Qiaochu has a post up on “going beyond your comfort zone” in mathematics. Towards the end of it, he lists many of the most commonly given good reasons for learning new areas of mathematics — and all of them are very good reasons. But there’s one that I want to talk about now which I don’t think I’ve seen mentioned before; it’s not so much a pedagogical principle as an autodidactical one.
Mathematical research is largely driven by insights — seeing that trick X can be applied to problem Y, for instance, or formulating conjecture C based on an interesting similarity between areas P and Q. Of course, top-quality work (especially theory-building work) is often focused around “big” insights, like the Langlands conjectures, or Serre and Grothendieck’s radical restructuring of algebraic geometry, or almost everything Riemann ever wrote. But as anyone who’s ever done serious mathematical research knows, those “big” insights are virtually always the culmination of a series of “small” insights that grew progressively larger, backed up by calculation or experimenting with special cases.
When you learn new mathematics, you get to have these “little insights” in spades — because the foundational level of any theory is built on them, and it’s much more fun to try to anticipate them in advance than to sit back and learn the material passively. Examples of these “little insights” might include:
- If you know that a vector space is in a sense “an abelian group over a field,” and you’ve learned the definition of a ring, you might start thinking about the properties of “a vector space over a ring.” Congratulations — you’ve rediscovered module theory.
- The realization that group presentations really just specify quotient groups of the free group — and the subsequent realization that every group is a quotient group of the free group on some number of generators.
- The realization that group representations and their characters can tell us a lot about the structure of the group, including, often, whether it is simple or solvable.
- The realization that powers don’t matter when we’re considering the solution set of a system of polynomials. (This leads directly to the Nullstellensatz.)
- Noticing that every group action is continuous under the discrete topology.
- Various and sundry other “hey, I’ve seen this before!” moments, for instance: seeing that taking the radical of an ideal is in many ways formally similar to taking the topological closure of a set; seeing that tensor products and/or direct sums are “essentially the same thing”; and many of the lower-level tricks in combinatorics that pop up time and again (e.g., counting mod 2).
I believe that every math course above perhaps freshman calculus should be taught in such a way as to maximize the number of these “aha!” moments where these little insights are gained, and something that’s taught in the next chapter or the chapter after that is anticipated, even if very informally. The insight muscle is among the mathematician’s most powerful tool, and we should exercise it early and often.