No, we are probably not living in a simulation.
The simulationist argument goes like this:
First off, note that the above is not an argument. Nowhere is the crucial premise that "it is possible to simulate a world in sufficient detail to both evolve intelligence and to run in a just-in-time manner" made even remotely plausible.
There is no basis whatsoever to believe this might be possible, and a great deal of reason to believe it isn't possible, particularly with regard to the "just-in-time" nature of the simulation that is required to get the argument off the ground at all, and I will show below just how wildly implausible these ideas are based on the only thing we actually know about: our universe.
An important constraint on any valid argument that I will employ is: NO MAGIC. By "magic" I mean waving your hands and saying, "MAYBE THERE IS A WAY OUT!" Whenever I ask "What might that way be?" the answer is always: "Magic!" That is, "something that isn't known to be even slightly plausible, and which is invoked here solely to save the simulationist idea."
I distinguish two kinds of magic: weak magic (or just "magic") and strong magic. The difference is one of specificity: strong magic requires a specific idea to be true that we know is almost certainly false, like the possibility of just-in-time simulation that is physically accurate, or infinite variations on physics that all just happen to be able to sustain intelligent life.
This is an almost trivially true statement, but it gets ignored by people who say, "Assuming infinite computing power is available..." when it is a matter of fact that infinite computing power is not available in our universe because infinite universe is not available in our universe.
Computation takes matter, time, and energy. We don't have infinite computing power, and if we're going to talk about simulating universes our first task must be to ask, "How much of our universe could we simulate if we used all of our universe to simulate it?"
It would be fairly silly to try to bootstrap an argument from a claim that we can simulate our own universe or something remotely resembling it when we can't.
That is, as applied to this case it means there is no magical means by which finite matter can encode infinite information or perform infinite computations in zero time. "Well maybe there is!" is not an argument: it is an invocation of magic, and--for reasons I'll get to below--invoking it makes complete nonsense of any pretence to holding Bayesian beliefs.
If proponents of magic in the simulated universe argument are going to allow it in their own "arguments", then I insist on being allowed to use it in counter-arguments, which will result in duelling infinities of a particularly absurd kind.
So, how much of our universe can we simulate in our universe? I'm going to use this estimate to show that "just in time" simulation is necessary to make the "simulated universe" argument go, and then argue that under the NO MAGIC constraint "just in time" claims fail, because they are nothing but magic... and the alternative of alternative physics in the universe simulating us is even stronger magic. So without magic, there is no way to sustain the claim that our universe could plausibly contain a simulation of our universe or anything like it. And with magic... well--spoiler alert--why would we bother to constrain our arguments to anything as boring as simulation?
Data storage is one physical requirement of computation, and since the simulationists are motivating their entire argument on what is possible to us today--they observe that people like me write simulations they don't understand and assume that implies some quite different kind of person could write a simulation for an entire universe--it would be leave the argument completely without motivation--it wouldn't even get off the ground--if we have to assume the currently impossible from the off. In particular, it would leave a large and very similar class of arguments equally well motivated as the simulationist one.
What is possible in data storage today is not much, but I'm going to be generous to the simulationists and assume that every single fermion in the universe can be used to store one bit. Since there is no magic allowed, and "Maybe quantum computing will allow us to store more" is an invocation of magic, this is what we're limited by: one bit per particle (proton, neutron, electron... I'm not bothering with unstable particles and neutrinos because they would require magic to do anything with).
Let's assume we can turn the whole universe into storage with a trivial amount left over for computing. What kind of precision to we need to simulate the universe we see?
I'm again going to be generous and assume only static properties put this limit on things, and ignore the well-known issues with numerical chaos that would cause significant deviations from strict Newtonian physics on very short timescales without much higher precision than what we're talking about here.
The anomalous dipole moment of the electron agrees with theory to about 32 bits. This is one the most precise physical measurements we have, so it is reasonable to use it as a standard of required precision in our imaginary simulation. That is, if the motions of every electron in the universe (in our non-just-in-time version of the simulation) were such that we couldn't detect anything wrong at the level we can measure the anomalous dipole moment of the electron at, we'd need about 32 bits per co-ordinate, of which there are six (three position, three momentum).
That means that we'd need 6*32=192 particles to represent the motion of one particle at 32 bit precision, so only half of one percent of our universe could be simulated in detail by such a machine. And remember, I'm being incredibly generous here: I'm assuming we have somehow managed to use the entire mass of our universe for storage, and still have enough left over for computation, somehow.
As Niel deGrasse Tyson points out in the article linked above, most of our universe is made in accessible by the speed of light--perhaps, he suggests implausibly, to prevent us from being able to go there so it doesn't have to be simulated in detail--so suggesting that we could somehow access it would mean that it is not, in fact, being simulated in any kind of just-in-time manner.
A just-in-time simulation would have far, far less of the universe available to its inhabitants to use in their own simulations, so any proponent of just-in-time simulation has two big problems: to show that it is possible at all, and to show that it can simulate an interesting fraction of a whole universe using only locally available resources.
We've established that it is impossible to simulate our universe from within our universe at a level of precision that is of the right order to get the anomalous dipole moment of the electron right.
"But maybe we don't have to!" say the simulationists. "Maybe there's MAGIC!"
There are several kinds of magic that might be invoked.
The first is to assume that the universe that is simulating us is much larger than ours. Problem solved! Except...
That is, anyone who says "Us being unable to simulate another universe doesn't mean we aren't in a simulation because the universe simulating us could have totally different physics from our universe that would allow it to survive even though our physics would cause to to collapse" is invoking strong magic. They are supposing a wild flight of fancy is true for no other reason than that it allows the simulationist hypothesis to survive.
Since this analysis of how much of a universe can be simulated within itself applies to any universe, we can conclude that--absent loopholes that will also turn out to be strong magic--no universe is capable of simulating itself, so the chain of simulations, if one exists, must be a chain of universes with radically different physics. And since we know the physics of the universe can't be just any which-way and still allow life to exist, the odds of any simulated universe being welcoming to life, much less intelligence, is vanishingly small.
The usual loop-hole that is poked to avoid having different physics in every simulated universe is the idea that "maybe the simulation uses some kind of just-in-time mechanism to restrict it to a tiny fraction of the universe in adequate detail." This too, turns out to require very strong magic.
It's very easy to causally say, "Well, video games do all kinds of clever approximations, maybe the beings simulating us do as well! Maybe those stars are just lights in the sky (until we look) and those clouds of gas and dust just look like they're undergoing turbulent... Oh."
The problem of turbulent flow is very simple: all scales matter. There is no way of computing large-scale flows that are physically realistic without also computing the structure of the flow at all scales. If we could, turbulence would not be the intractable computational problem it is.
"Well maybe there is such a way of doing the large-scale computation without doing all other scales!", is again, strong magic: it assumes the existence of a very specific state of affairs--the easy computational tractability of the Navier-Stokes equation--whose sole motivation is that if such a state of affairs existed, simulationism would be plausible.
But "this must be plausible to motivate my argument, therefore you should treat this as plausible" is not an argument. If you can't do a better job of motivating your argument than that, you need a better argument.
There is absolutely no reason to believe that physically realistic simulations of fluid flows--as opposed to wildly unrealistic video-game approximations--can be performed on anything resembling a "just in time" basis.
Asking anyone to grant you "just-in-time simulation" as a premise is pure question begging: if it was possible for the universe to be a simulation, it would be possible for the universe to be a simulation. True! But also: not interesting!
The "simulation argument is usually stated in terms of a false "trilemma" that was designed by philosopher Nick Bostrom for the apparent purpose of distracting everyone from the underlying implausibility argument given above. The argument claims the following:
"The fraction of human-level civilizations that reach a posthuman stage (that is, one capable of running physically implausible high-fidelity ancestor simulations) is very close to zero", or
"The fraction of posthuman civilizations that are interested in running physically implausible simulations of their evolutionary history, or variations thereof, is very close to zero", or
"The fraction of all people with our kind of experiences that are living in a physically implausible simulation is very close to one."
Since all of the arms of the "trilemma" involve a physically implausible condition, the only purpose of posing the problem with three branches is to exhaust our limited attention in the hope that no one notices that all three branches involve that assumption. None of the branches can ever be more plausible than that assumption, which is no more than epsilon, the mathematician's fictional "smallest non-zero number there is".
The whole "trilemma" is grounded in sand.
It is worth, however, considering the more distraction-free version of the argument, also helpfully stated by the Wikipedia article above and modified by me into an equally plausible alternative argument:
Many works of fantasy as well as some forecasts by serious magickal practitioners and and students of mytholgy predict that enormous amounts of magickal power will be available in the future. Let us suppose for a moment that these predictions are correct. One thing that later generations might do with their super-powerful spells is create detailed glamours of their forebears or of people like their forebears. Because their spells would be so powerful, they could create a great many such glamours. Suppose that these illusionary people are conscious (as they would be if the glamours were sufficiently fine-grained and if a certain quite widely accepted position in the philosophy of spirit is correct). Then it could be the case that the vast majority of minds like ours do not belong to the original race but rather to people ensorcelled by the advanced descendants of an original race.
By then setting up a supposed "trilemma" on the foundation of this baseless supposition, he goes on to argue that the alternative to us being the product of a sorcerer's spell is that we become extinct before our magicks become sufficiently powerful, an unpalatable and threatening alternative that is nicely manufactured to focus the mind on the "trilemma" rather than the implausible supposition upon which it is founded.
For literally nothing in my restatement of the argument in the "quote" above is less plausible than the actual argument: once upon a time it was widely believed that "magickal" forces allowed worlds to be glamoured into existence, and there is an extensive literature of beings in those glamoured worlds beginning to doubt their own reality relative to the protagonist, who is typically from the "real" world. It is such a stock device in a certain kind of fantasy that it's a a trope.
Replacing "computation" with "magick" neither increases nor decreases the plausibility of the argument, since as I have shown above there is no universe remotely like ours in which any universe remotely like ours can be simulated, and the scope of assumptions required to create a patina of plausibility are so broad and outrageously unlikely that claiming "magick" as the basis for the argument is literally just as plausible.
"If something that is almost certainly false is true, we almost certainly live in some kind of synthetic universe." Because simulation. Or magick. Take your pick.
Ergo: we probably don't live in a glamoured world.
Or a simulated one either.