Ars Mathematica

Vis - a - vis the comments in sigfpe’s last post, Krzystof Burdzy has a booklet online: Probability is Symmetry. On Foundations of the Science of Probability which introduces, discusses and critiques the Frequentist and Subjectivist foundational positions of Probability Theory. For any of our readers unfamiliar with either the terms or the arguments behind these philosophies of Probablility, this book forms an excellent primer on the subject, in addition to arguing for Burdzy’s own interpretation.

Saul Youssef has a collection of links to papers on exotic variations to probability theory. These are forms of probability theory that share many of the usual axioms of probability theory but in which the probabilities themselves lie in a set other than the non-negative reals eg. the complex numbers, the quaternions, or even the p-adics. The primary motivation is that classical mechanics plus complex probabilities looks a lot like quantum mechanics, and so if you believe in complex probabilities you no longer have to worry about things like wavefunction collapse. Unfortunately it’s all a bit confusing if you’re a frequentist.