Could we be living in a simulation, or does physics rule that out?

There's an argument, due to philosopher Nick Bostrom, that under some not-too-wild assumptions it's likely that we are living in a computer simulation. The simulation argument and its provocative conclusion has some high-profile supporters, such as Neil DeGrasse Tyson and Elon Musk. But it also has many detractors, among them the theoretical physicist and science communicator Dr. Sabine Hossenfelder, who runs a popular blog and YouTube channel with many interesting pieces on physics and philosophy. 

Hossenfelder recently published a video with the provocative title The Simulation Hypothesis is Pseudoscience, and I would like to address her critique. Her objections can be summarized as follows:

It's hard (or impossible) to simulate a world like ours because:

1. It's very hard and nobody currently knows how to reproduce the laws of physics in a computer algorithm. 
2. To avoid a computational nightmare of simulating every particle, it needs to be possible to not care much about carefully simulating places where no one is currently looking, and just fill them in in case someone looks. But it's very hard and nobody knows how.

But let Dr. Hossenfelder speak for herself, she says (and please check out her full piece at the link above):

The problematic part of Boström’s argument is that he assumes it is possible to reproduce all our observations using not the natural laws that physicists have confirmed to extremely high precision, but using a different, underlying algorithm, which the programmer is running. I don’t think that’s what Bostrom meant to do, but it’s what he did. He implicitly claimed that it’s easy to reproduce the foundations of physics with something else.

But nobody presently knows how to reproduce General Relativity and the Standard Model of particle physics from a computer algorithm running on some sort of machine. You can approximate the laws that we know with a computer simulation – we do this all the time – but if that was how nature actually worked, we could see the difference. Indeed, physicists have looked for signs that natural laws really proceed step by step, like in a computer code, but their search has come up empty handed. It’s possible to tell the difference because attempts to algorithmically reproduce natural laws are usually incompatible with the symmetries of Einstein’s theories of special and general relativity. I’ll leave you a reference in the info below the video. The bottomline is, it’s not easy to outdo Einstein.

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A second issue with Boström’s argument is that, for it to work, a civilization needs to be able to simulate a lot of conscious beings, and these conscious beings will themselves try to simulate conscious beings, and so on. This means you have to compress the information that we think the universe contains. Bostrom therefore has to assume that it’s somehow possible to not care much about the details in some parts of the world where no one is currently looking, and just fill them in in case someone looks.

Again though, he doesn’t explain how this is supposed to work. What kind of computer code can actually do that? What algorithm can identify conscious subsystems and their intention and then quickly fill in the required information without ever producing an observable inconsistency. That’s a much more difficult issue than Bostrom seems to appreciate. You cannot in general just throw away physical processes on short distances and still get the long distances right.

The objections Hossenfelder presents against the simulation argument are very interesting, but I believe not nearly as deadly as she takes them to be. I think there are good responses to them. First, let me give a quick summary of my responses and then I'll elaborate:

1. To disrupt the simulation argument we would need to conclude that it's likely impossible to simulate our laws of physics on a machine in enough detail. But not knowing how to do that is insufficient to warrant that conclusion.
2. We can present an argument that it's in fact possible to simulate our laws in enough detail, i.e. in a way that we, to borrow Hossenfelder's phrase, couldn't "see the difference".

Response 1. 

We have to distinguish two questions:

(a) is it probable that our laws could be simulated (in sufficient detail) by some algorithm?
(b) do we know how to do that?

For the simulation argument it's (a) that's important, not (b). Hossenfelder points out that the answer to (b) is currently no, but that doesn't do much to show that the answer to (a) is no. 

There are plenty of examples where we don't know how something might be done and yet do not infer from this that there is probably no way to do it. As one example, we currently don't know how to program a human-level AI, and yet most experts believe that it's highly probable that this is achievable within a matter of decades. Or here's another example, which might be more dear to Hossenfelder's heart: we don't know how to write down a theory that unifies gravity and quantum mechanics, but we don't conclude that it's likely impossible.

But Hossenfelder also provides some considerations that address (a), the important question, directly, so what I said above is not enough. To address it, let's look at the second response.

Response 2.

You can approximate the laws that we know with a computer simulation - we do this all the time - but if that was how nature actually worked, we could see the difference.

A reasonable argument can be made that the answer to (a) is yes, i.e. that it's likely possible to simulate our world in sufficient detail. What do I mean by "sufficient detail"? I mean, to borrow Hossenfelder's phrase, good enough so we don't "see the difference" between how a simulated and an unsimulated world would behave. The "see" part is crucial - for example, there's no need to simulate every particle in the middle of the Sun or in the air in your living room! A much much rougher approximation ("effective theory") will do perfectly fine without anybody seeing the difference. 

But what about when someone looks really closely at something, for example through a powerful microscope or in a particle accelerator? Since that someone is being simulated, the machine that's simulating them has access to everything they know and much more, so it's really easy for the machine to know what they're looking at and how closely, so it would know what to simulate with higher precision.

Once it knows the needed level of precision, can it achieve it? We can argue that the answer is very plausibly yes, since the goal, again, is for limited human minds not to see the difference, so the simulation just needs to outperform the human mind in predicting how that "something" someone's looking at should behave. And if we grant, as Hossenfelder does in her piece, that it's not particularly problematic to simulate human-like consciousness, then presumably the simulation can figure out anything the human mind can and much more. In other words, it can always be one step (or ten) ahead of us!

Here then is a compact form of the argument:
A. It's possible to simulate human level cognition.
B. If A then it's possible to simulate many human level agents.
C. If B then it's possible to outperform human level agents in predicting what the behavior of the external world should look like.
D. If C then it's possible to simulate many human level agents and the external world, and make sure the behavior of the external world looks to the agents like it should, to the best of their predictive ability.
E. Conclusion. It's possible to simulate our world in sufficient detail.