## Probability, Chaos and Randomness

Probabilities play an important role both in every-day life and in science. Windsurfers check the probability of wind before taking out their boards, and physicists calculate the probabilities for events like a system being in equilibrium. But what do we mean when we make probabilistic statements about things like having wind or being in equilibrium? In a collaborative project with Carl Hoefer, we develop a Humean interpretation of probability to answer this question:

**Humean Chance in Physics**, in Carl Hoefer: *Chance in the World. A Humean Guide to Objective Chance.* New York: Oxford University Press, 2019, 181-213, with Carl Hoefer.

**The Best Humean System for Statistical Mechanics**, *Erkenntnis* 80, 2015, 551–574, with Carl Hoefer.

Probabilities also occur in deterministic theories. On the face of it, this seems to be a contradiction because chance and determinism seem to be irreconcilable opposites: either something is chancy or it is deterministic, but not both. One of the advantages of the Humean approach is that it can reconcile chance and determinism:

**Chance and Determinism**, in Alan Hájek and Christopher Hitchcock (eds): *The Oxford Handbook of Probability and Philosophy*. Oxford: Oxford University Press, 2016, 460-474.

**Determinism and Chance from a Humean Perspective**, in Dennis Dieks, Wenceslao Gonzalez, Stephan Hartmann, Marcel Weber, Friedrich Stadler and Thomas Uebel (eds.): *The Present Situation in the Philosophy of Science*. Berlin and New York: Springer, 2010, 351-372, with Carl Hoefer.

Chaos is an important motivation, or indeed justification, for introducing probabilities into deterministic systems. Chaotic systems also exhibit random behaviour, and the so-called ergodic hierarchy is proffered as a tool to classify levels of randomness in chaotic systems. In a joint paper with Joseph Berkovitz and Fred Kronz we discuss the relation between chaos, randomness, and the ergodic hierarchy:

**The Ergodic Hierarchy, Randomness and Hamiltonian Chaos**, *Studies in History and Philosophy of Modern Physics *37, 2006, 661-691, with Joseph Berkovitz and Fred Kronz. (An addendum to this paper can be found **here**.)

An accessible introduction to these issues can be found in our piece on the ergodic hierarchy in the Stanford Encyclopedia of Philosophy:

**The Ergodic Hierarchy**, in Edward N. Zalta (ed.): *Stanford Encyclopedia of Philosophy* (first published April 2011; substantive revision May 2016), with Joseph Berkovitz and Fred Kronz.

The relation between chaos, randomness and the so-called Kolmogorov-Sinai entropy is discussed here:

**Chaos and Randomness: An Equivalence Proof of a Generalised Version of the Shannon Entropy and the Kolmogorov-Sinai Entropy for Hamiltonian Dynamical Systems**, *Chaos, Solitons and Fractals* 28, 2006, 26-31.

**In What Sense Is the Kolmogorov-Sinai Entropy a Measure for Chaotic Behaviour? – Bridging the Gap Between Dynamical Systems Theory and Communication Theory**, *British Journal for the Philosophy of Science* 55, 2004, 411-434.

Probabilities lie at the heart not only of statistical mechanics and chaos theory, but also of quantum mechanics. The Humean interpretation of probability can also give a coherent account of probabilities in some versions of quantum mechanics:

**Probability in GRW Theory**, *Studies in History and Philosophy of Modern Physics* 38, 2007, 371-389, with Carl Hoefer.

The so-called GRW version of quantum mechanics is discussed further here:

**Properties and the Born Rule in GRW Theory**, in Shan Gao (ed.): *Collapse of the Wave Function: Models, Ontology, Origin, and Implications.* Cambridge: Cambridge University Press, 2018, 124-133.

**On the Property Structure of Realist Collapse Interpretations of Quantum Mechanics and the So-Called “Counting Anomaly”**, *International Studies in the Philosophy of Science* 17, 2003, 43-57.