I have a Ph.D. in a field of mathematics in which complex numbers are fundamental, but I have a real philosophical problem with complex numbers. In particular, they arose historically as a tool for solving polynomial equations. Is this the shadow of something natural that we just couldn't see, or just a convenience?
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.
HN once recommended a book
https://press.princeton.edu/books/paperback/9780691133911/ne...
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?
Absolutely, thanks for asking!
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".
I used to feel the same way. I now consider complex numbers just as real as any other number.
The key to seeing the light is not to try convincing yourself that complex number are "real", but to truly understand how ALL numbers are abstractions. This has indeed been a perspective that has broadened my understanding of math as a whole.
Reflect on the fact that negative numbers, fractions, even zero, were once controversial and non-intuitive, the same as complex are to some now.
Even the "natural" numbers are only abstractions: they allow us to categorize by quantity. No one ever saw "two", for example.
Another thing to think about is the very nature of mathematical existence. In a certain perspective, no objects cannot exist in math. If you can think if an object with certain rules constraining it, voila, it exists, independent of whether a certain rule system prohibit its. All that matters is that we adhere to the rule system we have imagined into being. It does not exist in a certain mathematical axiomatic system, but then again axioms are by their very nature chosen.
Now in that vein here is a deep thought: I think free will exists just because we can imagine a math object into being that is neither caused nor random. No need to know how it exists, the important thing is, assuming it exists, what are its properties?
Correct. And this is the key distinction between the mathematical approach and the everyday / business / SE approach that dominates on hacker news.
Numbers are not "real", they just happen to be isomorphic to all things that are infinite in nature. That falls out from the isomorphism between countable sets and the natural numbers.
You'll often hear novices referencing the 'reals' as being "real" numbers and what we measure with and such. And yet we categorically do not ever measure or observe the reals at all. Such thing is honestly silly. Where on earth is pi on my ruler? It would be impossible to pinpoint... This is a result of the isomorphism of the real numbers to cauchy sequences of rational numbers and the definition of supremum and infinum. How on earth can any person possibly identify a physical least upper bound of an infinite set? The only things we measure with are rational numbers.
People use terms sloppily and get themselves confused. These structures are fundamental because they encode something to do with relationships between things
The natural numbers encode things which always have something right after them. All things that satisfy this property are isomorphic to the natural numbers.
Similarly complex numbers relate by rotation and things satisfying particular rotational symmetries will behave the same way as the complex numbers. Thus we use C to describe them.
As a Zen Koan:
A novice asks "are the complex numbers real?"
The master turns right and walks away.
For me, the complex numbers arise as the quotients of 2-dimensional vectors (which arise as translations of the 2-dimensional affine space). This means that complex numbers are equivalence classes of pairs of vectors is a 2-dimesional vector space, like 2-dimensional vectors are equivalence classes of pairs of points in a 2-dimensional affine space or rational numbers are equivalence classes of pairs of integers, or integers are equivalence classes of pairs of natural numbers, which are equivalence classes of equipotent sets.
When you divide 2 collinear 2-dimensional vectors, their quotient is a real number a.k.a. scalar. When the vectors are not collinear, then the quotient is a complex number.
Multiplying a 2-dimensional vector with a complex number changes both its magnitude and its direction. Multiplying by +i rotates a vector by a right angle. Multiplying by -i does the same thing but in the opposite sense of rotation, hence the difference between them, which is the difference between clockwise and counterclockwise. Rotating twice by a right angle arrives in the opposite direction, regardless of the sense of rotation, therefore i*i = (-i))*(-i) = -1.
Both 2-dimensional vectors and complex numbers are included in the 2-dimensional geometric algebra, whose members have 2^2 = 4 components, which are the 2 components of a 2-dimensional vector together with the 2 components of a complex number. Unlike the complex numbers, the 2-dimensional vectors are not a field, because if you multiply 2 vectors the result is not a vector. All the properties of complex numbers can be deduced from those of the 2-dimensional vectors, if the complex numbers are defined as quotients, much in the same way how the properties of rational numbers are deduced from the properties of integers.
A similar relationship like that between 2-dimensional vectors and complex numbers exists between 3-dimensional vectors and quaternions. Unfortunately the discoverer of the quaternions, Hamilton, has been confused by the fact that both vectors and quaternions have multiple components and he believed that vectors and quaternions are the same thing. In reality, vectors and quaternions are distinct things and the operations that can be done with them are very different. This confusion has prevented for many years during the 19th century the correct use of quaternions and vectors in physics (like also the confusion between "polar" vectors and "axial" vectors a.k.a. pseudovectors).
Also, with elementary math: y+ as positive exponential numbers, y- as negative. Try rotating 90 deg the axis, into the -x part. What happens?
A question I enjoy asking myself when I'm wondering about this stuff is "if there are alien mathematicians in a distant galaxy somewhere, do they know about this?"
For complex numbers my gut feeling is yes, they do.
> I doubt anyone could make a reply to this comment that would make me feel any better about it.
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.
I can't entirely follow the details, but apparently quantum mechanics actually doesn't work for fields other than C, including quaternions. https://scottaaronson.blog/?p=4021
In my view nonnegative real numbers have good physical representations: amount, size, distance, position. Even negative integers don't have this types of models for them. Negative numbers arise mostly as a tool for accounting, position on a directed axis, things that cancel out each other (charge). But in each case it is the structure of <R,+> and not <R,+,*> and the positive and negative values are just a convention. Money could be negative, and debt could be positive, everything would be the same. Same for electrons and protons.
So in our everyday reality I think -1 and i exist the same way. I also think that complex numbers are fundamental/central in math, and in our world. They just have so many properties and connections to everything.
> In my view nonnegative real numbers have good physical representations
In my view, that isn’t even true for nonnegative integers. What’s the physical representation of the relatively tiny (compared to ‘most integers’) Graham’s number (https://en.wikipedia.org/wiki/Graham's_number)?
Back to the reals: in your view, do reals that cannot be computed have good physical representations?
Good catch. Some big numbers are way too big to mean anything physical, or exist in any sense. (Up to our everyday experiences at least. Maybe in a few years, after the singularity, AI proves that there are infinite many small discrete structures and proves ultrafinitist mathematics false.)
I think these questions mostly only matter when one tries to understand their own relation to these concepts, as GP asked.
> In my view nonnegative real numbers have good physical representations: amount, size, distance, position
I'm not a physicist, but do we actually know if distance and time can vary continuously or is there a smallest unit of distance or time? A physics equation might tell you a particle moves Pi meters in sqrt(2) seconds but are those even possible physical quantities? I'm not sure if we even know for sure whether the universe's size is infinite or finite?
> I believe real numbers to be completely natural
You 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.
well it's hard to formally define them, but it's not hard to say "imagine that all these decimals go on forever" and not worry about the technicalities.
An infinite decimal expansion isn't enough. It has to be an infinite expansion that does not contain a repeating pattern. Naively, this would require an infinite amount of information to specify a single real number in that manner, and so it's not obvious that this is a meaningful or well-founded concept at all.
The way I think of it is this:
Imagine you have a ruler. You want to cut it exactly at 10 cm mark.
Maybe you were able to cut at 10.000, but if you go more precise you'll start seeing other digits, and they will not be repeating. You just picked a real number.
Also, my intuition for why almost all numbers are irrational: if you break a ruler at any random part, and then measure it, the probability is zero that as you look at the decimal digits they are all zero or have a repeating pattern. They will basically be random digits.
A long time ago on HN, I said that I didn't like complex numbers, and people jumped all over my case. Today I don't think that there's anything wrong with them, I just get a code smell from them because I don't know if there's a more fundamental way of handling placeholder variables.
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..
A decent substitute for i is R, an explicit rotation operator. Just a change of symbol but it clears a lot of things up.
I have MS in math and came to a conclusion that C is not any more "imaginary" than R. Both are convenient abstractions, neither is particularly "natural".
How do you feel about N?
> I believe real numbers to be completely natural,
Most of real numbers are not even computable. Doesn't that give you a pause?
Why would we expect most real numbers to be computable? It's an idealized continuum. It makes perfect sense that there are way too many points in it for us to be able to compute them all.
It feels like less of an expectation and more of a: the "leap" from the rationals to the reals is a far larger one than the leap from the reals to the complex numbers. The complex numbers aren't even a different cardinality.
> for us to be able to compute them all
It's that if you pick a real at random, the odds are vanishingly small that you can compute that one particular number. That large of a barrier to human knowledge is the huge leap.
Maybe I'm getting hung up on words, but my beef is with the parent saying they find real numbers "completely natural".
It's a reasonable assumption that the universe is computable. Most reals aren't, which essentially puts them out of reach - not just in physical terms, but conceptually. If so, I struggle to see the concept as particularly "natural".
We could argue that computable numbers are natural, and that the rest of reals is just some sort of a fever dream.
The idea is we can't actually prove a non-computable real number exists without purposefully having axioms that allow for deriving non-computable things. (We can't prove they don't exist either, without making some strong assumptions).
You can go farther and say that you can't even construct real numbers without strong enough axioms. Theories of first order arithmetic, like Peano arithmetic, can talk about computable reals but not reals in general.
> The idea is we can't actually prove a non-computable real number exists without purposefully having axioms that allow for deriving non-computable things.
Sorry, what do you mean?
The real numbers are uncountable. (If you're talking about constructivism, I guess it's more complicated. There's some discussion at https://mathoverflow.net/questions/30643/are-real-numbers-co... . But that is very niche.)
The set of things we can compute is, for any reasonable definition of computability, countable.
I hold that the discovery of computation was as significant as the set theory paradoxes and should have produced a similar shift in practice. No one does naive set theory anymore. The same should have happened with classical mathematics but no one wanted to give up excluded middle, leading to the current situation. Computable reals are the ones that actually exist. Non-computable reals (or any other non-computable mathematical object) exist in the same way Russel’s paradoxical set exists, as a string of formal symbols.
Formal reasoning is so powerful you can pretend these things actually exist, but they don’t!
I see you are already familiar with subcountability so you know the rest.
What do you really mean exists - maybe you mean has something to do with a calculation in physics, or like we can possibly map it into some physical experience?
Doesn't that formal string of symbols exist?
Seems like allowing formal string of symbols that don't necessarily "exist" (or well useful for physics) can still lead you to something computable at the end of the day?
Like a meta version of what happens in programming - people often start with "infinite" objects eg `cycle [0,1] = [0,1,0,1...]` but then extract something finite out of it.
I am talking about constructivism, but that's not entirely the same as saying the reals are not uncountable. One of the harder things to grasp one's head around in logic is that there is a difference between, so to speak, what a theory thinks is true vs. what is actually true in a model of that theory. It is entirely possible to have a countable model of a theory that thinks it is uncountable. (In fact, there is a theorem that countable models of first order theories always exist, though it requires the Axiom of Choice).
I think that what matters here (and what I think is the natural interpretation of "not every real number is computable") is what the theory thinks is true. That is, we're working with internal notions of everything.
I'd agree with that for practical purposes, but sometimes the external perspective can be enlightening philosophically.
In this case, to actually prove the statement internally that "not every real number is computable", you'd need some non-constructive principle (usually added to the logical system rather than the theory itself). But, the absence of that proof doesn't make its negation provable either ("every real number is computable"). While some schools of constructivism want the negation, others prefer to live in the ambiguity.
We have too much mental baggage about what a "number" is.
Real numbers function as magnitudes or objects, while complex numbers function as coordinatizations - a way of packaging structure that exists independently of them, e.g. rotations in SO(2) together with scaling). Complex numbers are a choice of coordinates on structure that exists independently of them. They are bookkeeping (a la double‑entry accounting) not money
I don't know if this will help, but I believe that all of mathematics arises from an underlying fundamental structure to the universe and that this results in it both being "discoverable" (rather than invented) and "useful" (as in helpful for describing, expressing and calculating things).
> but I believe that all of mathematics arises from an underlying fundamental structure to the universe and that this results in it both being "discoverable" (rather than invented) and "useful" (as in helpful for describing, expressing and calculating things).
That is an interesting idea. Can you elaborate? As in, us, that is our brains live in this physical universe so we’re sort of guided towards discovering certain mathematical properties and not others. Like we intuitively visualize 1d, 2d, 3d spaces but not higher ones? But we do operate on higher dimensional objects nevertheless?
Anyway, my immediate reaction is to disagree, since in theory I can imagine replacing the universe with another with different rules and still maintaining the same mathematical structures from this universe.
Why do you believe that the same mathematical properties hold everywhere in the universe?
Not OP but I think they are making a slightly different claim — that the universe sort of dictates or guides the mathematical structure we discover. Not whether they hold everywhere or not.
Not the person you're replying too, but ... because it would be weird if they didn't.
There are legitimate questions if physical constants are constant everywhere in the universe, and also whether they are constant over time. Just because we conceive something "should" be a certain way doesn't make it true. The zero and negative numbers were also weird yet valid. How is the structure of mathematics different from fundamental constants, which we also cannot prove are invariant.
The constants don't have to be the same everywhere. It is sufficient that everywhere in the universe follows some structure and rules, that's all.
Otherwise we have a random universe, which does not seem to be the case.
One nice way of seeing the inevitability of the complex numbers is to view them as a metric completion of an algebraic closure rather than a closure of a completion.
Taking the algebraic closure of Q gives us algebraic numbers, which are a very natural object to consider. If we lived in an alternative timeline where analysis was never invented and we only thought about polynomials with rational coefficients, you’d still end up inventing them.
If you then take the metric completion of algebraic numbers, you get the complex numbers.
This is sort of a surprising fact if you think about it! the usual construction of complex numbers adds in a bunch of limit points and then solutions to polynomial equations involving those limit points, which at first glance seems like it could give a different result then adding those limit points after solutions.
It's not like I have a real answer, of course, but something flipped inside of me after hearing the following story by Aaronson. He is asking[0], why quantum amplitudes would have to be complex. I.e., can we imagine a universe, where it's not the case?
> Why did God go with the complex numbers and not the real numbers?
> Years ago, at Berkeley, I was hanging out with some math grad students -- I fell in with the wrong crowd -- and I asked them that exact question. The mathematicians just snickered. "Give us a break -- the complex numbers are algebraically closed!" To them it wasn't a mystery at all.
Apparently, you weren't one of these math grad students, and, to be fair, Aaronson is starting with the question that is somewhat opposite to yours, but still, doesn't it intuitively make sense somehow? We are modeling something. In the process of modeling something we discover functions, and algebra, and find out that we'd like to use square roots all over the place. And just that alone leads us naturally to complex numbers! We didn't start with them, we only imagined an algebra that allows us to describe some process we'd like to describe, and suddenly there's no way around complex numbers! To me, thinking this way makes it almost obvious that ℂ-numbers are "real" somehow, they are indeed the fundamental building block of some complex-enough model, while ℝ are not.
Now, I must admit, that of course it doesn't reveal to me what the fuck they actually are, how to "imagine" them in the real world. I suppose, it's the same with you. But at least it makes me quite sure that indeed this is "the shadow of something natural that we just couldn't see", and I just don't know what. I believe it to be the consequence of us currently representing all numbers somehow "wrong". Similarly to how ancient Babylonian fraction representations were preventing ancient Babylonians from asking the right questions about them.
P.S. I think I must admit, that I do NOT believe real numbers to be natural in any sense whatsoever. But this is completely besides the point.
[0] https://www.scottaaronson.com/democritus/lec9.html
I think you would enjoy (and possibly have your mind blown) this series of videos by the “Rebel Mathematician” Prof Norman Wildburger. https://youtu.be/XoTeTHSQSMU
He constructs “true” complex numbers, generalises them over finite and unbounded fields, and demonstrates how they somewhat naturally arise from 2x2 matrices in linear algebra.
Unsure this would help, but maybe thinking in English Prime could be an interesting exercice. https://en.wikipedia.org/wiki/E-Prime
As a math enjoyer who got burnt out on higher math relatively young, I have over time wondered if complex numbers aren’t just a way to represent an n-dimensional concept in n-1 dimensions.
Which makes me wonder if complex numbers that show up in physics are a sign there are dimensions we can’t or haven’t detected.
I saw a demo one time of a projection of a kind of fractal into an additional dimension, as well as projections of Sierpinski cubes into two dimensions. Both blew my mind.
Might you mean an n-dimensional concept in n/2 dimensions?
I wonder off and on if in good fiction of "when we meet aliens and start communicating using math"- should the aliens be okay with complex residue theorems? I used to feel the same about "would they have analytic functions as a separate class" until I realized how many properties of polynomials analytic functions imitate (such as no nontrivial bounded ones).
Complex numbers are just a field over 2D vectors, no? When you find "complex solutions to an equation", you're not working with a real equation anymore, you're working in C. I hate when people talk about complex zeroes like they're a "secret solution", because you're literally not talking about the same equation anymore.
There's this lack of rigor where people casually move "between" R and C as if a complex number without an imaginary component suddenly becomes a real number, and it's all because of this terrible "a + bi" notation. It's more like (a, b). You can't ever discard that second component, it's always there.
We identify the real number 2 with the rational number 2 with the integer 2 with the natural number 2. It does not seem so strange to also identify the complex number 2 with those.
If you say "this function f operates on the integers", you can't turn around and then go "ooh but it has solutions in the rationals!" No it doesn't, it doesn't exist in that space.
You can't do this for general functions, but it's fine to do in cases where the definition of f naturally embeds into the rationals. For example, a polynomial over Z is also a polynomial over Q or C.
The movement from R to C can be done rigorously. It gets hand-waved away in more application-oriented math courses, but it's done properly in higher level theoretically-focused courses. Lifting from a smaller field (or other algebraic structure) to a larger one is a very powerful idea because it often reveals more structure that is not visible in the smaller field. Some good examples are using complex eigenvalues to understand real matrices, or using complex analysis to evaluate integrals over R.
I hate when people casually move "between" Q and Z as if a rational number with unit denominator suddenly becomes an integer, and it's all because of this terrible "a/b" notation. It's more like (a, b). You can't ever discard that second component, it's always there. ;)
1. Algebra: Let's say we have a linear operator T on a real vector space V. When trying to analyze a linear operator, a key technique is to determine the T-invariant subspaces (these are subspaces W such that TW is a subset of W). The smallest non-trivial T-invariant subspaces are always 1- or 2-dimensional(!). The first case corresponds to eigenvectors, and T acts by scaling by a real number. In the second case, there's always a basis where T acts by scaling and rotation. The set of all such 2D scaling/rotation transformations are closed under addition, multiplication, and the nonzero ones are invertible. This is the complex numbers! (Correspondence: use C with 1 and i as the basis vectors, then T:C->C is determined by the value of T(1).)
2. Topology: The fact the complex numbers are 2D is essential to their fundamentality. One way I think about it is that, from the perspective of the real numbers, multiplication by -1 is a reflection through 0. But, from an "outside" perspective, you can rotate the real line by 180 degrees, through some ambient space. Having a 2D ambient space is sufficient. (And rotating through an ambient space feels more physically "real" than reflecting through 0.) Adding or multiplying by nonzero complex numbers can always be performed as a continuous transformation inside the complex numbers. And, given a number system that's 2D, you get a key topological invariant of closed paths that avoid the origin: winding number. This gives a 2D version of the Intermediate Value Theorem: If you have a continuous path between two closed loops with different winding numbers, then one of the intermediate closed loops must pass through 0. A consequence to this is the fundamental theorem of algebra, since for a degree-n polynomial f, when r is large enough then f(r*e^(i*t)) traces out for 0<=t<=2*pi a loop with winding number n, and when r=0 either f(0)=0 or f(r*e^(i*t)) traces out a loop with winding number 0, so if n>0 there's some intermediate r for which there's some t such that f(r*e^(i*t))=0.
So, I think the point is that 2D rotations and going around things are natural concepts, and very physical. Going around things lets you ensnare them. A side effect is that (complex) polynomials have (complex) roots.
> Is this the shadow of something natural that we just couldn't see, or just a convenience?
They originally arose as tool, but complex numbers are fundamental to quantum physics. The wave function is complex, the Schrödinger equation does not make sense without them. They are the best description of reality we have.
The schroedinger equation could be rewritten as two coupled equations without the need for complex numbers. Complex numbers just simplify things and "beautify it", but there is nothing "fundamental" about it, its just representation.
But if you rewrite it as "two coupled equations", you are still using complex numbers, just in another guise.
Complex numbers are just two dimensional numbers, lol
The author mentioned that the theory of the complex field is categorical, but I didn't see them directly mention that the theory of the real field isn't - for every cardinal there are many models of the real field of that size. My own, far less qualified, interpretation, is that even if the complex field is just a convenient tool for organizing information, for algebraic purposes it is as safe an abstraction as we could really hope for - and actually much more so than the real field.
The real field is categorically characterized (in second-order logic) as the unique complete ordered field, proved by Huntington in 1903. The complex field is categorically characterized as the unique algebraic closure of the real field, and also as the unique algebraically closed field of characteristic 0 and size continuum. I believe that you are speaking of the model-theoretic first-order notion of categoricity-in-a-cardinal, which is different than the categoricity remarks made in the essay.
I believe the author does talk about the first-order model theoretic perspective at one point, but yes, I was referring to that notion.
Given that you have a Ph.D. in mathematics, this might seem hopelessly elementary, but who knows--I found it intuitive and insightful: https://betterexplained.com/articles/a-visual-intuitive-guid...
Related: https://news.ycombinator.com/item?id=18310788
I've always been satisfied with the explanation "Just as you need signed numbers for translation, you need complex numbers to express rotation." Nobody asks if negative numbers are really a natural thing, so it doesn't make sense to ask if complex ones are, IMO.
> Is this the shadow of something natural that we just couldn't see
In special relativity there are solutions that allow FTL if you use imaginary numbers. But evidence suggests that this doesn’t happen.
How does your question differ from the classic question more normally applied to maths in general - does it exist outside the mind (eg platonism) or no (eg. nominalism)?
If it doesn't differ, you are in the good company of great minds who have been unable to settle this over thousands of years and should therefore feel better!
More at SEP:
https://plato.stanford.edu/entries/philosophy-mathematics/
I like to think of complex numbers as “just” the even subset of the two dimensional geometric algebra.
Almost every other intuition, application, and quirk of them just pops right out of that statement. The extensions to the quarternions, etc… all end up described by a single consistent algebra.
It’s as if computer graphics was the first and only application of vector and matrix algebra and people kept writing articles about “what makes vectors of three real numbers so special?” while being blithely unaware of the vast space that they’re a tiny subspace of.
Clifford algebras are harder to philosophically motivate than complex numbers, so you've reduced a hard problem to a harder problem.
There is such a thing of using overly simple abstractions, which can be especially tempting when there's special cases at "low `n`". This is common in the 1D, 2D and 3D cases and then falls apart as soon as something like 4D Special Relativity comes along.
This phenomenon is not precisely named, but "low-dimensional accidents", "exceptional isomorphisms", or "dimensional exceptionalism" are close.
Something that drives me up the wall -- as someone who has studied both computer science and physics -- is that the latter has endless violations of strong typing. I.e.: rotations or vibrations are invariably "swept under the rug" of complex numbers, losing clarity and generality in the process.
Maybe the bottom ~1/3, starting at "The complex field as a problem for singular terms", would be helpful to you. It gives a philosophical view of what we mean when we talk about things like the complex numbers, grounded in mathematical practice.
Maybe it is a notation issue.
What is a negative number? What is multiplication? What is a complex "number"? Complex are not even orderable. Is complex addition an overloading of the addition operator. Same with multiplication?
What i squared is -1 ? What does -1 even mean? Is the sign, a kind of operator?
The geometric interpretation help. These are transformations. Instead of 1 + i, we could/should write (1,i)
The AI might be clearer: https://gemini.google.com/share/6e00fab74749
A lot of math is not very clear because it is not very well taught. The notations are unclear. For instance, another example is: what is the difference between a matrix and a tensor? But that is another debate for anyone who wants to think about it. The definition found in books is often kind of wrong making a distinction that shouldn't really exist more often than not.
> I believe real numbers to be completely natural, but far greater mathematicians than I found them objectionable only a hundred years ago
I believe even negative numbers had their detractors
Even the counting numbers arose historically as a tool, right?
Even negative numbers and zero were objected to until a few hundred years ago, no?
I'm presuming this is old news to you, but what helped me get comfortable with ℂ was learning that it's just the algebraic closure of ℝ.
And why would R be "entitled" to an algebraic closure?
(I have a math degree, so I don't have any issues with C, but this is the kind of question that would have troubled me in high school.)
Why would N be entitled to it? We made up negative numbers and more just to have a closure. You just learn about them at an age when you don't question it yet.
When it doesn't, we yearn for something that will fill the void so that it does. It's like that note you yearn for in a musical piece that the composer seems to avoid. One yearns for a resolution of the tension.
Complex numbers offers that resolution.
> And why would R be "entitled" to an algebraic closure?
It's the birthright of every field.
The good news is that Q is not really entitled to a closure either.
Personally, no number is natural. They are probably a human construct. Mathematics does not come naturally to a human. Nowadays, it seems like every child should be able to do addition, but it was not the case in the past. The integers, rationals, and real numbers are a convenience, just like the complex numbers.
A better way to understand my point is: we need mental gymnastics to convert problems into equations. The imaginary unit, just like numbers, are a by-product of trying to fit problems onto paper. A notable example is Schrodinger's equation.
The complex numbers is just the ring such that there is an element where the element multiplied by itself is the inverse of the multiplicative identity. There are many such structures in the universe.
For example, reflections and chiral chemical structures. Rotations as well.
It turns out all things that rotate behave the same, which is what the complex numbers can describe.
Polynomial equations happen to be something where a rotation in an orthogonal dimension leaves new answers.
> In particular, they arose historically as a tool for solving polynomial equations.
That is how they started, but mathematics becomes remarkable "better" and more consistent with complex numbers.
As you say, The Fundamental Theorem of Algebra relies on complex numbers.
Cauchy's Integral Theorem (and Residue Theorem) is a beautiful complex-only result.
As is the Maximum Modulus Principle.
The Open Mapping Theorem is true for complex functions, not real functions.
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Are complex numbers really worse than real numbers? Transcendentals? Hippasus was downed for the irrationals.
I'm not sure any numbers outside the naturals exist. And maybe not even those.
I've been thinking about this myself.
First, let's try differential equations, which are also the point of calculus:
Now algebraic closure, but better: Next, general theory of fields: I think maybe differential geometry can provide some help here.