Readit NewsYou can teach middle school children how to define complex numbers, given real numbers as a starting point. You can't necessarily even teach college students or adults how to define real numbers, given rational numbers as a starting point.
As the "evidence" piles up, in further mathematics, physics, and the interactions of the two, I still never got to the point at the core where I thought complex numbers were a certain fundamental concept, or just a convenient tool for expressing and calculating a variety of things. It's more than just a coincidence, for sure, but the philosophical part of my mind is not at ease with it.
I doubt anyone could make a reply to this comment that would make me feel any better about it. Indeed, I believe real numbers to be completely natural, but far greater mathematicians than I found them objectionable only a hundred years ago, and demonstrated that mathematics is rich and nuanced even when you assume that they don't exist in the form we think of them today.
I am also a complex number skeptic. The position I've landed on is this.
1) complex numbers are probably used for far more purposes across math than they "ought" to be, because people don't have the toolbox to talk about geometry on R^2 but they do know C so they just use C. In particular, many of the interesting things about complex analysis are probably just the n=2 case of more general constructions that can be done by locating R inside of larger-dimensional algebras.
2) The C that shows up in quantum mechanics is likely an example of this--it's a case of physics having a a circular symmetry embedded in it (the phase of the wave functions) and everyone getting attached to their favorite way of writing it. (Ish. I'm not sure how the square the fact that wave functions add in superposition. but anyway it's not going to be like "physics NEEDS C", but rather, physics uses C because C models the algebra of the thing physics is describing.
3) C is definitely intrinsic in a certain sense: once you have polynomials in R, a natural thing to do is to add a sqrt(-1). This is not all that different conceptually from adding sqrt(2), and likely any aliens we ever run into will also have done the same thing.
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Some specific cases: Wissel (1984) derived critical slowing down for ecology independently and was ignored for 20 years. The actual import to ecology came via economist Buz Brock, not a physicist. Nolasco & Dahlen (1968) derived period-doubling for cardiac tissue before Feigenbaum's universality result. Jaeger (2001) derived the edge-of-chaos condition for recurrent neural networks without citing Bak, Kauffman, or Langton.
The complex systems movement you reference existed. The paper documents that it didn't actually solve the transfer problem. The cross-citation analysis shows the gaps persisted through the 2000s and 2010s.
You're right that some domains imported rather than reinvented. The paper maps where each transfer was independent, where it was imported, and where it was partial. That's the point — the pattern is messier and more interesting than either "all independent" or "all imported."
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I get the same feeling when I think about monads, futures/promises, reactive programming that doesn't seem to actually watch variables (React.. cough), Rust's borrow checker existing when we have copy-on-write, that there's no realtime garbage collection algorithm that's been proven to be fundamental (like Paxos and Raft were for distributed consensus), having so many types of interprocess communication instead of just optimizing streams and state transfer, having a myriad of GPU frameworks like Vulkan/Metal/DirectX without MIMD multicore processors to provide bare-metal access to the underlying SIMD matrix math, I could go on forever.
I can talk about why tau is superior to pi (and what a tragedy it is that it's too late to rewrite textbooks) but I have nothing to offer in place of i. I can, and have, said a lot about the unfortunate state of computer science though: that internet lottery winners pulled up the ladder behind them rather than fixing fundamental problems to alleviate struggle.
I wonder if any of this is at play in mathematics. It sure seems like a lot of innovation comes from people effectively living in their parents' basements, while institutions have seemingly unlimited budgets to reinforce the status quo..