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SarahC comments on Vanity and Ambition in Mathematics - Less Wrong Discussion

9 Post author: multifoliaterose 12 October 2010 05:49AM

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Comment author: [deleted] 12 October 2010 12:55:47PM 21 points [-]

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.

Comment author: John_Baez 14 October 2010 05:39:07AM *  10 points [-]

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."

Comment author: minhyongkim 17 October 2010 01:22:41AM 3 points [-]

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.'

Comment deleted 12 October 2010 02:57:41PM [-]
Comment author: [deleted] 12 October 2010 03:26:21PM 0 points [-]

No, that seems to be true. "Understanding" in a thorough sense is pretty darn rare and usually confined to specialized fields of study.