Mapping the Community V: Those Mathematicians

Every April, the International Society for Nanoscale Science, Computation and Engineering meets in Snowbird, Utah, for the annual Foundations of Nanoscience (FNANO) conference – and some skiing during the last week of the season. In 2007, Omar Yaghi was invited to talk about chemistry’s “change from ‘shake and bake’ to rational design.” Rational design leads inevitably to mathematics, and Yaghi spent a few minutes complaining about mathematicians who were unwilling to work on problems that were not “interesting”.

Chemists are not alone: physicists have been complaining for centuries. But the reality is that mathematicians are pretty much like any other group of academics: they went into their own field because that was what they were interested in. Just as chemists and physicists are often more interested in chemical or physical problems, mathematicians are often more interested in mathematical problems. One difference is the physicists – and more recently chemists – depend on mathematics for their toolkit, but before the Twentieth century, mathematics did not need much help from other disciplines for solving problems. The rise of computer-aided proofs and experimental mathematics suggests that that may be changing, but even now mathematicians can be fairly self-reliant.

Except when it comes to looking for dragons to slay: mathematics has a long history of getting its problems from outside. So mathematical crystallography should be interesting to mathematicians – after all, crystallography provided mathematicians with the crystallographic groups. But still, the asymmetry resulted in cultural differences that may create difficulties for collaboration.

Despite several years of collaboration, I don’t really know enough about chemists to comment on them, so I will concentrate on the problems with collaborating with mathematicians.

1. There is an agenda problem. Non-mathematicians are interested in the goodies that mathematics can provide, while mathematics has a long tradition of being an enterprise undertaken for its own sake.
2. Mathematics is notoriously difficult. In fact, many students abandon more mathematical sciences (like physics) in favor of less mathematical ones (like chemistry) out of mathematics anxiety.
3. Abstraction. During the last two centuries, mathematics has grown increasingly abstract, and that impulse can make mathematics and mathematicians less accessible.
4. The fragmentation of the field. Mathematics is now fragmented into about six thousand fields, and few mathematicians have mastery over more than a few of these (for example, back in the 1990s when I was pretty much a normal mathematician, I was quite happy in my four fields).

Three things to keep in mind.

• In those societies where academic mathematics was substantial (and where surviving documentation is substantial) – like ancient Greece, China, India, the Middle East) – mathematics either had religious roots or at least religious entanglements, and there were strong connections and sympathies between mathematicians and metaphysical philosophers.
• Ever since Egypt and Mesopotamia, mathematics has been a middle class activity. Every urban society has had an enormous demand for mathematics and logic (considered broadly), so there was always a certain status accorded to the Queen of the Sciences.
• Mathematics has gone through several crises when it wasn’t clear what the standard should be for accepting a mathematical fact as true. A sequence of Greeks from Thales of Miletus and Hippocrates of Chios to Aristotle and Euclid developed and popularized the Axiomatic Method: one starts with a set of clear definitions, unequivocal assumptions, and Rules of Inference and, starting from the assumptions, one draws conclusions, one by one, using the rules. Such a sequence, from assumptions through successive conclusions up to the desired fact, is called a proof. This form of verification is extremely labor-intensive, but mathematicians rely on it because of many unhappy episodes when mathematicians evaded or fudged it. Much of the mathematical literature, and much of intermediate and upper level mathematical courses, are exercises in the axiomatic method.

Now let’s look at the four issues more closely.

1. The Agenda. This is probably the issue that bothers non-mathematicians the most: they complain that mathematicians do not find their work “interesting.” But again, as murder mystery writer Emma Lathen observed, “People … [are] basically not interested in [other people’s] problems; they [are] interested in their own.” The National Research Council, in its 1995 report on Mathematical Challenges from Theoretical / Computational Chemistry, recognized that chemists are interested in chemical problems while mathematicians are interested in mathematical problems:
   • From page 110, “… interdisciplinary work may be regarded … as “not real mathematics …” and most “Most academic mathematicians would agree that it is difficult to [evaluate] ‘interdisciplinary’ work …,” and at any rate, it is unlike to constitute “new mathematics.” “Such issues are particularly worrying for junior mathematicians” because of the tenure and promotion processes.
   • From page 111, “… analogous principles of departmental autonomy can affect chemists seeking to work with mathematicians.” While the report does not worry about the status of theoretical chemists – in computer science at least, theoreticians are often marginalized, and that is true to some extent in physics as well (it is the experimentalists Arno Penzias and Robert Wilson, not the theoreticians George LeMaitre, George Gamow and Ralph Alpher, who are went to Stockholm for the Big Bang) – it does remark that “Because theoretical/ computational chemists must often demonstrate the applications of their work to experimental areas of chemistry [something not even computer science requires!], fundamental work of a mathematical nature … may be undervalued.” On the other hand, “… chemistry departments have more experience evaluating multidisciplinary research …”

   There is a sort of political spectrum in mathematics, ranging from a mathematical Right that values mathematics for mathematics’ sake to a mathematical Left that values mathematics for its contributions to society.

   To the Right. Pythagoras and Plato associated numbers and geometry, respectively, with the overarching metaphysical reality of the universe. Leading Twentieth century advocates of the Right include the great mathematician Godfrey Hardy, whose Mathematician’s Apology can seem like a description of a recreational activity (“I have done nothing useful,” wrote the founder of mathematical genetics), and the mathematician (and science fiction writer) Eric Temple Bell, whose Development of Mathematics describes the history of the subject as if it was an art.

   To the Left. Despite their occasional Rightwing rhetoric, both Isaac Newton and Albert Einstein were to the mathematical Left, and their primary interest in mathematics was what they could do with it in physics. Leading Twentieth century advocates of the Left include the biologist (and political Leftist) Lancelot Hogben, whose Mathematics for the Million is an attempt to bring the first semester of calculus to the public, and mathematical educator Morris Kline, whose Mathematical Though from Ancient to Modern Times presents a more mathematical Left wing view of history (and whose mathematical Left-wing diatribe Why the Professor Can’t Teach: Mathematics and the Dilemma of American Undergraduate Education condemns modern mathematics – and modern art).

   Most mathematicians are moderates, but during the last century, the incentive structure for academic mathematics was Right-wing, although this may be changing.

2. Math is hard. Barbie got in trouble for saying that math class is tough, but it seems to be true. In fact, perennial mathematics education reform efforts are motivated (at least in part) by the perception that students are not learning adequately learning mathematics. Exactly what makes mathematics hard is unclear – it could be that students are taught that math is either easy or impossible and thus do not work hard at it, or it could be that there is something intrinsically unnatural about mathematical thinking that many people cannot master, or even (as Frank Smith argues in The Glass Wall: Why Mathematics Can Seem Difficult) the abstract language of mathematics. The former theory was the most popular until recently (and even now, the math gene is a favorite alibi for students who don’t associate homework scores with TV watching habits); nowadays, educators like Sheila Tobias have advanced the idea that aversion to mathematics, and not lack of math genes, is the problem. (There is also the theory that the problem is an aversion to hard work – for mathematics is hard work.) Whether the issue is the difficulty of mathematics, or an aversion to mathematics, many people who work with mathematicians have issues with the subject in itself, and that limits their ability to collaborate effectively.
3. Abstract Mathematics. Mathematics has a tendency to drift towards general statements of as universal applicability as possible. In other words, mathematics has a tendency towards the abstract. One can see this in numbers:
   • By repeated encounters with queues, one gets the notion of the first, the second, the third, and so on: from these one abstracts the notion of ordinals, giving the position of a bird on a pecking order, or landmarks on a route.
   • By repeated encounters with collections, one gets a notion of one object, two objects, three objects, and so on: from these one abstracts the notion of cardinals, giving the size of one’s herd of cows or collection of cowry shells.
   • Somehow these two notions were merged together to produce the notion of a (natural) number. Experiences with pieces of pies and shares of spoils led someone to the notion of a ratio, which was soon seen as another kind of number – which mathematicians call rational. (Indeed, the Greeks wrestled with a menagerie of kinds of numbers.)

   During the last two centuries, the rise of abstract algebra has put wheels on abstraction. For example, groups were originally actions that permuted solutions of polynomials, but these solutions were numbers so we had groups of numbers, but then there were groups of symmetries acting on polyhedra so we had groups of actions, and so on. By the time the crystallographic groups were enumerated, a group was any collection G of objects (together with an identity e) and a “binary operation” * such that the following are true:

   • For any x and y in G, x * y was in G. (Think of the integers with * being addition, or the positive rationals with * being multiplication, or crystallographic groups with * being composition.)
   • For any x, y, and z in G, (x * y) * z = x * (y * z). This is called associativity; notice that on numbers, addition and multiplication are associative while subtraction and division are not.
   • There is an element in G, call it e, such that for any x in G, x * e = e * x = x. If * is addition on numbers, then e is 0, while if * is multiplication on numbers, then e is 1.
   • For each x in G, there exists exactly one x ^-1 in G (called the inverse of G) such that x * x ^-1 = x ^-1 * x. On numbers, if * is addition then x ^-1 is –x, while if * is multiplication then x ^-1 is 1/x.

   The advantage of this level of abstraction is that once you verify a fact about groups, you verify it for all applications. The disadvantage is that abstractions are not as readily apprehended as concrete examples. The aesthetics and metaphysics suggested that abstraction allowed a mathematician to get at the mathematical content of the issue. Mathematics is now so abstract that a book on crystallographic groups can come out and not even mention screws, glides, or inversions at all.

4. The Fragmentation of Mathematics. As mentioned in a previous posting, there are about six thousand fields listed by the joint committees of the American Mathematical Society and Zentralblatt fur Mathematik. Since mastering a field entails dealing with the proofs of major results in the field (!), it is extraordinarily difficult to master more than a few of them. Then looking at the number of fields listed in that posting as being relevant to crystal structures, and how scattered those fields are, one cannot expect a given mathematician to be able to handle any crystal structure problem that comes up. (I have a unique alibi: I am not particularly competent in any of the listed fields on that posting.)

So there are likely to be challenges in collaborating with mathematicians. And as one might guess, mathematicians face challenges dealing with chemists. The most obvious is the language barrier.

• If you look at a mathematical article or text, you will see bricks – definitions and propositions, often with big block-like proofs – held together by motivating mortar. One can scan such a paper, but actually reading it is a Zen-like experience, since (unlike a chemist) the reader is supposed to actually verify the work by recapitulating the proof. (Certainly, this is what is expected of referees – which is why the refereeing process in mathematics journals can take months, or even years.)
• Non-mathematicians do not define their terms, so it’s hard to tell what they are talking about. While it is true that chemists will define terms like “phenol”, “aromatic”, and “organic”, they will tend to be mushy when talking about, say, polyhedra (which are defined very carefully in math books). This means that mathematicians have to work to nail down what the non-mathematicians are talking about.
• Non-mathematicians do not know what they want. An industrial mathematician (speaking at a seminar at USF) once said that a mathematical consultant should not just solve the problem that some industrialist asks about, for that solution often isn’t that helpful. The first thing to do, said this mathematician, is to figure out what the real problem is.
• Non-mathematicians have lower standards of verification. Mathematicians are so concerned with verification that they actually have fields devoted to verification in itself. Compare that to natural scientists, who are prone to optimistic generalization. This is partly Galileo’s fault – he was the one who said that one starts with a simple model and then adapt it to experimental results – and experimental results can establish a theory only within a limited domain. Lord Kelvin’s teapot on a burner is not the Earth warmed by the Sun, so it was perhaps unwise of him to use his teapot to model the cooling of the Earth with the cooling of his teapot – and worse, rejecting empirical evidence that his calculations were off.

So in comparison with natural scientists, mathematicians can be obsessive-compulsive. But we’ve learned to be careful the hard way, and we have the scars to prove it.

There is an additional problem. On page 112, the National Research Council wrote that “the mathematics curriculum is structured like a tree, with courses of potential interest to chemists at the end of a very long branch of prerequisites; the effect is to discourage chemists from obtaining any knowledge of advanced topics.” As we shall see in a later post, the emergence of a new field depends critically on recruitment, so curriculum is a very important matter. But there is a lot more to it than the Council suggests.

• Unlike other science departments – in fact, to an extent matched only by English – mathematics is a service department. Many science and engineering departments require that their students take then entire calculus sequence (up through multiple integration and elementary vector calculus), linear algebra, and differential equations. Many universities then offer a year course on “applied mathematics”: this course is usually generic, satisfying the needs of many science and engineering departments with quite different needs. This common approach is a reflection of limited resources.
  • Mathematics departments have heavy teaching duties, especially at the lower division (calculus and below). Resources are limited for upper division and graduate courses.
  • Students who want to complete their degrees in four years have only so many courses that they can put in their schedules. This problem is compounded by engineering schools with overstuffed major requirements.

  So a mathematics department would not have the resources to offer a panoply of advanced courses, one for each discipline. Chemistry students would have to make do with the applied mathematics course, and that means that a chemistry department would have to lobby for the content it wants.

• Mathematics is not the only curriculum shaped like a tree. For example, at USF, a student who wants to take practically any advanced topic must take the three semesters of calculus, and then an indoctrination course called Bridge to Abstract Mathematics, and only the Elementary Abstract Algebra. Chemistry, meanwhile, has two semesters of chemistry (plus lab) and two semesters of organic chemistry (plus lab), and some advanced topics do not require the latter: the tree is shallower, but it is there. As a result, a mathematics student who wanted to learn crystallography would also face a lot of courses.

  This becomes an issue when mathematicians and non-mathematicians work on interdisciplinary programs, especially on graduate interdisciplinary programs, which have to contend with college-level restrictions on how many undergraduate courses graduate students may take for credit.

There are many levels of collaboration, from researchers working on a common problem to departments articulating their curricula. The National Research Council makes a number of recommendations, like setting up joint seminars. But notice what this recommendation entails: people from both departments would have to organize the thing, and then faculty in both departments would have to find the time to attend it. Collaboration will require labor as well as stretching.