Any sufficiently advanced technology is indistinguishable from magic.

— Sir Arthur C. Clarke (1917 – 2008)

The above quote is quite famous, at least amongst certain types of people. And the core idea is a pretty idealistic and hopeful one: technology will one day get so advanced that it will look like magic.

Or maybe it’s actually quite realistic, under another lens. If you brought a peasant from the Middle Ages to the present and showed them fast-moving gigantic flying metal contraptions, thin screens that show people on the other side of the world, and little gadgets that let you scry the past and communicate with your loved ones no matter where they are, the peasant would run away screeching: “WITCHCRAFT!” They wouldn’t run very *far*, they’d probably be hit by a car, but they’d run alright.

Sufficiently analysed magic is indistinguishable from science (warning: TV Tropes link). This sentence is similar to the quote starting the post, but it’s not nearly as deep or meaningful. Science is, after all, just the method. If a thing *exists*, then it falls under the scope of science. So if magic exists and works then it can be science’d. Let’s try to science it. Exactly *how magical* does magic have to be before it goes beyond the boundaries of what’s achievable by technology? Exactly how advanced does technology have to be before it’s far enough from our suspension of disbelief that we’re willing to call it magic?

A more practical question might be: what should you conclude about the universe once you observe magic in it?

**Magical Universes**

In The Finale, one of the characters mentions two kinds of universes: natural universes, and magical universes.

The former are those that are describable by mathematically simple, exceptionless low-level laws. Our own universe looks like it works that way, though we don’t know yet what the true laws are. We know quite a bit about it, given that the places the boundaries of current physics are at are the subnanometric scale or the cosmological one. We have a feeling that the laws are in fact unifiable into one simple framework, though even if that’s not the case it still looks like the world is, at its core, maths.

Magical universes, on the other hand, are those whose rules *aren’t* like that. They’re universes that are complicated, full of exceptions for high-level phenomena, particularly minds. Whenever you find minds that are irreducibly complex and that physics bends this way or that to satisfy their whims, you have one of those universes. Most major religions believe we live in a magical universe, what with souls and miracles and stuff like that.

And this division *is* in fact natural and clear-cut: either the laws are universal, or they’re not. If there’s some law that applies in certain domains but not in others, then we’re in a magical universe. As soon as there’s a single exception, then that’s a magical universe. Of course, it might be a fairly *boring* magical universe, one where the speed of light in a vacuum is exactly *c* everywhere except within a three-meter sphere radius about five light-years from the Sun where it’s actually 0.9943*c* instead. In fact, for all we know *we’re* living in such a magical universe. So then maybe we could make these definitions more flexible to allow for a continuous gradation of magic, with more or less magical universes depending on how big the exceptions are.

At any rate, Clarke’s third law still applies: most appearance of magic can *still* be created by technology. Light travels slower in means other than a vacuum, so maybe we could use nanotechnology to create that sphere in such a way that it *looks* like a vacuum, but there’s in fact material there that makes light travel slightly slower. If you want a more complete example of exactly how far technology can simulate magic, I have a piece of fiction to offer you. However, it’s very spoilerish of me to tell you *what* piece of fiction it is, because it’s not apparent from the start that that universe’s magic is brought by technology, so I’m going to leave a link, and if you don’t want to know what it is, just don’t click it or mouse over it. It’s this one.

So, let’s see. Conjuration? Sure, nanotechnology. Flight? Easy peasy. Souls? People coming back from the dead? No problemo. Teleportation? Piece of cake. Violating the laws of thermodynamics? Well, as long as you *think* you’re violating them…

Is there any magic that can’t be created directly by technology?

**Time Magic and Harry Potter**

How would you go about going back in time? How would you go about making precise prophecies of the future? How would you deal with Time-Turners?

Suppose at time you heard a prophecy about something that would happen at time . Then ‘s state depends on ‘s state and on ‘s. Then ‘s state depends on ‘s state, but that one depends on ‘s and ‘s, which means ‘s state depends on itself, and we get a causal loop. How would you go about simulating this in a straightforward step-by-step way?

You wouldn’t. You might suggest that you could simulate it many times until the timeline converged to one where the loop was stable, but as anyone who has studied analysis can tell you, there is no guarantee of convergence after repeated application of a function many times, so this simulation may very well simply never converge.

Or suppose, for example, that I took a time-turner and made a decision: if in five minutes I don’t see a time-turned version of myself, then an hour after that I’m going to time-turn myself; if I *do* see me, then I won’t. How would that paradox resolve itself? There is no stable time loop that contains this.

And the answer is that… well, here’s a barrier technology can’t cross easily. You *can’t* in fact step-by-step compute a time-turner. The reason time-turners work is because Rowling *knew* what was going to happen in the future, so she structured the entire timeline to include that.

So, the way technology would have to create time-turners would be by… mind-controlling everyone involving, and reality, too. There’s no way around it, really. There’s no way, without knowledge of what you *want* the future to contain, to include “free will” (in the sense that only agents within the system can influence the system) *and* stable time loops. The way you’d simulate a universe with stable time loops would be simulating all possible universes given the starting conditions, then deleting all of those that *don’t* contain stable time loops and keeping only the ones that do. In fact, stable time loops can be used to solve the P vs. NP problem. In fact, stable time loops are completely *impossible* under quantum mechanics, in the sense that they’d *disprove* Q.M. if they were discovered.

**Occam’s Razor**

I mentioned, in my post about Occam’s Razor, that the best mathematical formalisation of the Razor we’ve been able to find was Solomonoff Induction, which penalises hypotheses based on their description length. So if that’s the case…

…then magical universes can more-or-less be ruled out *a priori*. Not the boring kind, but the kind of magical universes most religions believe we live in, where minds are ontologically basic and capable of breaking and twisting the laws of physics. Deities, souls, if taken at face-value, are *possible* in some worlds, but those worlds are so *vastly* more complicated than ours that the probability of finding ourselves in them is completely negligible.

I mean, you’d need a program that defines our entire world, and *then* it also had an exception clause that described *an entire human brain* and said “that thing has a soul” or “whenever that thing wishes something/prays/whatever, reality’s laws change.” It’s possible, but enormously unlikely.

But unlike stable time loops, most other magic is actually *feasible in our universe*, via technology. So if we find magic, we should *still* conclude technology. If we find stable time loops… maybe the world doesn’t behave as nicely as we’d previously thought.

**So… what?**

Okay, so… what does it mean, then, to find yourself in a magical universe? What do you conclude?

Maybe my phrasing was misleading when calling a magical universe “magical.” *Obviously* all laws are necessarily mathematical in some sense, since mathematics is… well, everything. Literally everything can be described by mathematics.

But a magical universe?

What kind of universe has exceptions for minds?

What kind of mechanism can systematically create things that are really unlikely by sheer accident?

What kind of force consistently beats formulating the absolutely simplest solutions, and can in fact search the solution space for something with very low probability that satisfies certain constraints?

Intelligence, of course.

Magical universes, if they exist, will probably be in their *vast* majority embedded in natural ones. Computer simulations, artificial environments *created* by an intelligence that inhabits a natural universe. Beating Solomonoff Induction, creating an exceedingly unlikely possible world so that conscious observers can find themselves *special*?

Magical universes are artificial. Magical universes, universes whose laws have exceptions… they’re, well, not natural. That’s why the other kind of universe is called Natural Universe. Because a magical one is, almost without a doubt, artificial, too.

So, sufficiently advanced technology is indistinguishable from magic. And magic is, all the time, sufficiently advanced technology. All magic is, necessarily, artificial.

The key assumption here seems to be: It is much less likely that time travel is possible in a natural universe, than that we don’t live in a natural universe.

That doesn’t seem right to me. Closed timelike curves are a thing in general relativity. There are certainly plenty of theoretical obstacles to time travel, but they tend to brush up against parts of physics that are still relatively speculative. Something like Hawking’s chronology protection conjecture might be a true law of physics, but then again, it might not be. The lawful nature of the universe, on the other hand, has held up consistently throughout centuries of scientific discovery. I know which assumption I’d abandon more quickly if time travel were discovered.

(Disclaimer: I don’t actually know very much about physics.)

The problem is that we have now to big theories that describe the universe, quantum theory and general relativity, which are hard to reconcile, as we all know, and they make markedly different predictions here. General Relativity says CTCs are possible, whereas Q.M. would be completely disproven by their discovery, and not in a “Oh I guess we need to find something that looks like Q.M. now.” way, but in a “This is completely diametrically opposed to actual observations and predictions made by quantum theory which have been revealed true again and again.”

Furthermore, G.R. is in some meaningful sense “less fundamental” than Q.M., so if I were to bet which one of them has the last word, it’d be Q.M. Besides, the places where G.R. fails are usually exactly those singular ones like the ones containing CTCs – for example, we know for a fact that Black Holes don’t actually have singularities at their center, and the latest developments and experimental data seem to point to the nonexistence of even the Event Horizon as it’s understood.

And finally, if it turns out that we find CTCs, then it turns out that our Universe isn’t actually deterministically Turing Computable (sorta), which means that we will have absolutely no idea how to make Occam’s Razor mathematically precise, and we already have a

lotof confidence in our mathematical formulation of the razor, both because of empirical validation and other more abstract philosophical/mathematical arguments. So yes, I expect with very high probability that CTCs will be found to not be possible in the final theory.Like Yudkowsky said, causal universes have the nice property of being able to be computed step-by-step by a Turing Machine, whereas universes with CTCs in them have to be brute-forcedly computed and then have all those inconsistent universes discarded, which doesn’t

looklike the Tao.These are all good arguments against CTCs being possible. But I don’t claim that CTCs are probably possible; I claim that if CTCs do turn out to be possible, then this would be very weak evidence at best for the universe being magical. It would be far more likely, in such a scenario, that there is some set of simple mathematical laws that governs the physics of our universe, and those laws happen to be ones that allow CTCs.

I disagree. If I were to find a CTC, I’d probably conclude I’m in a computer simulation of some sort, rather than a natural universe.

A computer simulation with simple mathematical laws, or one that has something like a user who occasionally engages in divine intervention? If the former, then I can’t imagine what kinds of mathematical laws could in principle underlie a simulated universe but not a real one, and in any case a simulated natural universe is still a natural universe. If the latter, I don’t see how you can conclude that from the presence of a CTC.

A computer simulation with laws that allow CTCs. In the former case, indeed they’re exactly the same laws, a simulated universe that has natural laws is indistinguishable from a natural universe. But I’d conclude that from the presence of a CTC by the argument above, because CTCs are

reallyunlikely according to Solomonoff.Or, actually, if I found a “natural” CTC in the sense of a simple static CTC that just

existed, then I’d probably be about equally doubtful about Solomonoff as a good guide to the magical reality fluid, and about living in a natural universe; if I found a time-turner, I’d believe I was in a magical universe with very very high confidence.GR is deterministic, and computable, and allows CTCs. Therefore your idea that a universe with a CTC would not be turing computable is wrong, by counterexample.

The uncomputability of CTCs isn’t a “local” property of a program, it’s a “global” one. Each step in a CTC is perfectly deterministic and locally computable, provided you have the information that comes from the future to compute the next step in that computation. So, yeah, assuming CTCs are possible and computable, CTCs are possible and computable.

My point is that GR very clearly allows solutions that are CTCs. This is a feature of GR. They might not be physical solutions, but they exist. Heck, the interior of something as simple as the Kerr metric has CTCs.

But GR is just a set of differential equations. You’ll have cauchy boundary/horizon somewhere, so you’ll need to specify conditions appropriately, but its still computable.

I think when you say “not computable” what you might mean is that the future can’t be predicted with perfect knowledge of the past, but that is a different question.

Why do you think stable time loops would disprove quantum mechanics? In quantum field theories, most higher order calculations involve loops in diagrams, which would be a very specific form of a stable time loop.

Also, stable loops can only solve arbitrary P=NP if you allow infinitely large stable loops. Scott Aaronson has some nice notes on this.

Also worth noting, some forms of quantum gravity also have closed time like curves.

Because then you could bring information about the outcome of an experiment to the past, and act on it. I wrote something more careful about this on the latest chapter (or latest but one? idk, chapter 9 or 10) of my Death Note fanfic.

What do you mean by infinitely large stable loops?

As for loops in diagrams, afaik those loops are equivalent to antiparticles going forward in time.

If your CTCs are of finite time T, then P = NP only for problems you can check in finite time T.

As for quantum mechanics and CTCs:

1. There are string theories that admit non-pathological CTCs. String theory is fully quantum, so nothing in the structure of quantum mechanics precludes CTCs. This

is proof by counterexample.

2. Sure you CAN recast loops as a particle/anti-particle pair. But its just as valid to think of a loop as a closed, time like particle loop. These are mathematically equivalent. So this is an example of a CTC that is computable.

Finally, I think you have weird understanding of computability theory. You seem to picture “computability” as being entirely equivalent to stepping a differential equation forward in time. This really isn’t the case- in GR we can imagine all sorts of coordinate changes, such as using information from the past and the future to calculate the local metric throughout space or something like that.

Its pretty easy to show that if you have a theory thats calculable in P, you can calculate the self-stable solutions you get from inserting all possible instances of time travel in NP.

Honestly I have to confess ignorance here. I don’t know enough GR to properly respond. This is the best my mind comes up with when trying to explain what I mean:

GR describes this 4-dimensional mathematical manifold, depending on parameters like energy density and matter distribution and whatnot. That’s all well and nice, but it looks like a very different thing from what a universe-simulation would look like, with a step-by-step computation of cause-and-effect which is what our experience currently looks like. You’re correct that I’m using “computability” in a maybe-overly-narrow way… but what I’m trying to get at here is that a deterministic TM would have to simulate all possible universes respecting the same boundary conditions and discard those that don’t contain stable CTCs

Like what you suggested about NP.

Like… nondeterminism is pretty much magic? In that something being NP just means that “there is provably some optimal solution to this problem and if you ran this machine, there would be some nonzero probability that it would arrive there.” Nondeterministic TMs are mathematical constructs, and “there is some possible computation this machine performs which outputs the correct result” is unsatisfying from a “physical” perspective.

I’m probably blabbering right now. Anyway.

I guess what I mean is that CTCs aren’t straightforwardly step-by-step deterministically computable, in the way Q.M. (while interpreting loops as particle/anti-particle pairs) is.

Something something.

So I think you need to be careful about what you mean by step-by-step in time. In special relativity, you can’t define a truly unique time direction/plane-of-simultaneity. This means that there isn’t really a universal idea of “step-by-step in time.” Different observers at the same point will have different ideas of time, so the idea that the turing machine can hold some “current state” and step forward through it isn’t physically realistic anyway. So locally, someone can always calculate what will happen forward in time if they know their local state, thats probably not what “the universe” itself is doing.

In GR, its even worse because the fact that space is curved means that you can’t even define a time dimension that is appropriate everywhere at once, just different patches of coordinates in different areas.

So this step-by-step in time idea isn’t a good way of thinking about things. Best to think about the differential equations themselves and whether a turing machine can solve them. The answer is “yes” even if you allow certain types of CTCs.

I don’t know what “the universe” itself is doing; I

dobelieve that when people say that the laws of physics are computable, they mean it in that way, locally step-by-step computable. I’m not sure what it would even mean to say the universe as awholeis computable. The main thing here is: quantum mechanics looks more real/close to the truth than general relativity, quantum mechanics is locally timewise step-by-step computable. As far as I know, anyway.But yes you’re right that if you allow certain types of CTCs, the anwer is “yes.” I’m just not sure Time-Turners are these 😛

When people say the universe is “computable” they general mean turing-computable. This is very different from what you are suggesting.

If you think about things as a graph, what you are suggesting is that only acyclic-graphs are computable, which is clearly not the case.

I also don’t think that quantum mechanics “looks more real” than general relativity. I’m not even sure that statement is meaningful.

Um. Correct me if I’m wrong, but as far as I know Turing Machines are as I described, step-by-step computable in time. Unless I’m missing something? How is this very different from what I am suggesting? What

exactlyis computed here?And it’s meaningful in the same sense quantum mechanics “looks more real” than classical mechanics. Both are clearly only emergent laws, not the true fundamental ones, but classical mechanics is in some sense more steps removed from the true fundamental laws than quantum mechanics.

You have a huge disconnect. The steps in a turing machine don’t have to be equivalent to steps in time, and usually they aren’t. They are steps forward in whatever algorithm you are running.

Imagine doing finite element analysis of a static problem- there will be lots of turing steps even though the problem itself doesn’t evolve with time.

For a thermodynamics problem, each step would update the state of some position in space until the system hits equilibrium.

For a special relativistic problem, there would be many more turing-updates of spatial positions than temporal positions, and infinitely many ways to pick a time direction. You could pick a zig-zag through time (forward along the time direction of observer X, backward along the time direction of observer Y, then forward again,etc) and things would work perfectly.

Also, quantum mechanics is a bad choice for the “fundamental laws” since quantum mechanics by itself isn’t even relativisticaly invariant.

When I said “quantum mechanics” I meant something more like QFT. As for the rest of it, yeah, I think you’re probably right.

But that does raise a question. My argument on this post was seeking to find some form of Magic that would serve as evidence that we live in a simulated universe, on the (purely philosophical) basis that Occam’s Razor “means” simpler universes are “more likely to be natural” whereas more complex universes are “more likely to be artificial” in some sense. Do you agree with this idea? What kinds of Magic do you think would complicate the laws of physics to that point? It still seems to be that, even if CTCs in general aren’t that Kolmogorov complex, mentally caused CTCs (such as Time-Turners) should be there.