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What makes a mathematician? A satisfying answer for me was given over three decades ago in the Kalahari Desert by an elderly professor, Robert Colgrave, who, after retirement at age 65 in England, had gone to Botswana to continue teaching mathematics. His definition: one who is solving a problem to which he/she does not yet know the answer. As a young want-to-be-mathematician on sabbatical, this seemingly simplistic perspective resonated with me. Why not be like Einstein and Poincaré, as they themselves had claimed, ever able to focus on a grain-of-sand-like irritant within an oyster-like mind, even while asleep and even over a period of years while doing many other things, and see what ensues.

But let us expand the question for this review by considering three accomplished mathematicians and their books—G. H. Hardy and his 1940 A Mathematician’s Apology; the statutory chair of numerical analysis at Oxford, Lloyd N. Trefethen, and his 2022 An Applied Mathematician’s Apology; and 2010 Fields medalist Cédric Villani and his 2020 Mathematics Is the Poetry of Science. We ask:

Why is it really worthwhile to make a serious study of mathematics, [and] what is the proper justification of a mathematician’s life ... [and h]ow do pure and applied mathematics differ from one another? (Hardy, pp. 65, 121).

Prefacing his response, Hardy warns us that his answer will be “egotistic,” and accordingly, he goes on to make extraordinary claims:

  1. i.

    There is the real mathematics of the real mathematicians, and there is what I call the trivial mathematics ... which includes its practical applications, the bridges and steam engines and dynamos (Hardy, pp. 64, 139).

  2. ii.

    Engineering is not a useful study for ordinary men (Hardy, p. 117).

  3. iii.

    Most people can do nothing at all well (Hardy, p. 67).

  4. iv.

    Exposition is for second-rate minds (Hardy, p. 61).

As help in understanding Hardy’s personality, George Pólya, who often took long country walks with Hardy [2, p. 73], observed, “Hardy liked to shock people mildly by stating unconventional views ... because he liked arguing” [6, p. 418]. To exemplify this whimsical/iconoclastic side, here is his ranking of mathematical talent: Ramanujan scored 100, full points; Hilbert trails by twenty points at 80; Hardy’s fellow collaborator on many works, John Littlewood, tallies but 30; and Hardy himself straggles in at 25 [2, p. 70].

How has the extended mathematical community responded to Hardy’s critique? Hardy’s friend and Oxford colleague Frederic Soddy, a 1921 Nobel laureate in chemistry, called the discussion in the Apology “cloistral clowning" [7, p. 3]. Another friend and Cambridge colleague, Norbert Wiener, sometimes referred to as the father of cybernetics, confessed [1, p. 158]:

When I returned to Cambridge after working with engineers for many years, Hardy used to claim that the engineering phraseology of much of my mathematical work was a humbug, and that I had employed it to curry favor with my engineering friends at MIT. He thought I was really a pure mathematician in disguise, and that these other aspects of my work were superficial. This, in fact, has not been the case.

My own friend, emeritus professor of electrical engineering and mathematical expositor extraordinaire Paul Nahin—in the same vein—originally subtitled his newest work An Engineer’s Reply to G. H. Hardy [5] (although I may have persuaded him otherwise in my foreword to his book).

Similarly, Nick Trefethen responded with an apology of his own. His analysis is both a memoir of life as a numerical analyst and reflections on the chasm that exists between pure and applied mathematics. Among his fun milestones: he was the first to buy a license for the computer algebra system MatLab and was the third ever to format a research report in \(\text{TeX}\). He defines his field as “the study of algorithms for the problems of continuous mathematics” (Trefethen, p. 1).

Although this definition includes developing both models of phenomena and methods of analysis, Trefethen evaluates his life’s work to date as having been focused on discovering new methods (Trefethen, p. 20). Of such accomplishments, Hardy would give hearty approval, since real mathematics “above all is technique” (Hardy, p. 134). Hardy goes on to include within real mathematics partial differential equation problems involving hydromechanics (fluid flow) and electromagnetism (Maxwell’s equations), areas rife with numerical analysis algorithms, what with turbulence and chaos and the butterfly effect. Hardy admits, however, that “most of the finest products of an applied mathematician’s fancy must be rejected [from being real mathematics]” (Hardy, p. 135). Such colorful language might tempt those who consider themselves applied mathematicians to conflate Hardy’s points (i) through (iv), and interpret him as saying that applied mathematics is for second-rate minds.

Moreover, contrary to “a widely held view ... that mathematics is one, that the difference between pure and applied is illusory ... an opinion [generally] held by pure mathematicians,” Trefethen testifies that the gulf between pure and applied mathematics is palpable (Trefethen, pp. 8, 46). For example, consider the office assignments in the new (as of 2013) Andrew Wiles Building at Oxford. Initially, “there had been talk of assigning offices randomly to encourage interactions between disparate fields” but “the pure people ended up on the north side ... and the applied ones on the south” (Trefethen, p. 19). Furthermore, with respect to the most prestigious award in mathematics, “Officially, Fields medals are for ‘mathematics,’ but it is understood that applied math doesn’t count, even though you’ll never find such a statement in print” (Trefethen, p. 8).

Trefethen adds, “I have long marveled at how most mathematicians prove their theorems without ... numerical experiments. Whatever the topic, I use the computer to guide me” (Trefethen, p. 23). Indeed, imagine what Newton might have done with a computer! (For such a fantasy, see the short story in [4, Section 8.3].) To help bridge this disconnect, as the book title itself might suggest, “the mission of [Trefethen’s] essay is to encourage better communication between the many parts of mathematics without pretending they are all the same” (Trefethen, p. 46).

Amidst this apparent fragmentation of mathematics, Trefethen wonders, “pure and applied alone isn’t enough to explain the weakness of the link between numerical people like me and [the Fields medalists from 1936 through 2018]; in mathematics the impact of these people on me has been epsilon” (Trefethen, p. 9). As he describes it, one imagines that the text of a typical Fields medalist treatise might as well be formatted in Tolkien’s old Entish, “a lovely language, [taking] a very long time to say anything in it, because [one does] not say anything in it, unless it is worth taking a long time to say” [8, p. 465]. So, too, after attending, for example, a lecture by the multiple prize-winner Yitang Zhang for his 2013 celebrated work on the twin prime conjecture, I will admit that my increased understanding of his approach was but a modest epsilon. Perhaps the avalanche of mathematical knowledge since the days of Hilbert—albeit a grand cornucopia—tempts each of us to emigrate to island specialities, thereby estranging us from other specialties. Littlewood encountered this phenomenon generations ago, writing, “I have tried to learn mathematics outside my fields of interest; after any interval I had to begin all over again” [3, p. 191].

What can be done to counter this isolation—this empty sense that “I am drifting away from” the center of current mathematics, as Trefethen hauntingly laments? (Trefethen, p. 66). Aha! Villani suggests that a mathematician is much like a poet. In fact, Hardy indirectly prefigures this analogy. His opening Apology story is a dinner conversation with the Cambridge poet and classicist A. E. Housman, during which they argued all evening at high table.

Villani’s thesis is that “only two things ... capture the attention of people everywhere: games and stories” (Villani, p. 36). And poems are the abstract essence of word games and stories—of birth, loss, death, isolation, surprise, joy. Just recall Keats’s “Ode on a Grecian Urn.” Recall Macbeth’s soliloquy of each of us being but a “poor player that struts and frets his hour upon the stage and then is heard no more.” And recall a tragic glory retold by Tennyson as “Half a league, half a league, half a league onward, all in the valley of Death rode the six hundred.” At times, philosophic poets, such as Plato, have strung reasoned words together “so great[ly] that some feel reality itself must be, at bottom, an abstract mathematical construction!” (Villani, p. 5).

Mathematics, argues Villani, is a game of describing and understanding—and doing it all as men and women, fashioning stories that we tell over and over, almost as poetry or prayer (Villani, p. 2). Our subject: the elusive notions of number and space. Our story: the strutting and fretting of us doing mathematics, of being creative with what we know, of being inspired by silly triggers, of thought experiments, of failures and successes, of setting standards of style and convention.

To illustrate, consider Trefethen as an example. On first looking at a standard definition of numerical analysis, as the study of methods of approximation and their accuracy, etc.—“How dreary,” he thought; whereupon he crafted his more poetic version (Trefethen, p. 14). Yet with a little more poetry—and apologies to Trefethen—perhaps we could embellish his definition with a Villani-ism, so that it becomes:

Numerical analysis is the study of algorithms for the problems of continuous mathematics such as modeling when a non-viscous, incompressible fluid can suddenly become violently agitated without any external force having been applied to it (Villani, p. 14).

Poetry adds energizing spice to mathematics. As a personal example, the subject matter for a semester-long 1977 graduate algebraic topology course under Joe Martin is fully and magically characterized and fondly remembered as the poetic query Might one peel an orange without breaking its skin?

As for stories, imagine this plot: “an obscure [nearly sixty-year-old] mathematician work[s] alone, or very nearly so, solv[ing] an important problem that has long stymied the greatest experts” (Villani, p. 3). Enter Yitang Zhang, as mentioned earlier!

For a longer gem of a story, Villani tells us afresh how Poincaré as a young man won the 1885 three-body prize sponsored by Sweden’s King Oscar II. After his entry was published in 1889, he realized that he had made a mistake. Oh, no! At his own expense, he recovered all the printed copies of his failed paper and emended it, replete with the stunning result that “a deterministic system could nonetheless give birth to unpredictability” (Villani, p. 14). As an extra appendix treat, Villani includes an excerpt from Poincaré describing his intuitive discoveries connecting differential equations with hyperbolic geometry, all due to seemingly trivial triggers such as drinking coffee late in the day and stepping down from a horse-drawn carriage.

Each of Villani’s tales “turns high-level mathematical research into a brilliant detective story, with breathtaking twists and turns on every other page” (Villani, p. 25). Probably because his own discoveries were inspired by the Nash–Moser theorem of nonlinear partial differential equations, Villani selected his book’s last tale to feature Nash. To summarize the story, in 1954, John Nash submitted a paper to the Annals of Mathematics. However, when the submission was reviewed, it “was [found to be] a meandering argument of staggering complexity, an opaque jumble of ideas whose main themes could barely be made out,” yet via “heroic efforts of the journal’s referee, Herbert Federer ... Nash’s manuscript was put into publishable form” (Villani, pp. 44-45). Again, we as mathematicians tend to “strut and fret our hour upon the stage,” and sometimes, together, we do well.

In this spirit, let me finish with a second Nash story. On January 9, 1997, at the Joint Mathematical Meetings in San Diego, California, Nash was slated to give a talk in a small theater with tiered seating. Every chair was filled. Some were standing. We all were excited to hear a Nobel laureate. In front of a curtained stage was a raised platform. Now at 68 years old, John was thin, almost wispy, soft-spoken, yet animated, all but tap-dancing on the dais. His overhead slides were inscrutable cursive swishes; his hands, too, waved this way and that while he was describing cooperative game coalition theory. Stepping back to accentuate a subtle point, he stepped too far, falling off the dais and backward into the curtains, disappearing from view. Rising in alarm as one, we all thought, “Oh, no! He’s killed himself.” Yet in the next moment, the curtains parted. He leapt spryly, unbelievably, onto the platform—like a veritable mythic Beowulf rising from the depths after slaying a monster, oblivious to battle trauma—missing but a beat in his explanations, ever focused on mathematics until the end.