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Concepts at the Heart of Mathematics—Through the Centuries

Katalin Bimbó

Communicated by Notices Associate Editor Emily Olson

book cover — ***The Story of Proof***
*Logic and the History of Mathematics*
By John Stillwell. Princeton University Press, 2022, 456 pp.

The preliminary title of this book was How Mathematics Works—as we learn from the preface. The author views the current title as less ambitious. It is really hard to give a concise and descriptive title to this very unique book, and the title of this review does not describe the content precisely either. Proofs are essential to mathematics. However, what was accepted as a proof changed through the history of mathematics. M. Kline in 5 mentions examples when famous mathematicians (e.g., J. J. Sylvester) proved “theorems” from false assumptions. For the sake of this book, Stillwell considers a proof to be what is presented in a paper or a book on mathematics, which is labelled as “proof” and is a correct demonstration. For example, Newton’s method of inversion of a power series was a heuristic argument at the time with no rigorous justification. I would guess Stillwell’s view on proofs would appeal to many readers, because it strikes a balance between formalized proofs (that logicians would consider to be rigorous proofs) and hand-waving arguments (that might be sufficient to convince a layperson).

Before going into some details about the content, it might be useful to clarify what this book is not about. The research area of proof theory deals with formalized proofs that may be couched in one or another proof system in one or another logic. A sample problem from this area would be whether the cut rule is admissible in S. C. Kleene’s sequent calculus formulation of first-order intuitionistic logic. The book is not about proof theory, although there is a mention of a formal system reminiscent of K. Schütte’s sequent calculus for first-order classical logic. Stillwell’s book is not a history of mathematics; historical details are complementary to the mathematical content. It should be noted that anybody who would like to follow a strand of history touched upon in the text will find a slew of references in the bibliography. The title of the book refers to logic too, and a branch of mathematical logic deals with formalized mathematical theories. D. Hilbert called investigations of certain properties of formalized theories (especially, of consistency and decidability) metamathematics. Stillwell gives a synopsis of Gödel’s incompleteness theorems, however, the study of formalized mathematical theories using the tools of symbolic logic (or the study of systems of formal logic) is not the main topic of this book.

To outline what the book is about, we can say that it describes the development of the core ideas of mathematics and their connections. Stillwell pays special attention to when a new proof method appears, and whether it emerges due to conceptual pressure in a theory, or it brings about new developments in a theory. The thread of the story runs through examples, though not all bits and pieces of proofs are labelled and categorized. It starts with proofs that are needed to deal with infinity and ends with proofs that secure the foundations of mathematics and reveal the comparative strength of famous theorems.

What is in this book?

A very simple view of mathematics is to say that mathematics is a study of numbers and shapes. Of course, in the 21st century, even a high-school student could give examples of mathematical statements or formulas that are neither about numbers nor about shapes such as the general form of the derivative of a polynomial function. But it is reasonable to look back at the work of ancient Greek thinkers and see that they elevated mathematics from practical know-how to a discipline by creating a theory of geometry and by proving theorems about numbers. Pythagoras’s theorem is a natural starting point for this book; it allows for the introduction of rational and irrational numbers, Pythagorean triples, and Euclid’s algorithm to find the greatest common divisor (gcd). Stillwell locates the motivation behind the invention of (deductive) proofs by the Greeks in the “fear of infinity” (or put less dramatically, in the need to justify statements about infinity).

The ten self-evident statements of Euclid’s Elements are often translated as “postulates” and “common notions.” These axioms deal with different aspects of the subject, and they are listed, which is helpful for readers who wish to follow the quite informal proofs of Euclid’s Propositions 4 and 5. Seeing the axioms makes it easier to understand why mathematicians tried to deduce Postulate 5 (P5) from the rest of the axioms. A presentation of Euclid’s proof of infinitely many prime numbers is a good excuse to introduce induction as a proof technique and to mention concepts such as perfect numbers, prime factorization, Mersenne primes, and geometric series.

The next chapter jumps to Hilbert’s axiomatization of geometry from 1899. Stillwell gives in parallel the geometric axioms and an axiomatization of complete ordered fields, thereby, describing two categorical formalizations of the reals. Hilbert’s axiomatization of geometry goes beyond Euclid’s geometry not only by filling a couple of gaps (e.g., the one found by M. Pasch), but by expanding the range of points, for instance, in a plane, from the set of constructible points to all points $\langle x,y\rangle$ ($x,y\in \mathbb{R}$). Perspectival drawings and paintings led to the study of projections and to theorems such as those of Pappus and Desargues—well before Hilbert’s axiom system. Stillwell not only states the latter two theorems, but he also lists an axiomatization of projective planes.

Archimedes gave surprisingly accurate approximations (in terms of fractions) for the value of $\pi$ in the 3rd century BCE. But algebra took off in earnest in the 9th century CE, then, it was propelled by the quest to solve cubic, quartic, and quintic equations. Notation for polynomials was nonexistent, and imaginary numbers were not even talked about at the time of H. Cardano, in the 16th century. Stillwell quotes Newton to illustrate a change within a 100 years or so, namely, that calculation with variables had become an accepted method to obtain results. He goes on to state and prove the factor theorem. R. Dedekind’s dimension theorem applied to a numerical formulation of the ancient Greek puzzle called “doubling the cube” gives an impossibility proof. The interplay between algebra and geometry is further illustrated by projective geometry. And another impossibility proof shows that there are no octonion projective spaces, which follows from the nonassociativity of the multiplication operation on octonions.

Algebraic geometry is a continuation of the development of the algebraic methods in the 16th and 17th centuries, including the introduction of coordinate systems. Stillwell starts with the conic sections and their equations, and he quickly introduces tangents, singularities, curves given by polynomials, and nonalgebraic curves. He illustrates that claims occasionally have a peculiar fate: Newton stated what is called Bézout’s theorem, which was proved much later by permitting complex coordinates, counting multiplicities of intersections, and placing the curves in a projective space. Thales’s theorem is also proved, but the proof is in the real vector space, which was introduced by H. Grassmann in the 19th century.

The content of Chapter 6 will likely be familiar to most readers, as it discusses the origins of calculus. But some might not know that the divergence of the harmonic series was proved by N. Oresme in the middle of the 14th century, or that several mathematicians gave infinite series and infinite products to approximate the value of $\pi$, which was proved irrational in 1761 by J. H. Lambert. Stillwell manages to explain many concepts and their connections from the binomial coefficient through the calculation of slope, area, and volume to infinitesimals. The latter require careful handling, and it was proved by A. Robinson (in the 1960s) that infinitesimals can be added consistently to the reals.

Chapter 7 begins with Euclid’s gcd algorithm, modular arithmetic, and the Pythagorean triples, and then the text moves on to rational points on a circle and parametric equations. Fermat’s little theorem and Fermat’s last theorem for the fourth power are proved. The latter can be translated into a proof about polynomials, which in turn leads to the parameterization of curves, elliptic integrals, and elliptic curves. Continuing the theme of divisibility into complex numbers and algebraic integers, the notion of primes is extended. E. Kummer’s determination to achieve unique prime factorization led to the notion of ideals. The latter are sets of numbers, and they may be seen to be related to R. Dedekind’s cuts (of $\mathbb{Q}$) that define real numbers. Ideals (i.e., cotheories) became very useful objects in lattice theory 1.

The fundamental theorem of algebra and its proof have a thought-provoking saga. Stillwell devotes a short chapter to this theorem alone; he gives several versions of the theorem, explains that early proof attempts (e.g., by C. Gauss) contained gaps and that filling the gaps produced a definition of real numbers. Having Dedekind’s cuts at hand, we get quick proofs of the least upper bound theorem and the intermediate value theorem.

The term “non-Euclidean geometry” usually refers to geometries that emerged once attempts to deduce P5 from Euclid’s other postulates failed. Stillwell has already made clear that projective geometry, at least in drawing manuals and scattered theorems, preceded the work of J. Bolyai and N. Lobachevskiǐ. Chapter 9 is devoted to non-Euclidean geometries starting with a treatment of the geometry of the sphere. This is an example where bits and pieces of a theory (non-Euclidean geometry) were worked out (due to the practical needs of astronomy and sailing) before a whole axiomatic system was formulated. The sphere has a constant positive curvature, while the Euclidean plane has curvature zero. In order to obtain an object (the pseudosphere) with constant negative curvature, transcendental curves, namely, the catenary and the tractrix, are introduced.

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzpicture}[scale=1.2] \begin{knot}[ clip width=5, clip radius=5pt, consider self intersections=no splits, ] \strand[blue,ultra thick] (0,1) to[out=144,in=108] (-.58779,-.809016) to[out=288,in=252] (.951057,.309017) to[out=72,in=36] (-.951057,.309017) to[out=216,in=180] (.58779,-.809016) to[out=0,in=-36] (0,1); \flipcrossings{4} \end{knot} \end{tikzpicture} — **Figure 1.**
Trefoil and knot $10_{123}$ (with 10 crossings).

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzpicture}[scale=1.3] \definecolor{teal}{rgb}{0.1,0.95,0.81} \definecolor{grey}{rgb}{0.75,0.75,0.75} \definecolor{lgr}{rgb}{0.7,0.9,0.7} \definecolor{dgr}{rgb}{0.05,0.6,0.05} \begin{knot}[clip width=5.5, clip radius=7pt, consider self intersections=no splits, only when rendering/.style={ lgr, line width=2.pt, double=dgr, double distance=1pt, }, ] \strand[teal,thick] (0,1) to[out=0,in=324] (-.58779,-.809016) to[out=144,in=108] (.951057,.309017) to[out=288,in=252] (-.951057,.309017) to[out=72,in=36] (.58779,-.809016) to[out=216,in=180] (0,1); \flipcrossings{1,2,5,6,8,9,10} \end{knot} \end{tikzpicture} — **Figure 1.**
Trefoil and knot $10_{123}$ (with 10 crossings).

The next chapter is on topology, which may bring very different images to one’s mind. Knots (from knot theory, see 4) have an accessible side to them such as the colorful diagrams in Fig. 1. But they belong to a modern field of mathematics that illustrates how various areas are tied together (pun intended!). Stillwell includes the Reidemeister moves and some knot invariants, and then moves onto graphs.

Chapter 11 turns back to the completeness of the real line and how this facilitated the development of the notions of limit and continuity. Stillwell defines continuity and uniform continuity, convergence and uniform convergence, and gives proofs for several theorems such as the Heine–Borel, the Riemann integrability of continuous functions, and the extreme value theorems. Some of these will reappear later in the book as theorems provable in certain subsystems of second-order Peano arithmetic.

The next three chapters belong together: first, basic set theory is presented, then the axiomatic approach is revisited via an axiomatization of Zermelo–Fraenkel set theory (ZF). Lastly, the axiom of choice (AC) is dealt with at some length. AC is often used without mention in mathematics, and here some of the set-theoretic equivalents such as Zorn’s lemma, the well-ordering principle, and some of its uses such as the Bolzano–Weierstraß theorem, the existence of non-Lebesgue measurable sets, and the Hausdorff–Banach–Tarski “paradox” are described. Finally, G. Cantor’s continuum hypothesis is mentioned, which is independent of ZFC (= ZF + AC) as was proved by K. Gödel and by P. Cohen.

The last two chapters quickly introduce predicate logic and computability, and the book is concluded with the incompleteness of arithmetic and set theory. Predicate logic proofs are certain trees here; a better-known proof system that uses trees is the method of analytic tableaux for which R. M. Smullyan’s 6 is a classic source. Computation has many models from recursive functions and combinatory logic to register machines and D. Scott’s model. Stillwell presents a version of Turing machines. A detailed proof of an incompleteness theorem is quite lengthy, as for example, G. Boolos and R. Jeffrey’s 2 illustrates. Stillwell sketches the proof together with Hilbert’s formalism and Brouwer’s intuitionism, which characterized acceptable mathematical proofs differently. Then he turns to reverse mathematics, the goal of which is to delineate sensible fragments of second-order arithmetic and to place theorems from analysis into these fragments according to their proof strength.

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzpicture}[scale=.5] \draw[red, fill=red!95!black] (-0.35,1.97) to (-1.88,-.68) to [out=135,in=240] (-1.73,1) to [out=60,in=180] (-0.35,1.97); \draw[green, fill=green!90!black] (0.35,1.97) to (1.88,-.68) to [out=45,in=-60] (1.73,1) to [out=120,in=00] (0.35,1.97); \draw[white] (0,-.7) -- (0,-1.2) node[anchor=north,text=black]{\textsf{X}}; \draw[black,fill=yellow!75] (0,-3) to [out=0,in=240] (2.6,-1.5) to [out=60,in=-60] (2.6,1.5) to [out=210,in=30] (1.73,1) to [out=-60,in=60] (1.73,-1) to [out=240,in=0] (0,-2) to [out=270,in=90] (0,-3); \draw[black,fill=green!90!black] (0,-3) to [out=180,in=-60] (-2.6,-1.5) to [out=120,in=240] (-2.6,1.5) to [out=-30,in=150] (-1.73,1) to [out=240,in=120] (-1.73,-1) to [out=-60,in=180] (0,-2) to [out=270,in=90] (0,-3); \draw[black,fill=blue] (-2.6,1.5) to [out=60,in=180] (0,3) to [out=0,in=120] (2.6,1.5) to [out=210,in=30] (1.73,1) to [out=120,in=0] (0,2) to [out=180,in=60] (-1.73,1) to [out=150,in=-30] (-2.6,1.5); \draw[black, fill=yellow!75] (-0.35,1.97) to [out=10,in=170] (0.35,1.97) to (1.18,0.53) to (0,-0.45) to(-1.18,0.53) to (-0.35,1.97); \draw[black, fill=blue] (-1.18,0.53) to (-1.88,-0.68) to [out=290,in=145] (-1.29,-1.54) to (0,-0.45) to (-1.18,0.53); \draw[black, fill=red!96!black] (1.18,0.53) to (1.88,-0.68) to [out=250,in=45] (1.29,-1.54) to (0,-0.45) to (1.18,0.53); \draw[black] (1.29,-1.54) to (-1.18,.53); \draw[black] (-1.29,-1.54) to (1.18,.53); \draw[black] (-1.88,-.68) to (-.35,1.97); \draw[black] (1.88,-.68) to (.35,1.97); \end{tikzpicture} — **Figure 2.**
A partial incompletable and a proper complete 4-coloring of a map.

In sum, this book contains multiple proofs, none of which comes from a formalized theory (in the sense of Hilbert). However, Stillwell underscores some components in proofs with names for them. The most prominent steps are the method of exhaustion (i.e., reasoning by cases) and (weak mathematical) induction. A method that is often used, but is not labeled here is reductio (i.e., proof by contradiction). Some steps that are specific to mathematical reasoning include renaming of variables (which is something else than the renaming of variables in $\lambda$-calculus) and translating between areas (e.g., rephrasing Fermat’s last theorem in terms of functions). Furthermore, some proofs seem to use analogical reasoning (e.g., the usual proof of Goodstein’s theorem). And the prolific use of diagrams suggests that pictures are vital to some proofs. A cautionary tale is A. Kempe’s “proof” of the four-color theorem (4CT), where both the “proof” and the counterexamples were diagrammatic. Figure 2 shows one of the smallest counterexamples to Kempe’s color-swapping algorithm derived from Soifer’s graph.

Graphic without alt text — **Figure 3.**
Kurt Gödel.

Arguably, the story of proofs starts around the time of Euclid. His axioms are self-evident, hence the deductive method provides secure foundations for geometry. As mathematics moved forward, occasionally, like a sputtering engine producing smoke rather than torque (or more prosaically, furnishing false claims supported by faulty justifications), not only new concepts were required but the language of mathematics and the proof techniques had to be clarified. The thrust to formalize mathematical theories wilted in the mid-20th century, partly because of Gödel’s theorems in 3. However, it seems that there have been new developments in the overarching story of proofs in the last 50 years or so. The proof of the 4CT is merely one of the examples that point to the use of computers. Stillwell allots a short section to the use of computers in checking proofs. Perhaps, an exploration of proof assistants, theorem provers, and proof checkers, which might lead to a new wave of rigorization in mathematics, should be the topic of an entire book.

A book for many readers

The book appears to have been produced with remarkable care. Although it is not teeming with references to online sources, there are some url’s mentioned for valuable resources such as an online version of O. Byrne’s colorful pictorial rendering of the first six books of Euclid’s Elements and a knot atlas.

This book will be a worthwhile read for anybody with some tertiary education in mathematics who would like to see how some core ideas developed and what was the driving force behind their evolution. A reader who does not have the time or patience to follow every detail in a proof can skip over it without losing an appreciation of the concepts involved. For instructors, this book could be a handy source to spice up a course with a bit of history or an outlook on the connections between disparate fields. Undoubtedly, anyone whose research area is touched upon in the book might become displeased, because it does not cover any of the topics at the level of detail that a research monograph or a research paper would. (I can attest to such feelings while reading the last chapters.) I imagine that experts will be able to overcome such perceptions and they will enjoy and profit from a more encompassing view of a significant portion of mathematics.

References

[1]: Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR0227053,

Show rawAMSref
\bib{Bi67}{book}{ author={Birkhoff, Garrett}, title={Lattice theory}, series={American Mathematical Society Colloquium Publications, Vol. XXV}, edition={3}, publisher={American Mathematical Society, Providence, R.I.}, date={1967}, pages={vi+418}, review={\MR {0227053}}, }
[2]: George S. Boolos and Richard C. Jeffrey, Computability and logic, 3rd ed., Cambridge University Press, Cambridge, 1989. MR1025336,

Show rawAMSref
\bib{BoJe92}{book}{ author={Boolos, George S.}, author={Jeffrey, Richard C.}, title={Computability and logic}, edition={3}, publisher={Cambridge University Press, Cambridge}, date={1989}, pages={xii+304}, isbn={0-521-38026-X}, isbn={0-521-38923-2}, review={\MR {1025336}}, }
[3]: Kurt Gödel. Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I. In Solomon Feferman, editor, Collected Works, volume I, pages 144–195. Oxford University Press, New York, NY, 1986. MR1549910 (MR0831941).
[4]: Louis H. Kauffman, Formal knot theory, Mathematical Notes, vol. 30, Princeton University Press, Princeton, NJ, 1983. MR712133,

Show rawAMSref
\bib{Ka2006}{book}{ author={Kauffman, Louis H.}, title={Formal knot theory}, series={Mathematical Notes}, volume={30}, publisher={Princeton University Press, Princeton, NJ}, date={1983}, pages={ii+168}, isbn={0-691-08336-3}, review={\MR {712133}}, }
[5]: Morris Kline, Mathematics: The loss of certainty, Oxford University Press, New York, 1980. MR584068,

Show rawAMSref
\bib{Kli80}{book}{ author={Kline, Morris}, title={Mathematics}, subtitle={The loss of certainty}, publisher={Oxford University Press, New York}, date={1980}, pages={vi+366}, isbn={0-19-502754-X}, review={\MR {584068}}, }
[6]: Raymond M. Smullyan, First-order logic, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 43, Springer-Verlag New York, Inc., New York, 1968. MR0243994,

Show rawAMSref
\bib{Sm68}{book}{ author={Smullyan, Raymond M.}, title={First-order logic}, series={Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 43}, publisher={Springer-Verlag New York, Inc., New York}, date={1968}, pages={xii+158}, review={\MR {0243994}}, }

Article DOI: 10.1090/noti2793

Credits

Book cover is courtesy of Princeton University Press.

Figures 1 and 2 and author photo are courtesy of Katalin Bimbó.

Figure 3 is by Marcel Natkin and courtesy of Stéphane and Laurent Natkin.

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Concepts at the Heart of Mathematics—Through the Centuries

What is in this book?

A book for many readers

References

Credits