Math is more like art than science. Natural sciences were just there to be discovered and figured out. We created math. We came up with the concept of zero, infinity, and numbers. The fact that something that we created can be as layered and infinite as math is mind boggling and beautiful. Just like a good piece of art is supposed to be inexhaustible in the meanings that one can extract from it, math is inexhaustible in the number of problems that it offers up for solution.
Most math problems are imagined. You start with a what if-then question. A solution is searched for. In the process, you might decide to tweak the question bit. Relax some restriction you previously imposed, come up with a new one instead, define something new along the way: create an environment conducive to a solution. You torture yourself for days, months, years. Then, hopefully, you come up with a proof.
“A proof, that is, a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted argument should feel like a splash of cool water, and be a beacon of light— it should refresh the spirit and illuminate the mind. And it should be charming.”
Then reality comes barging in with all its messy ugliness and sensibility. You know that a point with a certain property must exist in a certain kind of graph – you even know how to find it – but you cannot make your computer find it. Computers lack infinite memory or processing power. You know of no way to translate your proof in to an implementable algorithm that would work. You might even create one that seems to work pretty well, only you’re not sure if it actually will for all inputs, and you’re not exactly sure if it is providing you with the answer you want. Windows crashes.
And then there’s nothing left to do but to start torturing yourself again.