Letters to a Young Mathematician

by Ian Stewart

  You are about to join the first rung of the ladder by becoming a DXGS: Dr. X’s graduate student. From there you will, I’m sure, progress rapidly to PYR, promising young researcher, and thence to ER, established researcher. If you elect to remain in academia, the succeeding grades are SS (senior scientist), GOP (grand old person), and EG (emeritus guru).

  As a DXGS you will not yet have produced any ritual gifts of your own, and so cannot present them to anyone. You can request them, but normally only from your peers. When performing before the tribe—that is, giving seminars—you will repeatedly invoke two ancestors, a major theorist and your thesis adviser. The PYR is more relaxed and understands the rituals better. He or she will still invoke those two ancestors, but briefly and often as mere footnotes. Astute PYRs invoke SSs instead. They travel to tribal meetings (conferences) so laden with gifts that the journey is more like a pilgrimage, and dispense them liberally. They also feel able to request gifts from their seniors, though not too often and always politely. The ER seldom refers to a major theorist, preferring to mention only ancestors who are currently active, but—tellingly—an ER may also mention progeny, to prove that he or she has them. The ER does not bring gifts to the tribal meeting, having cleverly dispensed them in advance to the tribe’s inner circle.

  The SS invokes a major theorist frequently, with the goal of supplanting him or her by being seen to have made important advances over the major theorist’s ideas. The SS never gives or receives in public but expects to receive many gifts by more covert means. The GOP sits at the pinnacle of the gift-giving hierarchy, offering no gifts but requesting them from everyone, especially juniors. The EG is invoked as an ancestor by almost everyone, but takes part in absolutely no exchanges of gifts.

  The role that does not fit into this sequence—indeed, does not fit anywhere, which is its raison d’être, is the maverick guru (MG). Helen has this to say about MGs: “The Maverick Guru has an important symbolic role, having curious magical powers that cause fear and fascination in the community. The MG is outside the mainstream orthodoxy, but engaged in criticizing it. The MG cannot be invoked as an ancestor by junior members of the community who intend to stay within the mainstream . . . An erstwhile MG rapidly becomes a GOP.”

  I mention all this because you need to appreciate your place within the tribe, and because your progression from DXGS to PYR to ER depends heavily on your choice of X, who should be either an ER, an SS, or perhaps a GOP. Do not choose an MG, no matter how attractive that option may seem, unless you intend your entire career to operate outside the conventional ladder. And on the whole, I advise you to stay clear of GOPs. Trust me: I wanted to become an MG but I think I’ve ended up as a GOP instead. A GOP will have an impressive record, but much of it will date to the dim and distant past: five years ago, or even longer. The older an academic becomes, the more intellectual baggage he carries. GOPs’ minds tend to run along familiar grooves, and although they do this with impressive ease and confidence, their students may miss out on the really new ideas that are the lifeblood of research. Some GOPs, though, make excellent advisers despite that, usually those who are close to being MGs but aren’t quite.

  EGs never have students.

  My adviser was an SS in the field of group theory— the formal mathematics of symmetry—named Brian Hartley. He was young, but not too young. I didn’t choose him, and he didn’t exactly choose me either; I chose the field, and the system allocated me to him because he was in that field. There were four or five alternative choices. Any of them would have worked—I later got to know them all well, as colleagues—but my research would have been very different. I was very lucky to get Brian, who put me onto a problem—more an entire program—that really suited my interests and abilities. He was brilliant. He saw me regularly, was always available if I got stuck, and he was hardly ever stumped for an idea.

  Brian was, I think, slightly taken aback when I marched into his office on day one of my PhD course and demanded a research problem. Usually it takes students longer to get going. But within half an hour he had given me one—arising from one of his own papers, my first receipt of a gift—and it turned out to be a beauty. The program of research was to study a special type of group that a Russian mathematician, Anatoly Ivanovich Malcev, had associated with a different mathematical structure called a Lie algebra. This structure was first developed over a century ago by the Norwegian Sophus Lie, but (despite its name) it was mainly studied in the context of analysis, not abstract algebra. So Malcev’s purely algebraic version was a new point of view. Like many Russians at the time, he had sketched the ideas but not developed them in detail. My problem was to take Malcev’s thoughts and conjectures, and fill in the necessary proofs and other connections, in effect, to turn a set of sketches and renderings into a finished blueprint of a building. It took me three months, and I got hooked by Lie theory. I ended up writing my entire thesis on Lie algebras.

  Brian’s influence did not stop at research problems. He and his wife, Mary, entertained me and other PhD students at their home. Occasionally he invited me to join him at some jazz performance in a local pub. He was an academic father figure, a mentor, and a friend. In 1994 he died, unexpectedly, at the age of 55 while walking in the hills near Manchester. I wrote his obituary for The Guardian. It ended like this: “I last saw Brian a few weeks ago, at a meeting to mark the sixtieth birthday of a mutual friend. He had just taken up a much-prized fellowship that would relieve him of all teaching duties for five years to concentrate on research. He went out with his boots on, both literally, while walking in his beloved hills, and metaphorically. And that is how we shall all remember him.”

  I still find it hard to accept that he’s gone.

  As I say, I was lucky. The system assigned me the ideal adviser. But you can do better than that. Don’t leave it up to chance: choose your thesis adviser. Read the literature, talk to people in the field, find out who has a good reputation and—crucially—who is good with students. Draw up a short list. Visit them; in effect, interview them. Then trust your instincts. And remember, you don’t want a GOP who ignores you; you want a close personal relationship.

  Dare I add, not too close? The cliché of faculty sleeping with their students exists because it does happen. Someone once observed that the more subjective the discipline, the better the faculty dressed, and a similar principle seems to apply to illicit sex. Mathematicians, by and large, indulge in rather less of it, perhaps because we dress so badly. In any case, everyone knows it’s unprofessional, and nowadays there are sexual harassment laws. Enough said. For recreation and affection, restrict yourself to fellow students, or people from off campus, please.

  A standard joke states that mathematical ability is typically passed from father to son-in-law (or, these days, from mother to daughter-in-law). The point was that a male PhD student often married his adviser’s daughter. It’s one way to meet people off campus. So your real ancestry can be affected by your mathematical ancestry.

  Mathematicians are proud to trace their academic lineage through thesis advisers. Brian was my mathematical father, and Philip Hall my grandfather. Hall was of a generation for whom a PhD was unnecessary as a qualification for a university profession, but the most significant influence on his early work was William Burnside. Burnside can similarly be considered a mathematical son of Arthur Cayley, one of the most famous English mathematicians of Victorian times.

  I remember these things and value them. I know where and how I fit into the family tree of mathematical thought. Arthur Cayley is as important an ancestor to me as any of my biological great-great-grandfathers.

  Talent must be passed to succeeding generations. I’ve been thesis adviser to thirty students so far, twenty men and ten women. Since 1985, the proportions are fifty percent men, fifty percent women. I know women are just as good at math as men because I’ve watched both at close quarters. I am particularly proud of my mathematical daughters, most of whom hail from Portugal, where mathematics has long be
en viewed as a suitable activity for women. All of my Portuguese daughters have remained in mathematics. In fact, most of my graduate students have remained in mathematics, and every single one of them earned a PhD. However, one is now an accountant, several work in computing, and one owns an electronics company, or at least he did the last time I heard from him.

  The rest of the world is now following Portugal’s lead. In July 2005 the American Mathematical Society released the results of its 2004 Annual Survey of the Mathematical Sciences. Since the early 1990s, women have been receiving around 45 percent of all first degrees in math. Women received almost one-third of all U.S. doctorates in the mathematical sciences in the academic year 2003–2004, and one-quarter of those awarded in the top forty-eight math departments. In all, 333 women received math PhDs that year, the largest number ever recorded.

  The idea that math is not a suitable subject for women is stone-cold dead. The career ladder is open to both sexes, though it is still unbalanced at the top end.

  15

  Pure or Applied ?

  Dear Meg,

  When you’re choosing a subject area as a first-year grad student in mathematics, many people will tell you that your biggest choice is whether to study pure or applied mathematics.

  The short answer is that you should do both. A slightly longer version adds that the distinction is unhelpful and is rapidly becoming untenable. “Pure” and “applied” do represent two distinct approaches to our subject, but they are not in competition with each other. The physicist Eugene Wigner once commented on the “unreasonable effectiveness of mathematics” for providing insight into the natural world, and his choice of words makes it clear that he was talking about pure mathematics. Why should such abstract formulations, seemingly divorced from any connection to reality, be relevant to so many areas of science? Yet they are.

  There are many styles of mathematics, and while these two styles happen to have names, they merely represent two points on a spectrum of mathematical thought. Pure mathematics merges into logic and philosophy, and applied math merges into mathematical physics and engineering. They are tendencies, not the extremes of the spectrum. By an accident of history, these two tendencies have created an administrative split in academic mathematics: many universities place pure mathematics and applied mathematics in different departments. They used to fight tooth and nail over every new appointment and committee representative, but lately they are getting on rather better.

  As caricatured by applied mathematicians, pure math is abstract ivory-tower intellectual nonsense with no practical implications. Applied math, respond the pure-math diehards, is intellectually sloppy, lacks rigor, and substitutes number crunching for understanding. Like all good caricatures, both statements contain grains of truth, but you should not take them literally. Nevertheless, you will occasionally encounter these exaggerated attitudes, just as you will encounter throwbacks who still believe that women are no good at math and science. Ignore them; their time has passed. They just haven’t noticed.

  Timothy Poston, a mathematical colleague whom I have known for thirty-five years, wrote a penetrating article in 1981 in Mathematics Tomorrow. He observed—to paraphrase a complex argument—that the “purity” of pure mathematics is not that of an idle princess who refuses to sully her hands with good, honest work, but a purity of method. In pure math, you are not permitted to cut corners or leap to unjustified conclusions, however plausible. As Tim said, “Conceptual thinking is the salt of mathematics. If the salt has lost its savor, with what shall applications be salted?”

  A middle ground, dubbed “applicable mathematics,” emerged in the 1970s, but the name never really got established. My view is that all areas of mathematics are potentially applicable, although—as with equality in Animal Farm—some are more applicable than others. I prefer a single name, mathematics, and I believe it should be housed within a single university department. The emphasis nowadays is on developing the unity of overlapping areas of math and science, not imposing artificial boundaries.

  It has taken us a while to reach this happy state of affairs.

  Back in the days of Euler and Gauss, nobody distinguished between the internal structure of mathematics and the way it was used. Euler would write on the arrangement of masts in ships one day and on elliptic integrals the next. Gauss was immortalized by his work in number theory, including such gems as the law of quadratic reciprocity, but he also took time out to compute the orbit of Ceres, the first known asteroid. An empirical regularity in the spacing of the planets, the Titius–Bode law, predicted an unknown planet orbiting between Mars and Jupiter. In 1801 the Italian astronomer Giuseppe Piazzi discovered a celestial body in a suitable orbit and named it Ceres. The observations were so sparse that astronomers despaired of locating Ceres again when it reappeared from behind the sun. Gauss responded by improving the methods for calculating orbits, inventing such trifles as least-squares data fitting along the way. The work made him famous, and diverted him into celestial mechanics; even so, his greatest work is generally felt to have been in pure math.

  Gauss went on to conduct geographical surveys and to invent the telegraph. No one could accuse him of being impractical. In applied mathematics, he was a genius. But in pure mathematics, he was a god.

  By the time the nineteenth century merged into the twentieth, mathematics had become too big for any one person to encompass it all. People started to specialize. Researchers gravitated toward areas of mathematics whose methods appealed to them. Those who liked puzzling out strange patterns and relished the logical struggles required to find proofs specialized in the more abstract parts of the mathematical landscape. Practical types who wanted answers were drawn to areas that bordered on physics and engineering.

  By 1960 this divergence had become a split. What pure mathematicians considered mainstream—analysis, topology, algebra—had wandered off into realms of abstraction that were severely uncongenial to those of a practical turn of mind. Meanwhile, applied mathematicians were sacrificing logical rigor to extract numbers from increasingly difficult equations. Getting an answer became more important than getting the right answer, and any argument that led to a reasonable solution was acceptable, even if no one could explain why it worked. Physics students were told not to take courses from mathematicians because it would destroy their minds.

  Rather too many of the people involved in this debate failed to notice that there was no particular reason to restrict mathematical activity to one style of thought. There was no good reason to assume that either pure math was good and applied math was bad, or the other way around. But many people took these positions anyway. The pure mathematicians didn’t help by being ostentatiously unconcerned about the utility of anything they did; many, like Hardy, were proud that their work had no practical value. In retrospect, there was one good reason for this, among several bad ones. The pursuit of generality led to a close examination of the structure of mathematics, and this in turn revealed some big gaps in our understanding of the subject’s foundations. Assumptions that had seemed so obvious that no one realized they were assumptions turned out to be false.

  For instance, everyone had assumed that any continuous curve must have a well-defined tangent, almost everywhere, though of course not at sharp corners, which is why “everywhere” was clearly too strong a statement. Equivalently, every continuous function must be differentiable almost everywhere.

  Not so. Karl Weierstrass found a simple continuous function that is differentiable nowhere.

  Does this matter? Similar difficulties plagued the area known as Fourier analysis for a century, to such an extent that no one was sure which theorems were right and which were wrong. None of that stopped engineers from making good use of Fourier analysis. But one consequence of the struggle to sort the whole area out was the creation of measure theory, which later provided the foundations of the theory of probability. Another was fractal geometry, one of the most promising ways to understand nature’s irregularities. Prob
lems of rigor seldom affect immediate, direct applications of mathematical concepts. But sorting these problems out usually reveals elegant new ideas, important in some other area of application, that might otherwise have been missed.

  Leaving conceptual difficulties unresolved is a bit like using new credit cards to pay off the debts on old ones. You can keep going like that for some time, but eventually the bills come due.

  The style of mathematical thinking needed to sort out Fourier analysis was unfamiliar even to pure mathematicians. All too often, it seemed the objective was not to prove new theorems but to devise fiendishly complicated examples that placed limits on existing ones. Many pure mathematicians were disturbed by these examples, deeming them “pathological” and “monstrous,” and hoped that if they were ignored they would somehow go away. To his credit, David Hilbert, one of the leading mathematicians of the early 1900s, disagreed, referring to the newly emerging area as a “paradise.” It took a while for most mathematicians to see his point. By the 1960s, however, they had taken it on board to such an extent that their minds were focused almost exclusively on sorting out the internal difficulties of the big mathematical theories. When your understanding of topology does not permit you to distinguish a reef knot from a granny knot, it seems pointless to worry about applications. Those must wait until we’ve sorted this stuff out; don’t expect me to build a cocktail cabinet when I’m still trying to sharpen the saw.

  It did look a bit Ivory Tower. But collectively, mathematicians had not forgotten that the most important creative force in mathematics is its link to the natural world. As the theories became more powerful and the gaps were filled, individuals picked up the new kit of tools and started using them. They began wading into territory that had belonged to the applied mathematicians, who objected to the interlopers and weren’t comfortable with their methods.