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.
It's a bit easier in math than other subjects to know when you're right and when you're not. That makes it a bit easier to know when you understand something and when you don't. And then it quickly becomes clear that pretending to understand something is counterproductive. It's much better to know and admit exactly how much you understand.
And the best mathematicians can be real masters of "not understanding". Even when they've reached the shallow or rote level of understanding that most of us consider "understanding", they are dissatisfied and say they don't understand - because they know the feeling of deep understanding, and they aren't content until they get that.
Gelfand was a great Russian mathematician who ran a seminar in Moscow for many years. Here's a little quote from Simon Gindikin about Gelfand's seminar, and Gelfand's gift for "not understanding":
One cannot avoid mentioning that the general attitude to the seminar was far from unanimous. Criticism mainly concerned its style, which was rather unusual for a scientific seminar. It was a kind of a theater with a unique stage director playing the leading role in the performance and organizing the supporting cast, most of whom had the highest qualifications. I use this metaphor with the utmost seriousness, without any intention to mean that the seminar was some sort of a spectacle. Gelfand had chosen the hardest and most dangerous genre: to demonstrate in public how he understood mathematics. It was an open lesson in the grasping of mathematics by one of the most amazing mathematicians of our time. This role could be only be played under the most favorable conditions: the genre dictates the rules of the game, which are not always very convenient for the listeners. This means, for example, that the leader follows only his own intuition in the final choice of the topics of the talks, interrupts them with comments and questions (a privilege not granted to other participants) [....] All this is done with extraordinary generosity, a true passion for mathematics.
Let me recall some of the stage director's strategems. An important feature were improvisations of various kinds. The course of the seminar could change dramatically at any moment. Another important mise en scene involved the "trial listener" game, in which one of the participants (this could be a student as well as a professor) was instructed to keep informing the seminar of his understanding of the talk, and whenever that information was negative, that part of the report would be repeated. A well-qualified trial listener could usually feel when the head of the seminar wanted an occasion for such a repetition. Also, Gelfand himself had the faculty of being "unable to understand" in situations when everyone around was sure that everything is clear. What extraordinary vistas were opened to the listeners, and sometimes even to the mathematician giving the talk, by this ability not to understand. Gelfand liked that old story of the professor complaining about his students: "Fantastically stupid students - five times I repeat proof, already I understand it myself, and still they don't get it."
Although I agree on the whole, it might be worth recalling that 'I don't understand' can be agressive criticism in addition to being humility or a skill. Among many examples of this aspect, I rather like the passage on Kant in Russell's history of western philosophy, where he writes something like: 'I confess to never having understood what is meant by categories.'
My working assumption is that most people don't do this because they understand very little about anything, and don't know that there is a difference between "understanding" something and just reading something, or listening to what someone tells them.
Is that too pessimistic?
No, that seems to be true. "Understanding" in a thorough sense is pretty darn rare and usually confined to specialized fields of study.
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.
The author of this post pointed out that he said "t's noticeably less common for mathematicians of the highest caliber to engage in status games than members of the general population do." Somehow I hadn't noticed that.
I'm not sure how this affects my reaction, but I wouldn't have written quite what I wrote if I'd noticed that qualifier.
My experiences, as a kind of outsider who is just curious about some themes in math too and asks around for infos, explanations and preprints/slides, is that mathematicians are by far the easiest science community to communicate with. I conclude that status is of little relevance.
Mathematicians are like everyone else, a human, susceptible to the common people tendencies, unless there is 'something' in mathematical thinking that would put them into a different category, more human (the definition of to be a human even if in the strict sense of mathematical rigor would fail, wouldn't it?) or less human ( (a thinking machine).
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 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.
Excellent point.
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.
Thanks for your comment.
Also, this may not be true just for math but for other areas as well.
I completely agree. I wrote about math because it's what I know best, not to suggest that the phenomenon that I allude to is true for math more than for other fields. If I incorporate this discussion board posting into a top level posting, I'll mention this.
I addressed the rest of your comment in my revised post. (I accidentally posted to the discussion area prematurely before completing the post that I was working on - still getting used to the discussion area.)
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.)
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.
Thanks for your comment.
Also, this may not be true just for math but for other areas as well.
I completely agree. I wrote about math because it's what I know best, not to suggest that the phenomenon that I allude to is true for math more than for other fields. If I incorporate this discussion board posting into a top level posting, I'll mention this.
I addressed the rest of your comment in my revised post. (I accidentally posted to the discussion area prematurely before completing the post that I was working on - still getting used to the discussion area.)
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. This impression is consistent with the modesty that comes across in the writings of such mathematicians. I record some relevant quotations below and then discuss interpretations of the situation.
Acknowledgment - I learned of the Hironaka interview quoted below from my colleague Laurens Gunnarsen.
Edited 10/12/10 to remove the first portion of the Hironaka quote which didn't capture the phenomenon that I'm trying to get at here.
In a 2005 Interview for the Notices of the AMS, one of the reasons that Fields Medalist Heisuke Hironaka says
By the way, Mori is a genius. I am not. So that is a big difference! Mori was a student when I was a visiting professor at Kyoto University. I gave lectures in Kyoto, and Mori wrote notes, which were published in a book. He was really amazing. My lectures were terrible, but when I looked at his notes, it was all there! Mori is a discoverer. He finds new things that people never imagined.
(I'll note in passing that the sense of the "genius" that Hironaka is using here is probably different than the sense of "genius" that Gowers uses in Mathematics: A Very Short Introduction.)
In his review of Haruzo Hida’s p-adic automorphic forms on Shimura varieties the originator of the Langlands program Robert Langlands wrote
So ill-equipped as I am in many ways – although not in all – my first, indeed my major task was to take bearings. The second is, bearings taken, doubtful or not, to communicate them at least to an experienced reader and, in so far as this is possible, even to an inexperienced one. For lack of time and competence I accomplished neither task satisfactorily. So, although I have made a real effort, this review is not the brief, limpid yet comprehensive, account of the subject, revealing its manifold possibilities, that I would have liked to write and that it deserves. The review is imbalanced and there is too much that I had to leave obscure, too many possibly premature intimations. A reviewer with greater competence, who saw the domain whole and, in addition, had a command of the detail would have done much better.
For context, it's worthwhile to note that Langlands' own work is used in an essential way in Hida's book.
The 2009 Abel Prize Interview with Mikhail Gromov contains the following questions and answers:
Raussen and Skau: Can you remember when and how you became aware of your exceptional mathematical talent?
Gromov: I do not think I am exceptional. Accidentally, things happened, and I have qualities that you can appreciate. I guess I never thought in those terms.
[...]
Raussen and Skau: Is there one particular theorem or result you are the most proud of?
Gromov: Yes. It is my introduction of pseudoholomorphic curves, unquestionably. Everything else was just understanding what was already known and to make it look like a new kind of discovery.
In his MathOverflow self-summary, William Thurston wrote
Mathematics is a process of staring hard enough with enough perseverance at at the fog of muddle and confusion to eventually break through to improved clarity. I'm happy when I can admit, at least to myself, that my thinking is muddled, and I try to overcome the embarrassment that I might reveal ignorance or confusion. Over the years, this has helped me develop clarity in some things, but I remain muddled in many others. I enjoy questions that seem honest, even when they admit or reveal confusion, in preference to questions that appear designed to project sophistication.
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. More to the point, there's a compelling alternate explanation for why one would see lower than usual levels of status signaling among mathematicians of the highest caliber. Gromov hints at this in the aforementioned interview:
Raussen and Skau: We are surprised that you are so modest by playing down your own achievements. Maybe your ideas are naíve, as you yourself say; but to get results from these ideas, that requires some ingenuity, doesn’t it?
Gromov: It is not that I am terribly modest. I don’t think I am a complete idiot. Typically when you do mathematics you don’t think about yourself. A friend of mine was complaining that anytime he had a good idea he became so excited about how smart he was that he could not work afterwards. So naturally, I try not to think about it.
In Récoltes et Semailles, Alexander Grothendieck offered a more detailed explanation:
The truth of the matter is that it is universally the case that, in the real motives of the scientist, of which he himself is often unaware in his work, vanity and ambition will play as large a role as they do in all other professions. The forms that these assume can be in turn subtle or grotesque, depending on the individual. Nor do I exempt myself. Anyone who reads this testimonial will have to agree with me.
It is also the case that the most totally consuming ambition is powerless to make or to demonstrate the simplest mathematical discovery - even as it is powerless (for example) to "score" (in the vulgar sense). Whether one is male or female, that which allows one to 'score' is not ambition, the desire to shine, to exhibit one's prowess, sexual in this case. Quite the contrary!
What brings success in this case is the acute perception of the presence of something strong, very real and at the same time very delicate. Perhaps one can call it "beauty", in its thousand-fold aspects. That someone is ambitious doesn't mean that one cannot also feel the presence of beauty in them; but it is not the attribute of ambition which evokes this feeling....
The first man to discover and master fire was just like you and me. He was neither a hero nor a demi-god. Once again like you and me he had experienced the sting of anguish, and applied the poultice of vanity to anaesthetize that sting. But, at the moment at which he first "knew" fire he had neither fear nor vanity. That is the truth at the heart of all heroic myth. The myth itself becomes insipid, nothing but a drug, when it is used to conceal the true nature of things.
[...]
In our acquisition of knowledge of the Universe (whether mathematical or otherwise) that which renovates the quest is nothing more nor less than complete innocence. It is in this state of complete innocence that we receive everything from the moment of our birth. Although so often the object of our contempt and of our private fears, it is always in us. It alone can unite humility with boldness so as to allow us to penetrate to the heart of things, or allow things to enter us and taken possession of us.
This unique power is in no way a privilege given to "exceptional talents" - persons of incredible brain power (for example), who are better able to manipulate, with dexterity and ease, an enormous mass of data, ideas and specialized skills. Such gifts are undeniably valuable, and certainly worthy of envy from those who (like myself) were not so endowed at birth," far beyond the ordinary".
Yet it is not these gifts, nor the most determined ambition combined with irresistible will-power, that enables one to surmount the "invisible yet formidable boundaries" that encircle our universe. Only innocence can surmount them, which mere knowledge doesn't even take into account, in those moments when we find ourselves able to listen to things, totally and intensely absorbed in child play.
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. If this is true (as I strongly suspect to be the case based on my own experience, my observations of others, the remarks of colleagues, and the remarks of eminent figures like Gromov and Grothendieck), aspiring mathematicians would do well to work to curb their ambition and vanity and increase their attraction to mathematics for its own sake.