John Stillwell*Reverse Mathematics: Proofs from the Inside Out

Abstract

Reverse mathematics is a programme in mathematical logic, initiated in the mid-1970s, which seeks to determine which axioms are necessary to prove theorems in areas of ordinary mathematics such as real analysis, countable abstract algebra, countably infinite combinatorics, and the topology of complete separable metric spaces. Reverse Mathematics: Proofs From the Inside Out is the first popular book on the subject, aimed at advanced undergraduates in mathematics, but also a good introduction for philosophers of mathematics. The time is certainly ripe for such a book, bringing this fascinating area of contemporary mathematical logic to a broader audience. Stillwell motivates the study of reverse mathematics through the following extended analogy with geometry, developed in the first chapter. In Euclid’s Elements, the fifth (parallel) postulate is used to prove the Pythagorean theorem. But is the use of the parallel postulate necessary? In fact, it is, but demonstrating this necessity in a rigorous manner requires two things that are at the heart of reverse mathematics. The first requirement is a base theory that does not prove the Pythagorean theorem, but that is compatible with its truth. The second is a reversal: an implication, provable in the base theory, from the theorem (the Pythagorean theorem, in this case) to the axiom (the parallel postulate). The base theory in the case of Euclidean geometry is provided by the first four of Euclid’s postulates. The existence of non-Euclidean geometries that make the first four postulates true but the parallel postulate false demonstrates that the base theory does not prove the parallel postulate. This makes the reversal from theorem to axiom non-trivial, and shows that the axiom is indeed necessary in order to derive the theorem: whenever the theorem is true, the axiom must also be true.