Mathematical proofs, by their nature, are often very complex, and ascertaining whether they are, in fact, correct requires painstaking efforts by experts. A case in point is what happened on March 28, 2003, when Dan Goldston and Cem Yildirim, American and Turkish mathematicians respectively, believed they had achieved a major breakthrough with the so-called twin prime conjecture. Celebration turned into disappointment only a few weeks later when, on April 23, colleagues announced that they had found a hole in their argument. A year earlier the Englishman M. J. Dunwoody had presented a proof for the Poincaré conjecture. Here, too, not more than a couple of weeks went by before the proof turned out to be incomplete. A third case in point is Andrew Wiles’s proof for Fermat’s theorem. During the refereeing process it was discovered that the proof was incomplete. In this instance the error was repairable, but it took one and a half years and the help of a willing colleague to plug the hole.

Old, unsolved problems, especially those associated with the names of famous mathematicians, exert immense fascination. Pondering problems that experts from previous centuries dealt with has its own appeal. The 23 problems that David Hilbert, the famous mathematician from Göttingen, listed in 1900, which were to determine the direction of mathematical research for the better part of the next century, are engulfed by that same aura of mystery. By now solutions have been found for 20 of them, but numbers 6 (the axiomatization of physics), 8 (the Riemann conjecture), and 16 have so far eluded the mathematical world.

Indeed, numbers 8 and 16 are considered of such im-

portance that the mathematician Steve Smale listed them among the most important mathematical problems for the 21st century. But as in many cases, fascination is closely associated with danger. Famous problems work their magic also on people who do not possess the prerequisites to deal with them in a proper fashion. And, like falling in love with the wrong person, once one is hooked, the danger of self-deception regarding one’s own suitability is great.

Caution was therefore indicated when the news broke in November 2003 that a 22-year-old female student, Elin Oxenhielm from Sweden, had cracked part of Hilbert’s 16th problem. Her work had been checked by referees from the mathematics journal Nonlinear Analysis and accepted for publication. Elin Oxenhielm, immensely proud that her first piece of work was an absolute hit, immediately notified the media. Even though her department had advised her to let caution prevail, she proceeded to give interviews, announced plans for a book, and did not even rule out a film about Hilbert’s 16th problem. A brilliant career seemed to be in the bag, positions at leading institutes were in the offing, and a steady stream of funding was certain.

Hilbert’s 16th problem deals with two-dimensional dynamical systems. The solutions of such systems can reduce to single points or end in cycles. Hilbert investigated differential equations that describe such dynamical systems, equations whose right-hand sides were made up of polynomials.¹ He asked how the number of cycles depended on the degree of the polynomials. An answer would be of particular interest for complex or chaotic systems.

Oxenhielm’s eight-page paper begins with the observation that in simulations a certain differential equation behaves like the trigonometric sine function. Then she approximates the equation, without an estimation of even

the order of magnitude of the neglected terms. After recasting the equation a few times, a further approximation is made and justified by no more than numerical examples and computer simulations. Finally, in an unproven assertion, Oxenhielm claims that the results are not falsified by the approximations. Such nonchalant use of the mathematical rules of the game renders her work completely and utterly useless.

The media reports that Oxenhielm had brought about not only informed the public of her feat but also alerted professional circles. One incensed expert wrote an out-raged letter to Nonlinear Analysis with the urgent request to stop the intended publication. Elin Oxenhielm’s supervisor at the university, who had previously read and criticized her findings, asked the editors to delete her name from Oxenhielm’s list of acknowledgments. She did not want to be associated in any way with the paper. To add insult to injury, a technical university gave its freshman students a homework assignment to list the shortcomings in Oxenhielm’s paper.

The avalanche of criticism had its effects. On December 4, 2003, the publishers of Nonlinear Analysis announced they were postponing publication of Oxenhielm’s work pending further review. Shortly thereafter, the paper was withdrawn from publication.

How could things get that far? It is certainly not unusual for an inexperienced scientist to submit a paper that has errors or gaps to a journal. Usually the refereeing process of a journal ensures that mistakes come to light and inferior work does not get published. This is why journals with a good reputation often reject 90 percent or more of submitted papers. But in this particular case the process broke down completely. One expert who was interviewed believed that the journal’s referees, whose identities usually remain confidential, may have been engineers for whom approximations are common practice, so long as they do not cause problems. But to proceed in such a manner in the field of mathematics is unacceptable.

Furthermore, the fact that the young woman approached the media was quite unforgivable. It is the unfortunate

destiny of mathematicians that for most of their lives they sit all by themselves in little rooms, solving centuries-old problems. Only on rare occasions is the public notified of a success. Working under such circumstances, secluded from the hustle and bustle of the outside world, ensures a certain level of quality. Since mathematical proofs can only be considered correct after they have withstood the test of time, media fanfare is detrimental to the painstaking and protracted examination of a proof. Actively inviting this kind of public relations is unbecoming, to say the least.

Often the only satisfaction a mathematician may expect from a successful proof is acknowledgment from colleagues in the same field. Experts may number no more than a dozen, spread over the farthest corners of the globe. Receiving their e-mailed nods of approval often represents the height of approbation. On the rare occasions where solid applications for a mathematical theorem appear, they become public knowledge only many decades later. That young researchers, frustrated by the prospect of a life in the shadows of anonymity, seek out the public arena may be understandable. Nevertheless, most mathematicians avoid the limelight. To inform the media at the drop of a hat conveys an image from which the protagonists shy away. Mathematicians’ subtle trains of thought, minute considerations, and rigorous arguments do not lend themselves to infotainment. For better or worse, mathematics is a science with a low profile.