Quantified Ontology: The probability that we live in a simulation

What is the probability that we live in a simulation?

The very idea sounds absurd to most people, who only think of movies like "The Matrix" and similarly non-sensical plots when they hear about the idea of living in a simulation. However, anyone with a strong enough foundation in the relevant sciences should recognize that it is at least a possibility.

The most popular version of the Simulation Hypothesis that is being taken serious is the "Ancestor simulation" argument of Nick Bostrom. It states that one of the three following unlikely seeming statements has to be true:

The fraction of human-level civilizations that reach a posthuman stage (that is, one capable of running high-fidelity ancestor simulations) is very close to zero.
The fraction of posthuman civilizations that are interested in running ancestor-simulations is very close to zero.
The fraction of all people with our kind of experiences that are living in a simulation is very close to one.

(I make two assumptions in this article that are complex enough for their own books, so if you disagree with me on them don't bother reading the rest of the article: (1) Some variant of the multiverse hypothesis is true. (2) Functionalism is correct.)

A False Dichotomy

First of all, I believe that the question "do we live in a simulation or not?" is a false dichotomy.

From a mathematical and functionalist perspective, there is only processes running on processing substrates. There is no inherent difference between a process that is instantiated through physics directly, and one that is instantiated on a computer, since the computer is itself instantiated through physics.

So let's redefine what we mean. The question we really want to ask is this: Do we live in a processing substrate that is being overseen by an intelligence outside of that processing substrate?

You will note that this is a much more general way of phrasing things. In fact, even most religions would match these criteria: God is an intelligence that exists outside of our processing substrate (i.e. physics as we perceive them) and has ultimate authority over it.

On the other hand, if it turns out that reality at its most basic level is actually a simple Turing Machine that only indirectly simulates the laws of physics as we know them, then that wouldn't count as a simulation. This is relevant because most sensible variants of multiverse theories presuppose some bias towards universes with simpler mathematical structure, and the example universe that is a simple Turing Machine would thus actually be quite likely.

I believe that this redefinition of the question makes it much more meaningful. After all, the real question we want to know the answer to in practice is not some obscure physics question about different ways of instantiating an algorithm, but the very practical question "Is there something watching us? Something than can judge us like a god, and that can create an afterlife if it wants to?"

In that vein, I would like to introduce several possible reasons that a universe could be simulated:

The universe is being simulated by accident, as a side effect of something else. Maybe whoever is simulating this universe is only interested in the lifecycle of stars, and isn't even aware that we exist.
The universe is being simulated by post-humans as a documentary. The simulator is aware of us, but really just wants to see what happens next and has no plans to interfere. Nick Bostrom's "Ancestor Simulation" can be a type of this.
The universe is being simulated by post-humans to tell a story. Imagine it like the next step in the development of videogames. We all are NPCs, and somewhere in the world there are some people from outside the simulation who just want to have fun. Our reality exists for their amusement. This is not the nicest possible interpretation, but frankly given the popularity of video games and the inconsistency of human morality, an Ancestor Simulation of this type strikes me as very likely to happen.
There are likely a very large number of further explanations. After all, who knows what possible motivations a non-human simulator could have?

Depending on which of these hypotheses is true (if any), the implications for our lives would be very different.

Unfortunately I see no concrete ways to calculate the probabilities of any of these scenarios, only heuristics that might easily lead us wrong.

However, the question whether we live in a simulation at all, without going into the details of the type of simulation, can actually be considered largely independent of this using a concept I invented:

Quantified Ontology

I would like to introduce the concept of Quantified Ontology.

Ontology is normally concerned with the question whether or not something exists.

If we assume that the multiverse theory is correct, then I believe that this question is insufficient: In an infinite multiverse, everything exists an infinite number of times.

Quantified Ontology instead asks "how frequently does something exist, in comparison to something else?"

This is the only way to sensibly argue about things in an infinite multiverse, and as it turns out, it is also relevant for calculating the probability that we exist in a simulation.

We can rephrase the abstract question of the simulation hypothesis into a much more mathematically clear question using Quantified Ontology:

In an infinite multiverse, what is the relative frequency of "a human-like consciousness that is embedded in a processing substrate that is itself overseen by a human-like consciousness" compared to "a human-like conscious"?

To answer this question one can use different arguments, but as far as I am aware all of them are really just heuristics. There is no human-comprehensible formula that we could actually calculate to derive the answer. Example heuristics are:

Most civilizations will die out before they can run even one simulation. Therefore we are likely not in a simulation.
If a civilization is powerful enough to run one simulation, it will likely also be powerful enough to run an extremely large number of them. Therefore we are likely in a simulation.
Out of all possible entities that could be simulating a universe, those that do not pay close attention to the simulated universe are more efficient and therefore simulate more universes. Therefore we are likely in a simulation that is not being actively observed by a sapient entity, which is basically indistinguishable from not being in a simulation at all.

The main question I ask myself here is: Is this valid reasoning at all? Can you just count the number of universes, simulated or not, and let their frequency ratio determine their Quantitative Ontology index?

I have not found a satisfactory answer to this question. I suspect that it is impossible to derive an answer with the information we have. It may even be that I am not asking the right question and Ontology can not actualy be quantified in any meaningful way at all. There are some thought experiments that suggest this. Most interesting of these is the Sleeping Beauty Problem because it very succinctly pokes at the heart of the problem.

This ultimately comes back to the difference between two opposing philosophical viewpoints that give different interpretations of probability theory: The Self-sampling assumption and the Self-indication assumption.

Until and unless these fundamental questions in probability theory are resolved, it is impossible to quantify the degree of "Ontological strength" of a hypothetical universe. And until that is possible, it is impossible to give a sensible estimate for the probability that we live in a simulation.

In the absence of mathematical proof, all we have is heuristics.

Unofrtunately, whenever I think about this topic, I keep finding new arguments for why one side of the argument should be much more likely than the other. Each new argument always completely upsets the probabilities I assigned before. I find myself certain that we are in a simulation in one moment, then certain that we are at the base level of reality in the next moment, then back to the first opinion and short while later. There are simply too many chaotic elements to keep track of to make any kind of sensible estimate.

In conclusion, my answer to the question if we live in a simulation is "I have no idea one way or the other, and anyone else who claims they do is probably lying".