An experiment to perform in order to prove experimentally whether Pi exists independently from the mind

The idea came during discussion on embryophysics list

Below there is a description of the experiment that one could think of to check the relationships between Mathematics, Mind and Nature (the MMN experiment). In my view this could be done as a real experiment (so this is actually not a thought experiment) provided we find two mathematicians who agree to sacrifice their life for science. I believe that this should be not that difficult provided the importance of the experiment for the modern science.

Let us take a completely isolated bunker where the experiment begins. The initial conditions are enough so that mathematicians can comfortably chat for awhile with each other about Pi and prove that it exists. Eventually the oxygen in the bunker will run over and both mathematicians die. From a viewpoint of a natural science, we have a dynamical system that eventually comes to the equilibrium state. I assume that at the beginning when mathematicians prove that Pi exists we have a consequence of physical states where Pi exists indeed. If you are in doubt, please suggest any other physical states where you say that Pi exists. The goal of the experiment is to establish what happens with Pi at the end when the system reaches the stationary state.

Because of experimental settings, we can neglect the interaction with environment and I hope that this could be done even for the quantum mechanics treatment.

Before the experiment will be perform in real, you can take your bet on whether Pi is retained after the death of mathematicians or not.

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