The Paradigm of Quaternionic Probability: Transcending Chaos

What is it about?

This manuscript introduces a clear and easy-to-follow approach to probability that uses quaternions, a kind of number with four parts: one real part and three imaginary parts. Ordinary probability uses single numbers and works well for many problems. But since ordinary probability often misses important features, so Quaternionic Probability gives each event more structure. The heart of the paradigm is simple, instead of assigning a single number to how likely an event is, we assign a quaternion that carries four linked pieces of information on probability. This makes chaotic-looking behavior easier to explain because some apparent randomness comes from hidden directions and interactions. I will show in this book how basic probability concepts translate to this new setting. Notions like outcome, expectation, and independence are given quaternionic forms that keep the intuitive meaning while adding expressive power. The book gives a simple example and diagrams to show how quaternionic probabilities work in practice. In fact, my paradigm is applied here to the famous Two Boxes Problem. The paradigm also offers practical benefits, it suggests new tools for modeling in physics, engineering, and data science. For instance, filters and estimators can use quaternionic inputs to track orientation-dependent noise. Machine learning models can include quaternionic features to capture multi-directional patterns. These tools can make predictions more stable when systems exhibit layered or directional uncertainty. Accordingly, quaternionic probability increases expressiveness but requires careful interpretation. I will outline simple diagnostics and visualizations to help researchers adopt the paradigm. Hence, the goal is not to replace classical probability but to extend it where direction and influence matter. This extension helps us move past surface chaos to see clearer structure underneath. The work closes by discussing the next steps to apply my novel and innovative probability paradigm.

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Why is it important?

This book is titled “The Paradigm of Quaternionic Probability: Transcending Chaos”. It presents a creative and foundational work that formalizes my original and innovative Quaternionic Probability Paradigm (QPP), a successor and expansion of my Complex Probability Paradigm (CPP). The paradigm is applied to a historical significant problem in mathematics: The Two Boxes Problem, explored through a dedicated two chapters, beside the first chapter that develops and illustrates the QPP model, and where all of them couple rigorous derivations with stochastic simulations. My academic background – comprising PhDs in Applied Mathematics, Applied Probability and Statistics, and Computer Science – has enabled me to synthesize these disciplines into a unified epistemic framework. This paradigm is not merely a mathematical model; it is a philosophical and physical reimagining of probability itself. It builds upon the legacies of Pierre-Simon de Laplace, Jakob Bernoulli, Christiaan Huyghens, Andrey Kolmogorov, Blaise Pascal, Abraham De Moivre, and other giants whose contributions have shaped our understanding of randomness, determinism, and the universe. Probability theory has long served as a vessel of epistemic humility, quantifying uncertainty and modeling randomness. Yet within this humility lies a paradox: the desire to transcend stochastic opacity and unveil deeper determinism. The Quaternionic Probability Paradigm emerges precisely at this juncture – not as an extension of classical probability, but as a radical reconfiguration of its ontological and epistemological foundations. Rooted in Kolmogorov’s axioms, QPP does not discard the real probability space R, nor the complex extension C = R + M introduced in CPP. Instead, it quaternionizes the probability universe by introducing two additional imaginary components, beside the imaginary number i of CPP, which are j and k, yielding the quaternionic space: Q = R + M = R + (M1 + M2 + M3). This space is mathematically rigorous, philosophically resonant, and empirically generative. Within it, randomness dissolves into structured determinism, and uncertainty becomes a quantifiable projection of incomplete knowledge. The paradigm expands the axiomatic foundation of probability from Kolmogorov’s five to a total of eight axioms. These include the original five, and the three complex axioms of CPP are replaced by three new quaternionic axioms of QPP that govern the behavior of probability in a four-dimensional quaternionic hypercomplex space. Central to this framework are the Quaternionic Probability Vector Z, the Degree of Our Knowledge (DOK), and the Chaotic Factor (Chf), now extended to accommodate quaternionic structure. In QPP, every stochastic event in the universe Q attains unit probability – an assertion that redefines the very nature of chance.

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