The Paradigm of Complex Probability and Monté Carlo Methods: Taming Chaos

What is it about?

This book introduces a novel synthesis between the Complex Probability Paradigm (CPP) and Monté Carlo Methods, establishing a foundational framework for probabilistic simulation that transcends classical stochastic boundaries. This originated CPP reconceptualizes probability as a multidimensional epistemic field, wherein real, imaginary, and semantic components coalesce to encode uncertainty and observer entanglement. When applied to Monté Carlo Methods, CPP transcends classical randomness by embedding each stochastic trial within a complex-valued probability field. Additionally, CPP redefines the sampling space as a metarelational manifold, enabling simulations to incorporate not only statistical variance but also ontological depth and interpretive multiplicity. This paradigm shift invites a reexamination of randomness, convergence, and inference, positioning Monté Carlo Methods not merely as computational tools but as vehicles for epistemic exploration. By embedding CPP within the stochastic machinery of simulation, we inaugurate a generative modality for probabilistic reasoning – one that honors complexity and opens new pathways for transdisciplinary inquiry. Therefore, this paradigm shift opens new frontiers in applied mathematics, semantic physics, and the philosophy of simulation, positioning CPP as both a theoretical upgrade and a generative engine for scientific transmission.

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Why is it important?

In the year 1933, the Russian mathematician Andrey Nikolaevich Kolmogorov put forward the system of axioms of modern probability theory. By adding to Kolmogorov’s original five axioms an additional three axioms, this established system can be extended to encompass the imaginary set of numbers. Accordingly, the complex probability set C will be created and which is the sum of its corresponding real probability belonging to the real set R and of its corresponding imaginary probability belonging to the imaginary set M. Thus, all random phenomena do not occur now in the real set R but in the general complex set C that encompasses both R and M. Hence, we take into consideration supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory to evaluate the complex probabilities. This is consequently the objective of this novel paradigm. Subsequently, the outcome of the stochastic experiments that follow any probability distribution in R is now predicted perfectly and totally in C and the corresponding probability in the whole set C is always equal to one. Afterward, it follows that, luck and chance in R is substituted by absolute determinism in C. Therefore, we evaluate the probability of any probabilistic phenomenon in C by subtracting the chaotic factor from the degree of our knowledge of the random system. My groundbreaking Complex Probability Paradigm (or CPP) will be applied to the well-known theory of Monte Carlo Methods in order to express it perfectly and absolutely in a deterministic way in the universe C = R + M as well as to extend it to the probabilities’ universes M and C.

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