Math - what is it?
I never get tired of considering this question, because I think it says something about the nature of truth, reality, and our relationship to the universe. Here's my hot take 🔥
I believe there is an mathematical universe that exists independent of human reasoning. It is ordered, it is not chaos. Our way of using math is to construct increasingly elaborate frameworks by which to reason about our complicated universe. We build our maths in order to dig our way closer to the reality.
Are we capable of understanding that mathematical universe accurately? I don't think so. I think our capacity is limited, yet we've only scratched the surface of that limited capacity. Kurt Gödel's masterpiece, the "incompleteness theorem" was a brilliant proof putting hard limits on formal mathematical systems built from first order axiomatic logic. Does this finding might suggest limits to human reasoning? Namely that we cannot build a consistent, non-trivial mathematical framework that can answer all questions posed within that system.
By extension, this likely puts limits on computational models and what we can say about computer languages and programs. His work helped pave the way for the Church-Turing thesis that explores the limits of computation.
So why do we need increasingly complex mathematics? Because we use it to understand our universe, and this is our way of inching closer to the goal. We advance by building more robust models of our universe, and with each iteration we gain insights. This same reasoning provides the rationale for quantum computing - the need to model reality realistically.
"Nature isn't classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy." - Richard P. Feynman
Most computer scientists do not have a deep understanding of quantum mechanics, but the math of quantum computing is just math. What really happens under the hood when we can't see is not knowable to us, but a linear distribution of potential outcomes is an intuitive way to reason about it. The math gets us a closer approximation to an understanding of a reality we can never fully achieve.