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.”
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”: