Letters to a Young Mathematician (10 page)

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. Problems 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.

Mark Kac, a probabilist with interests in many other areas of application, wrote an amusing and penetrating
analysis of the pure mathematicians’ tendency to reformulate applied problems in abstract terms. He likened their approach to the invention of “dehydrated elephants”: technically difficult, but of no practical value. My friend Tim Poston pointed out that this is a poor analogy. It is actually rather easy to create a dehydrated elephant. The important technical issue is quite different: it is to ensure that when you add water, you get back a fully functioning elephant. Hannibal, he said, could have done with a cartload of dehydrated elephants when he marched on Rome.

Metaphors notwithstanding, Kac had a point: abstract reformulations are not an end in themselves. But he ruined it completely by offering an example. My wife’s father made the same mistake in the 1950s when he said, correctly, that most pop stars would never last, but then offered as his example Elvis Presley. Kac’s example of an archetypal dehydrated elephant was Steven Smale’s recasting of classical mechanics in terms of “symplectic geometry.” It would be too great a diversion to explain this new kind of geometry, but suffice it to say that Smale’s idea is now seen as a profound application of topology to physics.

Another vocal critic of abstraction in math was John Hammersley. A severely practical man and a consummate problem solver, Hammersley watched with dismay as the “new math” of the 1960s took over school curricula worldwide, and things like solving quadratic equations
were dumped in favor of gluing Möbius strips together to see how many sides and edges they had. In 1968 he wrote a celebrated diatribe called “On the Enfeeblement of Mathematical Skills by ‘Modern Mathematics’ and by Similar Soft Intellectual Trash in Schools and Universities.”

Like Kac, he had a point, but it would have been a much better point if he had not been so certain that anything he didn’t like was trash. “Abstract” is a verb as well as an adjective; generalities are abstracted from specifics. It is best to teach the specifics before performing the abstraction. But in the late 1960s, educators were throwing out the specifics. They had convinced themselves that it was more important to know that 7 + 11 = 11 + 7 than to know that either of them was 18, and even better to know that
a
+
b
=
b
+
a
without having a clue what
a
and
b
were. I can understand why Hammersley was livid. But . . . oh dear. From today’s vantage point he looks like a knee-jerk reactionary. It turns out that that “soft intellectual trash” consisted of useful and important ideas, but ideas best taught in a university, not high school. At its frontiers, mathematics had to become general and abstract: otherwise there could be no progress. Looking back on the 1960s from the twenty-first century, when the work of that period is bearing fruit, I think Hammersley failed to appreciate that new applications would need new tools, or that the theories being developed so assiduously by the pure mathematicians would be a major source of those tools.

What Hammersley denigrated forty years ago as “soft intellectual trash” is precisely what I use today to work on problems in fluid mechanics, evolutionary biology, and neuroscience. I use group theory, the fundamental language of symmetry, to understand the generalities of pattern formation and the application of those ideas to many areas of science. So do hundreds of others in math, physics, chemistry, astronomy, engineering, and biology.

People who are proud to be “practical” bother me just as much as those who are proud not to be. Both can suffer from blinkered vision. I am reminded of the chemist Thomas Midgley, Jr., who devoted much of his professional life to two major inventions: Freon and leaded gasoline. Freon is a chlorofluorocarbon (CFC), and this class of chemicals was responsible for the hole on the ozone layer and is now largely banned. Lead in gasoline is also banned, because of its adverse effects on health, especially that of children. Sometimes a narrow focus on immediate practicalities can cause trouble later.
It is easy to be wise after the event, of course, and the catalytic reactions on ice crystals that made apparently stable CFCs have such a damaging effect on the upper atmosphere were difficult to anticipate. But leaded gasoline was always a bad idea.

It’s fine for people to advocate their point of view on how math should be done. But they should not presume that there is only one good way to do math. I
value diversity, Meg, and I urge you to do the same. I also value imagination, and I encourage you to develop yours and use it. It takes a strong mixture of imagination and skepticism to see that what’s currently in fashion will not always be so, or that what your colleagues dismiss as a fad may be something considerably more. Today’s trendy fashion sometimes turns out to be threaded with pure gold.