Occam’s razor within Tegmark’s Mathematical Universe

An analysis of mathematical signatures of the Simulation Hypothesis.

The Monte Carlo Hypothesis, which emerges from a recent information-theoretic derivation of the Prime Number Theorem, makes the non-trivial assertion that the prime numbers have a maximum entropy distribution [1]. This hypothesis is significant as it may be deduced from an application of Occam’s razor to Tegmark’s Mathematical Universe Hypothesis, and it is experimentally verifiable using existing technology as it implies that prime encodings are incompressible relative to any machine learning model. Thus, we may view the experimental verification of the Monte Carlo Hypothesis using machine learning methods as a tabletop experiment for Tegmark’s Mathematical Universe Hypothesis.

Key assumptions:

1. Poincaré once argued that a physicist can only analyse the mathematical relations between things and not the things themselves. Hence, the Observable Universe is theoretically indistinguishable from its mathematical structure. By applying Occam’s razor, we may infer the plausibility of Tegmark’s Mathematical Universe Hypothesis which posits that the Observable Universe is indistinguishable from its mathematical structure.

2. The Universe emerged from a Singularity, which modern physicists call the Big Bang.

3. As established by Peano, a rigorous axiomatisation of the integers and arithmetical operations is necessary and sufficient to construct all of mathematics.

4. All the Quantum Information in the Universe is conserved.

5. Occam’s razor is generally applicable within Universes where information is conserved.

Number-theoretic consequences of Occam’s razor:

1. The prime numbers must have been specified as part of the initial Quantum State of the Universe. Hence, we may infer that the prime numbers initially had a maximum entropy distribution. Moreover, if we assume that all the Quantum Information in the Universe is conserved then a second application of Occam’s razor yields the prediction that the prime numbers have a maximum entropy distribution at any moment in time.

2. The prime numbers represent the First Cause within Peano Arithmetic. Hence, from an information-theoretic perspective they must represent a Microcosm of the Macrocosm. This yields the prediction that there ought to be a one-to-one correspondence between the mathematical structures in number theory and the mathematical structures in mathematical physics.

References:

1. Rocke (2022, Jan. 5). Kepler Lounge: Revisiting the unreasonable effectiveness of mathematics. Retrieved from keplerlounge.com

2. Rocke (2022, Jan. 12). Kepler Lounge: An information-theoretic derivation of the Prime Number Theorem. Retrieved from keplerlounge.com

3. Aidan Rocke (https://mathoverflow.net/users/56328/aidan-rocke), information-theoretic derivation of the prime number theorem, URL (version: 2021-04-08): https://mathoverflow.net/q/384109

4. Patrick Billingsley. Prime Numbers and Brownian Motion. The American Mathematical Monthly. 1973.

5. 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.

6. Shainline, Jeff. Does Cosmological Evolution Select for Technology? Arxiv. 2019.

7. Tegmark, Max. The Mathematical Universe. Foundations for Physics. 2007.

8. Rocke (2022, Aug. 22). Kepler Lounge: The secret life of the Cosmos. Retrieved from keplerlounge.com

9. Einstein, A; B Podolsky; N Rosen. “Can Quantum-Mechanical Description of Physical Reality be Considered Complete?” 1935.

10. James M. Robins, Tyler J. VanderWeele, Richard D. Gill. A proof of Bell’s inequality in quantum mechanics using causal interactions. Arxiv. 2012.