scitposting 6: ok undergraduate math is already a rabbit hole

Haha. You probably thought calculus or differential equations was the end of your journey down, down, down the rabbit hole. Lmao no. Ok, so maybe you also acknowledged the existence of matrices and vectors in linear algebra. But then you discovered that functions are also vectors, and this property was really useful in differential equations because the derivative behaved as a linear operator and the continuous functions as a vector space. So you were like, “what a nice connection!” (or you said “what the f*** i dont like this anymore”).

But then you discover that the treatment of functions as vectors (see previous scitpost) is on a very deep level, and that functions are basically infinite dimensional vectors. And differential operators acting on functions (one-variable or multivariable) are like matrices acting on Euclidean vectors in terms of its properties. THIS IS THE TRUE LINEAR ALGEBRA GUYS.

But then you discovered there was complex-valued calculus, i.e., complex analysis. Wow, interesting extension. And then you also discovered the applications of matrices as tensors to geometric problems. Ok, not too bad, that wasn’t too hard to connect. Then you discovered that these tensors had algebraic properties that formed a group, where associativity on binary operations applied and where identity and inverse elements existed (like in rotations, where the identity is rotation by 0 degrees, and the inverse of a rotation is rotating it backwards to its original position again). So you discovered abstract algebra existed, and there were groups, rings, fields, etc.

But then you also found out there was a more rigorous way to treat calculus, i.e., mathematical analysis where you traced all your conclusions back to the fact that the real numbers are an infinitely uncountable ordered field, or its set-theoretic properties.

Ordered field? That’s abstract algebra, huh. And the set-theoretic properties could be generalized (while taking inspiration from the real numbers) to the wild field of topology, which I think is spicy set theory. Then you also discover that the set of operators over a vector space also has its own topological properties NKDJASDJASKDNAJKWDAD WHAT.