Grothendieck views the mathematician and the problem as complimenting each other, the mathematician using the problem’s natural structure in its solution, rather than striking it with a foreign, invasive method. My view is that any problem that has resisted repeated direct attack from problem solvers, should naturally be of interest to theory builders. If you can’t solve a problem directly, then grow a crystal of theory around the problem and then hope that the solution you are looking for can be located somewhere inside the crystal. This is hard because the same person needs to know about both category theory and the problem domain, which is quite a heavy demand on a human brain.

I think we should levy a fee for category theorists to use the word “just”

reasoning about the system should be done recursively on its structure. ​ good software design is ultimately an art. ​ another example of reasoning via an interface. ​ I suspect that interfaces are in fact synonymous with compositionality. That is, compositionality is not just the ability to compose objects, but the ability to work with an object after intentionally forgetting how it was built. ​ can interact in complex ways that block understanding ​ More generally, I claim that the opposite of compositionality is emergent effects. The common definition of emergence is a system being ‘more than the sum of its parts’, and so it is easy to see that such a system cannot be understood only in terms of its parts, i.e. it is not compositional. Moreover I claim that non-compositionality is a barrier to scientific understanding, because it breaks the reductionist methodology of always dividing a system into smaller components and translating explanations into lower levels.