## 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.

That right! Another formulation: it is the things you _can_ prove that tell you how to think about tensor products. In other words, you let elementary lemmas and examples shape your intuition of the mathematical object in question. There’s nothing else, no magical intuition will magically appear to help you “understand” it.

Off topic: For those still struggling with when an element of a tensor product is zero, take a look at the criterion in Lemma Tag 04VX of the stacks project.

Nothing is more fruitful – all mathematicians know it – than those obscure analogies, those disturbing reflections of one theory in another; those furtive caresses, those inexplicable discords; nothing also gives more pleasure to the researcher. The day comes when the illusion dissolves; the yoked theories reveal their common source before disappearing. As the Gita teaches, one achieves knowledge and indifference at the same time.

— André Weil

In mathematics you don’t understand things. You just get used to them.

— John von Neumann

This is basically how I feel about a lot of analysis — practically a moral imperative, and Rudin is the bible.

Unrelatedly, the title of the post (before reading the post) reminded me of how a friend of mine compares life to a vector — it must have direction and magnitude.

The viewing of objects using their external relationships rather than their internal structure gives rise to the category theoretic view of mathematics and to object-orientation in computer programming.

The triumph of this viewpoint in computer science is so total that programming languages such as Haskell are explicitly modelled on category theory.

There are some excellent intermediate/advanced lectures on category theory at http://www.youtube.com/user/TheCatsters

A good introduction to categories is The Joy of Cats http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf

Dr. Gunnarson, I would like to quote some of the wonderful quotations in your post to Gowers weblog in 2009. Do you have references available? Thank you very much!

Reuben Hersh

rhersh@gmail.com

wow, thank you for this post!!! i wish you had written this ten years ago when i was scared of tensor products. It probably explains a large part of why I have found math so hard … because I have been trying too hard to understand everything and where everything comes from. For example, Serre duality…how did Serre discover it? Even the people in the 1800 working on curves … what made them notice sere duality ? Sometimes I suppose it is just better to accept serre duality, etc as a part of life one isn’t going to understand and just move on with life. i got a lot from johan’s comment above, too.