The book begins with a historical chapter, introducing the reader to the main players in the development of probability theory starting in the seventeenth century, including many names which will be familiar to all mathematicians: Fermat, Pascal, the Bernoullis and Fisher. The second chapter is the most mathematical, and discusses many of the axioms of probability and how to calculate the probability of complex events (such as getting dealt a full house in a game of poker) once one knows the probabilities of various primitive events (such as the probabilities of drawing a given card from a given deck), as well as fundamental theorems such as Bayes' Rule. The first two chapters of the book are very accessible, and I could easily imagine sharing them with students who are learning probability for the first time. Unfortunately, the same cannot be said of the rest of the book, which is far less accessible.
The remaining five chapters of the book are dedicated to the various methods for determining the probabilities of primitive events. Galvotti writes that "the determination of initial probabilities is a highly controversial issue, at the core of the debate on the interpretation of philosophy." The method that mathematicians will most likely be most comfortable with is that of Pierre Simon de Laplace, which Galvotti calls the 'Classical' method, and which calculates the probabilities of simple events as "the ratio of the number of favourable cases to the number of equally possible cases." There are several problems with this method, such as the idea that all cases are equally possible and even the idea that one could enumerate all of the cases in the first place, and this has led various philosophers to propose other interpretations. Galvotti dedicates a chapter to discussing the 'Classical' interpretation, and follows with a chapter each on the 'Frequency' interpretation favored by Venn, Reichenbach, and others, Popper's 'Propensity' interpretation, the 'Logical' interpretation developed by people such as DeMorgan, Boole, and Keynes, and the 'Subjective' interpretation. These chapters are quite interesting, but also quite dense as it describes many of the subtle differences in the various forms of each of the interpretations, and I am sure that many of these distinctions went over this reviewer's head due to his lack of a background in philosophy.
This leads me to my main criticism of the book, which I will admit up front is an unfair criticism. I thought that the subjects covered in the book were really interesting, but this is certainly a book written for philosophers of science rather than for scientists, and I wish that Galvotti had written the book with a slightly different target audience in mind. While she several times in the introduction refers to the book's "introductory character" and her aims to keep it accessible, it seems to me that she is trying to keep the mathematics accessible to the philosophers rather than keeping the philosophy accessible for the mathematicians.
The book concludes with a section on current thoughts in philosophy, and I wished this section had been expanded. It is easy for us to forget that, across campus, there are philosophers thinking and writing about the foundations of the tools which we scientists use every day, and I think that if an author would write a survey of the research that those philosophers are doing that was easily accessible to scientists that it would be something that many of us would enjoy reading, and also possibly change the way that we think about our own work.
Darren Glass is Assistant Professor of Mathematics at Columbia University.