I’m very much not a mathematician myself, but my sister is, and to the best of my understanding (which is not very far) she works with combinatorics and Schubert varieties. She tried to explain pipe dreams to me recently and the diagrams are cool but I still don’t really get what they mean. (But I can relay questions to her if this is meaningful to others.)
My own mathematical journey was mostly on the CS side of things, with proving the correctness and efficiency of algorithms and all that. But there were two moments I remember that felt like glimpses into the field as a whole. First was reading Hofstadter’s Gödel, Escher, Bach, which introduced me to the idea of Peano arithmetic and axiomatizing everything. We don’t actually need to take it on faith that the natural numbers work the way we expect them to. We can actually build them from the ground up! We can build an entire system of number theory this way! We can show from first principles that 1+1=2 (or rather S0+S0=SS0)!
The other was when I took a class on computer graphics taught by a famously eccentric math professor, whose goal for the class was for all of us to find something mathematical that interested us and then somehow make graphics about it. I ended up looking into geometric algebra and then somewhat falling down the rabbit hole.
If you’ve done any physics or engineering, you’re probably used to the right-hand rule for rotations. Apply a force F at a distance r and the resulting torque is the cross product r × F, a vector pointing along the axis of rotation.
But this always felt a little weird. What if you’re in two dimensions? Obviously rotations can happen in two dimensions. But there’s no longer an axis of rotation, there’s a point of rotation. And the torque is a scalar, not a vector. What about in four dimensions? How do you even do that?
It turns out there’s a beautifully simple and elegant solution. Instead of treating rotations as vectors, you can treat them as bivectors. If a vector is a directed length, a bivector is a directed area. And the dimensionality of the space of bivectors, in n spatial dimensions, is (n choose 2).
It’s pure coincidence that in three dimensions, (3 choose 2) is also three, and we can create a mapping between bivectors and standard vectors. But this mapping requires imposing a handedness on the universe (the right hand rule instead of the left hand rule) that can lead to wrong intuitions in physics. If you instead consider rotations to be bivectors then you don’t need a handedness and your intuitions will work in any number of spatial dimensions!
And then you can start working with multivectors, which are a linear combination of scalars and vectors and bivectors and so on. And once you do that (through a proof that’s absolutely beautiful and too long for me to derail this thread with) it turns out that both the complex numbers and the quaternions are special cases of this general system, in 2 and 3 dimensions respectively. And there’s a clear and intuitive reason why there’s not an equivalent of the quaternions in four dimensions (which would have seven elements in its basis). It’s really cool.
…so yeah, that’s why I like mathematics, though it’s not the path I ended up taking in life!