Honestly, the rigid conception is the correct one. Im of the view that i as an attribute on a number rather than a number itself, in the same way a negative sign is an attribute. Its basically exists to generalize rotations through multiplication. Instead of taking an x,y vector and multiplying it by a matrix to get rotations, you can use a complex number representation, and multiply it by another complex number to rotate/scale it. If the cartesian magnitude of the second complex number is 1, then you don't get any scaling. So the idea of x/y coordinates is very much baked in to the "imaginary attribute".
I feel like the problem is that we just assume that e^(pi*i) = -1 as a given, which makes i "feel" like number, which gives some validity to other interpretations. But I would argue that that equation is not actually valid. It arises from Taylor series equivalence between e, sin and cos, but taylor series is simply an approximation of a function by matching its derivatives around a certain point, namely x=0. And just because you take 2 functions and see that their approximations around a certain point are equal, doesn't mean that the functions are equal. Even more so, that definition completely bypasses what it means to taking derivatives into the imaginary plane.
If you try to prove this any other way besides Taylor series expansion, you really cant, because the concept of taking something to the power of "imaginary value" doesn't really have any ties into other definitions.
As such, there is nothing really special about e itself either. The only reason its in there is because of a pattern artifact in math - e^x derivative is itself, while cos and sin follow cyclic patterns. If you were to replace e with any other number, note that anything you ever want to do with complex numbers would work out identically - you don't really use the value of e anywhere, all you really care about is r and theta.
So if you drop the assumption that i is a number and just treat i as an attribute of a number like a negative sign, complex numbers are basically just 2d numbers written in a special way. And of course, the rotations are easily extended into 3d space through quaternions, which use i j an k much in the same way.
To be clear, this "disagreement" is about arbitrary naming conventions which can be chosen as needed for the problem at hand. It doesn't make any difference to results.
In particular, the core disagreement seems to be about whether the automorphisms of C should keep R (as a subset) fixed, or not.
The easy solution here would be to just have two different names: (general) automorphisms (of which there might be many) and automorphisms-that-keep-R-fixed (of which there are just the two mentioned.
If you make this distinction, then the approach of construction of C should not matter, as they are all equivalent?
To the ones objecting to "choosing a value of i" I might argue that no such choice is made. i is the square root of -1 and there is only one value of i. When we write -i that is shorthand for (-1)i. Remember the complex numbers are represented by a+bi where a and b are real numbers and i is the square root of -1. We don't bifurcate i into two distinct numbers because the minus sign is associated with b which is one of the real numbers. There is a one-to-one mapping between the complex numbers and these ordered pairs of reals.
Knowledge is the output of a person and their expertise and perspective, irreducibly. In this case, they seem to know something of what they're talking about:
> Starting 2022, I am now the John Cardinal O’Hara Professor of Logic at the University of Notre Dame.
> From 2018 to 2022, I was Professor of Logic at Oxford University and the Sir Peter Strawson Fellow at University College Oxford.
Also interesting:
> I am active on MathOverflow, and my contributions there (see my profile) have earned the top-rated reputation score.
Basically C comes up in the chain R \subset C \subset H (quaternions) \subset O (octonions) by the so-called Cayley-Dickson construction. There is a lot of structure.
The way I think of complex numbers is as linear transformations. Not points but functions on points that rotate and scale. The complex numbers are a particular set of 2x2 matrices, where complex multiplication is matrix multiplication, i.e. function composition. Complex conjugation is matrix transposition. When you think of things this way all the complex matrices and hermitian matrices in physics make a lot more sense. Which group do I fall into?
My biggest pet peeve in complex analysis is the concept of multi-value functions.
Functions are defined as relations on two sets such that each element in the first set is in relation to at most one element in the second set. And suddenly we abandon that very definitions without ever changing the notation! Complex logarithms suddenly have infinitely many values! And yet we say complex expressions are equal to something.
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.
How are there real numbers real? They're certainly not physical in a finite universe with quantised fundamental fields. I would say that natural numbers are there only physically represented ones and everything else is convenience.
My naive take is we discovered it as a math tool first but later on rediscovered it in nature when we discovered the electromagnetic field.
The electromagnetic field is naturally a single complex valued object(Riemann/Silberstein F = E + i cB), and of course Maxwell's equations collapse into a single equation for this complex field. The symmetry group of electromagnetism and more specifically, the duality rotation between E and B is U(1), which is also the unit circle in the complex plane.
> I have a real philosophical problem with complex numbers
> I believe real numbers to be completely natural
I have to say I find this perspective interesting but completely alien.
We need to have a way to find x such that x^2-2 = 0, and Q won’t cut it so we have R. (Or if you want, we need a complete ordered field so we have R)
We need to have a way to find x such that x^2+2 = 0, and R won’t cut it so we have C. (Or if you want, we need algebraic closure of R by the fundamental theorem of algebra so we need C)
I don’t really think any numbers (even “natural” numbers) are any more natural than any other kind of numbers. If you start to distinguish, where do you stop. Negative numbers are ok or not? What about zero? Is that “natural”? Mathematicians disagree about whether 0 is in N at least.
It reminds me of the famous quote from Gauss:
That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.
The real numbers have some very unreal properties. Especially, their uncountable infinite cardinality is mind boggling.
A person can have a finite number of thoughts in his live. The number of persons that have and will ever live is countably infinite, as they can be arranged in a family tree (graph). This means that the total thoughts that all of mankind ever had and will have is countably infinite. For nearly all real numbers, humankind will never have thought of them.
You can do a similar argument with the subset of real numbers than can be described in any way. With description, I do not just mean writing down digits. Sentences of the form "the limit of sequence X", "the number fulfilling equation Y", etc are also descriptions. There are a countably infinite descriptions, as at the end every description is text, yet there are uncountably many real numbers. This means that nearly no real number can even be described.
I find it hard to consider something "real" when it is not possible to describe most of it. I find equally hard when nearly no real number has been used (thought of) by humankind.
The complex extension of the rational numbers, on the other hand, feel very natural to me when I look at them as vectors in a plane.
I think the main thing people stumble over when grasping complex numbers is the term "number". Colloquially, numbers are used to order stuff. The primary function of the natural numbers is counting after all. We think of numbers as advanced counting, i.e., ordering. The complex "numbers" are not ordered though (in the sense of an ordered field). I really think that calling them "numbers" is therefore a misnomer. Numbers are for counting. Complex "numbers" cannot count, and are thus no numbers. However, they make darn good vectors.
If you view all of math as just a set of logic games with the axioms as the basic rules, then there's nothing unnatural about complex numbers. Various mathematical constructs describe various phenomena in the real world well. It just so happens that many physical systems behave in a way that can be very naturally described using complex numbers.
People thought negative numbers were weird until the 1800s or so, they arose in much the same way as a way to solve algebraic equations (or even just to balance the books, literally).
Complex numbers were always going to show up just so we could diagonalise matrices, which is an important part of solving (linear) differential equations.
I always wondered in the higher levels of maths, theoretical physics etc how much of it reflects a "real" thing and how much of is hand-wavey "try not to think about it too much but the equations work".
EG complex numbers, extra dimensions, string theory, weird particles, whatever electrons do, possibly even dark matter/energy.
Perhaps of your interest might be this work https://arxiv.org/abs/2101.10873v1 on why quantum physics needs complex numbers to work. Interesting noting though that as for solving polynomials, quantum physics might be also considered a “convenience” within the Copenhagen interpretation
C is the only way to make a field out of pairs of reals. Also (or rather just another facet of the same phenomenon) we might be interested in polynomials with integer coefficients, but some of those will have non integral roots. And we might be interested in polynomials with rational coeffs but some will not have rational roots. Same with the reals but the buck stops with the complex numbers. They are definitely not accidental they are the natural (so to speak) completion of our number system. That they exist physically in some sense is "unreasonable effectiveness" territory.
All of logic and math is a convincence tool. There are no, circles, quantities. Reality just is. We created these tools because they're a convinent way to cope with complexity of reality. There are no "objects" in a sense that chair is just atoms arranged chair-like. And atoms are just smaller particles arranged atom-like and yet physics operate in these objects treating them as something that exist.
So, now we have created these mental tools called mathematics that are heavily constrained. Then we create models that are approximately map 1:1 to some patterns that exist in reality (IE patterns that are roughly local so that we can call them objects). Due to the fact that our mental tools have heavy constrains and that we iteratively adjust these models to fit reality at focal points, we can approximately predict reality, because we already mapped the constrains into the model. But we shouldn't mistake model for the reality. Map is not territory.
I am with you on this (the challenge, not (yet) the phd), however, I myself have a far greater problem.
I do not see what’s the deal about prime numbers which seems to be more of a limitation on our end, similar to our shortage in understanding to a point we call e, π, √2 etc Irrationals.
We simply did not get the actual mathematical structure of the universe and we came up with something “good enough” that helps moving forward.
In the universe the perfect circle has perfect symmetry, hence perfect ratio, hence well-defined sweet heaven balanced harmonic entity.
Exponentials are natural phenomena. The very fact that e is its own derivative tells us we are all wrong here.
We are in an infinite escape that no matter how long we will play, and how many riddles we will solve, we will never get the entire picture.
Yes, primes are nice structure when you deal with us humans counting potatoes. But e, just e, let alone √2 or π are far more fascinating to me.
The e point cuts deep. e being its own derivative isn’t a curiosity. It’s saying that there’s a growth process so fundamental that its rate is indistinguishable from its state. That’s not a number — it’s a signature of how change works.
And yet: π, e, √2 — we only name them, define them, catch them using the integers. π is the ratio of circumference to diameter. Ratio of what to what? e is lim(1 + 1/n)^n. The integers sneak in.
Is that just our access route? Or is discreteness also woven into the fabric, alongside continuity?
My intuition led me to the following: we think our counting units (1, 2, 3, …) and fractions are the “numbers”, and when we want to refer to multi-dimensional phenomena, we use vectors or matrices or any other logical structure.
However, this is a very superficial aspect of the business, since the actual math is multi-dimensional inherently. The natural math is not linear, nor is it a plane. It is simply a multi-dimensional number system (imagine our complex numbers, but many other dimensions). Perhaps tensors or even more.
This is why we experience quantum mechanics as statistical states, results of specific measurements. We think in units, and we don’t understand things are happening in parallel across all directions.
Once we figure this out, we will understand why e, π and others are as natural as it gets, while our natural numbers are barely a dot, a point in the real math universe.
Sorry for the length but you triggered me with a long time pain point.
Ok... How about this? All (human) models of the universe are "Ptolemaic" to some degree. That is, they work but don't necessarily describe the true underlying structure ().
So it is a mistake to assume that any model is actually true.
Therefore complex numbers are just another modeling language, useful in certain contexts. All mathematics is just a modeling language.
() If you doubt this, ask yourself the question: Will the science of particle physics have changed in 100 years?
Stepping out of pure maths and into engineering we find complex numbers indispensable for describing physical systems and predicting system change over time.
I don’t have a list to hand, but there are so many areas of physics and engineering where complex numbers are the best representation of how we perceive the universe to work.
Also a PhD in math, where complex numbers are fundamental, and also part of large swaths of similar structures that are also fundamental. They fit in nicely among a ton of other similar structures and concepts, so they seem about as fundamental as sets or addition or groups or fields (and there it is).
They also seem fundamental to physical reality in a way most math concepts do not: they're required (in structure) for quantum mechanics, in many equations that seem to be part of the universe. The behavior of subatomic particles (and more precisely, QFTs), require the waveforms to evolve as complex valued functions, where the probability of an event is the magnitude of the complex value.
This has been tested between theory and experiment to about 14 decimal digits precision for QED.
I'd guess they should be considered as real as radio waves (which we don't see), as the fact things we think are solid are mostly empty space (which we don't feel), or that time flows at different rates under different situations (which we also don't experience). Yet all those things are more real than stuff our limited senses experiences.
> I believe real numbers to be completely natural, but far greater mathematicians than I found them objectionable only a hundred years ago
I suspect, as you may as while, that this quote is at the core of the matter. Identifying what you find the difference between real and complex numbers are. You are inclined to split them into separate categories. I suspect you must identify the platonic (Or HTW, if that is your metaphor) property of the real numbers which the complex lack.
I don't understand what it means for something to feel "natural". You can formally define the real numbers in multiple ways which are all isomorphic and coherent. These definitions are usually more complicated than people expect which nicely show that the real set is not a very intuitive object. Same thing for C.
There is not evidence for C. It's a construction. Obviously it shows up in physics models. They are built using mathematical formalism.
If multiple definitions turn out to be isomorphic, that's generally because there is an underlying structure linking the properties together.
Are real numbers not just "a convenience" in a sense? I do not see anything "fundamental" or "natural" about dedekind cuts or any other construction of the real numbers. If anything real numbers, to me, are more built out of the convenience of having a complete field extension of the rational numbers. We could do just fine with computable numbers and avoid a lot of problems that this line of convenience leads to.
[Obligatory: Engineering background. Not an expert]
I've always found it a bit odd that we DO define "i" to help us express complex numbers, with the convenient assumption that "i = sqrt(-1)"... but we DON'T have any such symbols to map between more than 2 dimensions.
I felt a bit better when I found out about
- (nth) roots of unity (to explore other "i"-like definitions, including things like roots of unity modulo n, and hidden abelian subgroup problems which feel a bit to me like dealing with orthogonal dimensions)
- tensors (e.g. in physics, when we need a better way to discuss more than 2 dimensions, and often establish syntactic sugar for (x,y,z,t))
IDK if that helps at all (or worse, simply betrays some misunderstanding of mine. If so, please complain- I'd appreciate the correction!)
> I doubt anyone could make a reply to this comment that would make me feel any better about it.
You may be right, but just to have said it : the Fast Fourier Transform requires complex numbers. One can write a version that avoids complex numbers, but (a) its ugliness gives away what's missing, and (b) it's significantly slower in execution.
In mathematics and physics, complex numbers aren't just "imaginary" values—they are the secret language of 2D rotation. While real numbers live on a 1D line, complex numbers inhabit a 2D plane, and multiplying them acts as a bridge between dimensions.
1. The Geometry of i
To understand how we switch dimensions, look at the imaginary unit i. In a standard real-number system, you only move left or right. Adding i introduces a vertical axis.
* The 90-degree turn: Multiplying a real number by i is geometrically equivalent to a 90° counter-clockwise rotation.
* The Dimension Switch: If you start at 1 (on the x-axis) and multiply by i, you land at i (on the y-axis). You have effectively "switched" your direction from horizontal to vertical.
2. Rotation via Euler’s Formula
The most elegant link between complex numbers and rotation is Euler’s Formula:
This formula places any complex number on a unit circle in the complex plane. When you multiply a vector by e^{i\theta}, you aren't changing its length; you are simply rotating it by the angle \theta.
Why this matters:
* Algebraic Simplicity: Instead of using messy rotation matrices (which involve four separate multiplications and additions), you can rotate a point by simply multiplying two complex numbers.
* Phase in Physics: This is why complex numbers are used in electrical engineering and quantum mechanics. A "phase shift" in a wave is just a rotation in the complex plane.
3. Beyond 2D: Quaternions
If complex numbers (a + bi) handle 2D rotations by adding one imaginary dimension, what happens if we want to rotate in 3D?
To handle 3D space without hitting "Gimbal Lock" (where two axes align and you lose a degree of freedom), mathematicians use Quaternions. These extend the concept to three imaginary units: i, j, and k.
> The Rule of Four: Interestingly, to rotate smoothly in three dimensions, you actually need a four-dimensional number system.
>
Summary Table
| Number System | Dimensions | Primary Use in Rotation |
|---|---|---|
| Real Numbers | 1D | Scaling (stretching/shrinking) |
| Complex Numbers | 2D | Planar rotation, oscillations, AC circuits |
| Quaternions | 4D | 3D computer graphics, aerospace navigation |
They can be treated as vectors, but they have "superpowers" that standard vectors do not.
1. The Similarities (The 2D Map)
In a purely visual or structural sense, a complex number z = a + bi behaves exactly like a 2D vector \vec{v} = (a, b).
* Addition: Adding two complex numbers is identical to "tip-to-tail" vector addition.
* Magnitude: The "absolute value" (modulus) of a complex number |z| = \sqrt{a^2 + b^2} is the same as the length of a vector.
* Coordinates: Both represent a point on a 2D plane.
2. The Difference: Multiplication
This is where complex numbers leave standard 2D vectors in the dust.
In standard vector algebra (like what you'd use in an introductory physics class), there isn't a single, clean way to "multiply" two 2D vectors to get another 2D vector. You have the Dot Product (which gives you a single number/scalar) and the Cross Product (which actually points out of the 2D plane into the 3D world).
Complex numbers, however, can be multiplied together to produce another complex number.
The "Rotation" Secret
When you multiply two complex numbers, the math automatically handles two things at once:
* Scaling: The lengths are multiplied.
* Rotation: The angles are added.
Standard vectors cannot do this on their own; you would need to bring in a "Rotation Matrix" to force a vector to turn. A complex number just "knows" how to turn naturally through its imaginary component.
3. When to use which?
Mathematically, complex numbers form a Field, while vectors form a Vector Space.
* Use Vectors when you are dealing with forces, velocities, or any dimension higher than 2 (like 3D space).
* Use Complex Numbers when you are dealing with things that rotate, ...
I guess I don't even really understand the objection. That's how ALL mathematics works. You specify some axioms or a construction and then reason about objects that satisfy those constraints. Some of them like the complex numbers turn out to be particularly useful.
But it's not fundamentally any different than what we do with the natural numbers. Those just feel more familiar to you.
Most commenters are talking about the first part of the post, which lays out how you might construct the complex numbers if you're interested in different properties of them. I think the last bit is the real interesting substance, which is about how to think about things like this in general (namely through structuralism), and why the observations of the first half should not be taken as an argument against structuralism. Very interesting and well written.
Real men know that infinite sets are just a tool for proving statements in Peano arithmetic, and complex numbers must be endowed with the standard metric structure, as God intended, since otherwise we cannot use them to approximate IEEE 754 floats.
I began studying 3-manifolds after coming up with a novel way I preferred to draw their presentations. All approaches are formally equivalent, but they impose different cognitive loads in practice. My approach was trivially equivalent to triangulations, or spines, or Heegaard splittings, or ... but I found myself far more nimbly able to "see" 3-manifolds my way.
I showed various colleagues. Each one would ask me to demonstrate the equivalence to their preferred presentation, then assure me "nothing to see here, move along!" that I should instead stick to their convention.
Then I met with Bill Thurston, the most influential topologist of our lifetimes. He had me quickly describe the equivalence between my form and every other known form, effectively adding my node to a complete graph of equivalences he had in his muscle memory. He then suggested some generalizations, and proposed that circle packings would prove to be important to me.
Some mathematicians are smart enough to see no distinction between any of the ways to describe the essential structure of a mathematical object. They see the object.
I was interested in how it would make sense to define complex numbers without fixing the reals, but I'm not terribly convinced by the method here. It seemed kind of suspect that you'd reduce the complex numbers purely to its field properties of addition and multiplication when these aren't enough to get from the rationals to the reals (some limit-like construction is needed; the article uses Dedekind cuts later on). Anyway, the "algebraic conception" is defined as "up to isomorphism, the unique algebraically closed field of characteristic zero and size continuum", that is, you just declare it has the same size as the reals. And of course now you have no way to tell where π is, since it has no algebraic relation to the distinguished numbers 0 and 1. If I'm reading right, this can be done with any uncountable cardinality with uniqueness up to isomorphism. It's interesting that algebraic closure is enough to get you this far, but with the arbitrary choice of cardinality and all these "wild automorphisms", doesn't this construction just seem... defective?
It feels a bit like the article's trying to extend some legitimate debate about whether fixing i versus -i is natural to push this other definition as an equal contender, but there's hardly any support offered. I expect the last-place 28% poll showing, if it does reflect serious mathematicians at all, is those who treat the topological structure as a given or didn't think much about the implications of leaving it out.
the title is a bit clickbait - mathematicians don't disagree, all the "conceptions" the article proposes agree with each other. It also seems to conflate the algebraic closure of Q (which would contain the sqrt of -1) and all of the complex numbers by insisting that the former has "size continuum". Once you have "size continuum" then you need some completion to the reals.
anyhow. I'm a bit of an odd one in that I have no problems with imaginary numbers but the reals always seemed a bit unreal to me. that's the real controversy, actually. you can start looking up definable numbers and constructivist mathematics, but that gets to be more philosophy than maths imho.
> But in fact, I claim, the smooth conception and the analytic conception are equivalent—they arise from the same underlying structure.
Conjugation isn’t complex-analytic, so the symmetry of i -> -i is broken at that level. Complex manifolds have to explicitly carry around their almost-complex structure largely for this reason.
Does anyone have any tips on how I would fundamentally understand this article without just going back to school and getting a degree in mathematics? This is the sort of article where my attempts to understand a term only ever increase the number of terms I don't understand.
PhD math guy here. Personally, I don't think this is a great article for a lay person to approach or appreciate. You might be able and interested to follow along with the various constructions of C, but without appreciation for formal logic and category theory it'll seem like distinctions without a difference (and to working mathematicians, they are). The model theory I found quite interesting, but was high effort even for me. And the philosophical points are also rather specific.
The square root of any number x is ±y, where +y = (+1)*y = y, and -y = (-1)*y.
So we define i as conforming to ±i = sqrt(-1). The element i itself has no need for a sign, so no sign needs to be chosen. Yet having defined i, we know that that i = (+1)*i = +i, by multiplicative identity.
We now have an unsigned base element for complex numbers i, derived uniquely from the expansion of <R,0,1,+,*> into its own natural closure.
We don't have to ask if i = +i, because it does by definition of the multiplicative identity.
TLDR: Any square root of -1 reduced to a single value, involves a choice, but the definition of unsigned i does not require a choice. It is a unique, unsigned element. And as a result, there is only a unique automorphism, the identity automorphism.
I really know almost nothing about complex analysis, but this sure feels like what physicists call observational entropy applied to mathematics: what counts as "order" in ℂ depends on the resolution of your observational apparatus.
The algebraic conception, with its wild automorphisms, exhibits a kind of multiplicative chaos — small changes in perspective (which automorphism you apply) cascade into radically different views of the structure. Transcendental numbers are all automorphic with each other; the structure cannot distinguish e from π. Meanwhile, the analytic/smooth conception, by fixing the topology, tames this chaos into something with only two symmetries. The topology acts as a damping mechanism, converting multiplicative sensitivity into additive stability.
I'll just add to that that if transformers are implementing a renormalization group flow, than the models' failure on the automorphism question is predictable: systems trained on compressed representations of mathematical knowledge will default to the conception with the lowest "synchronization" cost — the one most commonly used in practice.
67 comments
[ 2.8 ms ] story [ 78.5 ms ] threadThis disagreement seems above the head of non mathematicians, including those (like me) with familiarity with complex numbers
I feel like the problem is that we just assume that e^(pi*i) = -1 as a given, which makes i "feel" like number, which gives some validity to other interpretations. But I would argue that that equation is not actually valid. It arises from Taylor series equivalence between e, sin and cos, but taylor series is simply an approximation of a function by matching its derivatives around a certain point, namely x=0. And just because you take 2 functions and see that their approximations around a certain point are equal, doesn't mean that the functions are equal. Even more so, that definition completely bypasses what it means to taking derivatives into the imaginary plane.
If you try to prove this any other way besides Taylor series expansion, you really cant, because the concept of taking something to the power of "imaginary value" doesn't really have any ties into other definitions.
As such, there is nothing really special about e itself either. The only reason its in there is because of a pattern artifact in math - e^x derivative is itself, while cos and sin follow cyclic patterns. If you were to replace e with any other number, note that anything you ever want to do with complex numbers would work out identically - you don't really use the value of e anywhere, all you really care about is r and theta.
So if you drop the assumption that i is a number and just treat i as an attribute of a number like a negative sign, complex numbers are basically just 2d numbers written in a special way. And of course, the rotations are easily extended into 3d space through quaternions, which use i j an k much in the same way.
The easy solution here would be to just have two different names: (general) automorphisms (of which there might be many) and automorphisms-that-keep-R-fixed (of which there are just the two mentioned.
If you make this distinction, then the approach of construction of C should not matter, as they are all equivalent?
> Starting 2022, I am now the John Cardinal O’Hara Professor of Logic at the University of Notre Dame.
> From 2018 to 2022, I was Professor of Logic at Oxford University and the Sir Peter Strawson Fellow at University College Oxford.
Also interesting:
> I am active on MathOverflow, and my contributions there (see my profile) have earned the top-rated reputation score.
https://jdh.hamkins.org/about/
Basically C comes up in the chain R \subset C \subset H (quaternions) \subset O (octonions) by the so-called Cayley-Dickson construction. There is a lot of structure.
Functions are defined as relations on two sets such that each element in the first set is in relation to at most one element in the second set. And suddenly we abandon that very definitions without ever changing the notation! Complex logarithms suddenly have infinitely many values! And yet we say complex expressions are equal to something.
Madness.
More explicitly, it returns an equivalence class whose members are complex numbers that differ by integer multiples of 2*pi*i.
When it's important to distinguish members of the class, we speak of branches of the logarithm.
Also note the very cool and fun topology connection here. The keyword to search for is Riemann surface.
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.
https://press.princeton.edu/books/paperback/9780691133911/ne...
I ask as someone who doesn't understand as much as you, but who is charmed by such visual explanations :)
The electromagnetic field is naturally a single complex valued object(Riemann/Silberstein F = E + i cB), and of course Maxwell's equations collapse into a single equation for this complex field. The symmetry group of electromagnetism and more specifically, the duality rotation between E and B is U(1), which is also the unit circle in the complex plane.
> I believe real numbers to be completely natural
I have to say I find this perspective interesting but completely alien.
We need to have a way to find x such that x^2-2 = 0, and Q won’t cut it so we have R. (Or if you want, we need a complete ordered field so we have R)
We need to have a way to find x such that x^2+2 = 0, and R won’t cut it so we have C. (Or if you want, we need algebraic closure of R by the fundamental theorem of algebra so we need C)
I don’t really think any numbers (even “natural” numbers) are any more natural than any other kind of numbers. If you start to distinguish, where do you stop. Negative numbers are ok or not? What about zero? Is that “natural”? Mathematicians disagree about whether 0 is in N at least.
It reminds me of the famous quote from Gauss:
A person can have a finite number of thoughts in his live. The number of persons that have and will ever live is countably infinite, as they can be arranged in a family tree (graph). This means that the total thoughts that all of mankind ever had and will have is countably infinite. For nearly all real numbers, humankind will never have thought of them.
You can do a similar argument with the subset of real numbers than can be described in any way. With description, I do not just mean writing down digits. Sentences of the form "the limit of sequence X", "the number fulfilling equation Y", etc are also descriptions. There are a countably infinite descriptions, as at the end every description is text, yet there are uncountably many real numbers. This means that nearly no real number can even be described.
I find it hard to consider something "real" when it is not possible to describe most of it. I find equally hard when nearly no real number has been used (thought of) by humankind.
The complex extension of the rational numbers, on the other hand, feel very natural to me when I look at them as vectors in a plane.
I think the main thing people stumble over when grasping complex numbers is the term "number". Colloquially, numbers are used to order stuff. The primary function of the natural numbers is counting after all. We think of numbers as advanced counting, i.e., ordering. The complex "numbers" are not ordered though (in the sense of an ordered field). I really think that calling them "numbers" is therefore a misnomer. Numbers are for counting. Complex "numbers" cannot count, and are thus no numbers. However, they make darn good vectors.
Complex numbers were always going to show up just so we could diagonalise matrices, which is an important part of solving (linear) differential equations.
EG complex numbers, extra dimensions, string theory, weird particles, whatever electrons do, possibly even dark matter/energy.
So, now we have created these mental tools called mathematics that are heavily constrained. Then we create models that are approximately map 1:1 to some patterns that exist in reality (IE patterns that are roughly local so that we can call them objects). Due to the fact that our mental tools have heavy constrains and that we iteratively adjust these models to fit reality at focal points, we can approximately predict reality, because we already mapped the constrains into the model. But we shouldn't mistake model for the reality. Map is not territory.
I do not see what’s the deal about prime numbers which seems to be more of a limitation on our end, similar to our shortage in understanding to a point we call e, π, √2 etc Irrationals.
We simply did not get the actual mathematical structure of the universe and we came up with something “good enough” that helps moving forward.
In the universe the perfect circle has perfect symmetry, hence perfect ratio, hence well-defined sweet heaven balanced harmonic entity.
Exponentials are natural phenomena. The very fact that e is its own derivative tells us we are all wrong here.
We are in an infinite escape that no matter how long we will play, and how many riddles we will solve, we will never get the entire picture.
Yes, primes are nice structure when you deal with us humans counting potatoes. But e, just e, let alone √2 or π are far more fascinating to me.
The e point cuts deep. e being its own derivative isn’t a curiosity. It’s saying that there’s a growth process so fundamental that its rate is indistinguishable from its state. That’s not a number — it’s a signature of how change works. And yet: π, e, √2 — we only name them, define them, catch them using the integers. π is the ratio of circumference to diameter. Ratio of what to what? e is lim(1 + 1/n)^n. The integers sneak in. Is that just our access route? Or is discreteness also woven into the fabric, alongside continuity?
My intuition led me to the following: we think our counting units (1, 2, 3, …) and fractions are the “numbers”, and when we want to refer to multi-dimensional phenomena, we use vectors or matrices or any other logical structure.
However, this is a very superficial aspect of the business, since the actual math is multi-dimensional inherently. The natural math is not linear, nor is it a plane. It is simply a multi-dimensional number system (imagine our complex numbers, but many other dimensions). Perhaps tensors or even more. This is why we experience quantum mechanics as statistical states, results of specific measurements. We think in units, and we don’t understand things are happening in parallel across all directions. Once we figure this out, we will understand why e, π and others are as natural as it gets, while our natural numbers are barely a dot, a point in the real math universe.
Sorry for the length but you triggered me with a long time pain point.
Thanks for your comment.
So it is a mistake to assume that any model is actually true.
Therefore complex numbers are just another modeling language, useful in certain contexts. All mathematics is just a modeling language.
() If you doubt this, ask yourself the question: Will the science of particle physics have changed in 100 years?
I don’t have a list to hand, but there are so many areas of physics and engineering where complex numbers are the best representation of how we perceive the universe to work.
They also seem fundamental to physical reality in a way most math concepts do not: they're required (in structure) for quantum mechanics, in many equations that seem to be part of the universe. The behavior of subatomic particles (and more precisely, QFTs), require the waveforms to evolve as complex valued functions, where the probability of an event is the magnitude of the complex value.
This has been tested between theory and experiment to about 14 decimal digits precision for QED.
I'd guess they should be considered as real as radio waves (which we don't see), as the fact things we think are solid are mostly empty space (which we don't feel), or that time flows at different rates under different situations (which we also don't experience). Yet all those things are more real than stuff our limited senses experiences.
There's some string of research on if/how fundamental complex numbers are to QM, e.g., https://www.scientificamerican.com/article/quantum-physics-f...
I suspect, as you may as while, that this quote is at the core of the matter. Identifying what you find the difference between real and complex numbers are. You are inclined to split them into separate categories. I suspect you must identify the platonic (Or HTW, if that is your metaphor) property of the real numbers which the complex lack.
There is not evidence for C. It's a construction. Obviously it shows up in physics models. They are built using mathematical formalism.
If multiple definitions turn out to be isomorphic, that's generally because there is an underlying structure linking the properties together.
Are real numbers not just "a convenience" in a sense? I do not see anything "fundamental" or "natural" about dedekind cuts or any other construction of the real numbers. If anything real numbers, to me, are more built out of the convenience of having a complete field extension of the rational numbers. We could do just fine with computable numbers and avoid a lot of problems that this line of convenience leads to.
I've always found it a bit odd that we DO define "i" to help us express complex numbers, with the convenient assumption that "i = sqrt(-1)"... but we DON'T have any such symbols to map between more than 2 dimensions.
I felt a bit better when I found out about - (nth) roots of unity (to explore other "i"-like definitions, including things like roots of unity modulo n, and hidden abelian subgroup problems which feel a bit to me like dealing with orthogonal dimensions) - tensors (e.g. in physics, when we need a better way to discuss more than 2 dimensions, and often establish syntactic sugar for (x,y,z,t))
IDK if that helps at all (or worse, simply betrays some misunderstanding of mine. If so, please complain- I'd appreciate the correction!)
You may be right, but just to have said it : the Fast Fourier Transform requires complex numbers. One can write a version that avoids complex numbers, but (a) its ugliness gives away what's missing, and (b) it's significantly slower in execution.
Oh -- also --
e^(i Ⲡ) + 1 = 0
Nevertheless, you may be right.
In mathematics and physics, complex numbers aren't just "imaginary" values—they are the secret language of 2D rotation. While real numbers live on a 1D line, complex numbers inhabit a 2D plane, and multiplying them acts as a bridge between dimensions. 1. The Geometry of i To understand how we switch dimensions, look at the imaginary unit i. In a standard real-number system, you only move left or right. Adding i introduces a vertical axis. * The 90-degree turn: Multiplying a real number by i is geometrically equivalent to a 90° counter-clockwise rotation. * The Dimension Switch: If you start at 1 (on the x-axis) and multiply by i, you land at i (on the y-axis). You have effectively "switched" your direction from horizontal to vertical. 2. Rotation via Euler’s Formula The most elegant link between complex numbers and rotation is Euler’s Formula: This formula places any complex number on a unit circle in the complex plane. When you multiply a vector by e^{i\theta}, you aren't changing its length; you are simply rotating it by the angle \theta. Why this matters: * Algebraic Simplicity: Instead of using messy rotation matrices (which involve four separate multiplications and additions), you can rotate a point by simply multiplying two complex numbers. * Phase in Physics: This is why complex numbers are used in electrical engineering and quantum mechanics. A "phase shift" in a wave is just a rotation in the complex plane. 3. Beyond 2D: Quaternions If complex numbers (a + bi) handle 2D rotations by adding one imaginary dimension, what happens if we want to rotate in 3D? To handle 3D space without hitting "Gimbal Lock" (where two axes align and you lose a degree of freedom), mathematicians use Quaternions. These extend the concept to three imaginary units: i, j, and k. > The Rule of Four: Interestingly, to rotate smoothly in three dimensions, you actually need a four-dimensional number system. > Summary Table | Number System | Dimensions | Primary Use in Rotation | |---|---|---| | Real Numbers | 1D | Scaling (stretching/shrinking) | | Complex Numbers | 2D | Planar rotation, oscillations, AC circuits | | Quaternions | 4D | 3D computer graphics, aerospace navigation |
They can be treated as vectors, but they have "superpowers" that standard vectors do not. 1. The Similarities (The 2D Map) In a purely visual or structural sense, a complex number z = a + bi behaves exactly like a 2D vector \vec{v} = (a, b). * Addition: Adding two complex numbers is identical to "tip-to-tail" vector addition. * Magnitude: The "absolute value" (modulus) of a complex number |z| = \sqrt{a^2 + b^2} is the same as the length of a vector. * Coordinates: Both represent a point on a 2D plane. 2. The Difference: Multiplication This is where complex numbers leave standard 2D vectors in the dust. In standard vector algebra (like what you'd use in an introductory physics class), there isn't a single, clean way to "multiply" two 2D vectors to get another 2D vector. You have the Dot Product (which gives you a single number/scalar) and the Cross Product (which actually points out of the 2D plane into the 3D world). Complex numbers, however, can be multiplied together to produce another complex number. The "Rotation" Secret When you multiply two complex numbers, the math automatically handles two things at once: * Scaling: The lengths are multiplied. * Rotation: The angles are added. Standard vectors cannot do this on their own; you would need to bring in a "Rotation Matrix" to force a vector to turn. A complex number just "knows" how to turn naturally through its imaginary component. 3. When to use which? Mathematically, complex numbers form a Field, while vectors form a Vector Space. * Use Vectors when you are dealing with forces, velocities, or any dimension higher than 2 (like 3D space). * Use Complex Numbers when you are dealing with things that rotate, ...
But it's not fundamentally any different than what we do with the natural numbers. Those just feel more familiar to you.
I showed various colleagues. Each one would ask me to demonstrate the equivalence to their preferred presentation, then assure me "nothing to see here, move along!" that I should instead stick to their convention.
Then I met with Bill Thurston, the most influential topologist of our lifetimes. He had me quickly describe the equivalence between my form and every other known form, effectively adding my node to a complete graph of equivalences he had in his muscle memory. He then suggested some generalizations, and proposed that circle packings would prove to be important to me.
Some mathematicians are smart enough to see no distinction between any of the ways to describe the essential structure of a mathematical object. They see the object.
It feels a bit like the article's trying to extend some legitimate debate about whether fixing i versus -i is natural to push this other definition as an equal contender, but there's hardly any support offered. I expect the last-place 28% poll showing, if it does reflect serious mathematicians at all, is those who treat the topological structure as a given or didn't think much about the implications of leaving it out.
anyhow. I'm a bit of an odd one in that I have no problems with imaginary numbers but the reals always seemed a bit unreal to me. that's the real controversy, actually. you can start looking up definable numbers and constructivist mathematics, but that gets to be more philosophy than maths imho.
Conjugation isn’t complex-analytic, so the symmetry of i -> -i is broken at that level. Complex manifolds have to explicitly carry around their almost-complex structure largely for this reason.
So we define i as conforming to ±i = sqrt(-1). The element i itself has no need for a sign, so no sign needs to be chosen. Yet having defined i, we know that that i = (+1)*i = +i, by multiplicative identity.
We now have an unsigned base element for complex numbers i, derived uniquely from the expansion of <R,0,1,+,*> into its own natural closure.
We don't have to ask if i = +i, because it does by definition of the multiplicative identity.
TLDR: Any square root of -1 reduced to a single value, involves a choice, but the definition of unsigned i does not require a choice. It is a unique, unsigned element. And as a result, there is only a unique automorphism, the identity automorphism.
The algebraic conception, with its wild automorphisms, exhibits a kind of multiplicative chaos — small changes in perspective (which automorphism you apply) cascade into radically different views of the structure. Transcendental numbers are all automorphic with each other; the structure cannot distinguish e from π. Meanwhile, the analytic/smooth conception, by fixing the topology, tames this chaos into something with only two symmetries. The topology acts as a damping mechanism, converting multiplicative sensitivity into additive stability.
I'll just add to that that if transformers are implementing a renormalization group flow, than the models' failure on the automorphism question is predictable: systems trained on compressed representations of mathematical knowledge will default to the conception with the lowest "synchronization" cost — the one most commonly used in practice.
https://www.symmetrybroken.com/transformer-as-renormalizatio...