Stillwell, the author of this exceptional work, states: “This book grew from an article I wrote in 2008 for the centenary of Felix Klein’s Elementary mathematics from an advanced standpoint” [1], and that “what is new in this book ... is a serious study of what it means for one theorem to be ‘more advanced’ or ‘deeper’ than others.” The book covers: arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic. Chapters 2 through 9 develop these topics by elaborating their fundamental principles and making connections among them.

Within the first chapter, there is the very accurate observation that “logic is the heart of mathematics, yet logic is not viewed as a mathematical topic by many mathematicians.” The treatment of geometry here shows how an inner-product vector space “capture[s] all of Euclid’s geometry,” and imparts real insight. The discussion on potential versus actual infinity certainly focuses this abiding issue (see also Epstein [2]).

A subtopic of chapter 2 (“Arithmetic,” number theory) is the continued fraction version of Euclid’s greatest common divisor (GCD) algorithm. The uniqueness of prime factorization is deemed “not obvious.” The section on Gaussian integers a + bi is the closest thing to a royal road to understanding them that I’ve encountered.

Chapter 3, “Computation,” points out that “problems whose answers are hard to find, by any method, but easy to verify are not yet proved to exist.” This provides real insight into the unanswered P = NP question. The roles of Post, Church, and Turing in the development of models of computation are made very clear, as is the unsolvability of the halting problem.

Chapter 4, “Algebra,” includes development of the “algebraic side” of vector spaces. The solution by radicals ultimately yielded to Galois/group theory. The motivation for, and nature of, fields and rings is clearly explained using their axioms. As a practitioner of computing, I found the discussion of constructive versus nonconstructive proofs of the fundamental theorem of algebra particularly interesting. The author admits that “there is something problematic about” nonconstructive proofs, namely their dependence on properties of the real number system (to put my interpretation simply, if not simplistically).

Chapter 5 “Geometry,” proceeds from Euclid’s straightedge/compass constructions to the Euclidean geometry of inner-product vector spaces. Section 5.1, “Numbers and Geometry,” having to do with Hilbert’s axiomatization, describes a most subtle and powerful truth: the set of constructible numbers, a subset of R, suffices to model (my word) Euclidean geometry. This set is a “potential” infinity, as contrasted with the “actual” infinity of R. There is a natural segue to the latter-day field of reverse mathematics [3].

Chapter 6 is about calculus and infinite processes. The processes used are conventional and familiar, except that many are executed in several ways. Conventional and constructivist notions of continuity are contrasted. “Concepts that worry constructivists probably belong to advanced mathematics.”

Chapter 7 covers combinatorics. The notion of a regular polyhedral graph as a projection of a regular polyhedron onto the plane is used to prove (graph-theoretically) that there are only five regular polyhedra, the Platonic solids. The proof of the famous Bolzano-Weierstrass theorem is made perspicuous by use of an infinite binary tree of subintervals.

The evolution of probability from elementary to advanced is treated in chapter 8, and is illustrated by “describing the outcome of n coin tosses.” The 1D random walk is illustrated as closely related to coin tossing, and the Chebyshev inequality and weak law of large numbers are reached in short order. Narratives that include Cardano, Jakob Bernoulli, (especially) De Moivre, Gauss, and Laplace are instructive of the subject. A dart with an “infinitely sharp point” introduces measure theory, which is admittedly advanced.

Chapter 9, “Logic,” begins, “Proof, and hence logic, is essential to mathematics, but the logic used in mathematics has several distinctive features.” The chapter discusses predicate logic in conjunction with the axioms of Peano arithmetic (PA) and Zermelo-Fraenkel (ZF) set theory. ZF set theory is deemed an extension of PA via the axiom of infinity. The author cites, to my great pleasure, reverse mathematics as a “more refined view” than the admittedly advanced ZF. The predicate calculus/modulo-2 arithmetic illustrations of the tautology and satisfiability problems are superb and are also put in the language of P and NP.

Chapter 10, “Some Advanced Mathematics,” is an apt conclusion to this remarkable book. The brief treatment of projective geometry finally made the subject manageable for me. And then there is the wonderful characterization of incompleteness: “The incompleteness lies in the mathematics, not the logic. What we lack is a universal system of mathematical axioms, not universal rules of logic.”

This excellent book is definitely for mathematicians and fellow travelers. It filled several gaps in areas of my own purported expertise, and opened vistas in others. This book encouraged me to read all of the author’s works, some of which I had already perused.

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