*The Elements of Mathematics*Stillwell attempts to give both "a bird's eye view of elementary mathematics" and an explanation of what "elementary" means in mathematics, exploring its boundary with "advanced". He sets this in its historical and philosophical context, highlighting the relationships and dependencies between different areas of mathematics, explaining how some topics have only become elementary with modern methods, and generally emphasising material of broader interest.

Inspired by Felix Klein's 1908 *Elementary Mathematics from an
Advanced Standpoint*, Stillwell takes Klein's arithmetic, algebra,
geometry and calculus and adds to those computation, combinatorics,
probability and logic.

"Arithmetic" picks up one strand of number theory: the Euclidean algorithm, quadratic and Gaussian integers, and the Pell equation.

"Computation" looks at the complexity of basic arithmetic operations, the P-NP distinction, and the concept of a Turing Machine. It also touches on universal machines and unsolvable problems, which lie just outside "elementary",

"Algebra" considers rings, fields, vector spaces, and algebraic number fields; Stillwell argues that group theory is not elementary, with the sticking point being the non-commutativity of obvious examples.

"Geometry" encompasses Euclidean geometry, coordinate geometry, the use of vector spaces and inner products, and constructible number fields; it stops short of non-Euclidean geometry or projective geometry.

"Calculus" covers the differentiation and integration of "elementary functions", the rational functions plus logarithms and basic trigonometric functions; it stops before confronting the full complexity of the real numbers and deeper notions of continuity.

"Combinatorics" focuses on graph theory, but also looks at how combinatorial approaches connect to other areas, with proofs of the infinitude of primes and Fermat's little theorem, and (advanced rather than elementary) the Brouwer Fixed Point theorem.

The shortest chapter, "Probability" looks at the Gambler's Ruin problem, random walks, and the basic ideas of mean, variance and standard deviation, only touching on the law of large numbers.

And "Logic" introduces basic propositional calculus, quantifiers, and Grassman's inductive formalism for the natural numbers. It also touches on advanced topics such as Peano arithmetic, the real numbers and uncountability, and reverse mathematics, the determination of axioms from results.

Several of the chapters contain advanced material and a final chapter "Some Advanced Mathematics" takes up one advanced topic in each area.

*Elements of Mathematics* has no exercises and no great depth,
but it is a brisk presentation of a huge range of material.
Stillwell suggests that "readers with a good high school training
in mathematics should be able to understand most of the book",
which seems about right.

It would be useful for a high school student contemplating further
study in mathematics, giving them both a feel for the areas not
covered in the school curriculum and some idea "what comes next".
Another audience would be teachers of mathematics, at high school
or university, who are making decisions about a curriculum and
its sequencing. But anyone curious not just about mathematics but
about its conceptual structure should find something to mull over
in *Elements of Mathematics*, even if the actual mathematics is
mostly familiar.

October 2020

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