What tensor products taught me about living my life
When I was a junior in college, I went to the Budapest Semesters in Math. I got really bummed while I was there, and I was thinking of leaving math, when a friend of mine back home sent me Silverman and Tate’s book on elliptic curves. That book restored my faith in math and I decided to become a number theorist. I went back to Berkeley and enrolled in Hendrik Lenstra’s Class Field Theory class, which was the second semester of a grad number theory class, and in Ken Ribet’s second semester grad algebra class. Since I’d missed the first semester of each, I pretty much got my ass kicked. I lived and breathed algebra and p-adics and local-glocal principles for the next three months. It was pretty awesome and incredibly challenging. The moment of my biggest frustration happened when we learned about tensor products over arbitrary rings with zero divisors.
I kept trying to understand these rings, and in particular the elements of these rings. I wasn’t asking much: I just wanted to figure out the most basic properties of tensor products. And it seemed like a moral issue. I felt strongly that if I really really wanted to feel like I understand this ring, which is after all a set, then at least I should be able to tell you, with moral authority, whether an element is zero or not. For fuck’s sake!
I couldn’t do it. In fact all the proofs I came up with involved the universal property of tensor products, never the elements themselves. It was incredibly unsatisfying, it was like I could only describe the outside of an alien world instead of getting to know its inhabitants.
After a few months, though, I realized something. I hadn’t gotten any better at understanding tensor products, but I was getting used to not understanding them. It was pretty amazing. I no longer felt anguished when tensor products came up; I was instead almost amused by their cunning ways.
Every now and then something like that happens in my life. Something that I start out desperately wanting to understand, to analyze, and to own. It’s practically a moral imperative! And I consider myself a person who gets stuff done! How can I let this lie unexplained?
Then after a few days it turns out, no, I still don’t understand it, but it actually makes me like it more. In fact now I look forward to things like that; little puzzles of human existence, where, for perhaps small examples (like when you work over a field) you can understand the issue entirely, but overall you realize it’s harder than that, and moreover you shouldn’t kill yourself over it. You can remain content maybe knowing how to describe some of its properties, while allowing it to maintain its secrets, because life is actually more interesting that way.