introduction
"Do not go gentle into that good night,
Old age should burn and rave at close of day;
Rage, rage against the dying of the light."
"Do not go gentle into that good night," Dylan Thomas.
In general there are two biological limits that restrict the productivity of the mathematician. First, it takes a substantial period of time to develop the skills and knowledge base necessary for research. We look on fondly at the history of mathematicians past; people like Evariste Galois were able to revolutionize the field of their time before turning twenty-five (though in the modern era, Australian mathematician Terence Tao certainly tried). Aside from these outliers, I would hazard to guess that the average mathematician requires a solid twenty-five to thirty years to reach both a working level of skill in the major fields of modern study, and to specialize in some subfield.
On the other hand, it is commonly said that it is very rare for a mathematician to produce substantially new work past the age of forty. It seems that this rumor started with Hardy's famous essay, "A Mathematician's Apology" [1], in which he cites several mathematicians whose real work was finished by that age, including Galois as we have above. However, there is a confounding variable at work here: in the misty past, mathematicians (along with everyone else) just didn't live as long as they do now. The tragedy of Galois is not that his work was finished at twenty-one. It is that we know he (probably) had so much more to offer, and was cut down before his time.
If we subscribe to Hardy's estimate, then the average mathematician these days is productive for only ten or fifteen years before he or she is relegated to administrative matters and sent out to pasture until their 60th birthday festschrift. The difficulty here is that as the world of mathematics expands, it takes longer and longer to train new mathematicians, if only because the amount of content that needs learning keeps increasing. The current solution to this problem is Robert Heinlein's bane: specialization.
The difficulty with specialization is that it fragments the population of mathematicians. As the heavily-opinionated and controversial Doron Zeilberger recently put it, "I just came back from attending the 1052nd AMS (sectional) meeting at Penn State, last weekend, and realized that the Kingdom of Mathematics is dead. Instead we have a disjoint union of narrow specialties, and people who know everything about nothing, and nothing about anything (except their very narrow acre). Not only do they know nothing besides their narrow expertise, they tive researchdon't care!" [2] My experiences at various sectional, regional, and national conferences in the United States support this observation. At current population levels, and with the glory of the Internet, one would hope to be able to discuss one's research with tens of thousands of mathematicians. In reality, once one begins research in earnest, there are perhaps two to three hundred mathematicians that understand the particular dialect of mathematics needed.
But how can we hope to restore the Kingdom of Mathematics, as Zeilberger calls it, when already it takes far too long to train a mathematician to become knowledgeable in just one subfield? In part, this problem can be solved by better pedagogical techniques, a subject I will return to later. For the other half of the problem, we need to find ways to allow mathematicians to remain productive at least into their late fifties, if not later. There is currently a small cottage industry devoted to video games like Brain Age that claim to improve mental agility, but we need real cognitive science to back these things up. If someone out there is strapping mathematicians into fMRIs, I want to know about it! If no one is, well, they should start.
Until then, we will have to rely on some anecdotal evidence. There is at least enough evidence to suggest that it is possible for a mathematician to be productive past the mythical barrier of forty. For our purposes we will take for example Donald Knuth, Paul Erdős, and Jean Dieudonné.
The only living member of this somewhat arbitrary list, Donald Knuth has solemnly promised to finish the seven volume of his magnum opus, The Art of Computer Programming, before he dies. Currently, he's on volume four. As to how he maintains this level of output, he writes,
"My full-time writing schedule means that I have to be pretty much a hermit. The only way to gain enough efficiency to complete The Art of Computer Programming is to operate in batch mode, concentrating intensively and uninterruptedly on one subject at a time, rather than swapping a number of topics in and out of my head. I'm unable to schedule appointments with visitors, travel to conferences or accept speaking engagements, or undertake any new responsibilities of any kind. I'm glad that the WWW makes it possible for me to respond to questions that I don't have to see or hear." [3]
Paul Erdős took a somewhat different path toward attaining the intense and uninterrupted state of mind necessary for mathematics. He was a habitual user of amphetamines, using Benezedrine and/or Ritalin for the latter part of his career. The famous anecdote about this period of his life runs something like this:
"Colleagues worried that Erdös might have become addicted. In 1979, he accepted a $500 bet from his friend Ronald Graham. Graham challenged Erdös to abstain from speed for 30 days. Erdös met the challenge, but his output sank dramatically. Erdös felt the progress of mathematics had been held up by a stupid wager.
In an article by Paul Hoffman published in November 1987, Atlantic Monthly profiled Erdös and discussed his Benzedrine habit. Erdös liked the article, '...except for one thing...You shouldn't have mentioned the stuff about Benzedrine. It's not that you got it wrong. It's just that I don't want kids who are thinking about going into mathematics to think that they have to take drugs to succeed.'"[4]
Finally, we recount the story of Jean Dieudonné, one of the founding members of the group of French mathematicians that became Nicolas Bourbaki. As the de facto secretary of Bourbaki, he was almost forced to become a generalist. Borel notes in [5] ("Twenty-Five Years with Nicolas Bourbaki, 1949-1973", A. Borel, Notices of the AMS, March 1998)
"As a by-product, so to say, the activity within Bourbaki was a tremendous education, a unique training ground, obviously a main source of the breadth and sharpness of understanding I had been struck by in my first discussions with Bourbaki members.
The requirement to be interested in all topics clearly led to a broadening of horizon... acknowledged in particular... by Dieudonné:
'In my personal experience, I believe that if I had not been submitted to this obligation to draft questions I did not know a thing about, and to manage to pull through, I should never have done a quarter or even a tenth of the mathematics I have done.'"
Perhaps as a result of this role, Dieudonné was an active writer in the foundations of analysis until shortly before his death in 1992. His nine volume masterpiece, Traité d'Analyse, is one of the canonical textbooks we will mention later, and it was mostly written after Dieudonné turned fifty.
These three anecdotes give us some idea of what will prove necessary in the future of mathematics: cognitive discipline, performance-enhancing drugs, and collaboration as a means toward breaking free of (over-)specialization.
next: running up the mountain