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.
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.
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?
Pure probability focuses on developing fundamental tools to work with random elements. It's applied in the sense that it usually draws upon techniques found in other traditionally pure mathematical areas, but is less applied than "applied probability", which is the development and analysis of probabilistic models, typically for real-world phenomena. It's a bit like statistics, but with more focus on the consequences of modelling assumptions rather than relying on data (although allowing for data fitting is becoming important, so I'm not sure how useful this distinction is anymore).
At the moment, using probabilistic techniques to investigate the operation of stochastic optimisers and other random elements in the training and deployment of neural networks is pretty popular, and that gets funding. But business as usual is typically looking at ecological models involving the interaction of many species, epidemiological models investigating the spread of disease, social network models, climate models, telecommunication and financial models, etc. Branching processes, Markov models, stochastic differential equations, point processes, random matrices, random graph networks; these are all the common objects used. Actually figuring out their behaviour can require all kinds of assorted techniques though, you get to pull from just about anything in mathematics to "get the job done".
In my work in academia (which I’m considering leaving), I’m very familiar with the common mathematical objects you mentioned. Where could I look for a job similar to yours? It sounds very interesting
Sorry, I'm in academia too, but my ex-colleagues who left found themselves doing nearly identical work doing MFT research at hedge funds, climate modelling at our federal weather bureau, and SciML in big tech. I know of someone doing this kind of work in telecoms too, but I haven't spoken to them lately. Having said that, it's rough out there right now. A couple of people I know looking for another job right now (academia or otherwise) with this kind of training are not having much luck...
> 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?
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.
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.
Opposing view (that I happen to hold, at least if I had to choose one side or the other): not only is mathematics 'reality'; it is arguably the only thing that has a reasonable claim to being 'reality' itself.
After all, facts (whatever that means) about the physical world can only be obtained by proxy (through measurement), whereas mathematical facts are just... evident. They're nakedly apparent. Nothing is being modelled. What you call the 'model' is the object of study itself.
A denial of the 'reality' of pure mathematics would imply the claim that an alien civilisation given enough time would not discover the same facts or would even discover different – perhaps contradictory – facts. This seems implausible, excluding very technical foundational issues. And even then it's hard to believe.
> To the best of our knowledge, such cases are basically coincidence.
This couldn't be further from the truth. It's not coincidence at all. The reason that mathematics inevitably ends up being 'useful' (whatever that means; it heavily depends on who you ask!) is because it's very much real. It might be somewhat 'theoretical', but that doesn't mean it's made up. It really shouldn't surprise anyone that an understanding of the most basic principles of reality turns out to be somewhat useful.
I think you're not even disagreeing with me, we're just using different definitions of the word "reality". I meant it to use specifically "the physical world" - which you are treating as distinct from mathematics as well in your second paragraph.
Mathematics is an abstract game of symbols and rules invented by humans. It has nothing to do with reality. However it is quite useful for modelling our understanding of reality.
"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."
Would you have some examples?
(Only example that I know that might fit are quaternions, who were apparently not so useful when they were found/invented but nowdays are very useful for many 3D application/computergraphics)
Group theory entering quantum physics is a particularly funny example, because some established physicists at the time really hated the purely academic nature of group theory that made it difficult to learn.[1]
If you include practical applications inside computers and not just the physical reality, then Galois theory is the most often cited example. Galois himself was long dead when people figured out that his mathematical framework was useful for cryptography.
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.
> 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.
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.
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.