Logical/mathematical (non-physical) arguments against simulation hypothesis?

Question

So the Simulation Hypothesis, from what I've heard, is not scientifically testable because it is not falsifiable, which makes perfect sense to me and is the viewpoint I have always adopted.

Yet recently I have come across an argument from Michio Kaku on the topic of simulation hypothesis, and he has this to say about the concept:

I do not think we live in a computer simulation. No digital computer can compute all the motions of molecules in a simple object, e.g. the weather. The smallest object which can simulate the weather is the weather itself. When you add quantum corrections, then no digital computer on earth can simulate the quantum effects in the weather. So no digital computer can possibly simulate reality as we know it.

I mentioned that a digital computer cannot simulate even a simple reality, since there are too many molecules to keep track of, far greater than the capabilities of any digital computer. We need a quantum computer to simulate quantum reality, and hence, once again, the weather is the smallest object that can simulate the weather. Therefore, I don’t think we live in a simulation, unless the simulation is the universe itself.

Similar arguments are made regarding how the universe is too complex or large to simulate, such as this argument from Doug Bonderud:

Despite solid philosophical and theoretical arguments, a team from Oxford University has found reasonably solid evidence that our universe is more than a mobile application. The proof? Attempting to model specific quantum phenomena such as the Hall effect quickly gets out of hand — according to Cosmos, modeling just a few hundred electrons using the quantum Monte Carlo technique requires more atoms than exist in the universe. As noted by Fast Company, storing just 20 spins of one particle using this model would require one terabyte of RAM.

So I understand that, according to our known laws of physics (particularly the Bekenstein bound, which implies Bremermann's limit, which in turn is a consequence of the Uncertainty Principle) that trying to simulate an exact copy of our universe in our universe is impossible. But simulation hypothesis, at least how I understand it, is not asking about if it is possible to simulate our universe inside our own, but if there could be another reality "above" our own.

Why does the universe "above" ours have to conform to the same physical laws as our reality? Nothing in simulation hypothesis says this has to be the case; and it certainly doesn't seem like it is a logical impossibility. There doesn't seem to be any reason why, if we're in a simulation, the universe "above" our own couldn't have a much denser Bekenstein bound, or perhaps even none at all; the Uncertainty Principle may very well not apply to a higher reality. It may even be the case that such a reality is not "quantized" like ours, and is infinitely divisible, and thus an infinite amount of information can be "packed" in a finite space (in turn, allowing for processes like supertasks, possibly Super-Turing computation?). Obviously it would be impossible for anything like this to physically exist in our universe, but nothing seems to rule this out for a "higher" reality. The only requirement seems to be that any "lower" universe must be less powerful than the one above it.

However, could there be an argument that demonstrates the impossibility of the simulation hypothesis not with empirical testing, but with a proof by contradiction? i.e. Assume the universe is a simulation. Then A (rigorous proof details left as an exercise to the reader), which implies ~A, thus the universe cannot be a simulation. Does this kind of proof make sense when applied to a philosophical argument? It's definitely something I'm curious to know more about.

Answer 1

Math and logic can never prove anything about the world. To see this, consider Euclidean and non-Euclidean geometry. It was discovered in the 1840's that there are various flavors of non-Euclidean geometry that are logically consistent. And Euclidean geometry is logically consistent. But they both can't be true about the world, because they are inconsistent with each other.

What is the answer? Math can tell us, "If you assume this, then you can conclude that." Math can tell you when such an implication is logically correct. But math can never tell you what is true about the world we live in.

There is no logical reason we couldn't be dreaming, or brains in vats, or Boltzmann brains, momentary regions of coherence in an otherwise random and formless world.

There's no purely mathematical reason we couldn't be living in a simulation. We're arrangements of atoms that have self-awareness. There's no mathematical reason some race of beings couldn't have figured out how to organize atoms to create self-aware creatures. Call such a being God. Call it the Great Programmer in the Sky. Call it the "next level up." What's the difference? How could you use pure logic or math to say it's not possible? After all, some of us chunks of meat are self-aware. Were we made? Or did we develop by pure chance? In the end this is a religious argument, a matter of faith.

Math and logic are not enough to establish truth. We learned this in the 1840's, and the ramifications of that discovery are still percolating through society. If math and logic can't establish truth, then what can?

Answer 2

The real question is what a simulation is. Only from knowing what you mean by that it is possible to answer logically.

If you think that a simulation is a simulation on a computer than the only logical answer is no. Because the stuff simulated simply is'nt the stuff that ìs simulated. The things simulated, even when in every possible detail, simply are not the same. You can make a simulation of an atom but this simulation is not the atom itself. It's just a process consisting of the same things it simulates.

But what about the stuff itself? Can't this be itself a simulation? In other words, are elementary particles simulations by some other stuff which is not the elementary partiicles particle stuff and which behaves according to different laws as the laws to which the particles behave?

In other words, can the stuff we see and experience be a simulation by some other stuff or by the same stuff we are made of? Well, when using the same stuff dreaming gets close. What you see in dreams is the same as what you see in waking time. Buþ still you know that you are dreaming when awake. And this is a sound logical argument that you can't find youself in a simulation. Knowing that you are not in a simulated world. That you are not dreaming so to speak.

But how do you know? What is the waking state as compared to the simulated state? I think that the very fact that we can't wake up in the real world (the one outside the supposed simulation) and only wake up in the real world we are in (the supposed simulated world) is proof that our real world is not simulated. You need always beings to observe the world. These make the world come alive. Even a dream world they let come alive. How can these observers ever be simulated? It takes the real stuff to make them exist and not a simulation of the real sruff.

So the very distinction between simulated stuff and real stuff is already proof that you, and me, and the whole world, are not simulated. Even if a computer that sophisticated that it could simulate every particle inside you when you are dreaming then still the computer is not dreaming. The only computer who can do this is yourself. And with history reaching back to the start of the big bang you cant (not even in principle) make a computer simulate all particles inside you. Only real particles could do that. But even then you could't create a new you because of the same fact that the history of the whole universe (or at least in your past lightcone) is of importance for the state of the particles that make up you. Maybe a simulation of some very (artificially) limited collection of particles can be made but that still ain't the real stuf nor will it feel like the real stuff (on the inside).

How would you make a simulation of a universe if the stuff you simulate with is much smaller in amount than all the stuff you are simulating? Also this is impossible. You can't make a simulation of all particles of the universe if you have only a small part of them available for your simulation.

Answer 3

Assume that the likelihood of a given simulation to be run is inversely correlated with the computational complexity of the simulation, in the space of all the simulation ever run (Simplicity Assumption).

Also assume that we are equally likely to be in one of the many simulations.

It then follows that, if we are in a simulation, we are very likely to be in a simple simulation.

Therefore, every experience of "optional complexity" that we see around is a strong argument against us being in a simulation. It doesn't really matter how easy it is to simulate our universe, what matters is that it would be easier, and therefore more likely, to be in a simpler universe.

But we do observe optional complexity around us, therefore we can conclude that we are not in a simulation. If you are not convinced that we observe optional complexity around us, we can just wait more to get additional complexity. For instance a long living universe or a universe in which we space travel is more complex than in one in which we don't.

This argument is elaborated more in this paper or in the related blogpost.