Thanks!
Roger

614y

One thing I've noticed that is probably covered somewhere in LW archive (I hope!) is that really smart and rational people can sometimes just be really good at hiding the truth from themselves. In other words, the smarter you are, the better you are at Dark Arts, and the easiest person to trick with Dark Arts is sometimes yourself.
[Heads up: your comments are displaying as one long line requiring side-scrolling instead of with natural line breaks.]

414y

If I take two rocks, and put them in a cup where you can't see, and then put two more rocks in the cup, and then rattle it around and dump out the contents of the cup on the table, just look at it: two plus two equals four inside the cup.
If I flip this switch on the left, two lights come on, and then if I flip this other switch the other half come on, and then there are four lights.
I could list more examples, and you should have no trouble verifying them experimentally. Were you maybe expecting a stone tablet somewhere, that, if modified, would cause plusOf(2,2) to output five?

214y

That doesn't prove that 2+2=4. That proves that rocks obey a regularity that is concisely describable by reference to an axiom set under which 2+2=4. That's not the same thing. Math still "exists" only as a (human) representation of other real things.
(I will note in passing that Steven Landsburg, who promotes the "Math exists independently" belief -- indeed, pretty much defines his worldview by it -- argues for the position in his book The Big Questions essentially by cheating and slipping in the definition that "Math exists iff math is consistent." Go fig.)
No, but I can think of experiences under which I would keep the belief
a) "rocks obey a regularity that is concisely describable by reference to the standard axiom set for math"
but discard the belief
b) "under that axiom set, 2+2=4"
However, to get better insight into why this would happen, you should replace "2+2=4" with "5896 x 5273 = 31089508".

414y

These discussions about whether 2+2=4 are confusing because 2+2 is 4 by definition of this operation of addition, and then we see in what cases real world phenomena are described by this operation. If you define 2+2 as anything but 4, then you're just describing a different operation. There are many operations where 2 and 2 give 5.
The problem, always, is that there's no causal connection between mathematics and reality. Suppose you try to force one and say that however many rocks you have in a cup when you combine two cups (each with two rocks), that is going to be THE operation of addition. Then asking what 2+2 has to be is asking how many rocks you can have in the final cup. Well, it's not logically impossible for rocks to follow a rule that every four rocks in a certain small area will make a new rock and increase their number to 5. It just happens, in our reality, that rocks satisfy Peano arithmetic (and don't resonate daughter rocks).

314y

Well, it's complicated by the fact that rocks can break, which means that you need to go into a lot more detail to say the extent to which the standard axioms of math map to rocks. This is why simplistic proofs of 2+2=4 by reference to rock behavior are so misleading and unhelpful.
It's important to unpack exactly what is meant by "2+2=4". The most charitable unpacking I can give is that it means both:
a) There exists an axiom set under which (by implication, not definition), 2+2=4.
b) That axiom set has extremely frequent isomorphisms to (our observations of) physical phenomena.
But most people's brains, for reasons of simplicity, truncate this to "2+2=4". The problem arises when you try to take this representation and locate it somewhere in the territory, in which case ... well, you get royalties from The Big Questions, but you're still committing the mind-projection fallacy :-P

014y

Under a sufficiently high temperature, they will coalesce into one rock.
I don't quite have the argument framed, but it's something like arithmetic applies in our world, but only under circumstances which have to be specified separately from arithmetic.

114y

Those interested in a rigorous proof are advised to examine Principia Mathematica.

I personally take the so-called Tegmark Hypotheses (levels 1 and 3 being the ramifications of our universe, 2 being a subset of 4) to be de-facto true. If I am not mistaken, Tegmarkianism means that quantification over "individuals" is impossible, and thus consistent utilitarism will most likely be about anthropic probabilities.

-310y

How was that fact proved?

-2[anonymous]10y

It isn't. That would be de-jure truth. My assumption is based on the fact that with my internal states of knowledge it is an obvious deduction (not inference, mind, this is mathematics). I also have some nice conversational arguments for it.
Also, the "proof" of the Tegmark 4 hypothesis tastes like it might run into some Gödelian complications.
ETAL Other things I would call de-facto is P =/= NP, multiverse/multi-mind/mangled worlds interpretation of Quantum Mechanics and (before the LHC experiemnt) the existence of the Higgs field,

010y

So you are personally persuaded of it, and have unstated arguments for it.
Another remark which is much less interesting than production of the proof itself.
These are things I would call "punting", or personally comitting to.

0[anonymous]10y

Is it your informed opinion that I should edit my comments in order to make my communications clear(er)?

I find this terribly interesting, but rather than agree or disagree with it, I think I'd like some clarifications.

2 + 2 will always be 4 whether somebody is computing it or not.

Is this implying a "yes" to the Tree Falls in the Forest question? Has that question been decided, and if not, does this theory not fall apart without it?

On a related note, the core idea here seems to be that things that COULD exist (in the sense that they are mathematically possible) therefore DO, even if there may be no context for them to exist in (doesn't matter if...

814y

First off, welcome to Less Wrong! You should take a look at some of the sequences, as this exact question has been addressed (see: Disputing Definitions). To be brief, the consensus around here is that the "Tree Falls in the Forest" question is a wrong question, and should be dissolved.

0[anonymous]14y

Thanks nhamann. And fair enough. "Tree Falls" has always been a miserable example of what I'm getting at, so I'll try to be more general.
My question is more like "to what extent does perceiving or not perceiving something determine whether and to what degree it can be said to exist?". Clearly "Tree Falls" isn't the right way to ask this, but I can only assume that was the motive of whoever first asked it.
Anyway, the last I'd heard it was still (understandably) an undecided question. It is known though that observation affects reality so I would think it should be given some consideration.
My point as it relates to the original post though, is that if there's a chance that perception is a prerequisite for reality, then we may still have a chicken-and-egg problem with the idea of a "simulation no one is bothering to run".

414y

1. Be careful about how you interpret news stories about quantum physics - quantum physics is a very confusing subject, and is often distorted severely by the reporting process.
2. Affect and effect have tremendously different meanings in this context - please don't mix them up.

314y

I recommend looking through the Quantum Physics Sequence, or at least the "Quantum Physics Revealed As Non-Mysterious" and/or "And the Winner is... Many-Worlds!" subsequences. Aside from the general matters of map versus territory, our specific knowledge of quantum physics indicates that the observer/collapse effects you may have heard about are not part of what the universe is really doing.
As for self-perception... for all the problems with Descartes's philosophy of mind, "cogito ergo sum" is still a pretty good standard, at least for setting a bare minimum baseline for a definition of existence. (That is, if your definition of existence doesn't allow you to be pretty confident that you yourself exist, it can't be a very good definition.) Further, based on the assumptions of materialism and reductionism (see Zombies? Zombies! and GAZP), I concluded that if a being (whether a normal human you're interacting with, an AI, a person in a simulated universe, etc.) says that they feel conscious, real, etc., and you are confident that they have some mechanism for actually acquiring such a belief that is at least as good as your own (e.g. their program has to actually be mindlike, not just printf("I experience qualia! How mysterious!"); exit(0)), then you should take their word for it.

114y

The Simple Truth is another good place to start when considering these map-territory questions.

A good argument in this vein when telling people about this idea is to talk about fractals. The famous Mandelbrot set has literally uncountably infinite complexity, but arises out of very simple rules, and then ask them to characterize the difference between the abstract set of complex numbers and the rules defining it (barring model theory).

I have also (disappointingly/validatingly) thought of this and then read Tegmark. (It's even more disappointing/validating than that, though, since as well as Tegmark, you appear to have invented Syntacticism. You even have all my arguments, like subverting the simulation hypothesis and talking about 'closure'). However, I have one more thing to add, which *may* answer the problem of regularity. That one thing is what I call the 'causality manifold': Obviously by simulating a universe we have no causal effect upon it (if we are assuming the mathematical ...

I'm late to the thread, but I just had to leave a few thoughts.

Firstly, a well written post and I've thought about something similar too, as I suspect many have. I look forward to reading the Tegmark paper, which I have not yet done.

At least a couple of commenters opposed the jump from one to n to zero simulations, but I find it very intuitive. One can think about it like this: Suppose there is a non-iterative non-recursive formula for calculating the state of a deterministic universe at time t. This is analogous to having a digit extraction formula for pi...

814y

I don't think this argument works, because any interesting universe would have physics that allow the implementation of arbitrary Turing machines, and there is no non-iterative non-recursive formula for calculating the state of an arbitrary Turing machine at time t.

111y

Doesn't this imply that no finite universe is interesting?

It's an interesting idea, with some intuitive appeal. Also reminds me of a science fiction novel I read as a kid, the title of which currently escapes me, so the concept feels a bit mundane to me, in a way. The complexity argument is problematic, though--I guess one could assume some sort of per-universe Kolmogorov weighting of subjective experience, but that seems dubious without any other justification.

014y

Suppose we had a G.O.D. that takes N bits of input, and uses the input as a starting-point for running a simulation. If the input contains more than one simulation-program, then it runs all of them.
Now suppose we had 2^N of these machines, each with a different input. The number of instantiations of any given simulation-program will be higher the shorter the program is (not just because a shorter bit-string is by itself more likely, but also because it can fit multiple times on one machine). Finally, if we are willing to let the number of machines shrink to zero, the same probability distribution will still hold. So a shorter program (i.e. more regular universe) is "more likely" than a longer/irregular one.
(All very speculative of course.)

I think your theory predicts a single self-consistent universe -- the one we're in. (What do you think?)

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.

Instead, first consider only all internally-consistent mathematical structures. That is, consider the set of all mathematically true statements. Our intuition is that they 'exist' in some way independently. These exist, and nothing else. (Everything 'el...

014y

Are you saying that our universe's laws of physics may turn out to be the only self-consistent set of laws defining a universe-like structure, or that we should expand the definition of "the universe" to encompass all such structures? It looks like you may be saying both, but I'm not sure they're the same thing.

114y

I meant the former, that it would be interesting to consider the possibility that there might be only a single set of self-consistent laws that self-generate from a void, and that these result in our (single) universe.
Alternatively, there might be a few or infinitely many independent sets of self-consistent laws resulting in a multiverse of entirely independent universes.
My point is that we need to consider the question, because it's a completely different question to ask, 'how many independent sets of truth self-generate from a void?' and, 'how many models satisfy a set of limited observations?'.
When we remark that something different could have happened (I could have chosen a different major in college), we mean this in a limited logical sense. If everything is entangled with everything else, and especially if the universe is deterministic, things possible in this logical sense might not really be possible. That is, that they're not actually the case in any universe.

I was originally going to start, "I believe I understand this to level X," from the scale, but when I looked back to find what level I'm at, I found it hard to say beyond level 0 when there doesn't seem to be a way to make useful predictions about such a metaverse. Indeed, with the mention of Boltzmann brains and ignoring the 3^^^3 universes, it's almost as if you're telling us *not* to make predictions so we don't get further confused, and instead continue to treat our universe "as normal."

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”.

Wrong. The only "truth" is the actual getting "4" when someone applies those rules, and things like that you think that someone will get "4" when they apply the rules to "2 + 2".

The phrase "if A then B" generally just means something like "I tried putting A and not B into my idea of how the universe works, and it didn't make any sense".

114y

He showed that a structure of axioms will not be able to prove or disprove all possible theorems. You could interpret what you commented that way, but it has nothing to do with the post.

114y

Not quite. Godel's theorems are a bit more subtle. The first incompleteness theorem says that any consistent, complete axiomatic system which can model the natural numbers must contain statetements which cannot be proved or disproved in that system. (Even that is a bit of an imprecise statement. One needs to be careful about what one means by an axiomatic system. For our purposes, this can be phrased as a set of axioms and rules of inference which can be listed and applied using an effective method).
The second, closely related theorem, is that any axiomatic system obeying the conditions of the first theorem contains a statement equivalent to its own consistency iff the system is inconsistent. This also needs a fair bit of unpacking. The rough idea here is that any axiomatic system that can contain the natural numbers and is powerful enough to talk about an analog of itself cannot prove that analog's consistency.

012y

And anyway, if you can find a statement that can neither be proved nor disproved, then you can create two greater axiomatic systems where it's defined as true or false, respectively.
There's nothing more mysterious about this than the other axioms; math holds predictive power to the degree the axiomatic system we choose to use has axioms that match our universe.

114y

...no? How do Gödel's incompleteness theorems relate to this at all?

And the territory is not the campaign

''Instrumentalism is an interpretation within philosophy of science that a successful scientific theory reveals nothing known either true or false about unobservable aspects of nature.[1] By instrumentalism, then, scientific theory a tool whereby humans predict observations in a particular domain of nature by organizing laws, which state regularities, but theories do not unveil hidden aspects of nature to explain the laws.[2] Initially a novel perspective introduced by Pierre Duhem in 1906, instrumentalism is largely th...

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

quitethe 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 causedthatto 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

thoselaws? 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

aboutthe 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

canexist in this way — as one giant subjunctive — then we must accept that that is how and why our universedoesexist; 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 thathaschanged 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

thisuniverse, 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

entirelyfree 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 ifthe structure within the simulation(s) is not being causally influenced by any ongoing circumstances inthisuniverse. 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 Simsbecause 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 thatitis affected byour actionsbeyond 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

thisuniverse’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

doesturn 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

isas 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, soshouldwe 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 certainlynotimply 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 absurdumagainst the idea that a being within a simulationcouldfeel real. As we say, one person’smodus ponensis another person’smodus tollens. Since the conclusion I’m arguing for is merely unusual, not inconsistent (as far as I can tell), that takes out theabsurdum; 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 thereissomething 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.