Nick Bostrom's simulation argument is impressively simple and clearly formulated. it doesn't try to prove that we live in a simulation, but instead elegantly formulates three possibilities, one of which must be true. Elon Musk has a similar thesis, which has made the idea known to a broad public. the official paper is already over 14 years old and just as many pages short. yet the central statement is easy to understand and compact.

The following symbols are introduced first:

- \(f_P\): Fraction of human civilizations that survive and reach a posthuman stage
- \(f_I\): Fraction of human civilizations interested in ancestral simulations
- \(f_{sim}\): Fraction of human civilizations living in ancestral simulations
- \(\overline{H}\): Average number of people living in a pre-posthuman civilization
- \(\overline{N_I}\): Average number of ancestral simulations performed by a posthuman civilization interested in ancestral simulations

Then:

- \(f_I \cdot \overline{N_I}\): Average number of ancestor simulations performed by a posthuman civilization
- \(f_P \cdot \overline{H}\): Average number of persons who have reached a posthuman stage
- \(f_P \cdot \overline{H} \cdot f_I \cdot \overline{N_I}\): Average number of persons in ancestor simulations (one simulates exactly the fraction \(f_P \cdot \overline{H}\)
- \(f_P \cdot \overline{H} \cdot f_I \cdot \overline{N_I} +

\overline{H}\): Average number of people living either in ancestor simulations or in a pre-posthuman civilization

Now the quotient of the last two terms corresponds exactly to the fraction of people living in simulations:

$$f_{sim} = \frac{f_P \cdot \overline{H} \cdot f_I \cdot \overline{N_I}}{f_P \cdot \overline{H} \cdot f_I \cdot \overline{N_I} + \overline{H}}$$

We exclude \(\overline{H}\) and shorten (this is also the core of the argument):

$$f_{sim} = \frac{f_P \cdot f_I \cdot \overline{N_I}}{f_P \cdot f_I \cdot \overline{N_I} + 1}$$

Bostrom now assumes an extremely large \(\overline{N_I}\), which he justifies with the exponential, technological progress based on conservative estimates.

This results in the following cases:

- Case: \(f_P \approx 0\)
- a) \(f_I \approx 0 \Rightarrow f_{sim} \approx 0\)
- b) \(f_I > \epsilon \approx 0 \Rightarrow f_{sim} \approx 0\)

- Case: \(f_P > \epsilon \approx 0\)
- a) \(f_I \approx 0 \Rightarrow f_{sim} \approx 0\)
- b) \(f_I > \epsilon \approx 0 \Rightarrow f_{sim} \approx 1\)

In summary, at least one of the following three cases is met:

- \(f_P \approx 0\): Humanity is dying out before a posthuman stage has been reached
- \(f_I \approx 0\): No posthuman civilization is interested in ancestor simulations
- \(f_{sim} \approx 1\): The simulation hypothesis: We live in an ancestor simulation

In order to increase the probability of \(f_{sim} \approx 1\) (which Bostrom gives as approximately \(\frac{1}{3}\) ), we can reduce the other probabilities for \(f_P \approx 0\) and \(f_I \approx 0\) by observation in the future.

If the simulation hypothesis is true, the following applies: Since \(f_{sim} \approx 1 \neq 1\) , this case does not exclude that we (or our descendants) in the set \(\overline{H}\) and are, for example, one of the first people to conduct ancestral simulations. The whole point, however, is that this is extremely unlikely. So if we don't live in a simulation now, there is a high probability that our offspring will never do an ancestral simulation.

One of the prerequisites for Bostrom is the so-called substrate independence (ie consciousness can not only be implemented in carbon-based, biological neural networks in the brain, but also on a silicon basis in a computer). Also interesting: Independent of the idea (and at no time a prerequisite) is the possibility of simulations in simulations (with any level of nesting depth).

In my opinion, the meaning of the truth of the simulation hypothesis for our lives can be summarized under the motto “Who cares?” - but the simulation argument is definitely fascinating evidence.

At this point, the other exciting works by Nick Bostrom on the sleeping beauty problem , the doomsday argument and general self-indication assumption should also be mentioned.