It is well known that famous names such as Gauss, Euler and Newton were polymaths as well as their main fields of study and contributed from optics to ship building. Why was this the case in the past? It is known, as far as I am aware, to exist since the Greeks. Why are there so few modern polymaths?

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Really, it's because that was the social protocol at the time.

From the Wikipedia article on polymaths,

Many notable polymaths lived during the Renaissance period, a cultural movement that spanned roughly the 14th through to the 17th century and that began in Italy in the late Middle Ages and later spread to the rest of Europe. These polymaths had a rounded approach to education that reflected the ideals of the humanists of the time. A gentleman or courtier of that era was expected to speak several languages, play a musical instrument, write poetry, and so on, thus fulfilling the Renaissance ideal.

So you could say it was one of the primary tenets of Renaissance humanism. This approach stressed a person being proficient in many subjects, specifically the humanities. The philosophy was laid down in a book, The Book of the Courtier, written by Baldassare Castiglione. It laid down the ideas that the optimal person (symbolized by the main characters, a group of courtiers) should be extremely well-rounded.

Why are there so few modern polymaths?

I suspect a combination of apathy and the fact that society no longer values people who have a wide range of talents (unless we count colleges!). Today, we typically only major in one subject in college (although we can focus on a minor, as well). The major is in a specific field, which the student hopes to go into upon leaving college. Our education is utilitarian, but in a different sense than in the Renaissance: we don't need to learn shipbuilding if we're going to work at, say, an art museum, and society no longer expects us too.

This bit is a wee bit dubious, but hopefully my logic makes sense. The various fields of study today, especially in the sciences, are much broader than they were in the time of the Renaissance. It was much easier to learn physics back in pre-Newtonian times (and during Newtonian times, too!) because learning physics didn't entail learning everything from Lagrangian mechanics to tensor calculus. True, the equivalent then of a "physicist" would have to have a broad knowledge of philosophy and metaphysics (as well as possibly alchemy), but probably not as much as a physicist needs to know today.

Finally, it takes time nowadays to become an expert in something. Here's how to become a physicist - in only a decade or two:

  1. Work hard for 4 years in high school and get good grades; show an interest in science, especially physics, to attract colleges.
  2. Spend 4 years in college; majoring in physics, with a possible minor, often in a related field.
  3. Spend 4 to 8 years to get a PhD.
  4. Work as a postdoc at a university for 5 or so years.
  5. Become an assistant professor; work at a university for 5 more years.
  6. Look for a job as a physicist.

Why are there so few modern polymaths?

Because it is pretty much impossible nowadays to master many fields at a level to be able to make significant contributions due to the incredible size of knowledge we have now reached. E.g. David Hilbert was probably one of the last universal mathematicians. The time investment of becoming an expert in just one narrow field is such that one does not have the time to become expert at many.

The last great polymaths were John von Neumann and David Hilbert. After that we perhaps do not see examples in their class. Some may say that Terence Tao can be considered one considering the fact that he has contributed to so many different mathematical fields, but I don't think his diversity can rival say Gauss' or Euler's.

The main reason is that the length and breadth of human knowledge has expanded many folds now than in the time of the great polymaths we know about. Nowadays to get a PhD we work on a sub field of a sub field of a subfield of a field, and many a times we cannot even consider ourselves to be a expert in that particular sub-sub-sub field. In practice thus, it would be nearly impossible now to be a polymath, but one can always try.

Polymaths today exist. For example, the (late) Clifford Truesdell, Roger Penrose, etc.

Fred Hoyle, Paul Dirac's student, wrote on everything from physics to science fiction, to economics, to astronomy. Eddington dabbled in philosophy.

William Clifford, although he died in his thirties, wrote on almost every subject. So did William Strutt, Lord Raleigh. So did James Hutton.

Carl (von) Menger, the economist, father of the famous mathematician Karl Menger, had a library of over 30,000 books.

Condillac wrote over forty volumes. So did Wolff. Cauchy was a master of everything and anything except economics and history.

Waterston gave the modern kinetic theory of heat in a book about a neural network connectionist explanation of the brain (in 1840's!) prior to publishing in the Philosophical Journal and submitting his physics and thermodynamics work to other journals and presenting it to the royal society.

I suspect the real issue is quite basic.

Today's physics requires substantial time to learn. But, then again, we have better tools. In ten pages, using fiber bundles and groups and modern methods of integration, one can discuss dynamical histories with more precision and detail than a thousand pages in the nineteenth century. It's not true that one cannot know, for example, physics if one is a (mathematical) biologist, or that a physicist cannot know biology and economics.

We learn so much more, in depth and in breadth, and in terms of empirical knowledge, besides mathematical concepts. But our greater human capital makes the process much easier. We easily solve problems that would require months of correspondence and effort a hundred years earlier.

Compare the enormous literature prior the 1930's, on special functions, made useless by advances in basic methods in analysis, including use of operator methods.

Furthermore, although more is required to be known in each field, access to literature is much easier and faster than any time in the past, where one had to spend vast sums of money to obtain rare monographs several times a year, and this often by advance subscription or by chance purchase.

Only about 60 copies of one of Euler's major calculus books was sold in his lifetime. Within fifty years, all continental mathematics was taught using his methods.

No, the issue is elsewhere.

1) There is general lack of respect for a scientist, at least relative to the past in Western Europe.

As Truesdell wrote once, people who became scientists in the past gained tremendously in social "rank", status, income, if they succeeded. This is no longer the case. Scientists were very rare, and interesting persons, with whom the nobility liked to meet. Recall how the king of England, George, invited and met with Lichtenberg, Gauss's teacher.

Today, several order of magnitude more people are scientists, engineers, and most of these are per se, as is statistically necessary, unexceptional individuals. So each one is less valuable to the public, unless the public can understand what exactly one can do that another cannot.

2) There are far more opportunities to do other things today than in the past, so FEWER people devote as MUCH of their time to study and writing exclusively, despite our population being much larger. In the past it was done partly to entertain oneself, today it is partly work, compared to other things one could be doing.

Consider this: the foregone opportunities, the cost of spending AS MUCH TIME AS EULER on science, for example, is much greater today.

(So is the cost of having children by the way, since it reduces time that could be spent working or making use of all modern leisure goods, which is why people have three kids, not thirteen.)

To make their life interesting, the polymaths of the past, they sat around and read and read, and wrote, wrote, studied, studied, and corresponded, and sometimes, rarely, they also met. There was neither the television nor the internet nor rapid travel nor many stores nor even many restaurants, and social gatherings were in private homes or at the court. Few products existed. Few books were easily obtained. There were few industries willing to pay them to work on challenging problems with good pay. They filled their entire day in study. Of course they knew everything that was known, and were able to contribute something as well. They devoted their entire lives to knowledge for its own sake.Today, very few people are willing to do this, even inside one profession. It's too costly, unless you really like reading and writing.

On my opinion, this is an example of usual aberration which often happens when statistics is applied without careful thinking. The percentage of polymath's is probably the same. To the names already mentioned, let me add Terence Tao, the most famous modern case. But there are many others.

The reason for this aberration is the following. We remember now only few 18 century mathematicians. I doubt that an average modern mathematician will immediately list 20, not speaking of the "general public". These are the best of the best. Most of the rest are not remembered. Not surprisingly, percentage of polymaths among them is large.

Modern mathematicians are not that famous yet; their biographies are not written, general public does not know them yet:-) So many polymath's among them are simply not so well known to the general public. But I suspect that the percentage is the same.

In the "old days" (probably up to the eighteenth century), when the base of knowledge was narrow, the search for greater knowledge largely involved discovering hidden commonalities and "synergies" between mathematical ideas (e.g., the laws governing gravitation and electric fields are similar; imaginary numbers govern trigonometric calculations through DeMoivre's Theorem, etc.) In that kind of a world, "getting the big picture" or being an "expert" meant knowing a little bit about a lot of different math fields (and connecting them together).

Nowadays, the "lowing hanging fruit" has been picked, the basic knowledge (mostly) discovered, and the exploration into further connections goes "deeper" in "narrower" fields. Barring someone who is exceptional at 19th century-style "horizontal" thinking, the trend is for greater specialization, and fewer "poly"maths, or interdisciplinary people.

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