Apologies for pasting all of this, but the excerpt has always stuck with me. It seem correct. Are there alternative explanations other than mental processing speed and so on? (For example, later in life, you're less likely to be in a position to do the same sort of work. But that seems to have been tested by e.g. the institute of advanced study.)
As far as I can tell, this section might be one of those facts that people try not to think about too much. I don't worry about it, but I end up thinking about it a lot.
There was recently some headline news about an older mathematician that made a significant breakthrough. Other than that one outlier, have there been many important contributions made by people after the age of, say, 45?
--
"I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. To take a simple illustration at a comparatively humble level, the average age of election to the Royal Society is lowest in mathematics.
We can naturally find much more striking illustrations. We may consider, for example, the career of a man who was certainly one of the world's three greatest mathematicians. Newton gave up mathematics at fifty, and had lost his enthusiasm long before; he had recognized no doubt by the time that he was forty that his great creative days were over. His greatest ideas of all, fluxions and the law of gravitation, came to him about 1666, when he was twenty-four—'in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since'. He made big discoveries until he was nearly forty (the 'elliptic orbit' at thirty-seven), but after that he did little but polish and perfect.
Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself.
On the other hand the gain is no more likely to be substantial; the later records of mathematicians who have left mathematics are not particularly encouraging. Newton made a quite competent Master of the Mint (when he was not quarrelling with anybody). Painlevé was a not very successful Premier of France. Laplace's political career was highly discreditable, but he is hardly a fair instance, since he was dishonest rather than incompetent, and never really 'gave up' mathematics. It is very hard to find an instance of a first-rate mathematician who has abandoned mathematics and attained first-rate distinction in any other field.1 There may have been young men who would have been first-rate mathematicians if they had stuck to mathematics, but I have never heard of a really plausible example. And all this is fully borne out by my own very limited experience. Every young mathematician of real talent whom I have known has been faithful to mathematics, and not from lack of ambition but from abundance of it; they have all recognized that there, if anywhere, lay the road to a life of any distinction.
First, a note: math is one of the specific areas where humans generally peak really quite young. Math (quantitative reasoning/logic) is not the only area of psychometric intelligence testing (e.g., IQ) and not even a majority of it. So, it may be that it really is a bit harder for older mathematicians to make breakthroughs? I don't know. At any rate, citing mathematics as an indicator is probably not ideal, because math ability does indeed generally peak early.
However, as of 2011[1] the mean age of physics Nobel winners at the time of their achievements across the entire period of the award was 37.2 and since 1985 the mean age was 50.3.
According to the same paper, by the year 2000, Nobel-level achievement in physics before age 40 was only 19% of cases. It also appears that awards in chemistry and medicine are similarly increasing in mean age.
Is this dispositive? Certainly not. Maybe the Nobel committee prefers to award old scientists because of some unknown bias?
However, it does indicate that high achievement is both possible and normal in middle age and beyond.
Specifically responding to the increasing average age of Nobel prize winners: this is in part due to the increasing complexity of problems to solve. With our current ways of solving problems, the new problems become harder and harder. The existing human knowledge is also becoming harder and harder to understand, requiring somebody working in a field to spend much longer studying and catching up to the state of the art before being able to make a significant contribution to the field.
This is one of the reasons that I'm personally so excited about (and working on) the potential of spatially immersive media like VR to understand complex concepts. Taking a step back, tools like a graph plot enabled humans to understand complex concepts like differentials and projectile motion at a much younger age. Could a breakthrough with new ways of understanding human knowledge effectively do the same with knowledge that is today considered complex (eg, quantum mechanics)? If such a breakthrough happens, could we bring the average age of significant contribution in subjects like physics back down?
I don't know, but I hope so. :)
Edit. I also remember reading thoughts by either Michael Nielsen on the increasing age of Nobel-worthy contributions in physics, but I can't find it in my current sleep deprived state. I shall tomorrow if somebody else hasn't pointed to that article by then.
Maybe, but I somewhat doubt it. Nobel-level work is usually derivative of only a few basic concepts, but is otherwise quite daring. The academic guild system has become nearly impossible to get through, and all corporate or government research depends on passing through that system first. Nobody can just get to work. First you have to get in. 99% of applicants are mostly concerned with prestige or career opportunities. 1% wants to do research. Then you need funding. This comes by helping professors on their ideas, not yours. Then you need a job. Better pick a popular field and find ‘business value’. Now it’s time to buy a house. Maybe you’ll get back to that big idea you had once you pay it off. Student loan availability turned the academic pipeline into a job requirement, and a job is then required to pay it off. We’re just entering the era of Nobel winners that started school during the Vietnam boom. It seems to me that the Nobel will cease to have any meaning in a decade or so. The trend is that contemporary prizes are given to politically valued choices from a huge field of contributors, or as an honorarium for ‘famous’ professors. The last 20 years of minted professors are far more focused on job security than great research, and it would be surprising to see a lot of individual breakthroughs at the Nobel level. And then there are corporate labs, which are run by professional managers handing out nebulous quarterly objectives with a side of panic. Forget about it. The biggest hope for ambitious research may be self-funded entrepreneurs. There must be, somewhere along that path to human colonization of space, a Nobel for Elon.
Good point. I call this problem the Giant's Shoulder Climbing Problem. Isaac Newton said he could only see farther because he was standing on the shoulders of giants. By giants he meant all the knowledge amassed by previous generations. The problem is, nowadays the giants got so big, that one can spend the better part of a life just climbing the damn giant, many failing to reach the ever receding shoulders.
I've pondered on this problem a bit before. To solve it, I reached your same conclusion, that we need some breakthrough with new ways of understanding human knowledge simplifying knowledge that is today considered complex, or in other words, we need at least try to build some sort of elevator.
IMHO, the most promising possible breaktroughs I could find were:
(I) a reform in math education, with early introduction of schoolchildren to computer algebra system (CAS) software, shifting curriculum away from tedious manual computations and trick learning. When in university, for example, I learned lots of integration tricks, and forgot most of them a few years later. Would my time had better invested just learning SymPy instead of all those tricks? This idea is pushed by Conrad Wolfram. For example. See his talk at https://youtu.be/jE9lU4E52Vg
(II). a reform of physics education to replace vector algebra with proper geometric algebra, as advocated by David Hestenes. Vector algebra as taught in physics today is actually a hack pushed by Gibbs, that only works well in 3D and demands a lot of shoehorning to work in problems with higher dimensionality. Geometric algebra scales well in any number of dimensions, and many problems become easier. The four Maxwell equations, for example, became one. See the discussion in https://physics.stackexchange.com/a/62822 :
"Now, the contention is that Clifford algebra is under-utilized in basic physics. Every problem in rigid-body dynamics is at least as easy when using Clifford algebra as anything else — and most are far easier — which is why you see quaternions being used so frequently. Orbital dynamics (especially eccentricity!) is practically trivial. Relativistic dynamics is simple. Moreover, once you've gotten practice with Clifford algebra in the basics, extending to electrodynamics and the Dirac equation are really simple steps. So I think there's a strong case to be made that this would be a useful approach for undergrads. This could all be done using different tools, of course — that's how most of us learned them. But maybe we could do it better, and more consistently.
No one is claiming that Clifford algebra is fundamentally new; just that it could be bundled into a neater package, making for easier learning. Try teaching a kid who is struggling with the direction of the curl vector that s/he should really be thinking in terms of the algebra generated by the (recently introduced) vector space, subject only to the condition that the product of a vector with itself is equal to the quadratic form. Or a kid who can't understand Euler angles that this rotation is better understood as a transformation generated (under a two-fold covering) by the even subalgebra of Cl3,0(R). No one here is arguing that that should happen. GA is just a name for a pedagogical approach that makes these lessons a whole lot easier than they would be if you sent the student off to read Bourbaki. Starting off with GA may be slightly harder at the beginning, but pays enormous dividends once you get to harder problems. And once teachers and textbooks get good at explaining GA, even the introduction will be easier."
>There was recently some headline news about an older mathematician that made a significant breakthrough. Other than that one outlier, have there been many important contributions made by people after the age of, say, 45?
There is one other effect beyond ability: people over 45 rarely take up new interests. There are plenty of cases of people who continue working in the same field through their 40s and 50s and continue to make advances, but there are many fewer cases of an individual beginning to work in a field after 40 and going on to make a major discovery. In pure math, possibly the most aesthetic-driven technical field (it is impractical by definition), this effect might be especially strong.
Richard Hamilton (not to be confused with William Rowan Hamilton of action-principle fame) initiated the application of the Ricci flow to the geometrization conjecture in 1982 at 39 and continued his work through the '90s, being credited by Perelman as making crucial contributions to the final solution of the Poincare conjecture in three dimensions. He's probably the most prominent example.
>It is very hard to find an instance of a first-rate mathematician who has abandoned mathematics and attained first-rate distinction in any other field.
As far as I can tell, this section might be one of those facts that people try not to think about too much. I don't worry about it, but I end up thinking about it a lot.
There was recently some headline news about an older mathematician that made a significant breakthrough. Other than that one outlier, have there been many important contributions made by people after the age of, say, 45?
--
"I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. To take a simple illustration at a comparatively humble level, the average age of election to the Royal Society is lowest in mathematics.
We can naturally find much more striking illustrations. We may consider, for example, the career of a man who was certainly one of the world's three greatest mathematicians. Newton gave up mathematics at fifty, and had lost his enthusiasm long before; he had recognized no doubt by the time that he was forty that his great creative days were over. His greatest ideas of all, fluxions and the law of gravitation, came to him about 1666, when he was twenty-four—'in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since'. He made big discoveries until he was nearly forty (the 'elliptic orbit' at thirty-seven), but after that he did little but polish and perfect.
Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself.
On the other hand the gain is no more likely to be substantial; the later records of mathematicians who have left mathematics are not particularly encouraging. Newton made a quite competent Master of the Mint (when he was not quarrelling with anybody). Painlevé was a not very successful Premier of France. Laplace's political career was highly discreditable, but he is hardly a fair instance, since he was dishonest rather than incompetent, and never really 'gave up' mathematics. It is very hard to find an instance of a first-rate mathematician who has abandoned mathematics and attained first-rate distinction in any other field.1 There may have been young men who would have been first-rate mathematicians if they had stuck to mathematics, but I have never heard of a really plausible example. And all this is fully borne out by my own very limited experience. Every young mathematician of real talent whom I have known has been faithful to mathematics, and not from lack of ambition but from abundance of it; they have all recognized that there, if anywhere, lay the road to a life of any distinction.
1 Pascal seems the best."