The art of definition
Definitions are basic objects in mathematics. Even so, I’ve never seen the art of definition explicitly taught, and I have rarely seen the need for a definition explicitly discussed.
Have you ever noticed how damn hard it is to make a good definition and yet how utterly useful a good definition can be?
The basic definitions inform the research of any field, and a good definition will lead to better theorems than a bad one. If you get them right, if you really nail down the definition, then everything works out much more cleanly than otherwise.
So for example, it doesn’t make sense to work in algebraic geometry without the concepts of affine and projective space, and varieties, and schemes. They are to algebraic geometry like circles and triangles are to elementary geometry. You define your objects, then you see how they act and how they interact.
I saw first hand how a good definition improves clarity of thought back in grad school. I was lucky enough to talk to John Tate (my mathematical hero) about my thesis, and after listening to me go on for some time with a simple object but complicated proofs, he suggested that I add an extra sentence to my basic object, an assumption with a fixed structure.
This gave me a bit more explaining to do up front – but even there added intuition – and greatly simplified the statement and proofs of my theorems. It also improved my talks about my thesis. I could now go in and spend some time motivating the definition, and then state the resulting theorem very cleanly once people were convinced.
Another example from my husband’s grad seminar this semester: he’s starting out with the concept of triangulated categories coming from Verdier’s thesis. One mysterious part of the definition involves the so-called “octahedral axiom,” which mathematicians have been grappling with ever since it was invented. As far as Johan tells it, people struggle with why it’s necessary but not that it’s necessary, or at least something very much like it. What’s amazing is that Verdier managed to get it right when he was so young.
Why? Because definition building is naturally iterative, and it can take years to get it right. It’s not an obvious process. I have no doubt that many arguments were once fought over whether the most basic definitions, although I’m no historian. There’s a whole evolutionary struggle that I can imagine could take place as well – people could make the wrong definition, and the community would not be able to prove good stuff about that, so it would eventually give way to stronger, more robust definitions. Better to start out carefully.
Going back to that. I think it’s strange that the building up of definitions is not explicitly taught. I think it’s a result of the way math is taught as if it’s already known, so the mystery of how people came up with the theorems is almost hidden, never mind the original objects and questions about them. For that matter, it’s not often discussed why we care whether a given theorem is important, just whether it’s true. Somehow the “importance” conversations happen in quiet voices over wine at the seminar dinners.
Personally, I got just as much out of Tate’s help with my thesis as anything else about my thesis. The crystalline focus that he helped me achieve with the correct choice of the “basic object of study” has made me want to do that every single time I embark on a project, in data science or elsewhere.