Are the Imaginary Numbers Real?

The complex numbers are vectors with a special kind of product which turns them into a field (an algebraic structure that behaves like a "number"). If you think about it that way, then it doesn't really seem so mysterious. There are all sorts of weird algebraic structures that have similar properties to "numbers". Not all structures have all the properties you want, nor do they all extend the real numbers though.

I don't know much about physics, so I actually have no idea if imaginary quantities are observable, or if they're just a purely formal mathematical device. Or I guess another question is: are there physical relationships we cannot state without the complex numbers? If you can rewrite a statement about a complex number in terms of a vector, you can imagine that a lot of work can just be rewritten to deal entirely with real number pairs instead? That's how I would judge if something is "real".

https://physics.stackexchange.com/questions/11396/can-one-do...

https://physics.stackexchange.com/questions/32422/qm-without...

i'm not sure about the whole of platonist philosophy or competing theories, but the platonist way of handling "numbers" seems exactly what i use internally:

- "imaginary" numbers are exactly as real as irrational, rational, integers, natural numbers, zero and the rest.

- all these kinds of "numbers" are real, but exist as symbolic abstractions. e.g. not observable in the physical universe.

for one thing, while you can observe two cows, you cannot observe more than 2*100 cows. but these are both natural numbers. it would be an unsatisfying (to me) definition of real if realness depended on the size of a natural number, and probably its application (cows vs. subatomic particles).

and while you can't really observe the number two by itself, it can still be used to communicate and reason about the number of cows and other objects. it's a symbol, a very useful abstraction. real in the platonic sense, but abstract - a distinct type of real outside time and the physical universe.

irrational numbers and even complex numbers are then simply other useful abstractions. it's useful to learn this about mathematics and physics in general because, e.g. "is velocity real?", "what about string theory? - is the 10th dimension real?"

the rest of the article dives further down the rabbit hole of structuralism, another philosophy of mathematics, apparently a reaction to platonism.

at this point, my interest in philosophy has dropped to a level that can not be represented by any number at all. meta-unreal (a level of real that is not real even in the abstract sense), and so rational as to be completely irrational.

i've concluded that numbers are also exactly as real as the theories and philosophies of reality, but far more useful.