My friend Julian Chadez mentioned to me (from a conversation with our friend Semon Rezchikov) the following two things that are vital to progress and quite taxing to constantly keep in mind.
- Never be ashamed of not knowing something.
- Never shame others for not knowing something.
I read Bill Thurston’s On Proof and Progress this morning. I figured I’d share a few things I’ve learned this summer about the sociology and psychology of being a part of the mathematical community, on the off chance that you might find it encouraging or helpful.
0. Communicating in a pedagogically correct manner
I have spent a large part of the summer learning how to speak to other mathematicians, and the standards generally enforced, and I have much more to learn. It is somewhat complicated to keep in mind what is commonly thought of in various subfields as “easy to understand” or “basic,” and what is “complex” or “impossible to grasp.” You musn’t allow it to affect what you view as natural. Just recognize that what you see as simple is not necessarily so in the eyes of others, and vice versa.
Keep track of what causes people to shut off, what causes people to feel like they are being talked down to; probe to find out what their mental models are and what they think of as important. Teaching in a way that feels collaborative involves a large amount of empathy and a change of language, for example:
No: I was explaining this, and your question confused me.
- Yes: We were doing this, and I got confused.
1. Articulating vague thoughts
It is very important to realize that there are many ways to make precise a vague question (e.g., an undeveloped feeling of connection between two things usually not thought of as connected, an uncertainty about a concept which you are not able to pinpoint).
Spending a large amount of time alone to develop things in your own mental sandbox until they are ready to be translated into words is healthy and natural. But do translate them into words, even if they are not fully formed, s.t. you don’t end up with a theory so removed and technical from the outside that it remains unabsorbed.
Trying to communicate the way you think about an idea in its purest form is not always useful or interesting to the other party, but sometimes it is.
2. Giving a lecture
Giving a good talk and controlling the room are, unfortunately, entirely different skills.
It seems that in a lecture setting, one must abandon hope of communicating formal information, and instead try to communicate key insights and mental models — usually just one or two — and back up philosophical statements with numerous simple examples!
(I gave a talk where each sentence I said aimed to convey things that had originally taken me months to even begin to appreciate. I also let a few people interrupt the flow of the talk in order to heckle wrt small vocabulary differences and technical details. Someone stood up in the middle of my talk and talked for 10 minutes. This is not the way to go.)
3. Building your own mental models
You are going to reinvent things, lots of things. This is good: it is important to practice making original discoveries! If you hold the belief that understanding = you could have invented it yourself, then reinvention is especially encouraging!
Most of my current mental models come from sitting alone and drawing in my notebook what concepts mean to me, the questions they lead to and flowed from, what images they invoke…
It is an intimate act, personally understanding a concept. For me, it involves a lot of doodling and staring out into space.
It also involves a lot of chunking (e.g., a CW-space is an indexed array with attaching maps; a functor is a generalized manifold; approximation of continuous processes via power series; reducing complexity by looking for the base objects and laws which generate the objects you care about).
I started reading histories of mathematics and found that some of the connections and motivations I’d come to myself, and many I hadn’t seen, were the historical reasons for their invention (H-spaces are more general versions of Lie groups, homotopy theory came from complex analysis so what comes from the calculus of variations, etc).
For this reason, there is incredible joy in reading original papers, or in historical and careful recounts of such papers (e.g., Dirichlet’s lectures on Gauss’s Disquisitiones Arithmeticae), as the life of concepts often seems to be lost through a game of paper telephone (the citations of old papers usually don’t convey the interests of the old author).
4. Vocab hunting
This is an incredibly fun and superficially rewarding game: coming across a technical term (e.g., pre-mouse) in a language in which you are not conversant (e.g., model theory), and then “chasing” (via wikipedia links and journal articles) the concept until you find/reformulate the definition into a language you speak. This is best done when you need a pick me up or can’t sleep.