Life begins at N=40
In this week's New Scientist there is an article by Marcus du Sautoy entitled Life begins at N=40, in which he ridicules the idea that if you are a mathematician then "after 40 you're past your best and will never make any worthwhile discoveries", which is reinforced by the fact that the award of the Fields Medal (the Nobel prize of mathematics, sort of) has a cut-off age of 40. As a counterexample he cites the case of Lennart Carleson who was productive into his later years. I agree with everything in his article, and I want to generalise what he says.
The assumption that people "will never make any worthwhile discoveries" past a certain age is not limited to mathematics, and is also profoundly misleading in fields of work where it takes many years to become proficient. It is true that a 20-something has reserves of stamina that enable them to focus their activities in ways that are impossible for a 40-something. But it is also true that a 40-something has much more experience and insight than a 20-something, and can finesse problems in ways that are not possible for the less experienced worker. There is a trade-off here: one's stamina declines with age, yet one's experience increases with age. There are other factors to consider, such as the lack of mental baggage that is carried by a 20-something (this is good), and the general decline of mental abilities with age (this is bad), but I believe that the stamina/experience trade-off is the key thing.
In my own experience, when I was a 20-something I used to do a prodigious amount of mathematical work almost none of which became a permanent part of my subsequent work, but I now do much less mathematical work of which a much higher proportion is incorporated into my subsequent work. I now don't follow up lines of work that I can intuitively see are going nowhere, whereas in my earlier work I followed up all lines of enquiry in detail. Basically, I now work much more efficiently than I used to.
There is another effect that I have observed, which has to do with the relationship between younger and older researchers. I distinctly recall that when I was a 20-something I used to think that older researchers didn't know what they were talking about, and that they should thus be ignored. My perceptions were only partially true, because older researchers carry a lot of baggage that gets in the way of clear thinking, but they do know what they are talking about. What I did not anticipate was that the same argument would be applied to me when I became an older researcher myself! There are vast tracts of knowledge that I see being reinvented from scratch (or, more usually, being ignored because they are considered to be "too difficult") by the next generation of researchers, because they assume that they are so clever that they are the first people to ever think about these problems. This is a waste of resources.
One way in which this "generation gap" between researchers manifests itself is in the analytic/numerical trade-off. People who were educated before computers were widely available find analytic calculations much easier to do than those who were educated after computers became widely available. Older researchers tend to calculate, whereas younger researchers tend to compute. There is a group of people who straddle the advent of computers who can freely and productively mix analytic calculation and numerical compution; I am one of these people. We need to encourage researchers to be "bilingual", so that they can mix numerical computation with analytic calculation.
How can this harmonious mix of analytic calculation and numerical computation be achieved? We need a single tool that seamlessly unifies the two approaches. In my own work I have found that Mathematica is the ideal tool for this, because with it I can do everything that I ever used to do (i.e. numerical and analytic work) and lots more besides. I have found that this way of working has completely upset the stamina/experience trade-off that I mentioned earlier, because now Mathematica provides the stamina and I provide the experience. It just gets better with time, and I look forward to being a highly productive 80-something.