Coming from a pure mathematics background I have yet to encounter a system that I would say is a fundamentally different kind of mathematics rather than a different practical application of mathematics. My first post in this thread was talking about how something like “commutativity of addition” might exist in the universe in a pre-thinking-thing state, but fundamentally the problem with the idea of commutativity of addition being in the universe is that even if it is that doesn’t show that the universe has math in it. Commutativity of addition isn’t in mathematics.
Sure, in many sets that we have defined an addition operator over that operator is commutative, but it certainly doesn’t commute when adding strings. “Hello” + " World" != " World" + “Hello”. If I created a mathematical system to describe putting together jigsaw puzzles, I highly doubt addition would be commutative. “That’s not addition” is a No True Scotsman argument. I can call it addition if I want. A professor I once had said, “People sometimes assume commutativity because that’s how they’ve been adding and multiplying their whole lives, but most operations do not commute. Taking off your clothes and getting into the shower are not commutative.”
If there’s anything that’s like a grand unifying theory of mathematics I guess it would be set theory, but even then that’s not really necessary to mathematics. Because of this thread I spent some time thinking about what I think is the very essence of mathematics, and I think I have it: Considering a set of assumptions in a vacuum. That’s really what mathematics is about.
So when Euclid says that you can draw a line between two points, you can’t import any external notions about what a “line” is or what a “point” is (That’s not to say that Euclid himself didn’t import such notions, he probably did, contemporary mathematics is not very old, I’d say it’s younger than the incompleteness theorem). In university we would have heated debates about whether ℕ contained 0. I would say I didn’t care if ℕ contained 0, but if it did then 0 + 0 = 1*. That’s mathematics, not caring what symbol you use to refer to the additive identity, not importing any assumptions from outside.
I think that’s the innovation that has made a system that is so good at helping us with difficult contemporary science problems that require us to look past our pre-conceived notions of how things work (which are increasingly useless for advanced science).
* I actually usually consider ℕ including an additive identity and talk about ℤ+ if I want a set that doesn’t for some crazy reason, this was a joke (Yes, it got laughs)
