A persistent error in students' attitude to the role of mathematics in physics is that it makes the subject more "rigorous" at the cost of making it more difficult. The opposite is true. The proper role of mathematics is to make things easy. When something can be expressed in the precise language of mathematics, results can be obtained by the application of given rules. Calculations are so simple that even computers can do them. Chris Isham's lectures on the mathematical and structural foundations of quantum theory, reproduced in this book, provide an excellent illustration of this truth.

Half of the book describes the theory of operators on linear vector spaces, and some applications of this theory. Here, everything is clear and straightforward. The remainder is concerned with what all this mathematics means; more precisely, with the relation between the quantities used in the mathematical description of world, and the experiences we have of that world. We have no precise language even to formulate the questions, far less to answer them. Some of these questions existed already in classical physics, but it was reasonable there to leave them to philosophers. In quantum mechanics, the questions are clearly a part of physics, essentially because the gap between the mathematics and the things we "see" is here larger.

Isham's undergraduates at Imperial College are doubly fortunate. They have the opportunity to meet these issues and they are led to them by an expert in both the mathematical structures and in the philosophical background who is also a superb teacher. The students will need to be dedicated, as there are few concessions to giving the course a popular appeal, for instance by giving examples of the strange quantum effects that the formalism yields. This avoidance of examples, prompted by the need to be fully general, has the effect of giving the book a curious balance. There is a lot on various theorems telling us how not to make a hidden-variable version of quantum theory, but no description of the actual way that it can be done, that is the Bohm model.

Also, the concept of realism often seems to mean classical-realism. In the book, realism is defined in terms of physical quantities having values, but this requires us to say what the physical quantities are; they need not be those of classical physics. Indeed the definition of a "physical quantity" could require it to have values.

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The explicit collapse models of Giancarlo Ghirardi, Alberto Rimini, Tulio Weber and Philip Pearle are examples that are fully realistic in the usual sense of that word, in that they describe what is, and do not need assumptions about observers (at least not at a naive level) - but they do not have particles; they are entirely about waves.

The importance of the Bohm model, and of collapse models, lies not only in the fact that one of them might be correct, but in the new ways of looking at quantum mechanics they provide. For example, they do not naturally fit in with the linear vector space structure. These, however, are small quibbles in a welcome addition to the modern literature on quantum theory.

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It is good to have a book that gives such an excellent description of the mathematical structure of quantum theory, written by an author who recognises the need to understand its meaning. When we know what it all means, we may even know why we cannot put it into the sort of mathematical language with which our computers can cope.

Euan J. Squires is professor of applied mathematics, University of Durham.