Aug 172013

Many of us in mathematical crystallography, or theoretical crystallography, or the foundations of crystallography, or whatever you want to call it, have the impression that this is a field, that it is emerging (or re-emerging), and perhaps it would be a good thing if this field became more active and popular.

But what is entailed by an emerging field? And how would we encourage its growth? It turns out that there are experts on this kind of thing: the sociologists and philosophers of science. Let’s begin at the beginning (philosophers, like mathematicians, like to define their terms): what do we mean by a field? This is not a silly question: in fact, I claim that it is central to what we want to do.

Lindley Darden wrote that a field as an area of science with “a central problem, a domain of items taken to be facts related to that problem, general explanatory factors and goals providing expectations as to how the problem is to be solved, techniques and methods, and concepts, laws and theories related to the problem which attempt to realize the explanatory goals.” A field is about a problem. Darden outlined the emergence of cytology and biochemistry, and claimed that what they had in common was the discovery of something new, the introduction of new techniques for studying it, and the successful explanation of the new thing.

This is the underlying economic reality: people are not interested in we are interested in; they are interested in what they are interested in. They have their own agenda. Getting involved in a new field – even paying attention to a new field – is a lot of work. Why should they do that? An emerging (or re-emerging) field will gain attention and support if it provides something useful. Here is Ralph Waldo Emerson’s famous mousetrap quote (the original quote didn’t mention mousetraps):

… If a man has good corn, or wood, or boards, or pigs, to sell, or can make better chairs or knives, crucibles or church or gans, than anybody else, you will find a broad hard-beaten road to his house, though it be in the woods. And if a man knows the law, people find it out, though he live in a pine shanty, and resort to him. And if a man can pipe or sing, so as to wrap the prisoned soul in an elysium; or can paint landscape, and convey into oils and ochres all the enchantments of Spring or Autumn; or can liberate or intoxicate all people who hear him with delicious songs and verses; tis certain that the secret cannot be kept: the first witness tells it to a second, and men go by fives and tens and fifties to his door. …

It may not be clear that there is something entirely new. For example, the notion of a group took several decades in the late Eighteenth century and early Nineteenth century to emerge: mathematicians from Leonhard Euler (in 1761) to Evariste Galois (in 1831) toyed with groups of permutations, but it was Galois who launched the word group and only later, in 1846, that Augustin Cauchy announced that for over eighty years, mathematicians had been exploring (drum-roll, please), groups of permutations (see Hans Wussing’s The Genesis of the Abstract Group Concept for details). And through those eighty years, and ever since, groups held mathematicians’ attention because they were helpful.

Here is an example I have been foisting on passersby. How did a growing knowledge of physics, and a growing sophistication in engineering drawing, transform architecture? There are many ways, and here is one. Before the Renaissance, domes were either shallow or small or buttressed.

Image posted on Wikimedia Commons by Anthony M. Image posted on Wikimedia Commons by idobi Image posted on Wikimedia Commons by Milos Radevic

Three famous pre-Renaissance domes. The Pantheon (left) is shallow, the Dome of the Rock (center) is a small wood dome, and the Hagia Sophia (right) is intermediate- sized and buttressed on all sides. (Images hotlinked from Wikimedia Commons.)

The problem was hoop force, the outwards force on the lower part of the dome: a big, tall, masonry dome was likely to collapse as the lower part burst outwards. So before the Renaissance, architects made the dome shallow (no lower part to burst outwards), or small (much smaller hoop force), or made of wood (which has greater tensile strength than stone), or buttressed, etc.

In the early Renaissance, the City of Florence decided to build a cathedral dedicated to St. Mary of the Flowers, with an immense octagonal dome 43 meters wide – wider than any dome since the Pantheon itself. They built the cathedral, but prepared no buttressing for the dome, and then … they had a church with no roof. After several decades, they were persuaded to let Filippo Brunelleschi solve the problem, which he did. The solution was a pair of concentric domes: the light outside one we see from afar, and a sturdy inner dome reinforcing the outer one (and itself reinforced with several bands) and helping support the lantern on top. Shortly afterwards, Michelangelo employed the same design when building the dome of St. Peter’s Basilica. And then came Christopher Wren‘s triple dome, with light inner and outer shells giving the hemispherical shape, and a sturdy catenary-shaped cone in between. Wren’s variant of Brunelleschi’s innovation was anticipated in Asia: the multiple dome / shell design makes onion domes (like the Taj Mahal, built before St. Paul’s cathedral) possible.

Image posted on Wikimedia Commons by Bob Tubbs Image posted on Wikimedia Commons by Amandajm Image posted on Wikimedia Commons of engraving produced in 1755 Image posted on Wikimedia Commons by Dhirad

From left to right, Brunelleschi’s double dome in Florence, then Michelangelo’s double dome in Rome, then cross section of Wren’s triple dome in London, and to the right, (possibly) Ustad Ahmad Lahauri‘s double dome in Agra. Images hotlinked from Wikimedia Commons.

This was also an era when the drawing techniques of secretive medieval guildsmen were gradually displaced by the more precise drawings of (often geometry-oriented) artists of the Renaissance realist movement – including Brunelleschi and Michelangelo’s successor for St. Peter’s, Jacopo Barozzi da Vignola, who took over Michelangelo’s project when he died, and completed it using Michelangelo’s drawings. The engineers of the fifteenth, sixteenth, and seventeenth centuries were able to build unprecedented structures not only because of the growing understanding of physics, but also because of the more careful and comprehensive design. (See Henry Cowan, The Master Builders : A History of Structural and Environmental Design from Ancient Egypt to the Nineteenth Century and Peter Jeffrey Booker’s A History of Engineering Drawing for more details.)

Science and technological progress is a field of study in itself these days. One of the major recent studies presents European technological innovation during the last millennium as a culture of improvement, in which we do not have one inventor suddenly inventing the steam engine, but instead several centuries of inventors successively improving on their predecessors’ steam engines (which, in turn, had been rediscovered from antiquity). The two most popular models of this kind of cultural process probably are:

  • Donald Campbell’s notion of evolutionary epistemology, which was an extension of his psychological theory of Blind Variation and Selective Retention (BVSR). A creative process – either individual or social – relies on an idea generator, and ideas are selected based on the needs of the moment. The selected ideas are remembered, and gradually a structure of knowledge (or of culture) is assembled. Campbell was motivated by Darwin’s theory of natural selection, and subsequently by the theory of punctuated equilibrium advanced by Niles Eldredge and Stephen Jay Gould: during the evolutionary process, there are long periods of relatively little change in a species, “punctuated” by brief periods of rapid change when new species emerged from the old.
  • Thomas Kuhn’s division of scientific research into two categories. Normal science, which is the “puzzle solving” that occupies most of industrial and academic science, and which operates under the auspices of a consensus “paradigm” accepted by most practitioners. On occasion, internal and external stresses on a paradigm leads to a transition from one paradigm to another, which we call a scientific revolution.

Notice that both models encompass scientific revolutions very well, but do not particularly address the problem of stretching to reach a notion just outside of the scientific community’s grasp.

Several years ago, I advanced a sort of merged metaphor. Imagine scientific truth (whatever that is) being an invisible edifice, and imagine that scientists are gardeners growing a vine up the edifice. We can’t see the edifice, but we can see the vine on the edifice creating an outline of the turrets, balconies, gargoyles, and other (invisible) features of the edifice. This metaphor has two important features for us, both based on the notion that an emerging field needs recruits, and thus the vine’s growth must be guided not only for exploring the edifice, but to make it navigable by novices.

  • The metaphor distinguishes Kuhn’s normal science – which consists of the vine filling in the gaps to reveal the reliefs on the wall (which is very important to industry, which needs to get things right) – from non-revolutionary frontier science, where gardeners try to persuade the vine to grope towards … something that they can’t see but that a few of them guess is there. This image was used to describe the difficulties some precocious mathematicians had in getting anyone to understand their work.
  • The metaphor does not address revolution as a research activity, but instead the result of revolution as a pedagogical one. The pedagogical problem at the base of the vine is how to get novices going; gardeners solve this problem by tending a few vast trunks, with numerous low branches to climb on and explore. The pedagogical problem near the frontier is to find an accessible path to the frontier, which means developing a bough or thick rope of tendrils to the desired location.

    In his history of group theory, Wussing (above) points to Galois’ almost wilful opacity as one of the reasons why his work took over a decade after his death to have an impact, and in my paper, I followed Michael Crowe’s account of Hermann Grassman burying his predecessor of vector algebra in difficult tomes, as opposed to Willard Gibbs’ more strategic marketing (with preprints). And this brings us to where all this sociology and philosophy can help us: if we are in an emerging field, what are we supposed to do?

    Let’s first look at how an innovation happens. The popular vision is of some hermit who invents a better mousetrap, and the world beats a path to his door. Probably the most popular example of this in animal behavior was that of a low-ranking female in a Japanese monkey troop that twice discovered more efficient food processing techniques that subsequently percolated up the (male-dominated) hierarchy. Unfortunately, the human innovators of this type that come to mind are often people like Galois, Grassman, Vincent van Gogh, and Gregor Mendel: while every once and a while, such a lonely genius (like Srinivasa Ramanujan) is discovered Hollywood-style by a perceptive master (like Godfrey Hardy), but very few masters seem as perceptive and receptive as Hardy.

    Kuhn thought that the social and institutional obstacles to a marginal figure making a major innovation and having the community respond positively were so difficult that most successful innovations would arise closer to the center of the action. Indeed, Bernard Barber enumerated a whole list of different reasons and ways that the scientific community resists innovation: some scientists cling to their old models and methodologies, many have a variety of attitudes towards mathematics, some are wary of anything outside of their immediate specialty, and following the stereotype, “… sometimes men of higher professional standing sit in judgment on lesser figures before publication and prevent a discovery’s getting into print,” and “That the older resist the younger in science is another pattern that has often been noted by scientists themselves and by those who study science as a social phenomenon.” An even gloomier view was advanced by Donald Hambrick and Ming-Jer Chen, who wrote that practitioners in neighboring fields may see a new field as a competitor for scarce resources.

    Dean Simonton additionally argued that a successful innovation would likely be made by someone who had already so mastered the subject that they were already known (I seem to remember one writer trotting out Isaac Newton, Charles Darwin or Maynard Keynes as examples).

    But there is an intermediate possibility suggested by the explosion of Cubism a century ago. Stoyan V. Sgourev notes that in 1907, when Pablo Picasso produced Les Demoiselles d’Avignon below, he was a minor figure but not marginal: he was already known for his Blue Period paintings. In addition, the growth of the middle class and the appearance of mass art had ironically produced niche markets for exotic art, and Picasso had connections to one of the niche marketers. After four years, Cubism was probably “the dominant avant-garde idiom in Paris,” but as Cubist art circulated in private markets (and not the conservative salons), the movement fragmented with no real leaders, but instead a growing cohort of practitioners. In essence, what Cubism needed was infrastructure, an accessible economic demand, and a network to exploit, but no stellar leaders (although Picasso and Georges Braque would be proclaimed the leaders later, after the critics recovered from the shock).

    Posted on Wikimedia Commons by Olpl as a work in the public domain Image posted on the Artchive

    Les Demoiselles d’Avignon by Picasso (hotlinked from Wikimedia Commons) and Houses iin L’Estaque, (hotlinked from the Artchives) the work described by art critic Louis Vauxcelles as “Bizarreries Cubiques,” thus giving “Cubism” its name.

    But mathematical crystallography already has senior and prominent participants and friends. So where do we go from here? Well, we are feeling our way, and considering the far-flung nature of the field (both as an interdisciplinary field and one whose practitioners are geographically scattered), one issue is communication: Diana Crane claimed that practitioners who are clustered together can support each other, while those lacking contacts in ancillary or outlying areas that they develop interest in will have as their primary resource journals, books, and conferences, but that can be difficult.

    So we are talking about infrastructure (and economic demand, which will be the subject of later posts). Conferences, journals, workshops, and so on. There is another consideration: Jan Fagerberg and Bart Verspagen noted several authors who argued that in order to established credibility among other scientific fields, a new field has to construct visible quality controls – although I suspect that a snooty exclusivity could be unhelpful, especially in this field, which is ancillary to so many amateurs and enthusiasts, whose presence can be felt by the success of conferences like the one described in Marjorie Senechal and George Fleck’s Shaping Space: a Polyhedral Approach and books like Herman Weyl’s Symmetry and John Conway, Heidi Burgiel and Chaim Goodman-Strauss’ Symmetries of Things; indeed, young amateurs and enthusiasts will be among our recruits (and older amateurs and enthusiasts will be among our community supporters – and in an era of intelligent design, climate denial, and ancient aliens, science can use all the community support it can get).

    One priority for infrastructure is recruitment. Luis Bettencourt et al constructed a model of new scientific fields and looking at cosmological inflation, cosmic strings, prions, H5N1 influenza, carbon nanotubes, and quantum computing, they found that the following formula is a good fit. Let ΔP be the number of new publications in a year, ΔA the number of new authors, and α a “scaling exponent”, then ΔP / (ΔA)α is nearly constant – and they call this the normalization constant. In other words, the number of publications corresponded well to the number of recruits (although there is also an α which reflects how well the field is doing).

    Notice that this is the number of new authors, not the number of young practitioners. But either way, we are talking about recruits, and we are back to the pedagogical issue. The field will need expository materials and tutorial programs – not unlike the workshops organized by the IUCr Commission on Mathematical and Theoretical Crystallography – and we need to think about students. How would a student learn mathematical crystallography? How would they become prepared to learn mathematical crystallography? One of the issues raised in the National Research Council’s Mathematical Challenges from Theoretical / Computational Chemistry is that mathematics students don’t learn chemistry and chemistry students don’t learn (much) mathematics, so neither cohort is prepared to do much in the intersection of chemistry and mathematics. This will have to change if mathematical crystallography is to progress.

    At the SIAM Mathematical Crystallography mini-symposiums, one of the participants worried that if we recruited graduate students, they might have difficulty getting employment. As someone who concentrated on mathematical logic as a student – the joke there was that one had to wait until someone died before one could get a job – I am sensitive to the issue. There seem to be two reactions.

    • One of my colleagues would tell his students not to go into mathematics unless they were committed to it. It was a calling, not a career. One went into it because it was beautiful. And if one would be practical, learning mathematics prepared one for many careers (like the humanities, which seems just as good – or possibly better, according to some studies – than business school in preparing students for business careers).
    • If we believe that mathematical crystallography is good for something, that it is useful in solving important problems, then we believe that our work will create an economic demand for our students. That puts the ball squarely in our court: it’s up to us to create a market for our students.

    If we are going to recruit people, we have an obligation to know what we are doing.

  • Aug 102013

    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.