Age and mathematical productivity

The relationship between age and mathematical productivity has been examined in the sociology of science, the history of mathematics, and the psychology of creativity. Mathematics has long carried a reputation as a field whose important work is done young. G. H. Hardy wrote in A Mathematician's Apology (1940) that mathematics, more than any other art or science, is "a young man's game", and that he knew of no major mathematical advance begun by a mathematician past fifty.^[1] Alfred W. Adler repeated the claim in The New Yorker in 1972.^[2] The same idea is built into the age limit of the Fields Medal and into the retirement rule of the Bourbaki group.^[3]^[4]

Empirical work has tested the claim with mixed results. Harvey C. Lehman's Age and Achievement (1953) concluded that notable contributions cluster in early and middle adulthood, in mathematics most often between the ages of 30 and 40,^[5]^[6] but his methods drew criticism.^[7] Nancy Stern found in 1978 that productivity and citation data on American mathematicians did not support the claim that the young do the most important work.^[8] Arthur M. Diamond Jr. reported in 1986 that mathematicians' current output declined with age even though citations to their accumulated work often peaked late in a career,^[9] and mathematicians surveyed by Reuben Hersh in 2001 described very varied experiences of aging.^[10] A 2014 review of the wider literature on age and scientific creativity found that great scientific output typically peaks in middle age, with wide variation among individuals and a rising average age of major contributions over the twentieth century.^[6]

Mathematical culture

The association between mathematics and youth has often rested on celebrated cases of mathematicians who did major work early or died young. Hardy's remarks appear in section 4 of the Apology, where he names Galois, Abel, Ramanujan, and Riemann among mathematicians who died young and states that he does not know of an instance of a major advance begun by a man over fifty.^[1]

Adler took a similar position in The New Yorker. He wrote that the mathematical life of a mathematician is short, that work rarely improves after roughly the age of twenty-five or thirty, and that a mathematician who has accomplished little by then is unlikely to accomplish much afterward.^[2]

Prizes and institutions

Fields Medal

The International Mathematical Union (IMU) describes the Fields Medal, awarded every four years at the International Congress of Mathematicians, as recognising "outstanding mathematical achievement for existing work and for the promise of future achievement". Under the medal's statutes, a candidate's fortieth birthday must not occur before 1 January of the year of the congress at which the medals are awarded.^[3]^[11]

Andrew Wiles was past this age limit by the time his proof of Fermat's Last Theorem was complete. Instead of a medal, the IMU presented him with a special silver plaque at the 1998 congress in Berlin.^[12]^[13]

Bourbaki

The Bourbaki group, a collective of mostly French mathematicians writing under a shared pseudonym, kept an internal age limit of its own. According to Encyclopædia Britannica, members agreed to retire from the group at age 50 while its ranks were refilled with younger recruits.^[4]

Empirical studies

Lehman and his critics

The first large quantitative treatment was Lehman's Age and Achievement, which tabulated the ages at which notable contributions were made across many fields. Lehman placed the peak for mathematics roughly between ages 30 and 40, earlier than his figures for fields such as astronomy but later than his figure for chemistry.^[5]^[6]

The psychologist Wayne Dennis challenged Lehman's methods in 1956. Dennis argued that Lehman had not controlled for age at death, so that short-lived contributors could only be represented by youthful achievements and the resulting curves overstated early work; he also objected that the age intervals Lehman used varied from discipline to discipline in ways that could inadvertently maximise the apparent peaks.^[7]^[14]

Benjamin F. Jones, E. J. Reedy, and Bruce A. Weinberg reviewed this literature in 2014. They reported that great scientific output typically peaks in middle age, that the classic orderings of fields by peak age are less stable than once thought, and that the average age at which major scientific contributions are made rose substantially over the twentieth century, with an especially sharp decline in contributions made before age 30.^[6]

Studies of mathematicians

Nancy Stern's 1978 article in Social Studies of Science set out to bring mathematics, a field she described as largely avoided by sociologists of science, within the scope of that discipline. Stern compared citations to the work of mathematicians elected to the US National Academy of Sciences with citations to the work of a random sample of university-based American mathematicians, using productivity and citation counts to test the claim that younger mathematicians are more apt to do important work; part of her analysis asked whether citation counts can serve even as a rough measure of quality in mathematics.^[8] Stern concluded that the evidence invalidated the claim, whether it is taken to mean that the most significant work comes from younger mathematicians or that individual mathematicians do their best work while young.^[8]

Arthur M. Diamond Jr. assembled longitudinal data on mathematicians and scientists at six major university departments, covering age, salaries, annual citations to each scholar's accumulated work, citations to current output, and the quantity of current output. In a 1986 article in the Journal of Gerontology he reported that the quantity and quality of current research output appeared to decline continuously with age, while annual citations peaked anywhere from age 39 to age 89 in the different departments, with a mean peak age of 59.^[9]

Reuben Hersh put the question to working mathematicians for a 2001 article in The Mathematical Intelligencer. The replies varied widely, but Hersh identified several statements he thought most respondents would accept: that no single pattern of aging describes all mathematicians, that many mathematicians have remained productive at advanced ages, and that aging brings most mathematicians some loss of memory and computing ability, losses which broader perspective and mature judgment may compensate for.^[10]

Gender and scholarly persona

The historian of mathematics Michael J. Barany has examined Hardy's "young man's game" remark through the history of the scholarly persona, historicising the norms of youth, gender, and play behind the image of the young mathematician and tracing how their importance changed in the international mathematical community of the mid-twentieth century. Bourbaki, and the mathematicians who wrote under its name, serve as his central example.^[15]

Claudia Henrion's Women in Mathematics: The Addition of Difference (1997), built around interviews with prominent American women mathematicians, treats the belief that mathematicians do their best work in their youth as one of several myths of mathematical culture. The chapter devoted to that belief, titled "Is Mathematics a Young Man's Game?", pairs its discussion with a profile of the topologist Joan Birman.^[16]