The Art of Proof is intended to be taught over one semester or two quarters. Calculus and maybe linear algebra are two subjects that a normal student will have covered, and they will have done so to a satisfactory level. The student’s prior intuitive knowledge is given a foundation on solid intellectual ground through the skillful combination of a chatty writing style and engaging examples. Integers, induction, algorithms, real numbers, rational numbers, modular arithmetic, limits, and uncountable sets are some topics discussed in this course. Methods like axioms, theorems, and proofs are taught in conjunction with the mathematics themselves instead of being presented in an abstract setting.

Short essays on additional themes are included at the end of the book. These are intended to be presented in a seminar-style format by small teams of students, either in a classroom or mathematics club environment. Some examples are continuity, cryptography, groups, complex numbers, ordinal numbers, and generating functions.