Beware the Casual Polymath

We live in times of great disaggregation, and yet, seem to learn increasingly from generalists.

In the past, an expert in one field of Psychology might have been forced to teach a broad survey class. Today, you could have each lecture delivered by the world’s leading expert.

Outside of academia, you might follow one writer’s account to learn about SaaS pricing, another to understand the intricacies of the electoral college, and yet another to understand personal finance. In economic terms, content disaggregation enabled by digital platforms ought to create efficiencies through intellectual hyper-specialization.

Instead, we have the endless hellscape of the casual polymath. A newsletter about venture capital will find time to opine on herd immunity. The tech blog you visit to learn about data science is also your source of financial strategies for early retirement. The Twitter account you followed to understand politics now seems more focused on their mindfulness practice. We have maxed out variety of interests within people, at the cost of diversity across them.

It’s not difficult to imagine how this happened. The flip side of disaggregation is that each would-be expert is able to read broadly as well. The world of atomized content through hyper-specialization isn’t a stable equilibrium. We are all casual polymaths now.

As romantic as the idea seems, I worry it’s grossly suboptimal. Sure, there are cases where combining ideas from disparate fields can lead to new insight, but today’s generalists are not curating a portfolio of skills so much as they are stumbling about. Behavioral Econ is the love child of economics and psychology, early AI researchers maintained a serious interest in cognitive science. What exactly are your cursory interests in space exploration, meta-science and bayesian statistics preparing you for?

I understand that we can sometimes only “connect the dots looking backwards”. Perhaps there is valuable work in a statistical meta-analysis of aerospace research. But that’s only true if you’re going into some degree of depth in each field.

None of this is meant to dissuade you from becoming a polymath, just to be a little more distrustful of others who claim to be.

1. Polymaths use status in one field to gain capital in another

In 1947, with financial support from his father, JFK won a seat in the US House of Representatives with no political experience. As his father would later remark: “With the money I spent, I could have elected my chauffeur.”

If you saw JFK in 1947, you might have thought “wow, he’s rich, his father was the Chairman of the SEC, and he’s a member of the US House of Representatives, what an impressive guy!” A decade later, you could have added “Pulitzer Prize winning author“ to that list.

But this reasoning is totally backwards. JFK was only able to become a politician because of his wealth. In fact, his father only became SEC Chairman after extensive political donations to FDR. And obviously, his book was ghost-written by his speechwriter.

So you’re justified in being impressed by exactly one accomplishment, and everything else ought to be discounted.

We already understand this intuitively, but only in a limited set of cases. If a pop star becomes an actor, we are not impressed by their wide range of talents. Instead, we understand that popularity is a semi-fungible good.

2. Polymaths are abusing Gell-Mann Amnesia

As I wrote earlier:
You go on Twitter, you read someone’s tweet on a subject you know something about, and see that the author has no understanding of the facts. So you keep scrolling and read their tweets about cancel culture, space exploration and criminal justice reform, totally forgetting how wrong they were before.

In this sense, every tweet is an option with asymmetric returns. If you’re right, you cash out; if you’re wrong, everyone forgets and you lose nothing. The incentive is to ramp up variance, make bold claims in a variety of areas, and hope you’re right some of the time.

Accordingly, authors will make bold claims in a variety of areas, and you may be inclined to believe them even after seeing how wrong they are. Unless you are also a polymath, and a polymath in the same domains, you will probably not be capable of evaluating their competence.

Of course, you might rely on external opinions, which brings us to the last point.

3. Polymaths are evaluated by non-polymaths

Leonardo da Vinci is the most famous polymath of all time and the model omni-competent Renaissance Man.

Leonardo da Vinci also didn’t know math. Issacson’s book details numerous episodes in which:

  • Leonardo comes up with a million dollar business idea, later realizes his basic arithmetic was off by more than an order of magnitude.
  • Leonardo claims to be a military engineer to gain acceptance at the Milanese court. In fact, he has never built any kind of weapon or siege device.
  • Leonardo claims to have solved the ancient puzzle of doubling the cube. Except his “solution” only works if you can’t tell the difference between the square root of 3 and cube root of 2.

This last example is especially notable because Issacson himself doesn’t seem to catch it, instead uncritically praising Leonardo’s discovery. [Details in Appendix]

And yet Wikipedia writes:
…many historians and scholars regard Leonardo as the prime exemplar of the “Renaissance Man” or “Universal Genius”, an individual of “unquenchable curiosity” and “feverishly inventive imagination.”[6] He is widely considered one of the most diversely talented individuals ever to have lived.[10] According to art historian Helen Gardner, the scope and depth of his interests were without precedent in recorded history

Of course the inflation of his mathematical and engineering ability makes sense when you consider the that judges in question are predominantly art historians. Rather than as a Renaissance Man, Leonardo would be better regarded as an exceptional painter with various hobbies.

While an expert in one domain may just be a savant with a “kooky knack”, mastering multiple unrelated skills feels like evidence of general intelligence, or in Leonardo’s case, “Universal Genius”. If someone is good at computer science, epidemiology and finance, surely we can trust their opinion on politics as well?

Except what’s really happening is that we’ve chosen to privilege certain combinations of skills as impressive, while taking others for granted.

A physicist who studies math, can write code for analysis and understand complex systems is not hailed as a polymath. They’re just seen as obtaining the basic set of skills required for their profession. Similarly, a basketball player who can run, shoot and block is not any kind of “polymath”.

You might object that this is because physics and basketball are specific clusters of skills. Running quickly is more closely related to throwing a ball than software engineering is to epidemiology.

This might be true in specific cases, but in general, it’s a coincidence of which skills cluster into occupations. A small business owner who manages their own books, handles sales and manufactures their product is not considered a polymath, no matter how distinct those fields might be. Computational social scientists are not considered polymaths, neither is an OnlyFans creator who single handedly runs everything from marketing to modeling, nor a translator who has to master ancient greek, dive deeply into historical context, and also be a great poet in their own right.

To be clear, there are still good reasons to learn a variety of skills. As Marc Andreessen put it:
All successful CEO’s [sic] are like this. They are almost never the best product visionaries, or the best salespeople, or the best marketing people, or the best finance people, or even the best managers, but they are top 25% in some set of those skills, and then all of a sudden they’re qualified to actually run something important.

He goes on to provide examples, listing Communication, Management, Sales, Finance and International Experience.

I’m broadly in agreement. Learning these skills will probably benefit your career. Just understand that no one has ever been hailed as a polymath because they’re good at both communication and management. They’re just considered basically competent at their job.

I don’t want to dissuade anyone from learning broadly and reading widely. Of course athletes should cross train, and intellectuals should read outside their domain, and software engineers might benefit from public speaking classes.

My point is that we should not trust or glorify people on the basis of their apparent “Universal Genius”. Having a variety of interests is no more a sign of generalized intelligence than being able to walk and chew gum. And if someone does appear to have accomplishments in a variety of domains with fungible currency, their total status should not be a sum or multiple, but merely the status of their single most impressive feat.

So go read your SaaS/Meta-Science/Aerospace blog and revel in the genuine joy of intellectual curiosity. As Tyler Cowen would say, I’m just here to lower the status of polymaths.

Appendix: Doubling the Cube

Here’s the full quote from Issacson:
These obsessions led Leonardo to an ancient riddle described by Vitruvius, Euripides, and others. Faced with a plague in the fifth century BC, the citizens of Delos consulted the oracle of Delphi. They were told that the plague would end if they found a mathematical way to precisely double the size of the altar to Apollo, which was shaped as a cube. When they doubled the length of each side, the plague worsened; the Oracle explained that by doing so they had increased the size of the cube eightfold rather than doubling it. (For example, a cube with two-foot sizes has eight times the volume of a cube with one-foot sides.) To solve the problem geometrically required multiplying the length of each side by the cube root of 2.

Despite his note to himself to “learn the multiplication of roots from Maestro Luca,” Leonardo was never good at square roots, much less cube roots. Even if he had been, however, neither he nor the plague-stricken Greeks had the tools to solve the problem with numerical calculations, because the cube root of 2 is an irrational number. But Leonardo was able to come up with a visual solution. The answer can be found by drawing a cube that is constructed on a plane that cuts diagonally through the original cube, just as a square can be doubled in size by constructing a new square on a line cutting it in half diagonally, thus squaring the hypotenuse.

To be clear, this was not an acceptable failing indicative of his time. From Wikipedia, cube roots date back to 1800 BCE, a method for calulating cube roots was given in the 1st century BCE.

The first part is right. The hypotenuse of a unit square has length SquareRoot(2), and the square constructed with side length SquareRoot(2) has area 2.

But this method doesn’t work with cubes. To get a cube with volume 2, each side needs to have length CubeRoot(2). But the hypotenuse of one side of the unit cube is still SquareRoot(2), and the diagonal is SquareRoot(3).

From Wikipedia:
doubling the cube is now known to be impossible using only a compass and straightedge

Issacson didn’t explicitly say that Leonardo was limiting himself to compass/straightedge constructions. Wikipedia does list several Solutions via means other than compass and straightedge that had already been discovered in ancient Greece. None of these solutions fit Issacson’s description of Leonardo’s method.

Searching for “Leonardo da Vinci doubling the cube” or “Leonardo da Vinci Delian Problem” turns up a few results:

A pamphlet for an exhibition at the Louvre:
Various Attempts at Doubling the Cube. The Extension of the Pythagorean Theorem to the Power of 3. Attempt at the Geometrical Construction of Square Roots from 1 to 9 Pen and brown ink About 1505 Here again, Leonardo endeavours to double the cube by various means – including applying the Pythagorean theorem to volumes rather than surface areas. Biblioteca Ambrosiana, Milan, Codex Atlanticus, fol. 428R

Which again, is not a solution since the Pythagorean theorem, even extended to n-dimensions, only results in square roots.

A blog matches up the Louvre exhibit with scans of Leonardo’s notebook:
Highlights a different section from the Louvre exhibit, called an “Doubling the Cube; An Empirical Solution”, which is just an approximation using “edge length very slightly greater than 5”.

An article titled Leonardo and Theoretical Mathematics:
Discussion of various attempts by Leonardo to double the cube, which writes:
If the diagonal (or diameter) of a square with a side of 1 is the graphic visualisation of
the incommensurable quantity of the square root of two, is the diagonal of a cube with a
side of 1 the graphic answer to the irrational number equal to the cube root of 2?
The answers are no. The diagonal of the cube is equal to the square root of 3, and not
the cube root of 2, which is a smaller number than the square root of 2.

Then goes on to discuss several other attempts that also proved fruitless:
In addition to looking for his own solutions to the duplication of the cube, Leonardo also studied the classical solutions of the ancient Greek mathematicians

Described in the same Wikipedia article, there is an elegant geometric solution, so long as you are able to mark the straightedge.

This is all to say, I’m moderately confident Issacson was not describing a valid solution.

Funny enough, when Plato originally proposed the problem, he was similarly frustrated. Described in Plutarch’s Quaestiones Convivales from Moralia:
And therefore Plato himself dislikes Eudoxus, Archytas, and Menaechmus for endeavoring to bring down the doubling the cube to mechanical operations; for by this means all that was good in geometry would be lost and corrupted, it falling back again to sensible things, and not rising upward and considering immaterial and immortal images, in which God being versed is always God.

[EDIT 10/15/2020]

Discussion of this post on Marginal Revolution.

Discussion of this post on Hacker News, mostly people accusing me of being a bot.