I RECENTLY attended a conference in honor of Ivo Babuska, a professor at the University of Texas, with whom I have written several mathematical papers. There were toasts with a crowd-pleasing (if prudently priced) malbec and puns riffing on “singular value decomposition” that elicited much mirth. After all, it was also Ivo’s 90th-birthday party.
Ivo remains passionately immersed in research, despite the dearly held popular belief that mathematicians are over the hill at 40.
Partly, this cliché stems from the stories about “Mathematician Early Death Syndrome” (MEDS): Galois shot in a duel at 20, Ramanujan felled by illness at 32. Turing waited until almost 42, but made his tragedy more striking by committing suicide.
Clearly, Ivo has survived this fate, by eschewing duels (a hurried exit from Czechoslovakia with his family the week after the Russians invaded in 1968 may have helped), sleeping early (a timer plunges all the rooms of his house into darkness at 10 p.m.) and eating right (who knew dumplings and tongue — which Mrs. Babuska once served me — were so healthy?).
The notion of premature decline has been cultivated by mathematicians themselves — most famously by G. H. Hardy, who lamented, in his 1940 memoir, “A Mathematician’s Apology” (published when he was in his early 60s, and still widely extolled as the definitive look into a mathematician’s soul), that “mathematics, more than any other art or science, is a young man’s game.”
Until recently mathematics has, indeed, been a “man’s game.” Strides have been made in addressing the sexism in the field, but what about the ageism?
Mathematicians are eligible for the Fields Medal, the discipline’s most venerable honor, up only to the age of 40. This led to an awkward situation in the 1990s, when Andrew Wiles finished proving Fermat’s last theorem — surely the most famous mathematical problem in history — just after turning 40. Denied the medal, he was awarded a silver plaque instead, as a consolation prize.
Professor Wiles will, however, be honored with the Abel Prize in Norway this May. Although named after another MEDS victim — Niels Henrik Abel, tuberculosis, 26 — this award recognizes lifetime achievement, with an average recipient in his mid-70s. Since just about all the winners have been vigorously engaged in research at the time of the award, they provide a string of counterexamples to Hardy’s declaration.
Several such exceptions also show up in a survey of 66 mathematicians over 50, published in The Mathematical Intelligencer. A 1978 studyfound no clear relationship between age and mathematical productivity, or age and quality of research.
And yet mathematicians can’t shake the creeping fear that the cliché is true; those still active may complain of their work being increasingly regarded as irrelevant.
John Tate, the 2010 Abel laureate, said in an interview that mathematicians did their best work at an age when they “don’t have a lot of baggage” and “haven’t worn grooves in their brains.” In other words, naïveté — even brashness — is needed for the most original moves in the “game” that is mathematics.
Hardy believed that the only important questions in the field arose internally from this game, that the sole purpose of a mathematician was to create beautiful and “almost wholly useless” theorems.
But ever since its inception, mathematics has also been driven by another powerful force: applications. From the early commerce and measurement needs that motivated the Sumerians to the subject’s symbiotic co-development with physics, mathematical inquiry has been spurred by questions from external fields. Although Hardy disparaged any math that could be applied to real life as “ugly,” “dull” and “trivial,” surely usefulness should be an additional measure for a mathematician’s worth?
This is where experience and maturity help. Ivo’s most profusely citedpaper, published when he was 70, contains one of those clarifying, deceptively simple-looking ideas that can emerge only with the deep and broad insight of a long career: a general mathematical method that can be (and is being!) used by engineers to design better machine parts.
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With doctorates in both engineering and mathematics, Ivo is also an example of someone who bridges two fields. In this, he embodies another skill enhanced by experience — the ability to interact with non-mathematicians, to interpret their questions mathematically and to explain solutions in their language.
Hardy dismissed exposition as “work for second-rate minds,” but such activity is critical for a field notoriously inept at communicating its results to outsiders.
It’s of course unfair to criticize Hardy, given how much the world has changed since his day. The division he created between “beautiful but useless” and “useful but ugly” mathematics has long been breached; even his own “useless” research area of number theory has become essential in cryptography and cybersecurity. Conversely, many elegant and aesthetically pleasing mathematical theories have emerged from the most utilitarian applications — even from the analysis of machine parts, as I can personally attest.
Let’s cherish Hardy’s theorems, not his opinions, and recognize mathematics as a field with diverse goals and needs, where people can expect to make useful contributions regardless of gender or age.
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