Revisiting the foundations of probability theory

Probabilities are ubiquitous in modern natural sciences. They represent the main conceptual and formal tool used to describe and quantify the likelihood of something to happen, as well as the confidence with which we (do not) know something in a prediction. In particular, since the advent of quantum theory, specifically with the verification of the violation of Bells inequalities, it is possible to make strong arguments in favor of fundamental indeterminism in physics. But are probabilities the straightforwardly appropriate mathematical tools to attribute measures of likelihood to events is of prime importance to model indeterministic theories? The aim of this project is to explore, both at the conceptual and at formal level, the possibility of indeterministic physics where the concept of becoming (thus passage of time) is fundamental, both within quantum theory (where indeterminism is expected) and in classical theories (where there is general consensus about they being fully deterministic). In particular, I will address the capabilities and the limits of using probabilities to describe fundamental indeterminacy in physical theories and develop novel mathematical and conceptual tools to overcome the problems encountered along the way. The proposed project is interdisciplinary in nature. It comprises a more formal mathematical part which will fully deal with the foundations of probability theory and a second more conceptual and interpretational part that aims to provide probabilities and the new tools developed with a clear interpretation. These two parts will allow to develop formal and conceptual tools to describe indeterminism in physics, which remains the main driving force of this project.

Our best physical theories rely on probabilities to describe the natural world. We use them to predict the behavior of microscopic quantum particles, complex systems, and even cosmological phenomena. Yet, despite their success, we still lack a clear understanding of what probabilities mean when applied to nature itself. Do they merely reflect our lack of knowledge, or do they express a fundamental indeterminacy in the physical world? This project addressed this question by re-examining the foundations of probability in physics. The research explored whether the standard mathematical framework of probability theory is sufficient to describe a genuinely indeterministic universe - one in which the future is not completely fixed by the past. Traditionally, physics assumes a sharp divide: classical physics is deterministic, while quantum mechanics introduces irreducible randomness. The project challenged this view by investigating whether indeterminism may be present at a deeper level across physical theories. A central line of research developed new conceptual and mathematical models in which physical quantities contain only finite information. In standard physics, quantities are described using real numbers, which encode infinite information. The project examined the consequences of replacing this assumption with finite-information descriptions, leading to novel forms of physical indeterminacy that arise even in classical systems. The work combined methods from theoretical physics, mathematics, and philosophy of science. It analyzed how probability relates to causality, time, and prediction, and whether alternative mathematical frameworks - including approaches in which time and "becoming" are fundamental - may provide more adequate tools to describe an indeterministic reality. The project resulted in numerous peer-reviewed publications in high-impact international journals across both physics and philosophy, reflecting its strongly interdisciplinary nature. Its findings contributed to ongoing debates on the interpretation of quantum mechanics, the nature of determinism, and the conceptual structure of physical laws. In the longer term, this research may influence how future fundamental theories - including approaches to quantum gravity - conceptualize randomness, information, and the evolution of the universe. Moreover, I was able to show that several features and conceptual problem typically ascribed to quantum physics stem from indeterminism and can be already found at the classical level if one assumes fundamental indeterminacy. By clarifying the meaning and limits of probability in physics, the project advances our understanding of how nature operates at its most fundamental level.