# Computations are observer-dependent

May 1, 2021

If the Planck-Church-Turing thesis is valid [1,2], the equivalence of work and computation would immediately follow:

\begin{equation} \text{work} \implies \text{computation} \end{equation}

\begin{equation} \text{computation} \implies \text{work} \end{equation}

However, this is not something we may take for granted as it has non-trivial implications. It would almost immediately follow that all physical theories are observer dependent.

Computations are observer-dependent in general as a theory of computation requires that the syntax and semantics of computations be expressed within a formal language. In fact, the epistemic limits of that formal system define a world model and so all computations are fundamentally Bayesian. Moreover, computations are the expression of mathematically precise relations between publicly available data so computations are meaningful only if they are simultaneously meaningful for a population of observers.

This means that a shared language must emerge through a population of biological
organisms and this language must reach a degree of sophistication by some process of linguistic
morphogenesis that is sufficient for the analysis of mathematical expressions. In particular, observations are both
*mathematical relations* and *computations* because these are necessarily carried out by information-processing devices.
But, what is the exact nature of these mathematical relations?

We would be wise to go back to Poincaré who noted that the scientist discovers the relations between things and not the ultimate nature of things themselves. In this context, the most fundamental physical observations are expressions of fundamental relations between humans and their environment. It follows that the general question of what is computable is a fundamental problem in epistemology and possibly the most important problem.

This will certainly require a deeper investigation of the three-way correspondence between Math, Physics and Computation or what is known as the Planck-Church-Turing thesis. I believe that an investigation into the foundations of mathematics which unifies math, physics and computation will require a deeper understanding of biological evolution and cosmological natural selection.

One approach to both cosmological natural selection and the foundations of math, physics and computation would be to analyse models of computation that are possible with black hole computers. Related work has also been done on the holographic principle.

## References:

\(1.\) Pauli Space. The Planck-Church-Turing thesis. 2021.

\(2.\) Aidan Rocke (https://cstheory.stackexchange.com/users/47594/aidan-rocke), Understanding the Physical Church-Turing thesis and its implications, URL (version: 2021-02-22): https://cstheory.stackexchange.com/q/48450

\(2.\) Hajnal Andréka et al. Closed Timelike Curves in Relativistic Computation. 2011.

\(3.\) Bertrand Russell. Human Knowledge: Its Scope and Limits. Routledge. 2009.

\(4.\) John A. Wheeler, 1990, “Information, physics, quantum: The search for links” in W. Zurek (ed.) Complexity, Entropy, and the Physics of Information. Redwood City, CA: Addison-Wesley.

\(5.\) Jeffrey M Shainline. Does cosmological evolution select for technology? Institute of Physics. 2020.

\(6.\) Poincaré. La Science et l’hypothèse. Champs sciences. 2014.

\(7.\) Chomsky, Noam (1956). “Three models for the description of language” (PDF). IRE Transactions on Information Theory (2): 113–124.