Alex Simpson: Synthetic probability theory

Abstract: I shall present an alternative to the standard set-theoretic formulation of probability theory. The alternative is based on an axiomatisation of the notion of "random variable" as a primitive concept rather than as a defined concept. The axioms are intended to capture certain key properties of random variables that underpin probabilistic reasoning. In this way, the axiomatisation is founded on our contingent experience of probability, and so may be described as providing a "synthetic" approach to probability theory. (In contrast, the usual set-theoretic formulation of probability theory is "analytic" in the sense that the theorems of probability theory are necessary truths that follow from the set-theoretic meaning ascribed to probabilistic
concepts.)

A main motivation for the proposed approach is that certain cumbersome aspects of probability theory are simplified. For example, sample spaces play no role, and one never needs to consider sigma-algebras and associated notions of measurability.

Although the axioms are intuitive, it is not trivial that they are consistent. This is established by showing that they are validated by a suitably constructed set-theoretic universe. However, the talk will not focus on this point.

Instead, in the talk, I shall present the axioms from first principles, motivating them intuitively. I shall also illustrate how several concepts from probability theory can be reformulated and redeveloped starting from the proposed axiomatisation.