Vanity and Ambition in Mathematics

The most impressive quality I've seen in mathematicians (including students) is the capacity to call themselves "confused" until they actually understand completely.

Most of us, myself included, are tempted to say we "understand" as soon as we possibly can, to avoid being shamed. People who successfully learn mathematics admit they are "confused" until they understand what's in the textbook. People who successfully create mathematics have such a finely tuned sense of "confusion" that it may not be until they have created new foundations and concepts that they feel they understand.

Even among mathematicians who project more of a CEO-type, confident persona, it seems that the professors say "I don't understand" more than the students.

It isn't humility, exactly, it's a skill. The ability to continue feeling that something is unclear long after everyone else has decided that everything is wrapped up. You don't have to have a low opinion of your own abilities to have this skill. You just have to have a tolerance for doubt much higher than that of most humans, who like to decide "yes" or "no" as quickly as possible, and simply don't care that much whether they're wrong or right.

I know this, because it's a weakness of mine. I'm probably more tolerant of doubt and sensitive to confusion than the average person, but I am not as good at being confused as a good mathematician.

In my 25 years of being a professional mathematician I've found many (though certainly not all) mathematicians to be acutely aware of status, particularly those who work at high-status institutions. If you are a research mathematician your job is to be smart. To get a good job, you need to convince other people that you are smart. So, there is quite a well-developed "pecking order" in mathematics.

I believe the appearance of "humility" in the quotes here arises not from lack of concern with status, but rather various other factors:

1) Most of us know that there are mathematicians much better than us: mathematicians who could, with their little pinkie on a lazy Sunday afternoon, accomplish deeds that we might struggle vainly for years to achieve.

2) Many of us realize that it's wiser to emphasize our shortcomings than boast of our accomplishments.

By the way: people quoted in this article are all extremely high in status, and indeed it's mostly such mathematicians who wind up talking about themselves publicly, answering questions like "Can you remember when and how you became aware of your exceptional mathematical talent?" Every mathematician worth his or her salt knows of Hironaka, Langlands, Gromov, Thurston and Grothendieck. So these are not typical mathematicians: they are our heroes, our gods.

It is nice having humble gods. But still, they're not stupid: they know they're our gods.

I think it's quite normal that if someone is acknowledged by their peers to be among the very best at what they do, they won't waste much time with status games.

There's an exception if doing what they do requires publicity to bring in sales or votes.

I was reading Ecology, the Ascendent Perspective by Robert Ulanowicz when I came across the following interesting passage which made me think of this thread:

In January 1684 Sir Christopher Wren and Edmund Halley met with Hooke in Oxford during a session of the Royal Society. Wren and Halley were both interested in establishing a rigorous connection between the inverse-square law of attraction and the elliptical shape of planetary orbits. When they inquired of Hooke whether such a connection was possible, Hooke told them he had already completed the demonstration but that he intended to keep the proof secret until others, by failing to solve the problem, had learned to value it (Westfall 1983).

Wren and Halley evidently were dissatisfied with Hooke's coyness, for in August of the same year, when Halley was in Cambridge, he sought out Newton in order to pose the problem to Hooke's rival. Newton told Halley that he too had already solved the problem but had mislaid the proof...Wren decided to call the bluff of the two enemies by publicly offering an antique book worth forty shillings as a prize to the individual who could provide him a proof within two months.

Newton was distraught when Halley told him of Hooke's claim to a proof. Apparently, Newton did not lie when he claimed that he had already proved the connection, for a copy of a proof that antedates Halley's visit has indeed been found among Newton's papers. Being cautious in the company of someone who communicated with Hooke, Newton probably feigned having misplaced the proof. We can only imagine his horror, then, when he looked up his demonstration only to discover that it was deeply flawed. The thought of leaving the field open to Hooke drove Newton to near-panic. He abandoned all other ongoing projects to rush into seclusion and attempt a rigorous exposition. Once thus engaged, the positive rewards of the creative process seem to have drawn him ever deeper into the project. He virtually disappeared from society until the spring of 1686, at which time he emerged, on the brink of mental and physical exhaustion, with three completed volumes of the Principia in hand.

I'm worried about the lack of a citation for the last paragraph, but if this is accurate then it is very interesting.

In my time in the mathematical community I've formed the subjective impression that it's noticeably less common for mathematicians of the highest caliber to engage in status games than members of the general population do.

Higher status will not in itself help you solve hard mathematical problems. You need economic security, good conditions in which to think, and so forth, and society can help or hinder you there. But when you face a problem that no-one else has ever solved, it's you versus the universe. There's no big brother around who will tell you the answer if only you can win his favor. So the psychology of how to make progress in such a situation is existential rather than social.

Hard to distinguish here between lack of status games and heavy countersignaling. Also, this may not be true just for math but for other areas as well.

If this is true for math more than other areas then the most probable explanation is that math is a low status but highly intellectual area to go in for many notions of status (especially in highschool and some parts of college). So you are more likely to get people who don't care about status as mathematicians.

The amount of focus on the subject itself which is required to do mathematical research of the highest caliber is very high. It's plausible that the focuses entailed by vanity and ambition are detrimental to subject matter focus.

This rings true in my experience as well. I've found that thinking about how other people will perceive what you're doing tends to derail mathematical thought. It would be interesting to look at other fields where excellence requires extreme focus and see if they also exhibit the same phenomenon.

aspiring mathematicians would do well to work to curb their ambition and vanity and increase their attraction to mathematics for its own sake.

I think this is good advice. Combining it with Grothendieck's idea about the importance of complete innocence gives you

Aspiring mathematicians would do well to work to curb their ambition and vanity, and increase their playful innocent attraction to mathematics for its own sake.

I haven't kept careful stock of my impressions on this matter, but while I don't recall noticing any high caliber mathematicians positively signaling status, I don't recall noticing high caliber scientists in other fields doing it either. I'm skeptical that this is a principle that applies particularly to mathematicians, rather than to highly esteemed researchers in general.

I think countersignaling is a pretty strong explanation here. When everyone already knows that you're a thinker of great prestige, you stand to gain more status by signaling that you're also humble about it than reminding people of how exceptional you are.

I would predict, and it's my impression that this is the case, that you would be more likely to find high profile figures positively signaling their status in factioned fields. When you're not the discoverer of some universally accepted principle or data, but merely the champion of some particular interpretation or school of thought, then you stand to gain by signaling how smart and right you are.

I interpret the above quotations (and many others by similar such people) to point to a markedly lower than usual interest in status. As JoshuaZ points out, one could instead read the quotations as counter-signaling, but such an interpretation feels like a stretch to me. I doubt that in practice such remarks serve as an effective counter-signal.

There is however a third alternative: noblesse oblige (fake) humility. Part of the standard role of a high-status person is to "show kindness toward their loyal subjects" (this is the "Gandalf" quality that Eliezer disdains). This differs from countersignaling in that it doesn't involve mimicking low-status behavior. (An appropriate analogue might be the very richest people starting philanthropic foundations -- this is different from both conspicuous consumption and dressing like poor people.)