[tuhgdetzhh]

One way to sharpen the question is to stop asking whether C is "fundamental" and instead ask whether it is forced by mild structural constraints. From that angle, its status looks closer to inevitability than convenience.

Take R as an ordered field with its usual topology and ask for a finite-dimensional, commutative, unital R-algebra that is algebraically closed and admits a compatible notion of differentiation with reasonable spectral behavior. You essentially land in C, up to isomorphism. This is not an accident, but a consequence of how algebraic closure, local analyticity, and linearization interact. Attempts to remain over R tend to externalize the complexity rather than eliminate it, for example by passing to real Jordan forms, doubling dimensions, or encoding rotations as special cases rather than generic elements.

More telling is the rigidity of holomorphicity. The Cauchy-Riemann equations are not a decorative constraint; they encode the compatibility between the algebra structure and the underlying real geometry. The result is that analyticity becomes a global condition rather than a local one, with consequences like identity theorems and strong maximum principles that have no honest analogue over R.

I’m also skeptical of treating the reals as categorically more natural. R is already a completion, already non-algebraic, already defined via exclusion of infinitesimals. In practice, many constructions over R that are taken to be primitive become functorial or even canonical only after base change to C.

So while one can certainly regard C as a technical device, it behaves like a fixed point: impose enough regularity, closure, and stability requirements, and the theory reconstructs it whether you intend to or not. That does not make it metaphysically fundamental, but it does make it mathematically hard to avoid without paying a real structural cost.

[hodgehog11]

This is the way I think. C is "nice" because it is constructed to satisfy so many "nice" structural properties simultaneously; that's what makes it special. This gives rise to "nice" consequences that are physically convenient across a variety of applications.

I work in applied probability, so I'm forced to use many different tools depending on the application. My colleagues and I would consider ourselves lucky if what we're doing allows for an application of some properties of C, as the maths will tend to fall out so beautifully.

[debatem1]

Not meaning to derail an interesting conversation, but I'm curious about your description of your work as "applied probability". Can you say any more about what that involves?

[jalapenos]

> Take R as an ordered field with its usual topology and ask for a finite-dimensional, commutative, unital R-algebra that is algebraically closed and admits a compatible notion of differentiation with reasonable spectral behavior.

No thank you, you can keep your R.

Damn... does this paragraph mean something in the real world?

Probably I've the brain of a gnat compared to you, but do all the things you just said have a clear meaning that you relate to the world around you?

[StopDisinfo910]

> does this paragraph mean something in the real world?

It's actually both surprisingly meaningful and quite precise in its meaning which also makes it completely unintelligible if you don't know the words it uses.

Ordered field: satisfying the properties of an algebraic field - so a set, an addition and a multiplication with the proper properties for these operations - with a total order, a binary relation with the proper properties.

Usual topology: we will use the most common metric (a function with a set of properties) on R so the absolute value of the difference

Finite-dimentional: can be generated using only a finite number of elements

Commutative: the operation will give the same result for (a x b) and (b x a)

Unital: as an element which acts like 1 and return the same element when applied so (1 x a) = a

R-algebra: a formally defined algebraic object involving a set and three operations following multiple rules

Algebraically closed: a property on the polynomial of this algebra to be respected. They must always have a root. Untrue in R because of negative. That's basically introducing i as a structural necessity.

Admits a notion of differentiation with reasonable spectral behaviour: This is the most fuzzy part. Differentiation means we can build a notion of derivatives for functions on it which is essential for calculus to work. The part about spectral behavior is probably to disqualify weird algebra isomorphic to C but where differentiation behaves differently. It seems redondant to me if you already have a finite-dimentional algebra.

It's not really complicated. It's more about being familiar with what the expression means. It's basically a fancy way to say that if you ask for something looking like R with a calculus acting like the one of functions on R but in higher dimensions, you get C.

[xanderlewis]

I'm sure you don't have the brain of a gnat, and, even if you did, it probably wouldn't prevent you from understanding this.

As for whether these definitions have a clear meaning that one can relate to 'the world': I think so. To take just one example (I could do more), finite-dimensional means exactly what you think it means, and you certainly understand what I mean when I say our world is finite (or three, or four, or n) dimensional.

Commutative also means something very down to earth: if you understand why a*b = b*a or why putting your socks on and then your shoes and putting your shoes on and then your socks lead to different outcomes, you understand what it means for some set of actions to be commutative.

And so on.

These notions, like all others, have their origin in common sense and everyday intuition. They're not cooked up in a vacuum by some group of pretentious mathematicians, as much as that may seem to be the case.

[brazzy]

Math and reality are, in general completely distinct. Some math is originally developed to model reality, but nowadays (and for a long time) that's not the typical starting point, and mathematicians pushing boundaries in academia generally don't even think about how it relates to reality.

However, it is true (and an absolutely fascinating phenomenon) that we keep encountering phenomena in reality and then realize that an existing but previously purely academic branch of math is useful for modeling it.

To the best of our knowledge, such cases are basically coincidence.

[deterministic]

Mathematics is a game of symbols and rules.

Each of the "ordered field", "inital R-algebra" etc. are the names of a set of rules and constraints. That's all it is. So you need to know those sets of rules to make sense of it. It has nothing to do with brain size or IQ :)

In other words, you define a new thing by simply enumerating the rules constraining it. As in: A Duck is a thing that Quacks, Flies, Swims and ... Where Quacks etc. is defined somewhere else.

[donkeybeer]

Yes, the point of mathematics is so that a gnat could do it. These abstractions are about making life easy and making things that previously needed bespoke solutions to be more mechanical.