At about this time each year for the past four years, the American Mathematical Society has published an attractive (full color, high gloss) pamphlet titled What's Happening in the Mathematical Sciences. Written by mathematics writer Barry Cipra, WHIMS sets out to do what its title suggests: It provides accounts of the most significant developments in mathematics over the previous twelve months, written at an accessibility level roughly the same as Scientific American.
Volume 4 of WHIMS appeared recently, and it's as good as the previous three issues. Barry Cipra is to be praised for his excellent mathematical writing, Paul Zorn has done a magnificent job as editor, and the AMS does the mathematical community a great service by publishing this annual summary.
One thing that struck me as I read this latest issue, however, is that all the developments described occurred in previous years. Now, don't get me wrong. I am not drawing any conclusions from this observation. For one thing, the acquisition of human knowledge proceeds at its own, often erratic pace. For another, there is more to mathematics than major breakthroughs that hit the front pages of daily newspapers.
The reason I noticed the absence of new results in the latest WHIMS was simply that 1998 has been the very year in which math has found its way onto network television, national radio, and the national press, not just occasionally but with some frequency. There were even both a movie and a deodorant called Pi.
Why this sudden interest? Probably a range of factors. Following my "Math Becomes Way Cool" article, a number of readers wrote to tell me their own explanation. I suspect that, as several of my correspondents suggested, the most significant factor leading to the sudden surge of interest was Andrew Wiles' proof of Fermat's Last Theorem.
Yes, it was over five years ago when Wiles first announced that he had found a proof, and four years since he was able to correct a major error in that first proof and produce an argument that the experts agreed was correct. But because of the media attention that Wiles' breakthrough attracted, public awareness of mathematics as an area of research was raised to a hitherto unprecedented level. That high level of interest was sustained by the subsequent appearance of a television documentary and an associated, best selling book on Wiles' dramatic breakthrough.
Add into the mix equally successful recent biographies of Paul Erdös and Nobel prizewinner John Nash, and you have a media environment in which math is floating around with everything else. (The thing to remember about radio and television is the degree to which they are parasitic on newspapers and magazines, and on each other. Once a topic is "in the loop," it tends to get a lot of coverage. It's getting into the loop in the first place that's the problem for the media hungry publicist.)
He follows that with a discussion of some recent work that suggests an amazing possible connection between the Riemann Hypothesis of analytic number theory and a problem in quantum physics. The connection would imply that the zeros of the zeta function can be interpreted as energy levels in the quantum version of some classically chaotic system.
After giving readers his own account of Erdös, Cipra goes on to tell us about some recent advances in the use of computers to do mathematics. First, the widespread availability of powerful computer algebra systems has led to a revival of some decidedly old-fashioned algebraic techniques in geometry.
Next, Cipra reports on the 1996 success of the theorem-proving program EQP in proving the Robbins Conjecture, a puzzle that had resisted all attempts at a solution since it was first proposed in the 1930s. The conjecture was that a particular collection of valid propositions of boolean algebra actually constitutes an axiom system for boolean algebra.
Taking the symbol + to denote logical disjunction (or) and N for negation (not), one way to axiomatize boolean algebras is by the following four axioms:
(1) p + q = q + p
(2) (p + q) + r = p + (q + r)
(3) p + N(q + N(q)) = p
(4) p + N(N(q) + N(r)) = N(N(p + q) + N(p + r))
In 1933, Edward Huntington showed that axioms (3) and (4) can be replaced by the single statement:
p = N(N(p) + q) + N(N(p) + N(q))
Huntington's student Herbert Robbins suggested the following alternative to his adviser's axiom:
p = N(N(p + q) + N(p + N(q)))
However, neither Robbins nor Huntington was able to prove that this alternative could be used in place of Huntington's axiom. Nor was anyone else. The question remained unresolved until EQP found a proof. The new proof derived a known axiom from the Robbins axiom, in just short of 50,000 steps.
After computers come flour beetles, when Cipra reports on some fascinating experimental work that shows that the population growth of a controlled colony of the insects shows all the familiar signs of chaotic dynamics: period doubling bifurcations, strange attractors, and the like. (Life imitating mathematics?)
Quantum computing -- still all promise, but a fascinating concept that will change the face of computing if it ever becomes a practical proposition -- is the topic of a chapter titled From Wired to Weird. That in turn is followed by a report on a new method for public-key cryptography that has the mathematical potential to rival the popular RSA system.
Cipra caps off his tour with a chapter on mathematics and art and a reprint of Henri Poincare's famous 1908 essay "Mathematical Discovery".
What's Happening in the Mathematical Sciences, Vol 4 is available directly from the AMS for $14 plus post and packing. To order by phone call toll-free 1-800-321-4AMS (4267) in the US and Canada, or 401-455-4000 worldwide. Ask for itemcode: HAPPENING/4. To order electronically, click here.
- Keith Devlin