Prof. Susan Colley, Mathematics

If mathematics isn't finished, then what's left to do?

Is mathematics finished?

Many perhaps wish it were so. To most of us, mathematics feels finished: its processes are ancient, its symbols arcane, and its language didactic, even dogmatic. And, of course, much of mathematics is very abstract. These qualities give mathematics a timelessness and remoteness that makes it appear to exist apart from anything human, hence eternal and complete. Moreover, mathematicians' preferred mode of expression, namely, rigorous formal proof, is precise (sometimes excruciatingly so) and highly stylized—nothing like our usual informal daily discourse.

Primarily, however, mathematics appears to us to be finished because almost all the mathematics we use or ever will use is old—really old. Even undergraduate majors can complete their degrees without seeing much if any twentieth-century mathematics. Calculus, for example, is more than three hundred years old (even older, if you includes some of the mathematics of antiquity that inspired Isaac Newton and Gottfried Leibniz to develop their calculus). Algebra is centuries older than calculus, and, for many of us, it's the last mathematics we'll ever think about.

This is an entirely different circumstance from other scientific fields. What beginning biology student doesn't know something about DNA, now a "classical" notion at age 50? Students of physics are made aware of a zoo of elementary particles that were unknown when their grandparents were in school. Moreover, older scientific ideas, such as the humorism of ancient Greek medicine or the phlogiston theory of combustion are simply discarded when observational evidence discredits them, or a newer theory provides better explanations. In fact, who beyond historians of science ever thinks about the phlogiston theory of combustion?

In contrast, mathematics engages in very little self-pruning. This is because it is concerned with understanding entirely general patterns and structures; mathematics qua mathematics is not really about explaining the natural world, even though it has proved to be enormously helpful in doing so. It also operates deductively rather than inductively, and hence, so long as the logic is correct, its results are necessarily and permanently true. Consequently, its ideas never need to be discarded, although, just like the rest of science, some areas fall out of fashion. That mathematics tends to accumulate material instead of regularly shedding some actually has been quite fortunate. Areas of the subject developed for one reason have proved to be just what was needed for another purpose. To take just one example, tensor analysis was originally created to understand questions in non-Euclidean geometry; it turned out to be exactly the mathematics Albert Einstein needed for general relativity. However, the bulk and intricacy of mathematics also makes it forbidding and intimidating. No one now living knows more than a small piece of it, so it's hard to imagine that there could be anything left to discover.

If mathematics isn't finished, then what's left to do?

Plenty, really. And there always will be. In 1900, David Hilbert addressed the Second International Congress of Mathematicians in Paris. In his lecture, he provided a list of twenty-three problems whose "deep significance...for the advance of mathematical science in general and the important role which they play in the work of the individual investigator are not to be denied." The Hilbert Problems, as they have become known, spanned most of the subject as it then existed. Collectively, they set the tone for mathematical research in the twentieth century. Some of the problems were quite specific; some were more like proposals for general research programs and thus it is difficult to determine precisely when they are actually solved. The general consensus is that only three of the original twenty-three problems remain open, including the Riemann Hypothesis, a celebrated conjecture about the location of the zero values of a particular function that turns out to have profound and far-reaching consequences regarding the distribution of prime numbers, and applications concerning encryption and decryption of data.

To mark the turn of this century, the Clay Mathematics Institute in Cambridge, Massachusetts announced the Millennium Prizes: $1 million awards for the solutions to seven outstanding mathematical questions. It's not an easy way to get rich, but one of the problems, the Poincaré Conjecture, does appear to have been established. Roughly, the conjecture states that any closed and bounded three-dimensional space with the property that any loop lying in the space can be continuously shrunk to a point must be, essentially, the same as a three-dimensional sphere. (In the purest ascetic style, the mathematician credited with proving the conjecture, Grigory Perelman, has eschewed all prizes and remunerations.) Proving or disproving the Riemann Hypothesis is also on the list of seven prize problems.

But famous unsolved problems do not provide the only fuel for the continued growth of mathematics. Every time you look at numerical patterns or geometric shapes, or use a quantitative procedure, it's natural for questions to arise. Some will have immediate answers, but even some questions that are easy to pose can lead to surprisingly deep and subtle ideas. It is precisely because mathematics is a human construction, and because people are unendingly creative in their wonderings and musings, that mathematics will never end.

Why didn't I know this?

Well, it's not a typical topic for the dinner table. (It has been discussed at my dinner table, but somehow I think that's an exception.) Since most of us only need to know and use very old mathematics, there's little reason for the subject to come up. Also, contemporary mathematical research is very specialized and layered with technicalities and does not always aid the digestion. Finally, mathematicians are nothing if not compulsively precise. As a rule, they just don't like describing mathematical ideas in a manner that isn't completely correct—after all, abstract concepts combined with perfect, pristine logic are their stock in trade. Years of study, training, and practice make certain habits of thought and expression instinctive to mathematicians, and they are consequently reluctant (sometimes even unable) to communicate in a manner that goes against their highly cultivated nature. In fact, some mathematicians do make strenuous efforts to explain their work to the general public, to convey some of its vibrancy and the love and excitement they feel for it. But, like mathematics itself, there is still more work to be done.