[ExI] Simulations and Fixed Points

[Stuart LaForge]

Lately I have been trying to come up with mathematical ways of testing
whether we live in a simulation or not. I still can't prove it one way or
another, although I have discovered some pretty interesting consequences
which result if we do in fact live in a simulation.

One such counter intuitive result involves the mathematical concept of
"fixed points". A fixed point of a function is a point in the domain of a
function where the output of the function exactly matches the input. For
example, in f(x)=x^2, there are two fixed points at x=0 and at x=1 because
0^2=0 and 1^2=1.

There are many so-called fixed-point theorems that depend on the nature of
the functions in question. These theorems are important because computer
simulations are themselves functions and so one or more of these
fixed-point theorems might shed some light on the question of whether our
reality is a simulated reality or the base reality.

For example, if we assume that we live in a simulation that is a
contraction mapping of the base reality, that is to say it is a
small-scale model of the base reality, then the Banach fixed-point theorem
would apply.

This would mean that the simulation that we live in would contain a single
fixed point that would be exactly the same point in the base reality. In
other words the base and simulated realities would coexist together at
that single mathematical point as if the two realities were tangent at
that single point.

One can see this must be true quite intuitively for simple two dimensional
spaces. For example imagine you have a map of your town at any scale. You
could throw that map down on the ground at any location in town, in any
orientation with regards to north-south axis and there is guaranteed to be
a single unique point on the map that is a fixed point. This fixed point
would have the property that if you stuck a pin through that point on the
map, it would also go through the exact point on the ground that is
represented by that point on the map.

What this implies for the simulation argument is that if we live in a
simulation that is a scale model of some larger reality, then there has to
be some unique event that happens at the exact same time and place in both
the simulated and base realities.

I wonder what such an event would look like to us? Would it stand out or
be completely mundane and uninteresting?

Curious indeed.

Stuart LaForge