Searching For New Mathematics

by Ivars Peterson


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Nonmathematicians interested in keeping up with contemporary mathematical research face daunting obstacles. The high level of abstraction, unenlightening notation, and the format of journal papers all conspire to make any foray into the mathematical literature frustrating and unrewarding. Mathematicians can do more to make their research accessible to a wider audience.

To most outsiders, modern mathematics represents unknown territory; its borders are protected by dense thickets of technical terms, its landscapes strewn with indecipherable equations and impenetrable concepts. Few realize that the world of modern mathematics is rich with vivid images, potentially useful notions, and provocative ideas.

Students of mathematics, faced with the chore of memorizing multiplication tables, lost in the labyrinths of geometric proofs, or pondering how long it takes a slowly leaking conical vessel to drain, often have the feeling that mathematics is little more than an unchanging body of knowledge that must be painstakingly and painfully passed on from generation to generation. Missing is a sense of how mathematics has evolved since its origins in the distant past and how new mathematics is constantly being discovered and created.

Even scientists and engineers often harbor an image of mathematics as a well-stocked warehouse from which to select ready-to-use formulas, theorems, and results to advance their own theories. Mathematicians, on the other hand, see their field as a rapidly growing endeavor, furnishing a rich and ever-changing variety of abstract notions.

It is the process of abstraction, and the language that goes with it, that makes mathematical terrain so difficult for nonmathematicians to penetrate. Outsiders find it hard to get at the information encoded in mathematical formulas because little or nothing in the actual patterns of the stark symbols on a printed page offers them a clue to a formula's meaning. To an outsider, the manipulation of such symbols looks like a private, almost magical game aimed toward mysterious, unworldly ends.

Complicating the situation further is the fact that mathematicians often appropriate simple, everyday words for their own purposes, using them in unexpected ways or assigning to them specific, technical meanings to express abstract concepts. Furthermore, the language used in mathematics is unusually dense. The precise meaning and position of every word and symbol makes a difference. Princeton mathematician William Thurston expressed the difference between reading mathematics and reading other subject matter in this way:

Mathematicians attach meaning to the exact phrasing of a sentence, much more than is conventional. The meanings of words are more precisely delimited. When I read articles or listen to speeches in the style of the humanities ...I find I have great trouble concentrating and comprehending: I think I try to read more into the phrases and sentences than is meant to be there, because of habits developed in reading mathematics. [I]
It is very easy for mathematicians to slip into their compact, symbolic notation. Comfortable with their specialized language, they too often fall into the trap of assuming their listeners or readers have equal facility with that language. So often it seems much easier and more efficient for them to state or write down an equation or some shorthand mathematical expression than to convey the idea in words.

To see how an outsider typically views the situation, imagine what would happen if I were to sprinkle this essay with a liberal dose of French words and phrases. A few readers would understand me; a few more would be able to guess my meaning. I suspect, however, that most readers would quickly lose the thread of my argument. That is what generally happens when outsiders listen to mathematicians talking mathematics or try to read a mathematical research paper.

But the levels of abstraction and the specialized vocabulary of mathematics are only part of the problem. What are some of the other factors that stand in the way of understanding and appreciating what mathematicians are up to? The answer requires a closer look at the places where one might expect to obtain a glimmer of what mathematical research is all about.

If you don't happen to have a mathematician around with whom you can strike up a conversation, then you might try the library - perhaps looking into a mathematics research journal. But the format of most journal papers seems to conspire against the broad communication of new mathematical ideas. That arid format, ruled by conventions established during the early part of this century, seems specifically designed to keep things within a small, exclusive club. The titles, abstracts, and introductions of many mathematical papers say: "Outsiders keep out! This is of interest only to those few already in the know."

Although similar problems occur in just about all the scientific literature I have seen, I find the situation in the sciences somewhat better than in mathematics. I can usually read a paper ( at least the first few paragraphs ) in, say, The Astrophysical Journal or Physical Review Letters to get a sense of what that topic is about and whether it is worth pursuing further. Try reading, say, the Bulletin of the American Mathematical Society to get a similar feel. It doesn't work The language gap is too great and the style too esoteric.

Papers in the Proceedings of the National Academy of Sciences (PNAN), one of the few journals that publishes original research covering a broad range of scientific and mathematical topics, reinforce that impression.
Here is how one mathematics paper begins:

Hypergraphs form a natural generalization of graphs in which (hyper)edges consist of k-element subsets of the vertices rather than pairs in the case of graphs. In refs. I and 2, we introduced the concept of a quasi-random graph property. [2]

In one sense, that introduction is not particularly formidable. The words themselves sound reasonably familiar. I think I know what a "graph" is. But the introduction doesn't tell me what I would really like to know. There is nothing I can use to judge the significance or usefulness of the result. Why did someone bother to look at this problem? I have no context in which to place this work.

Compare this with a biochemistry paper in the same issue. The title is certainly intimidating: "Localization of virus-specific and group- specific epitopes of plant poty-viruses by systematic immunochemical analysis of overlapping peptide fragments" [3]. But the introductory paragraph makes up for that mouthful of a title. It helps set the stage, although, like most scientific writing, it suffers from overuse of the passive voice:

Plant diseases are estimated to be responsible for economic losses worldwide of $60 billion per annum. The most important pathogens are fungi, with plant viruses the second most important group of infectious agents. Of the 28 plant virus groups or families the potyvirus group is the largest....

And it goes on in that vein for a few more paragraphs, gradually getting more specific.

Perhaps I have chosen extreme examples. Not all biochemistry papers are quite this lucid, and not all mathematics papers start so abruptly. For example, the same issue of the Proceedings contains the following introduction to a mathematics paper:

The arithmetic nature of classical constants of analysis and geometry is the main focus of transcendental number theory. Typical questions include: are the constants irrational, transcendental, algebraically independent? how well are they approximated by rational numbers? [4]

That is not necessarily easy for an outsider to grasp in detail, but this straightforward introductory paragraph establishes a historical context for what follows. And I can tell that it's a topic I want to pursue further.

But more often than not, a mathematics paper begins in this fashion (an example from another recent issue of PNAS): "Let A by a k-bialgebra with multiplication z and comultiplication A. We write A..." [5] . It is not my intention in these examples to denigrate the research involved. It may very well be interesting and important. I just can't tell from these papers. And PNAS is important because it is one of the few places where nonmathematicians can catch a glimpse of what's going on in mathematics.

Are mathematicians deliberately trying to keep their ideas to themselves and perhaps a few colleagues? It certainly looks that way. When authors or reviewers say, as they so often seem to, that "everybody" knows certain results or knows why certain problems are interesting, and that those results need not be explained nor put in context in papers, they are preaching to a very small group of the converted.

I believe it doesn't have to be that way. Research worth publishing should also be worth communicating. There is room in the mathematical literature for at least a small concession to a nonmathematical audience that may actually find the work of interest. And if mathematics is more than just a private game, if it is a consumer good, then mathematicians must take some responsibility for communicating their ideas in ways that convey the meaning of their work to broader audiences.

In a 1986 letter to the Notices of the American Mathematical Society, mathematician James A. Yorke of the University of Maryland expressed his frustration with the conventions of mathematical papers in these words:

As a mathematician who has many contacts with physicists, I sometimes encounter physicists who want to understand various theorems. The results are necessary for their research.... They find they cannot read papers in math journals....If mathematics is supposed to be useful in surprising ways, who is supposed to find the specific applications? [6]

Soon after the publication of The Mathematical Tourist [8], I received a telephone call from a chemist, who wanted further information about one of the topics in the book. He commented that he appreciated the book because it presented novel mathematical ideas in terms that he could readily picture and grasp. That glimpse persuaded him that certain new mathematical approaches may be of value to him, and he decided to invest the time required to learn the mathematics involved.

There is another factor that contributes to the forbidding image that many people have of mathematics and mathematicians. Professional mathematicians, in formal presentations and published papers, rarely show the human side of their work. Frequently missing, among the rows of austere symbols and lines of dense prose and among the barely legible mathematical formulas marching across transparencies projected on a screen, is the idea of what their work is all about - how and where their piece of the mathematical puzzle fits in, the sources of their ideas, their fountains of inspiration, and the images that carry them from one discovery to another.

Creative mathematicians testify to the importance of mental imagery in their work. The stark symbolic formulas of higher mathematics represent merely the final stage in a thought process more often than not beginning in the concrete and ending in the abstract. But how little of that gets conveyed!

James Yorke again writes:

When trying to glean from papers the authors' motivations for doing the work, we often get an impression that the authors might be saying "so and so worked on this problem and I can generalize those results." In other words, the goal is one - upmanship. There is very little discussion of goals in the literature. Graduate students - and outsiders - form their views of mathematics research in large part from the literature, so this lack of guidance encourages the beginner to do motivationless research. [6]

It is not surprising that outsiders rarely see mathematics as the exciting, human endeavor it is. What emerges instead is an elegant, remote architecture, with the scaffolding down and the blueprints stored away. The human element is hidden. Indeed, mathematicians seem to insist on presenting to the world a face of inhuman perfection. They try to sweep aside messy details and clearly prefer to keep any disputes within the family.

It's no wonder outsiders regard the laws of mathematics as akin to the Ten Commandments, created by forces beyond the comprehension of mere mortals. How else can someone convey the triumphs and tribulations of mathematical research - an endeavor that obviously captivates a significant number of people for very human reasons - without getting into the human elements that go into building the structure of mathematics?

Here's something else that is often missing. Few people outside of mathematics are aware of the field's empirical aspect. Much of the mathematics encountered by high school and college students seems carved in stone, passed on unchanged from one generation to another. Yet even the fundamental principles of arithmetic and plane geometry were once the subject of debate and speculation. It took centuries of constant questioning, brilliant guesses, and steady refinements to build the edifice now known as mathematics.

Indeed, experiment plays an important role in many fields of mathematical research, but you wouldn't know it from most mathematics textbooks. Although their work differs from the experimental research associated with, say, test tubes and noxious chemicals, mathematicians, like chemists and other researchers, often collect piles of data - whether prime numbers or diagrams of knots - before they can begin to extract and abstract the principles that neatly account for their observations. Strict reliance on deduction - the hops, steps, and jumps from one theorem or logical truth to another that we usually associate with mathematics - is not always sufficient. This side of the mathematical story rarely gets told.

Behind the apparently stolid, pristine, immutable public face of mathematics lies the exciting, turbulent, ever-changing world of mathematical research. Just as physics and other sciences go through episodes of both revolution and evolution, mathematics, too, changes and grows, not only in the way it is applied but also in its fundamental structure. New ideas are introduced; intriguing connections between old ideas are discovered. Chance observations and informed guesses develop into whole new fields of inquiry.

And more often than not, a piece of abstract mathematics worked out years before (and believed to be totally without practical value) finds a role in the "real" world. But, at the same time, it is worth remembering that scientists and engineers usually have to find or invent for themselves the mathematics they need, in the process often rediscovering concepts already in the mathematical literature.

It is encouraging to an outsider today that mathematics as a whole seems to show a renewed emphasis on applications, a return to concrete images, and an increasing and more explicit role for mathematical experiments. These changes are making mathematics generally more accessible to outsiders than before.

Moreover, mathematics has many different faces, and computers and computer graphics are making a real difference. Although many mathematicians still have the feeling that using a computer is akin to cheating and say that computation is merely an excuse for not thinking harder, computers are beginning to creep into mathematics By providing vivid images that suggest new questions, computation and computer graphics are helping to mend a rift that had developed between pure and applied mathematics and between mathematics and science.

Nevertheless, appreciation of the austere beauty of mathematics requires much effort and dedication. In 1930, German mathematician Wolfgang Krull reflected on the isolation mathematicians sometimes feel:

The more we ourselves are enraptured by the beauties of mathematics, the more we regret that we can bring so few people to share our pleasure. But at least those of us in the school of abstract mathematics have one consolation: as we make our presentations clearer and more transparent, they automatically become easier to understand. Bear in mind that four hundred years ago, arithmetic was a difficult art. So great an educator as Melanchthon [a sixteenth century scholar who reformed German education] did not trust the average student to penetrate the secrets of fractions. Yet now every child in elementary school must master them. Perhaps eventually the beauties of higher mathematics...will be accessible to every educated person. [7]

But people don't want to and should not have to wait hundreds of years to get a sense of what is happening in mathematics now. People are genuinely curious about mathematics, despite the overwhelming fear of the subject that so many seem to feel. And for many researchers, knowledge of recent mathematical work - of what mathematicians can and cannot accomplish - would be of immeasurable value.

Mathematics is full of unanswered questions, which far outnumber known theorems and results. I find it fascinating that mystery seems an inescapable ingredient of mathematics. It is its nature to pose more problems than it can solve. In fact, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs.

Far from being a domain of largely settled questions, mathematics is truly a wilderness. The well-mapped settlements lie few and far between, scattered across the continent and linked by a still skimpy network of highways and trails, some better traveled than others. With suitable guidance, it's a world worth exploring - a chance to catch a glimpse of a great intellectual adventure and perhaps to learn something useful. But it is sad that outsiders seem to have so few roads into this world.



Received by the editors April 30, 1990; accepted for publication April 30, 1990. This article was adapted from a presentation given at "Modern Perspectives of Mathematics: Mathematics as a Consumer Good, Math- ematics in Academia," Cornell University, March 30. 1990. t Science News, 1719 N Street N.W., Washington, D.C. 20036.

Posted with permission from SIAM Review, volume 33, number 1, pp. 37-42.
Copyright 1991 by the Society for Industrial and Applied Mathematics
All rights reserved.
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