What are top pure mathematicians experts in? How about top doctors? The easy answers would be “pure math” and “medicine”, but these are a bit too vague to be satisfying (What is pure math? What is medicine?). They also don’t capture all of what these experts excel at.

To know what an expert of a particular type is truly expert at, we need to understand the system that creates that type of expert. What are these experts rewarded for doing or knowing? What actions make them more or less likely to succeed? If in a particular field a behavior X is strongly rewarded with prestige, promotions or wealth (or not doing X is strongly punished) then we would expect many experts in that field to do X. Those that do it naturally or adapt to the system and do it consciously will be more likely to succeed, while those failing to do it will tend to remain obscure or move into other fields. What determines an expert’s actual expertise is not just what her field is ostensibly about, but what knowledge and behaviors she is rewarded or punished for having.

Let’s take the field of pure mathematics as a case study. Experts in this field are created through the University system. Top expertise is determined primarily through the impact and quantity of publications. To produce many publications with high impact, you need traits like an expansive knowledge of theorems and methods, a strong ability to reason logically, the inclination to work for long periods on difficult problems, and sufficient creativity to generate novel ideas. These are the things we typically assume a mathematician has to be good at. But top mathematicians must have another extremely important skill as well: a knack for producing work that other mathematicians will consider important. It is largely other pure mathematicians who determine what gets published, which grants get issued, and what jobs get offered, so it is largely other pure mathematicians that ultimately determine who the top experts in their own field are.

To pure mathematicians the importance of a theorem is generally not determined by whether it has real world applications (in some circles, applications may even be considered a negative). It also is only partially captured by how hard the theorem was to produce. Importance depends on things like whether:

- other pure mathematicians find the theorem aesthetically pleasing (it should be simple yet non-trivial, powerful, and of an appealing form)
- it has implications for or connections to the work of many other pure mathematicians
- it provides new insights related to something that mathematicians care about that was previously not well understood
- other respected mathematicians have tried and failed to produce something similar
- it is part of a mathematical subfield that currently happens to be trendy

Mathematicians who have a non-standard aesthetic (i.e. their definition of what is beautiful or interesting in math differs a lot from what other pure mathematicians think) are at a big disadvantage, as are ones who work in areas that happen to be unsexy at the moment, and ones whose work doesn’t tie nicely into work that many other mathematicians care about. We should expect therefore that top pure mathematicians don’t just have expert mathematical knowledge, but also are expert at playing the pure mathematics game. That means being excellent at producing work that fits certain subjective criteria that are highly valued by many other pure mathematicians.

The pure mathematics game is not a bad game. It is very intellectually stimulating, and it sometimes yields results that end up being extremely useful in practical applications (especially in physics and engineering). But it certainly is not the only game that could be played in mathematics, and its rules are not optimized to make it as beneficial to society as it could be.

The rules of the game are self-perpetuating to a significant extent. Those who play by different rules are less likely to succeed in the field, and those who are good at playing the existing game are more likely to succeed. Those who advance are eventually in a position to decide who else gets to advance, and generally choose people with tastes similar to their own, so the game continues for another round. The details of the game drift over the decades as fashions come and go, and professor die. But certain aesthetic values and notions of what is important can get preserved over long periods.

A similar analysis can be done for nearly any field. What are top doctors expert at? Not just medical knowledge and procedures, but also at satisfying those who reward them. One group that rewards doctors (with their money) is patients. If a doctor is really good it is obviously true that a patient’s condition is more likely to improve, and therefore the patient is more likely to come back next time they have a problem. But it can be very hard to tell whether your improvement was due to your doctor’s skill, or whether you would have gotten better just as fast if you had gone to an average doctor, or even to no doctor at all. Many conditions get better regardless of what treatment is given (and some get worse even when treated properly), so it may be very difficult for patients to accurately assess the true skill of a doctor.

Doctors don’t get rewarded just for making you well fast. They also get rewarded for doing things with you that you believe a good doctor would do, since those behaviors make you view them as a good doctor (and so you’ll be more likely to go back in the future and tell your friends about them). These “good doctor” behaviors might include spending extra time with you, using obscure but impressive sounding medical language, speaking with authority even when they aren’t very sure of themselves, running lots of tests that aren’t really necessary, or using expensive and complex looking machines. There are plenty of ways a doctor could give you the impression of having done a great job that have little or nothing to do with getting you well faster or at low-cost. A doctor who gets you well fast but doesn’t SEEM like a good doctor could in some cases keep fewer patients than a doctor who seems to beĀ better than he truly is. So we should expect to find that top doctors (of the type that rely on patients for almost all of their income) are not just medical experts, but are usually very good at meeting people’s expectations about how a good doctor should behave.

Some exercises for the reader:

- What are art critics truly expert at?
- What are lawyers truly expert at?
- What are contractors truly expert at?

When you consider what experts in a field are truly good at, don’t just ask “what is this field about?” but also, “what traits or behaviors get rewarded or punished in this field?” Behaviors that get rewarded or punished are not necessarily the ones we typically associate with the field, but considering them gives us information about how the experts truly behave.

Influences: Robin Hanson

Great post, Spencer. I think an important question beyond initial discussion is whether the game-playing reaches a level where it is detrimental to the advancement of the field as a whole. Or maybe not detrimental, but at least retarding. Perhaps in the grand scheme of things, these innovations will happen anyway and the window of skepticism is a positive outcome because it ensures that ideas are well-vetted before being propagated.

On a personal level, I found that one of the more frustrating aspects of academia (particularly in the hard sciences) was the need to accept many givens without being able to materially question assumptions in any sort of meaningful way. Not only that, but I often felt chastised for questioning assumptions.

I can certainly understand this line of thinking as a musician. It can be frustrating to talk about music with a non-musician because their questioning of the assumptions I took to be truth (or at least proved theorems) would often seem unfounded or at least ill-informed. But occasionally, either because of an extraordinary “outsider” perspective or because of my temporary willingness to try and hear out a layman, a rather beautiful interaction would occur and both of our minds would be opened a little bit more.