Kepler Lounge
What is an imaginary number?
In the following analysis, we trace the development of geometric algebra which allows a unified approach for tensors, quaternions, differential forms, spinors and lie algebras.
Bellman's Principle of Optimality and the Hamilton-Jacobi-Bellman equation
Any dynamical system that has a variational formulation admits an optimal controller within the framework of Hamilton-Jacobi-Bellman theory.
Money as a scalar field via Reinforcement Learning
By considering the general problem of multi-agent reinforcement learning, we show that money naturally emerges as an instrument for large-scale collaboration that inherits the scalar-field property of the value function in the Bellman equation.
Penrose and Weinstein on Spinors
A brief exposition on the central role of Spinors in modern Cosmology and modern physics, or proof of the importance of mathematical physics.
Mach's Principle and Einstein's theory of the Aether
In the following analysis we demonstrate that Einstein's theories of Special and General Relativity were constrained by his far-reaching insights concerning the Aether.
Paraconsistency and Evolvability
Reflections on the unreasonable effectiveness of mathematics and its implications for Artificial Intelligence.
A sublime proof of Euler's formula
A proof of Euler's formula using the Leibniz product rule.
Kolmogorov's theory of Algorithmic Probability
An introduction to Kolmogorov's theory of Algorithmic Probability, which clarifies the notion of entropy, and its application to the game of 20 questions.
A riddle on the 'Law' of Excluded Middle
A medieval riddle that exposes the false axiom embedded in the 'Law' of Excluded Middle.
The limits of mathematical induction
Using the theory of Algorithmic Probability and the Universal Distribution, we demonstrate that the definition of arbitrarily large integers is unsound. It follows that the Principle of Mathematical Induction, which depends upon the 'Law' of Excluded Middle, relies upon undefined terms.
Occam's razor within Tegmark's Mathematical Universe
An analysis of mathematical signatures of the Simulation Hypothesis.
The secret life of the Cosmos
A high-level summary of Cosmological Natural Selection as developed by Lee Smolin and Jeff Shainline which provides a naturalistic account for the Simulation Hypothesis.
The Spectral Geometry of the Prime Numbers
By exploring the correspondence between the Buckingham-Pi theorem and Unique Factorization Domains using Koopman Operators, we find that the Prime Numbers have an emergent Spectral Geometry.
Archimedes' Constant is absolutely normal
Using the theory of Algorithmic Probability, we demonstrate that Archimedes' Constant is absolutely normal.
Bell's theorem via counterfactuals
A proof of Bell's theorem via counterfactuals, which clarifies the nature of Quantum Randomness and its relation to Quantum Probability.
Random Walks and the Riemann Hypothesis
Starting from the observation that the statistical behaviour of the Mertens function is reminiscent of a one-dimensional random walk on non-zero values of the Möbius function, we demonstrate that the Riemann Hypothesis implies an entropy-bound on the Mertens function.
Gödel numbers and arithmetic functions
Gödel numbers may be used to define a programming language, where sets of propositions may be studied using arithmetic functions.
Quantum Probability and the Prime Number Theorem
We may demonstrate that the Prime Number Theorem emerges as a natural consequence of a thought experiment in Quantum Mechanics.
The Von Neumann Entropy and the Riemann Hypothesis
In the following analysis, the Riemann Hypothesis is presented as a natural Gedankenexperiment in Quantum Mechanics that reveals the super-deterministic nature of the Universal Wave Function. This forces us to reconsider claims that the theory of algorithmic probability, where probabilities are of a deterministic and frequentist nature, lack careful epistemic and physical justification.
Elements of Quantum Probability theory
An introduction to Lev Landau's approach to Quantum Probability theory.
Super-determinism via Solomonoff Induction
A super-deterministic theory of Quantum Mechanics is derived from Solomonoff's theory of Universal Induction, or what is generally known as Occam's razor.
The differential entropy of the Erdős-Kac distribution
Building upon the work of Billingsley, an entropy formula for the distribution of prime factors is carefully developed. This allows us to compare the entropy of the normal order of prime factors, which is constant, relative to extreme values where the entropy is unbounded.
Cramér's random model as a Poisson Process
In this article we demonstrate that Cramér's model is well-approximated by a Poisson Process, and that Cramér's conjecture may be deduced from an entropy bound on prime gaps.
Three master keys for Probabilistic Number Theory
Information-theoretic foundations for Probabilistic Number Theory.
An information-theoretic derivation of the Prime Number Theorem
An information-theoretic derivation of the Prime Number Theorem, using the Law of Conservation of Information.
An information-theoretic proof of the Erdős-Kac theorem
An information-theoretic adaptation of the Erdős-Kac theorem theorem, which informally states that the number of prime divisors of very large integers converges to a normal distribution.
Chebyshev's theorem via Occam's razor
An information-theoretic derivation of Chebyshev's theorem(1852), an important precursor of the Prime Number Theorem.
The Algorithmic Probability of a Prime Number
The Algorithmic Probability of a Prime Number, defined using Levin's Coding Theorem.
Revisiting the unreasonable effectiveness of mathematics
More than 60 years since Eugene Wigner's highly influential essay on the unreasonable effectiveness of mathematics in the natural sciences, it may be time for a re-appraisal.
Occam's razor
An information-theoretic formulation of Occam's razor as the Universal A Priori Probability.
The Law of Conservation of Information
Given that all physical laws are time-reversible and computable, information must be conserved as we run a simulation of the Universe forward in time.
Erdős' proof of Euclid's theorem
An information-theoretic adaptation of Erdős’ proof of Euclid’s theorem, which shows that the information content of finitely many primes is insufficient to generate all the integers.