When someone linked me to Ravi Vakil’s advice for potential graduate students, I was struck by the following passage:
…[M]athematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you’ll never get anywhere. Instead, you’ll have tendrils of knowledge extending far from your comfort zone [emphasis mine]. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning “forwards”. (Caution: this backfilling is necessary. There can be a temptation to learn lots of fancy words and to use them in fancy sentences without being able to say precisely what you mean. You should feel free to do that, but you should always feel a pang of guilt when you do.)
It’s great to hear this coming from an expert because this is exactly what I’ve been doing for the past year without realizing it. Without formally learning anything, I’ve begun extending tendrils into algebraic topology, category theory, and all sorts of subjects about which I still can’t say anything particularly intelligent. However, from my experience so far I have a tentative list of the benefits of this strategy:
- It becomes easier to recognize related concepts or constructions across different subjects, hence to tie them together.
- If you have a concept you don’t fully understand sitting in the back of your head, it may come to pass that once you learn the necessary tools to understand it you may re-derive the concept partially based on your memory. As Richard Feynman said, “what I cannot create, I do not understand.”
- Certain things become better motivated if you can say to yourself something like, “oh, I know why we’re learning about Theorem X; it’s an instance of Phenomenon Y which has lots of other nontrivial instances.” Here I’ll give an example: Pontryagin duality.
- You are naturally led to ask lots of questions, and questions are great. “This looks a lot like Theory Z,” you might ask your professor. “What’s the connection?”
The idea that constantly working outside your comfort zone is key to progress appears to me to be a general phenomenon; in two-player games and sports, for example, playing opponents who are better than you is a great way to improve.
What I’m curious about, though, is whether the undergraduate math curriculum explicitly encourages “tendril” behavior. Perhaps it’s just something every math major should be motivated to do independently, but I can’t help but think that Ravi’s advice, which I’ve never seen written down anywhere else, should be more widely acknowledged.
Annoying Precision 
There is a book, The Power of Full Engagement, by two psychologists who study the theory of motivation. They have examined thousands of examples of the world’s top performers in a variety of different fields (sports, business, etc.) and found the same pattern each time: the most successful are those who practice extreme bursts of stress (i.e., go beyond their comfort zone) interspersed with extreme periods of relaxation. (The thesis they present in the book is a little more complicated than that, but this is the general gist of it. Reading it has helped me understand mathematics more than any single math book has.)
Many professors will do this–in lectures, they’ll give mention of what’s to come, or of far-reaching applications, or even long digressions if there’s time. I remember my undergraduate honors calc 1 professor even spent a day talking about complex analysis, another talking about Cantor sets, and in subsequent semesters a day of calculus on manifolds. Not on a 100% rigorous level of course, but a taste.
One thing I do often that has had similar results to what you talk about (although perhaps more localized) is to read through various papers that sound interesting based on the title, but to skim rather than go through in detail.
This has the advantage of giving me a general impression of what people can do in a field, and what questions are currently of interest.
As I said, I think this is more localized, in that you have the “tendrils” going out into the current areas of research in a specific field, rather into general fields throughout all of mathematics.
Reading papers is definitely a great habit, and I’m glad I picked it up earlier rather than later. In practice, I only have the patience for papers whose basic arguments I can understand, so I haven’t tried to read through anything technical in a field other than combinatorics…
[…] By Harrison Qiaochu has a post up on “going beyond your comfort zone” in mathematics. Towards the end of it, he lists […]
In my experience, undergraduate math doesn’t really encourage this kind of seeking. But, of course, they try to teach you “what is right.” I feel that the department can certainly improve in this area. For one, having conversations with professors is a great way learn about things that you otherwise wouldn’t hear about, and going to seminars (and the UMA talks) is too. Unfortunately, I don’t think that this is a very high priority here…
I believe the advice is true. I don’t get anywhere in math since I never leave my comfort zone. Thank you for the post :).