The Princeton Companion To Mathematics

Bertrand Russell, in his book The Principles of Mathematics, proposes the following as a definition of pure mathematics. Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth. The Princeton Companion to Mathematics could be said to be about everything that Russell’s definition leaves out. Russell’s book was published in 1903, and many mathematicians at that time were preoccupied with the logical foundations of the subject. Now, just over a century later, it is no longer a new idea that mathematics can be regarded as a formal system of the kind that Russell describes, and today’s mathematician is more likely to have other concerns. In particular, in an era where so much mathematics is being published that no individual can understand more than a tiny fraction of it, it is useful to know not just which arrangements of symbols form grammatically correct mathematical statements, but also which of these statements deserve our attention. Of course, one cannot hope to give a fully objective answer to such a question, and different mathematicians can legitimately disagree about what they find interesting. For that reason, this book is far less formal than Russell’s and it has many authors with many different points of view. And rather than trying to give a precise answer to the question, “What makes a mathematical statement interesting?” it simply aims to present for the reader a large and representative sample of the ideas that mathematicians are grappling with at the beginning of the twenty-first century, and to do so in as attractive and accessible a way as possible.