[throwawaymath]

I think characterizing duality in this way is kind of superfluous, because the only way all those meanings of duality are the same is in the most abstract sense of the word.

In other words, lots of things have duals. But the duality between any given pair of things doesn't necessarily expose any deep, fundamental connection to another pair of things which have duality. So it's not that duality features so heavily throughout mathematics as its own concept; rather, we frequently build new theories to tie these things together. It's helpful to be able to translate things from one context to another context.

We could just as easily say that isomorphisms are insane because they feature heavily throughout mathematics. But I don't think that provides a deep insight, because it's not like an isomorphism is a special property that ties a bunch of mathematics together in a grand way. Specific pairs of things can be isomorphic. Likewise specific pairs of things can be duals.

Any given pair of dual things is its own duality. It doesn't necessarily have anything to do with the way another pair of objects is in duality. The terminology here is semantically convenient for intuition, but it's definitely overloaded. I think the commonalities you're seeing here are simply due to the vast utility of linearity in all of those disciplines.

[adamnemecek]

> I think characterizing duality in this way is kind of superfluous,

It's an analysis done out of necessity. These dualities might not be a 100% in every case, but maybe I care about the ways in which they are similar.

> because the only way all those meanings of duality are the same is in the most abstract sense of the word.

So is a monad. Do you think that in the future, the level of abstraction in mathematics is going to increase or decrease?

[throwawaymath]

It will increase, which I guess is sort of my point. We already know there's a lot of abstraction. If these things are only alike semantically (two pairs of dual things can be completely unrelated), what does it gain you to point out they've everywhere?

I don't mean to be obtuse, but it strikes me as saying that a city is full of concrete.

[adamnemecek]

You can think of it as a really nice intermediate language.

It might be hard to build a computer system that lets you reason about both probability and say quantum mechanics.

However it might be easier to build a system that phrases a probabilistic problem in terms of duality and then solves it.

Like why should each of this have it's own foundation when there's one that captures a lot of them?