A Whirlwind Tour of Mathematics and Its History

By Ernest Davis

The Story of Proof: Logic and the History of Mathematics. By John Stillwell. Princeton University Press, Princeton, NJ, November 2022. 456 pages, $45.00.

Mathematician John Stillwell is both a historian and philosopher of math. He is a remarkably prolific author of textbooks on subjects such as the geometry of surfaces, Lie theory, number theory, and the relation of set theory to analysis, as well as several rather advanced popular math books. Within the last five years, he published Reverse Mathematics: Proofs from the Inside Out (2018); A Concise History of Mathematics for Philosophers (2019); Algebraic Number Theory for Beginners: Following a Path from Euclid to Noether (2022); and his newest book, The Story of Proof: Logic and the History of Mathematics.

In the preface, Stillwell states his purpose in writing The Story of Proof:

It is about proof — not just about what proof is but about where it came from, and perhaps where it is going.…Even professional mathematicians will be enlightened, I believe, by seeing the evolution of proof in mathematics, because advances in mathematics are often advances in the concept of proof.

This is an ambitious goal, and I will soon return to the question of how well Stillwell achieves it. First, however, let me describe the book’s contents.

Stillwell offers a whirlwind tour across a huge panorama that encompasses most topics in undergraduate mathematics. Over the course of more than 400 pages, he covers the following subjects: the Pythagorean theorem and its proofs; the discovery of irrational numbers and the subsequent response of Greek mathematics; Euclid’s axiomatization of geometry, Moritz Pasch’s betweenness axioms, and David Hilbert’s axiomatization; projective geometry; quadratic and cubic equations; quaternions and octonions; groups, fields, rings, vector spaces, and Galois theory; algebraic geometry, conic sections, and cubic curves; infinite series and power series; differential and integral calculus; integrals of algebraic curves and elliptic integrals; number theory, Gaussian integers, and prime ideals; the fundamental theorem of algebra; non-Euclidean geometry; graph theory; Leonhard Euler’s polyhedron formula, the Euler characteristic of surfaces, and knot theory; the rigorous formulations of the real line, continuity, convergence, and the Riemann and Lebesgue integrals; Cantorian set theory, infinite ordinals and cardinals, the continuum hypothesis, and inaccessible cardinals; axiomatizations of the integers, reals, and set theory; the axiom of choice and its implications; propositional logic, predicate calculus, Kurt Gödel’s completeness theory, and Turing machines; Gödel’s incompleteness theorem; and the proof of the consistency of Peano arithmetic via \(\epsilon_0\) induction. These are just the major headings; many other topics appear more briefly.

Stillwell provides a myriad of proofs throughout the book, mostly in their original or early forms. He gives Pappus’ proof of the isosceles triangle theorem, Bonaventura Cavalieri’s proof of the volume of the sphere, Isaac Newton’s derivation of the power series for the inverse of the natural logarithm (i.e., \(e^x\)), Pierre-Simon Laplace’s proof of the fundamental theorem of algebra, Pierre de Fermat’s proof of Fermat’s last theorem for \(n=4\), and many others. Finally, Stillwell includes some discussion of the evolution of fundamental mathematical concepts throughout the centuries. His commentary on infinity and continuity is particularly well done.

The book strictly adheres to pure mathematics and contains almost no consideration of applications. The only biographical information is birth and death years, and Stillwell does not discuss any general or intellectual history outside of mathematics. As such, there is an awful lot of math and mathematical history in The Story of Proof. Stillwell is profoundly knowledgeable, and his writing is careful and clear. The book is also physically beautiful, with many vibrant figures and a pleasing use of color within the text (see Figures 1 and 2). Kudos to the publisher’s production staff.

Nonetheless, I found The Story of Proof less enjoyable and less satisfying than Stillwell’s earlier books. The problem with whirlwind tours is that they are rushed, which manifests in a number of general issues. First, Stillwell’s presentation is often so compressed that it will be unintelligible to readers who do not already know the subject, and useless to those who do. I do not see how anyone can glean many takeaways from a two-page presentation of Galois theory, a two-paragraph discussion of Felix Hausdorff’s definition of a topological space and continuity, or a two-page description of the Lebesgue measure and integral. Though Stillwell’s account of these items is logically complete and self-contained, true understanding requires more motivation, exposition, and examples.

Second, Stillwell skips important aspects of certain fields. It seems odd to write a chapter about the fundamental theorem of algebra for polynomials with real coefficients without mentioning that it also applies to polynomials with complex coefficients, and that the complex numbers are therefore algebraically closed — especially since Stillwell discusses algebraic closure in other contexts. Equally strange is his decision to write about projective geometry while barely mentioning projective transformations, and to write about linear algebra without referencing linear transformations or matrices. Stillwell describes Newton and others’ ad hoc derivation of the power series for specific functions without mentioning Brook Taylor’s 1715 discovery of the Taylor series, which subsumed them all. Finally, it is odd that Stillwell covers the axiomatization of set theory without naming Russell’s paradox, which was a large stumbling block in the process of axiomatization and necessitated a variety of workarounds.

On a larger scale, some important areas of math are excluded — even Stillwell can’t squeeze all of mathematics into less than 500 pages. I am personally more interested in probability theory and functions of a complex variable than some of the subjects in the text, but that is a matter of taste. However, the nearly complete omission of dynamical systems of every kind seems much less acceptable. The characterization of change and motion over time has been a major theme of mathematics for 350 years, and ignoring that entire field seriously distorts the history of math — particularly in the 17th and 18th centuries.

In terms of the book’s stated intentions, the examination of conceptual evolution is shortchanged. As I noted previously, Stillwell’s treatment of the evolution of the concept of infinity is excellent (he did this at greater length and detail in his 2010 book, Roads to Infinity: The Mathematics of Truth and Proof). His treatment of the real line, continuity, and integration is also good. However, his presentation of the history of differentiation, groups, functions, and axiomatization has major gaps.

The evolution of the concept of a proof is especially elusive and difficult to trace; it would require a much deeper and more careful historical analysis than Stillwell provides here. He describes many specific proofs across a range of historical periods, but these examples do not in themselves make a case for conceptual change. Proofs from 300 years ago look different from today’s proofs for a number of reasons; for instance, the available mathematical toolbox has expanded enormously, writing styles and terminologies have changed, and specific mathematical concepts have advanced. The concept of a proof itself has also undoubtedly transformed over time, but such change is particularly hard to tease out from the historical record.

The introduction of the purely symbolic proof in formal logic in Alfred North Whitehead and Bertrand Russell’s Principia Mathematica (and other works of that time) represented a major extension to the concept of proof at the start of the 20th century. Otherwise, the concept has remained remarkably constant over the last 2,000 years, especially when compared to other mathematical categories like number and function. Some of Euclid’s proofs had technical gaps (which Stillwell discusses), and some of the things that Euler did with infinite series make current mathematicians seasick (which he does not discuss). Even so, what these historical mathematicians considered to be proofs are, by and large, still recognizable as such to us. Identifying how the concept of proof has changed would be both interesting and valuable, but doing so would require a much more careful analysis and comparison than what Stillwell has carried out.

All in all, I think that The Story of Proof would have been better if Stillwell had covered fewer subjects in greater depth and analyzed conceptual change more carefully. But even though he has not quite accomplished the goal that he set for himself, many readers will learn a lot from this book. The ideal reader is likely a bright mathematics college student who has broad interests in both math and math history — a student who would enjoy learning that the theory of conics attains its most unified form when viewed in the context of the complex projective plane, and seeing how Euler solved the Basel problem of computing \(\Sigma^\infty_{n=1}1/n^2\). If Stillwell’s brief account inspires such a reader to seek out a fuller treatment in some other book, then so much the better.

Ernest Davis is a professor of computer science at New York University’s Courant Institute of Mathematical Sciences.