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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.

Elizabeth Henning
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    Most of such kind of story appears in the bibiliorgraphy. While different mathematician have different background, what make "serious investigation" difficult. – yaoxiao Jan 02 '15 at 01:07
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    why do you want to know? – Will Jagy Jan 02 '15 at 01:14
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    The question is not well defined: eg does it mean "all" mathematicians, "most" mathematicians, or "some" mathematicians? And who will make an objective assessment of what constitutes a particular mathematician's best work? – Geoff Robinson Jan 02 '15 at 01:22
  • As stated, it's not well-defined, but it could be--I'm casting a wide net to see what's out there. Have any studies like this yet been done? It's a question worth answering for the same reason analogous questions about women and "mathematical aptitude" are interesting. – Elizabeth Henning Jan 02 '15 at 01:50
  • I suspect studies have been done, but I am sufficiently Eeyorish that I wouldn't set much store by the methodology. There is the issue of controlling for the changing mores and contexts over the course of history, even within last 100 years. – Yemon Choi Jan 02 '15 at 02:04
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    Put another way (progressing from Eeyore to Benjamin) -- I don't see how there could be any credible data that help to give a meaningful answer. – Yemon Choi Jan 02 '15 at 02:05
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    @YemonChoi Meanwhile, I voted to close. Not fond of fishing expeditions. – Will Jagy Jan 02 '15 at 02:07
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    @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? – Elizabeth Henning Jan 02 '15 at 02:28
  • Elizabeth, not for me. I've often gotten overruled, of course. The worst one was some guy who asked about the philosophy behind Mochizuki's work, which is still unconfirmed, years later. – Will Jagy Jan 02 '15 at 02:32
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    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.) – Todd Trimble Jan 02 '15 at 02:47
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    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." http://www.hks.harvard.edu/fs/gborjas/publications/journal/JHR2015.pdf – jemmy.bruce Jan 02 '15 at 02:53
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    @DJBruce Aha! You should repost this as an answer. This is the kind of thing I'm looking for, although, as you say, it answers a somewhat different question. Thank you! – Elizabeth Henning Jan 02 '15 at 02:58
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    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. – arsmath Jan 02 '15 at 06:50
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    I suspect that the what the users who are trying to close this topic dislike isn't the style or tone of the question but the subtext of its content. However, the number of views strongly suggests that this topic deserves to be opened back up. – Elizabeth Henning Jan 02 '15 at 07:13
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    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. – Alexandre Eremenko Jan 02 '15 at 08:10
  • Related: http://mathoverflow.net/questions/25630/major-mathematical-advances-past-age-fifty and http://mathoverflow.net/questions/3591/mathematicians-who-were-late-learners-list – Timothy Chow Jan 02 '15 at 15:52
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    @TimothyChow If this question is to stay open, I think we should keep focused on actual studies/surveys rather than anecdotes and famous one-off cases, which seem to have been the outcome of the two questions you link to – Yemon Choi Jan 02 '15 at 16:11
  • This is not an answer to the question, but here is a link to a recent meta-survey about women in mathematics, which is a topic mentioned in the comments: http://arxiv.org/abs/1412.4104. It suggests that a similar survey about the requested topic may also exist. – Sam Hopkins Jan 02 '15 at 17:14
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    I like the question, but I would cut the words "extremely widespread and toxic" and "useful", for a more neutral formulation. Similarly I would cut "serious", "actually", "really", and the two pairs of "what...is" in the second paragraph, to reduce unneeded heat in a reasonable issue. Maybe people would be more open to the resulting question. –  Jan 02 '15 at 17:49
  • Elizabeth, I wouldn't read too much into the number of views or the presumed dislike of the question: the number of views might be people seeing what all the hubbub is about (and I think people often check back in to see what's happening with controversial questions), and also there is some built-in conservatism towards questions that are on the soft side, since many here want to retain the strong emphasis on research mathematics and not get overrun by "discussion-y" questions, even when they think the questions are otherwise of interest. – Todd Trimble Jan 02 '15 at 17:57
  • @Matt F. OK, I'll edit it a bit. Thanks for the feedback. – Elizabeth Henning Jan 02 '15 at 18:58
  • $P=Probability$.

    $$P(\mbox{young prof at tenure track})\gg P(\mbox{old prof at tenure track})$$ $$P(\mbox{tenure track profs work hard})\gg P(\mbox{tenured profs work hard})$$ $$P(\mbox{tenure track profs having fertile ideas})\gg P(\mbox{tenured profs having fertile ideas})$$ $$\mbox{ because techniques tenured profs might have pioneered is well assimilated in community}$$

    Seems like a reasonable explanation.

     – Turbo Jan 02 '15 at 21:10
  • @arsmath I'm afraid I think all those metrics are terrible and verge on meretricious, if used to judge quality rather than "productivity" – Yemon Choi Jan 03 '15 at 21:47
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    @Yemon Choi I'm not crazy about those kinds of metrics either. Simonton, whom I mention in my answer below, gets around the problem of quality by simply substituting quantity. He makes a not unreasonable argument that the two are strongly enough correlated to be a valid proxy, but I'm not convinced. Nonetheless, I still prefer careful results based on data known to be imperfect over speculation and anecdotally substantiated canards. That pretty much describes all science that isn't pure mathematics. – Elizabeth Henning Jan 04 '15 at 02:44
  • Well, as a very rough starting point you could look at winners of the Fields Medal and similar prizes, and look at their age at the time they did the work for which they won the prize. – Bob Jarvis - Слава Україні Jan 04 '15 at 03:50
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    If the question is declared out-of-scope, consider reposting at [Skeptics.SE] – Mast Jan 04 '15 at 16:48
  • The Young family have done great work in mathematics, under a different interpretation of "when they're Young". – Ben McKay Jan 09 '21 at 17:43
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    One important factor to keep in mind is that academic mathematics has been a highly competitive field for at least the past 50 years. In my experience, so competitive that small variations in productivity early on lead to significant variations in career opportunities. Moreover, the tenure track system at R1 institutions rewards mathematicians with early success. Doesn't that mean that people who peak later get put on a track where they have less direct access to the mathematical community (possibly paired with heavier teaching loads)? It would be a mistake to ignore this selection bias. – R. van Dobben de Bruyn Jan 10 '21 at 04:02
  • Following is common hearsay I learned. Older people cannot learn new but can dive deep on things they learned when they were younger. How true is it in mathematics or reallife even? – Turbo Apr 07 '21 at 17:03
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    @1.. Good thing you learned this "common hearsay" when you were young, so now you'll be able to dive deeper into it when you get older and realize how ridiculous it is. – Elizabeth Henning Apr 07 '21 at 17:06
  • @ElizabethHenning might be correct as mathematicians do not change fields, drs do not, engineers do not etc after certain age (do not is a measure 1-eps wording). – Turbo Apr 07 '21 at 17:57

6 Answers6

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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.

Carlo Beenakker
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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.

Elizabeth Henning
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  • 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: http://meta.stackexchange.com/questions/5234/how-does-accepting-an-answer-work – Todd Trimble Jan 03 '15 at 02:55
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    Simonton's 1997 article is available here. – Benjamin Dickman Jan 03 '15 at 03:51
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    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. – Elizabeth Henning Jan 03 '15 at 04:26
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    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. – Fred Daniel Kline Jan 03 '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

Friedrich Knop
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jemmy.bruce
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    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. – Yemon Choi Jan 02 '15 at 17:37
  • Possibly a less subjective question is the degree to which "cognitive mobility" (propensity for switching fields) explains the apparent decrease in output. The authors say that it accounts for "about half" that decrease but I'm not sure I buy all the assumptions involved in computing the "about half" figure. – Timothy Chow Jan 02 '15 at 18:31
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    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.) – Joel David Hamkins Jan 02 '15 at 19:17
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    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 (http://en.wikipedia.org/wiki/Sports_Illustrated_cover_jinx). – Qiaochu Yuan Jan 02 '15 at 21:47
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    There is an entertaining discussion of this article by Janos Kollar in the January 2015 Notices of the AMS http://www.ams.org/notices/201501/rnoti-p21.pdf. – Danny Ruberman Jan 02 '15 at 22:04
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    @Joel David Hamkins Two observations: (1) The stated purpose of the Fields, in part, is to recognize "the promise of future achievement." (2) If it is possible to demonstrate that it does significantly motivate high-level productivity, then the IMU should reconsider the age cutoff. – Elizabeth Henning Jan 02 '15 at 23:28
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    @Qiaochu Yuan: Wouldn't regression to the mean also apply to the control group that they use in the paper? – Timothy Chow Jan 04 '15 at 20:33
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    @Timothy: fair enough. My apologies for not looking at the paper before commenting. – Qiaochu Yuan Jan 05 '15 at 04:41
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    Isn't Jean-Pierre Serre a counter-example ? He was awarded the Fields medal (1954) for his work in Algebraic Topoloygy. Today, he is even more well known as an Algebraic Geometer and a Number theorist, among other qualities. – Denis Serre Jan 09 '21 at 16:17
  • @DenisSerre Deere was $26$ at Fields ceremony and people start their phd at 26. So Serre was a generation below old. – Turbo Apr 07 '21 at 22:53
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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."

Community
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Joseph O'Rourke
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    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 ... – Nik Weaver Jan 02 '15 at 21:58
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    I remember reading this when Slate first posted it. It's a nice piece, but it's purely anecdotal. – Elizabeth Henning Jan 02 '15 at 23:11
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    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. – JSE Jan 07 '15 at 20:14
  • @JSE: Ha! :-) ${}$ – Joseph O'Rourke Jan 07 '15 at 20:52
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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").

Arturo Ortiz Tapia
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This misconception that mathematics is a young person's game seem to have been pushed most forcefully by Hardy, but the list of examples he gives is very misleading because they all died young and so obviously could not go on to contribute anything further anyway! He also conveniently forgets some famous counterexamples like Euler and Weierstrass. Cramer, for example, could be classed as a young prodigy, since he obtained his PhD equivalent at age 20, but the work for which he is known today was done when he was into his 40s.

The question asks for research on this question so I will provide two physics examples from my own reading.

1.) In 1988, Aharonov and colleagues published an influential PRL article on quantum mechanics. In 2020, Aharonov and Rohlich published a PRL article on quantum mechanics which looks set to be just as influential and is no different in quality or insight to the 1988 paper.

2.) In 1974, Georgi and Glashow published a PRL article on field theory which has been very influential. In 2020, Georgi published a PRL article on field theory which does not look to differ from the paper published forty-five years earlier in terms of quality and insight.

Hollis Williams
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    The question asked for "references for serious (preferably academic rather than journalistic) research that actually crunched the data and produced interesting conclusions...." I don't think your answer does that. – Gerry Myerson Dec 22 '19 at 15:46