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There is no shortage of anecdotes and conjectures on both sides of this widespread belief, but good supporting data either way is harder to find. Can anyone provide any references for serious (preferably academic rather than journalistic) research that actually crunched the data and produced interesting conclusions about whether this bit of folklore is reality-based?

I put "mathematicians do their best work when they're young" in quotes because this is clearly not a well-posed question--it is only intended to be shorthand for any of a number of questions on this topic.

Historical studies (Évariste Galois, etc.) are OK, but studies on people born after, say, 1950, would be of greater interest and relevance.

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    $\begingroup$ @Will Jagy There are plenty of MO questions--especially soft questions--that solicit examples or information in precisely the way that this question does. I am most definitely not interested in opinions; I am seeking information and references. Would rephrasing help? $\endgroup$ – Elizabeth Henning Jan 2 '15 at 2:28
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    $\begingroup$ There is at least a factual question here: what studies if any have been done? Judging their seriousness or the usefulness of their conclusions could be tricky of course. Might it be worth asking over at Academia.SE? (I'm not too familiar with that site.) $\endgroup$ – Todd Trimble Jan 2 '15 at 2:47
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    $\begingroup$ The following paper provides perhaps one attempt at approaching a similar question. For example, the authors' note, "It turns out that Fields medalists are not only publishing fewer papers in the post-medal period, and that those papers are relatively less important, but they are also accepting fewer mentees under their wing." hks.harvard.edu/fs/gborjas/publications/journal/JHR2015.pdf $\endgroup$ – jemmy.bruce Jan 2 '15 at 2:53
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    $\begingroup$ This question seems obviously fine as is to me. This seems perfectly amenable to study, and someone may have study it. There is no perfect methodology, but there are various metrics that could be used to measure productivity, such as number of papers, prestige of journals the papers appear in, etc. This aren't perfect proxies, but they would tell us something. $\endgroup$ – arsmath Jan 2 '15 at 6:50
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    $\begingroup$ I see nothing wrong with this question: it is a legitimate "reference request" on the question which is of interest to many. I vote to re-open. $\endgroup$ – Alexandre Eremenko Jan 2 '15 at 8:10
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These two studies arrive at what seems to be a more sensible conclusion:

Age and Scientific Performance, Stephen Cole (1976).

The long-standing belief that age is negatively associated with scientific productivity and creativity is shown to be based upon incorrect analysis of data. Analysis of data from a cross-section of academic scientists in six different fields indicates that age has a slight curvilinear relationship with both quality and quantity of scientific output. These results are supported by an analysis of a cohort of mathematicians who received their Ph.D.'s between 1947 and 1950. There was no decline in the quality of work produced by these mathematicians as they progressed through their careers.

Age and Achievement in Mathematics: A Case-Study in the Sociology of Science, Nancy Stern (1978).

The claim that younger mathematicians (whether for physiological or sociological reasons) are more apt to create important work is unsubstantiated... I have found no clear relationship between age and achievement in mathematics.

For anecdotes and "advice to aging mathematicians", I might recommend Mathematical menopause, or, a young man's game?, by Reuben Hersh (The Mathematical Intelligencer, 2001).

Until we find a consensus about which advances are "major," we can't refute Hardy's claim that no major advance has been made by a mathematician over 50. But his slogan, "Mathematics is a young man's game," is misleading, even harmful.

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    $\begingroup$ I'd upvote twice if I could, thanks for those informative references! $\endgroup$ – Thomas Sauvaget Jan 2 '15 at 18:26
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    $\begingroup$ These first two are useful, but more recent work would be better still. $\endgroup$ – Elizabeth Henning Jan 2 '15 at 23:34
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    $\begingroup$ Up next in research papers: "Mathematics is a young man's game" considered harmful. $\endgroup$ – Nit Jan 3 '15 at 21:43
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I hope it's OK to post an answer to my own question since it's community-wiki. Here are a couple of things I found down this rabbit-hole.

Dean Simonton at UC Davis has done some work claiming that there is a slow age-related decline in quality and quantity of creative output, but the relevant variable is career age, not biological age. He also makes it clear that although he believes there is a clear aggregate trend, the individual variability is much greater than the aggregate variability. Furthermore, he attributes the decline mostly to factors other than biological aging.

Simonton, D. K. (1997). Creative productivity: A predictive and explanatory model of career trajectories and landmarks. Psychological Review, 104, 66-89.

This paper is behind a subscription paywall (but there is a link below in the comments), so instead I'm posting this link to the PowerPoint (sorry) of his 2005 talk at the Max Planck International Research Network on Aging:

http://psychology.ucdavis.edu/Simonton/MxAgCrProd.ppt

I couldn't find a good sound bite from Simonton's paper. Here is a quote from Arne Dietrich's 2004 paper The cognitive neuroscience of creativity:

Simonton (1997) has convincingly demonstrated that “creative productivity is a function of career age, not chronological age” (p. 70). Although career age and chronological age are highly correlated, latecomers to a discipline show the same career trajectories and landmarks, as well as conformity to the 10-year rule (Simonton, 1997, 2003). For instance, mathematicians peak on average at 26.5 years of career age, while historians peak at 38.5 (Simonton, 1997). Because prefrontal-dependent mental functions do not significantly decline until old age, the distinction between chronological and career age can be accommodated as long as the creator’s career onset is not at an advanced chronological age.

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  • $\begingroup$ It's certainly okay to post an answer of your own, community wiki or not. At some point you might want to officially accept an answer; see here for more on this: meta.stackexchange.com/questions/5234/… $\endgroup$ – Todd Trimble Jan 3 '15 at 2:55
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    $\begingroup$ Simonton's 1997 article is available here. $\endgroup$ – Benjamin Dickman Jan 3 '15 at 3:51
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    $\begingroup$ Thanks for the link, and thanks, Todd, for the info. I would feel bad accepting my own answer if it weren't community-wiki! But I'm not that thrilled with what I've found so far--here's hoping there's something better out there. $\endgroup$ – Elizabeth Henning Jan 3 '15 at 4:26
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    $\begingroup$ I think that as an individual becomes more accomplished in her specialty, her employer rewards her with promotions. Each step up that ladder requires more time for non-specialty duties. Therefore, less time for creativity and thus fewer papers. $\endgroup$ – Fred Kline Jan 3 '15 at 14:08
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As I originally said as a comment, the following paper provides perhaps one attempt at approaching a similar question. Namely the authors explore what effect winning the Fields medal has on mathematicians productivity. To do this they examine publication and citation rates of a select group of mathematicians over time. The authors' note, "It turns out that Fields medalists are not only publishing fewer papers in the post-medal period, and that those papers are relatively less important, but they are also accepting fewer mentees under their wing."

"Prizes and Productivity: How Winning the Fields Medal Affects Scientific Output"

George J. Borjas and Kirk B. Doran

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    $\begingroup$ Of course, one might argue that they are doing less work but "better" work... and the sheer subjectivity of the original phrase "their best work" makes me sceptical that any investigations of this maxim would stand up to scrutiny. $\endgroup$ – Yemon Choi Jan 2 '15 at 17:37
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    $\begingroup$ Even if those who win a Fields have reduced productivity, it doesn't mean that the Fields medal does not motivate productivity, since the existence of the medal presumably motivates those who haven't won it yet. (The objection is rather like pointing out that those who have just completed a marathon run slower than those who are about to win one.) $\endgroup$ – Joel David Hamkins Jan 2 '15 at 19:17
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    $\begingroup$ That just sounds like regression to the mean to me. It's the same effect that causes athletes that have just appeared on the cover of Sports Illustrated to have worse seasons afterwards (en.wikipedia.org/wiki/Sports_Illustrated_cover_jinx). $\endgroup$ – Qiaochu Yuan Jan 2 '15 at 21:47
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    $\begingroup$ There is an entertaining discussion of this article by Janos Kollar in the January 2015 Notices of the AMS ams.org/notices/201501/rnoti-p21.pdf. $\endgroup$ – Danny Ruberman Jan 2 '15 at 22:04
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    $\begingroup$ @Qiaochu Yuan: Wouldn't regression to the mean also apply to the control group that they use in the paper? $\endgroup$ – Timothy Chow Jan 4 '15 at 20:33
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Jordan Ellenberg (JSE on MO) wrote a nice article after Perelman announced his solution of the Poincaré conjecture:

"Is Math a Young Man's Game? No. Not every mathematician is washed up at 30." Slate, May 2003. (article link.)

The article ends with this:

"It's only in the presence of both conditions—deduction and inspiration, long experience and youthful audacity—that new math gets made, as it was made by Perelman, and as it was made on the day Poincaré wrote down his conjecture. He was 50 years old."

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    $\begingroup$ As I read this article, I kept noticing how free it was of gross misconceptions about mathematics. Of course, it turns out that the author is a mathematician ... $\endgroup$ – Nik Weaver Jan 2 '15 at 21:58
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    $\begingroup$ I remember reading this when Slate first posted it. It's a nice piece, but it's purely anecdotal. $\endgroup$ – Elizabeth Henning Jan 2 '15 at 23:11
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    $\begingroup$ I haven't read this in years. I have to say, I have learned a lot since then about how to construct a magazine article; this one is kind of confusing, especially where it suddenly veers from general math sociology into a discussion of algebraic topology with no transition. Still, since I am 11 years older now than I was when I wrote that, I heartily endorse its conclusions. $\endgroup$ – JSE Jan 7 '15 at 20:14
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If everybody assumes mathematicians to be "a different cup of tea" or that "they cook themselves apart from the rest of scientists" (or that they are not scientists at all, if you will), then I would have nothing to add to what has already been said. However, let us suppose for a moment that mathematics is comparable to any other scientific endeavor. Then we have this article of Packalen & Bhattacharya, 2015, "Age and the trying out of new ideas" (http://www.nber.org/papers/w20920.pdf) which I would say is closely related to your question. Not in the sense of bulk production, but in terms of originality, which in my humble opinion, it is also very, very important for mathematicians (perhaps even more than size of production). They studied biomedical articles since 1946, and one of the conclusions they arrive at, is that the best possible team in order to obtain an original paper, is a young scientist coupled with a "more experienced" one (read it as "older", maybe "much older").

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