1. No one is really a polymath.
2. No one is really a unimath, for that matter.
3. Many supposed polymaths apply a relatively small number of learning techniques to many fields. They remain specialized, although their modes of specialization happen not to line up with how the academic disciplines are divided. Say you apply non-parametric statistics to five different fields — do you have one specialization or five?
4. What to make of the economist who can run experiments, use computational methods, build models, run regressions, find new data sources, has mastered machine learning, can speak fluently about macroeconomics, and popularize for a lay audience. Is there any such person? (No.) Would he or she count as a polymath?
6. One of my views in talent search is that extremely talented people are almost always extraordinarily good at one or more entirely trivial tasks. “I can tell exactly how much people weigh just by looking at them.” That sort of thing. What is your claim in this regard? Polymaths also must encompass the trivial!
7. How many “polymaths” are great at say only seven very trivial tasks, and fail to excel at anything important. Should the polymath concept be held hostage to Jeremy Bentham?
8. Is Leibniz — amazing philosopher, an inventor of the calculus, mastery of languages, theologian, diplomacy, legal reform, inventor, political theorist, and supposed expert on China — the most amazing polymath of all time?
9. Leonardo seems a little thin in actual achievement (though not imagination) once you get past the visual arts. And there are fewer than fifteen paintings to his name.
10. I think of the 17th century as a peak time for polymaths. Enough chances to learn and create things, and read lots, but not so much knowledge that you could stand on only one frontier.
11. John Stuart Mill is the most impressive polymath economist.
12. Alan Turing contributed to virtually every field, but in some sense he did only one thing. Von Neumann did more than one thing, did he do two? He too contributed to virtually every field.
13. I am very much a fan of Susan Sontag, but I think of her as having done, in essence, “only one thing.”
14. Here is a good piece Beware the Casual Polymath.
I am very happy to recommend this book, especially to MR readers, the full title is The Polymath: A Cultural History from Leonardo da Vinci to Susan Sontag, by Peter Burke.