Ultimate Abstraction

From Eugenia Cheng's Beyond Infinity, Chapter 13:

Mathematics ... is like an unending series of mountain ranges. Every time you conquer one, you have a moment of elation and then you realize that conquering this peak has just enabled you to see the higher peaks ahead ....

This is inevitable. In fact, it's part of the power of mathematics. It's because the concepts we study can always be built into bigger ones. Those, in turn, can be built into bigger ones, and so on forever. This is because of the abstract nature of it. If we are climbing mountains, we don't build more mountains out of the ones we've already climbed — we just see more that are there. In math, we keep building bigger and bigger ones.

It's not that we do it willfully; it's that the methods we develop for studying things in math are actually new pieces of math. To study those, we create new pieces of math that then need to be studied. This doesn't happen if you're studying birds: the methods you develop for studying birds aren't themselves birds.

This is how category theory arose, as a new piece of math to study math. In a way, category theory is an ultimate abstraction. To study the world abstractly you use science; to study science abstractly you use math; to study math abstractly you use category theory. Each step is a further level of abstraction. But to study category theory abstractly you use category theory.

^z - 2017-08-24