Riemann went nuts working on higher dimensional mathematics and linear algebra. At the time there was not a clear use case for math higher than like 3 or 4 dimensions, but he drove himself crazy discovering it anyways. Today, this kind of math underlies all of artificial intelligence
this post was submitted on 26 Jun 2025
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Strangest? Functional analysis, maybe. I understand it's used pretty extensively in quantum field theory, although I don't actually know firsthand.
That's a body of mathematics about infinite-dimensional spaces and the operations on them. Even more abstract ways of defining those operations exist and have come up as well, like in Tseirlson's problem, which recently-ish had a shock negative resolution stemming from quantum information theory.
There's constructions I find weirder yet, but I don't think p-adic numbers, for example, have any direct application at this point.
Non-Euclidean geometry was developed by pure mathematicians who were trying to prove the parallel line postulate as a theorem. They realized that all of the classic geometry theorems are all different if you start changing that postulate.
This led to Riemannian geometry in 1854, which back then was a pure math exercise.
Some 60 years later, in 1915, Albert Einstein published the theory of general relativity, of which the core mathematics is all Riemannian geometry.
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The math fun fact I remember best from college is that Charles Boole invented Boolean algebra for his doctoral thesis and his goal was to create a branch of mathematics that was useless. For those not familiar with boolean algebra it works by using logic gates with 1s and 0s to determine a final 1 or 0 state and is subsequently the basis for all modern digital computing
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I work with a guy who is a math whiz and loves to talk. Yesterday while I was invoicing clients, he was telling me how origami is much more effective for solving geometry than a compass and a straight edge.
I'll ask him this question.
My disclaimer: I don't know what any of this means, but it might give you a direction to start your research.
First thing he came up with is Number Theory, and how they've been working on that for centuries, but they never would have imagined that it would be the basis of modern encryption. Multiplying a HUGE prime number with any other numbers is incredibly easy, but factoring the result into those same numbers is near impossible (within reasonable time constraints.)
He said something about knot theory and bacterial proteins, but it was too far above my head to even try to relay how that's relevant.
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Does this count? Because it really is wtf.
Don't put that cursed shit on mathematicians, lol.
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A brain teaser about visiting all islands connected by bridges without crossing the same bridge twice is now the basis of all internet routing. (Graph theory)
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It's imaginary numbers. Full stop. No debate about it. The idea of them is so wild that they were literally named imaginary numbers to demonstrate how silly they were, and yet they can be used to describe real things in nature.
I mean, quaternions are the weirder version of complex numbers, and they're used for calculating 3D rotations in a lot of production code.
There's also the octonions and (much inferior) Clifford algebras beyond that, but I don't know about applications.
Yeah but aren't quaternions basically just a weird subgroup of 2x2 complex matrices?
Would that make it less true? Complex numbers can be seen as a weird subgroup of the 2x2 real matrices. (And you can "stack" the two representations to get 4x4 real quaternions)
Furthermore, octonions are non-associative, and so can't be a subgroup of anything (although you can do a similar thing using an alternate matrix multiplication rule). They still show up in a lot of the same pure math contexts, though.
I just think complex vector spaces are a great place to stop your abstraction
Stopping while we're ahead? Never!
/s, but also I'm sort of in this picture.
Well who wants constraints anyway? The most inconvenient constraints in the wrong place can make certain things much more complicated to deal with... Now a nice, sensible normal Hilbert space, isn't that lovely?
I don't really get 'em. It seems like people often use them as "a pair of numbers." So why not just use a pair of numbers then?
They also have a defined multiplication operation consistent with how it works on ordinary numbers. And it's not just multiplying each number separately.
A lot of math works better on them. For example, all n-degree polynomials have exactly n roots, and all smooth complex functions have a polynomial approximation at every point. Neither is true on the reals.
Quantum mechanics could possibly work with pairs of real numbers, but it would be unclear what each one means on their own. Treating a probability amplitude as a single number is more satisfying in a lot of ways.
That they don't exist is still a position you could take, but so is the opposite.
I totally get your point, and sometimes it seems like that. Why not just use a coordinate system? Because in some applications the complex roots of equations is relevant.
If you square an imaginary number, it's no longer an imaginary number. Now it's a real number! That's not something you can accomplish with something like a pair of numbers alone.
Because the second number has special rules and a unit. It's not just a pair of numbers, though it can be represented through a pair of numbers (really helpful for computing).
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IIRC quaternions were considered pretty useless until we started doing 3D stuff on computers and now they're used everywhere
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