Jerry Coyne calls our attention to this abstract, from a recent issue of Proceedings of the National Academy of Sciences:
We show how to measure the failure of the Whitney move in dimension 4 by constructing higher-order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previously known link invariants such as Milnor, Sato-Levine, and Arf invariants. We also define higher-order Sato-Levine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney tower. Together with Milnor invariants, these higher-order invariants are shown to classify the existence of (twisted) Whitney towers of increasing order in the 4-ball. A conjecture regarding the nontriviality of the higher-order Arf invariants is formulated, and related implications for filtrations of string links and 3-dimensional homology cylinders are described.
He then remarks, “This shows how far removed mathematics is from even other scientists. Or are our own biology abstracts just as opaque to mathematicians?”
Now, the first thing I would point out is that this abstract would be opaque to most mathematicians as well. For myself, I can recognize it as having something to do with differential topology, and there are a few phrases in there with which I am familiar, but I’d be hard-pressed to tell you what the paper is actually about.
Of course, jargon is an affliction common to just about every academic discipline, and not just in the sciences. I would say, though, that math is probably among the worst offenders. The abstract of a typical research paper in mathematics is opaque not just to non-mathematicians, but to all mathematicians who are not specialists in the particular research area being addressed. And when I say opaque, I mean opaque. As in, you won’t make it past the first sentence.
Biology certainly is not as bad. In evolutionary biology I am definitely an amateur, but I find that I can often understand the introduction and discussion sections of a typical paper well enough to explain the gist to someone else. In math, it is usually impossible even to explain the problem to a non-mathematician.
Making matters worse is a mathematical culture that favors brevity and concision far, far more than it does clarity. Submit a paper with two consecutive sentences of exposition and watch how quickly the referee gets on you for it. A typical research journal in mathematics is just a stack of papers between covers. Compare this to a journal like Science or Nature. In addition to the formal research papers, they also have research summaries. These are typically written at a high level, but are also accessible enough so that a typical reader can understand what has been accomplished. And this is in addition to other non-research features, like editorials, book reviews and perspective articles. In mathematics there is none of that. Occasionally you get a survey article to keep you abreast of recent developments in some field or other, but these are just as opaque as the work they are describing.
Simply put, it is an awful, almost physically unpleasant experience to read a research paper in mathematics, at least if you want anything more than a superficial understanding of what was done. That is why it takes so damn long to get a paper through peer review. It’s because every time the referee glances over at the paper sitting on top of the filing cabinet, he thinks of something else he’d rather be doing. If you learn the fate of your paper within six months you’ve beaten the odds, but it’s even money that your paper will just disappear into the ether.
The fun part of doing research is when you go to a conference, meet the person who wrote the paper, and ask him to explain what he actually did. For one thing, usually the author is so delighted that anyone gives a crap about his paper that he will patiently spend hours, if necessary, explaining it to the thickest graduate student. For another, about ninety-five percent of the time the paper is ultimately pretty simple.
Sadly, the rot extends to math textbooks as well, which, with very few exceptions, are simply horrible. I mean really, really bad. It is commonly considered a great faux pas to actually explain what you’re doing. You will be accused of being overly wordy if you do anything other than produce an endless sequence of definition-theorem-proof. Mathematicians too often seem to take absolute delight in being as opaque as possible. I can’t tell you how many times I have heard friends and colleagues praise for their concision textbooks which, to my mind, are better described as harbingers of the apocalypse. If, as a textbook author, you place yourself in the student’s shoes and try to anticipate the sorts of questions he is likely to have approaching the material for the first time, a great many of your colleagues will say that you have done it wrong.
Let me turn the floor over to Morris Kline, who, in his 1978 book Why the Professor Can’t Teach nailed this issue perfectly:
But the decision is readily made. It is easier to say less. This decision is reinforced by the mathematician’s preference for sparse writing. If challenged, he replies, “Are the facts there?” This is all one should ask. Correctness is the only criterion and any request for more explanation is met by a supercilious stare. Surely one must be stupid to require more explanation. Though brevity proves to be the soul of obscurity, it seems that the one precept about writing that mathematicians take seriously is that brevity is preferable above everything, even comprehensibility. The professor may understand what he writes but to the student he seems to be saying, “I have learned this material and now I defy you to learn it.” …
A glaring deficiency of mathematics texts is the absence of motivation. The authors plunge into their subjects as though pursued by hungry lions. A typical introduction to a book or a chapter might read, “We shall now study linear vector spaces. A linear vector space is one which satisfies the following conditions…” The conditions are then stated and are followed almost immediately by theorems. Why anyone should study linear vector spaces and where the conditions come from are not discussed. The student, hurled into this strange space, is lost and cannot find his way.
Some introductions are not quite so abrupt. One finds the enlightening statement, “It might be well at this point to discuss…” Perhaps it is well enough for the author, but the student doesn’t usually feel well about the ensuing discussion. A common variation of this opening states, “It is natural to ask…,” and this is followed by a question that even the most curious person would not think to ask.
Exactly right. What’s tragic about this is that math, far more than most other subjects, really does make sense. You really can “figure it out” in a way that you often can’t in other branches of inquiry.
The inability of so many mathematicians to place themselves in the shoes of their students was brought home to me as an undergraduate. I was a sophomore, and was just starting to get serious about mathematics. Browsing through the course catalog I noticed an entry for Differential Geometry. I had no idea what that was. I had never even heard that phrase before. So I went to the open house the math department held for people considering a math major, and I asked one of the professors the following question: “What is differential geometry?” He answered, appearing to believe sincerely that he was being helpful, with a jargon-rich description of some open problems in the field.
Equally scathing (and eloquent!) is Gian-Carlo Rota, in his book Indiscrete Thoughts:
By and large mathematicians write for the exclusive benefit of other mathematicians in their own field even when they lapse into “expository” work. A specialist in quantum groups will write only for the benefit and approval of other specialists in quantum groups. A leader in the theory of pseudo-parabolic partial differential equations in quasi-convex domains will not stoop to being understood by specialists in quasi-parabolic partial differential equations in pseudo-convex domains….
The bane of expository work is Professor Neanderthal of Redwood Poly. In his time, Professor Neanderthal studied noncommutative ring theory with the great X, and over the years, despite heavy teaching and administrative commitments (including a deanship), he has found time to publish three notes on idempotents (of which he is justly proud) in the Proceedings of the American Mathematical Society.
Professor Neanderthal has not failed to keep up with the latest developments in noncommutative ring theory. Given more time, he would surely have written the definitive treatment of the subject. After buying his copy of Y. T. Lam’s long-expected treatise at his local college bookstore, Professor Neanderthal will spend a few days perusing the volume, after which he will be confirmed in his darkest suspicions: the author does not include even a mention, let alone a proof, of the Worpitzky-Yamamoto theorem! Never mind that the Worpitzky-Yamamoto theorem is an isolated result known only to a few initiates (or perverts, as graduate students whisper behind the professor’s back). In Professor Neanderthal’s head the omission of his favorite result is serious enough to damn the whole work. It matters little that all the main facts on noncommutative rings are given the clearest exposition ever, with definitive proofs, the right examples, and a well thought out logical sequence respecting the history of the subject.
I recall reading, the last time the Field’s Medals were awarded, descriptions of the work that earned the recipient’s their awards. They were written in the usual format, with thick, dense jargon starting right in the opening sentence. It is as though it never even occurred to the writer to make his description accessible to mathematicians outside his own, narrow research specialty. Now, I grant you that I am not in the forefront of research mathematics. Though I try to keep one foot in the research world, I do not see myself primarily as a researcher. But come on! I do have a PhD in the subject, and I have been a professional mathematician for fifteen years (starting the count when I entered graduate school.) And yet, I am unable to explain the accomplishments of the modern giants in my discipline. Very frustrating.
That, at any rate, is the bad news. That attitude is still very prevalent among the top research schools, and is even more oppressive in second-rate departments pretending to be among the first-tier. But my impression is that it is far less popular than it used to be. I think there has been a resurgence of interest in good expository writing, and of not being quite so arrogant and dogmatic about what’s important in mathematics. The mathematical world is vast, and there’s plenty of room for everyone. Research is important, of course, but so is teaching and pedagogy and outreach and attempts to let the rest of the world know why they should care about what we are up to.