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.

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.

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.

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.

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".

No thank you, you can keep your R.

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

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

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

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

Would you have some examples?

(Only example that I know that might fit are quaternions, who were apparently not so useful when they were found/invented but nowdays are very useful for many 3D application/computergraphics)

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?

Can you? I can only imagine world_state(t + ε) = f(world_state(t), true_random_number_source). And even in that case we do not know if such a thing as true_random_number_source exists. The future state is either a deterministic function of the current state or it is independent of it, of which we can think as being a deterministic function of the world state and some random numbers from a true random number source. Or a mixture of the two, some things are deterministic, some things are random.

But neither being deterministic nor being random qualifies as free will for me. I get the point of compatibilists, we can define free will as doing what I want, even if that is just a deterministic function of my brain state and the environment, and sure, that kind of free will we have. But that is not the kind of free will that many people imagine, being able to make different decisions in the exact same situation, i.e. make a decision, then rewind the entire universe a bit, and make the decision again. With a different outcome this time but also not being a random outcome. I can not even tell what that would mean. If the choice is not random and also does not depend on the prior state, on what does it depend?

The closest thing I can imagine is your brain deterministically picking two possible meals from the menu based on your preferences and the environment respectively circumstances, and then flipping a coin to make the final decision. The outcome is deterministically constraint by your preferences but ultimately a random choice within those constraints. But is that what you think of as free will? The decision result depends on you, which option you even consider, but the final choice within those acceptable options does not depend on you in any way and you therefore have no control over it.

Not sure what you mean here, but non-random + non-caused is the very definition of free will. It is closely bound up with the problem of consciousness, because we need to define the "you" that has free will. It is certainly not your individual brain cells nor your organs.

But irrespective of what you define "you" to be, free will is the "you"'s ability to choose, influenced by prior state but not wholly, and also not random.

And, No, I am not talking about compatibilism.

Now describe something that is non-random and not-caused. I argue there is no such thing, i.e. caused and random are exhaustive just as zero and non-zero are, there is nothing left that could be both non-(zero) and non-(non-zero). Maybe assume such a thing exists, how is it different from caused things and random things?

[...] free will is the "you"'s ability to choose, influenced by prior state but not wholly, and also not random.

I am with you until including influenced by prior state but not wholly but what does and also not random mean? It means it depends on something, right? Something that forced the choice, otherwise it would be random and we do not want that. But just before we also said that it does not wholly depend on the prior state, so what gives?

I can only see one way out, it must depend on something that is not part of the prior state. But are we not considering everything in the universe part of the prior state? Does the you have some state that the choice can depend on but that is not considered part of the prior state of the universe? How would we justify that, leaving some piece of state out of the state of the universe?

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.

I think it’s modern science’s use of math that made people forget this.

Lovecraft captured well that feeling with Cosmic Horror. But, you know, in the 20's, 30's, 40's, scifi writters evolved. Outdated, romantic foes (specially the French and German romanticism) keep bitching over and over about the pure and 'simple' past, as if the universe had a meaning per se. And they are utterly lost. Forever.

Archimedes and Euclides won over Aristotle. Guess why. Math itself it's the Logos.

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 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 Gauss quote is what made me finally "understand" Complex Numbers as the article states; "The complex numbers are an algebraically closed field with a distinguished real coordinate structure <C,+,.,0,1,Re,Im>".

Welch Labs on Youtube has an excellent series of videos titled "Imaginary Numbers are Real" graphing the geometrical implications - https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703... (book versions can be bought at - https://www.welchlabs.com/resources)

A Short History of Complex Numbers by Orlando Merino gives the historical context (pdf) - https://kleinex.mit.edu/~dunkel/Teach/18.S996_2022S/history/...

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...

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.

The problem is that it’s not a strict total order so doesn’t order them “enough”. For a field F to be ordered it has to obey the “trichotomy” property, which is that if you have a and b in F, then exactly one of three things must be true: 1)a>b 2)b>a or 3)a = b.

If you define the ordering by modulus, then if you take, say z_1 = 1 and z_2 = i then |z_1| = |z_2| but none of the three statements in the trichotomy property are true.

[1] For a complex number z=a + b i, the modulus |z|= sqrt(a^2 + b^2). So it’s basically the distance from the origin in the complex plane.

> There are a countably infinite descriptions, as at the end every description is text

seems to hide some nuance I can't follow here. Can't a textual description be infinitely long? contain a numerical amount of operations/characters? or am I just tripping over the real/whole numbers distinction

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).

For complex numbers my gut feeling is yes, they do.

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.

What about if I have a rock and I pick up another rock that is slightly bigger. Do I now have 2 rocks or a bit more than 2 rocks? Which one of my rocks is 1? Maybe the second rock, so when I picked up the first rock I was actually wrong - I didn’t have one rock I had a little bit less than one rock. So now I have a little bit less than 2 rocks actually. How can I ever hope to do arithmetic in this physical representation?

The more I think through this physical representation thing the less sense it makes to me.

OK so say somehow I have 2 rocks in spite of all that. The room I am in also has 2 doors. What does the 2-ness of the rocks have in common with the 2-ness of the doors? You could say I can put a rock by each door (a one-to-one correspondence) and maybe that works with rocks and doors but if you take two pieces of chocolate cake and give one to each of two children you had better be sure that your pieces of chocolate cake are goddam indistinguishable or you will find that a one-to-one correspondence is not possible.

To me, numbers only make sense as a totally abstract concept.

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?

I think these questions mostly only matter when one tries to understand their own relation to these concepts, as GP asked.

Rational numbers I guess, but real numbers? Nothing physical requires numbers of which the decimal expansion is infinite and never repeating (the overwhelming majority of real numbers).

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 searched what's the smallest time unit and its also planck's time constant

The smallest unit of time is called Planck time, which is approximately 5.39 × 10⁻⁴⁴ seconds. It is theorized to be the shortest meaningful time interval that can be measured. Wikipedia (Pasted from DDG AI)

From what I can tell there can be smaller time units from these but they would be impossible to measure.

I also don't know but from this I feel as if heisenberg's principle (where you can only accurately know either velocity or position but not both at the same time) might also be applicable here?

> A physics equation might tell you a particle moves Pi meters in sqrt(2) seconds but are those even possible physical quantities

To be honest, once again (I am not a physicist) but Pi is the circumference/diameter and sqrt(2) is the length of an isoceles triangle ,I feel as if a set of experiment could be generated where a particle does indeed move pi meters in sqrt(2) meters but the thing is that both of them would be approximations in the real world.

Pi in a real world sense made up of the planck's length/planck's time in my opinion can only be measured so much. So would the sqrt(2)

The thing is, it might take infinitely minute changes which would be unmeasurable.

So what I am trying to say is that suppose we have infinite number of an machine which can have such particle which moves pi meters in sqrt(2) seconds with only infinitely minute differences. There might be one which might be accurate within all the infinite

But we literally can't know which because we physically can't measure after a point.

I think that these definitions of pi / sqrt 2 also lie in a more abstract world with useful approximations in the real world which can also change given on how much error might be okay (I have seen some jokes about engineers approximating pi as 3)

They are useful constructs which actually help in practical/engineering purposes while they still lie in a line which we can comprehend (we can put pi between 3 and 4, we can comprehend it)

Now imaginary numbers are useful constructs too and everything with practical engineering usecases too but the reason that such discussion is happening in my opinion is that they aren't intuitive because they aren't between two real numbers but rather they have a completely new line of axis/imaginary line because they don't lie anyone in the real number plane.

It's kind of scary for me to imagine what the first person who thought of imaginary numbers to be a line perpendicular to real numbers think.

It literally opened up a new dimension for mathematics and introduced plane/graph like properties and one can imagine circles/squares and so many other shapes in now pure numbers/algebra.

e^(pi * i) = -1 is one of the most (if not the most) elegant equation for a reason.

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.

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.

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.

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 do feel like when I was young or when I tried to teach some of my neighbour's daughter something once.

At some point, one just has to accept it when they are young.

It's sort of a pattern, you really can't explain it to them. You can just show them and if they don't understand, then just repeat it. You really can't explain say complex numbers or philosophy or even negative numbers or decimals.

A lot of it is visual. I see one apple and then the teacher added one more and calls it two.

Its even hard for me to explain this right now because the very sentence that I am trying to say requires me to say one and two so on and this is the very thing that the children are taught to learn. So I can't really say one apple without saying one but I think that now my point is that I couldnt have said one without seeing one apple in the first place.

Then came some half bit apples which we started calling fractions and mixed fractions and then we got taught of a magic dot to convert fractions -> decimals -> rationals -> real numbers / exponents -> complex numbers -> (??)

A lot of the times atleast in schooling I feel like one just has to accept them the way they are because you really cant get philosophical about them or necessarily have the privilege or intellectual ability to do so.

We are systematically given mental baggage about what a number is because for 99.9% use cases that's probably enough (Accounting and literally even shopkeeping or just the whole world revolves around numbers and we all know it)

I honestly don't know what I am typing right now. I am writing whatever I am thinking but I thought about that we aren't the only ones like this.

We might think we are special in this but Crows are really intelligent as well (a little funny but I saw a cronelius shorts channel and If this sort of humour entertains you, I will link their channel as well)

I searched if crows can count numbers and found this article https://www.npr.org/2024/07/18/g-s1-9773/crows-count-out-lou...

And I Found this to be pretty interesting to maybe share. Maybe even after all of this/all development made, we are still made of flesh & still similar to our peers at animal kingdom and they might be as smart as some toddlers when we were first taught what numbers are and maybe they are capable of learning these mythical abstract baggage and we humans are capable of transferring/training others with this mental baggage not necessarily even being humans (Crows in this case)

It's always sad to see how humanity ignores other animals sometimes.

We might have created weapons of mass destructions, went to moon and back but we as a society are still restricted by basic human guilts/flaws which I feel like are inevitable whether the society is large & connected creating different types of flaws & also the same when its small & hunter-gathering oriented.

It's really these issues combined with whenever some real problems comes with us that we push for the next generation and so on and so on and then later we try to find scapegoats and do wars and just struggle but once again the struggle is felt the most by a middle class or the poor.

The rules of the game of life are still/might still be fundamentally broken but we are taught to accept it when we are young in a similar fashion to numbers which might be broken too if you stare too long into them.

But I guess there's hope because the system still has love and moments of intimacy and we have improved from past, perhaps we can improve in future as well. One can be sad and depressed about current realities or if the future looks bleak. Perhaps it is, perhaps not, only future can tell but the only thing we can do right now is to hopefully stay happy and smile and just pain/suffering is a universal constant in life but maybe one can derive their own meaning of existence withstanding all these hardships and having optimism for a better future and maybe even taking actions in each of our individual ways doing what we do best, doing what we enjoy, spending time with our family/community. Maybe its a cope for a world which is flawed but maybe that's all we need to chug along and maybe leave a footprint in this world when the days are feeling down.

I don't know but lets just be kind to each other. Let's be kind to animals and humans alike. Because I feel like most of us are similar than different and sometimes we feel empty for very minor reasons in which even minor gestures from others might be enough to make us happy again. Let's try to be those others as well and maybe reach out if there's something troubling anyone.

I am really unable to explain myself but my point is that there's still beauty and life's still good even with these flaws. It's kind of like a sine wave and if one would zoom enough they would only see things flat (whether at the top of the curve or at the bottom) but in a reality both are likely. Both are part of life as-is and if one can be happy in both, and still intend to do good just for the good it might do and the sake for it itself, then I feel as if that might be the meaning of life in general.

Can we be happy in just existing? and still do our best to improve our lives and potentially others surrounding us in a community whether its small or large that's besides the point imo

I feel as if we all are in a loop keeping the system of humanity alive while maybe going through some troubles in a more isolationist period at times. We are so connected yet so disconnected at the same time in today's world. This is really the crux of so many issues I feel. We as humanity have so many paradoxical properties but a system will still work as long as not all people question it simultaneously.

I hope this message can atleast make one feel more aware & more like not being in an automatic loop of sorts and sort of snapping out of it & perhaps using this awareness for a more deeper reflection in life itself and maybe finding the will to live or forging it for yourself and periodically going to it to find one's own sense of meaning in a world of meaninglessness.

This has been cathartic for me to write even though I feel as if I might not be able to make it all positive from perhaps despair to optimism but maybe that's the point because I do feel positive in just accepting reality as-is and leaving a foot print in humanity in our own way. Maybe this message is my way of shouting in the world that "hey I exist look at me" but I hope that the deeper reason behind this is because I feel cathartic writing it and perhaps maybe it can be useful to anyone else too.

Most mathematicians see N as fundamental -- something any alien race would certainly stumble on and use as a building block for more intricate processes. I think that position is likely but not guaranteed.

N itself is already a strange beast. It arises as some sort of "completion" [0] -- an abstraction that isn't practically useful or instantiatable, only existing to make logic and computations nice. The seeming simplicity and unpredictability of primes is a weird artifact of supposedly an object designed for counting. Most subsets of N can't even be named or described in any language in finite space. Weirder still, there are uncountable objects behaving like N for all practical purposes (see first-order Peano arithmetic).

I would then have a position something along the lines of counting being fundamental but N being a convenient, messy abstraction. It's a computational tool like any of the others.

Even that though isn't a given. What says that counting is the thing an alien race would develop first, or that they wouldn't immediately abandon it for something more befitting of their understanding of reality when they advanced enough to realize the problems? As some candidate alternative substrates for building mathematics, consider:

C: This is untested (probably untestable), but perhaps C showing up everywhere in quantum mechanics isn't as strange as we think. Maybe the universe is fundamentally wavelike, and discreteness is what we perceive when waves interfere. N crops up as a projection of C onto simple boundary conditions, not as a fundamental property of the universe itself, but as an approximate way of describing some part of the universe sometimes.

Computation: Humans are input/output machines. It doesn't make sense to talk about numbers we'll physically never be able to talk about. If naturals are fundamental, why do they have so many encodings? Why do you have to specify which encoding you're using when doing proofs using N? Primes being hard to analyze makes perfect sense when you view N as a residue of some computation; you're asking how the grammatical structure of a computer program changes under multiplication of _programs_. The other paradoxes and strange behaviors of N only crop up when you start building nontrivial computations, which also makes perfect sense; of course complicated programs are complicated.

</rant>

My actual position is closer to the idea that none of it is natural, including N. It's the Russian roulette of tooling, with 99 chambers loaded in the forward direction to tackle almost any problem you care about and 1 jammed in pointing straight down at your foot when you look too closely at second-order implications and how everything ties together. Mathematical structures are real patterns in logical space, but "fundamental" is a category error. There's no objective hierarchy, just different computational/conceptual trade-offs depending on what you're trying to do.

[0] When people talk about N being fundamental, they often talk about the idea of counting and discrete objects being fundamental. You don't need N for that though; you need the first hundred, thousand, however many things. You only need N when talking about arbitrary counting processes, a set big enough to definitely describe all possible ways a person might count. You could probably get away with naturals up to 10^1000 or something as an arbitrary, finite primitive sufficient for talking about any physical, discrete process, but we've instead gone for the abstraction of a "completion" conjuring up a limiting set of all possible discrete sets.

EG complex numbers, extra dimensions, string theory, weird particles, whatever electrons do, possibly even dark matter/energy.

() If you doubt this, ask yourself the question: Will the science of particle physics have changed in 100 years?

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.

No, it really is C, not R^2. Consider product spaces, for example. C^2 ⊗ C^2 is C^4 = R^8, but R^4 ⊗ R^4 is R^16 - twice as large. So you get a ton of extra degrees of freedom with no physical meaning. You can quotient them out identifying physically equivalent states - but this is just the ordinary construction of the complex numbers as R^2/(x^2 + 1).

> but rather, physics uses C because C models the algebra of the thing physics is describing.

That's what C is: R^2, with extra algebraic structure.

Maybe it’s just my math background shouting at me about what “model” means, but if object X models object Y, then I’m going to say that X is Y. It doesn’t matter how you write it. You can write it as R^2 if you want, but there’s some additional mathematical structure here and we can recognize it as C.

Mathematicians love to come up with different ways to write the same thing. Objects like R and C are recognized as a single “thing” even though you can come up with all sorts of different ways to conceive of them. The basic approach:

1. You come up with a set of axioms which describe C,

2. You find an example of an object which follows those rules,

3. That object “is” C in almost any sense we care about, and so is any other object following the same rules.

You can pretend that the complex numbers used in quantum mechanics are just R^2 with circular symmetries. That’s fine—but in order to play that game of pretend, you have to forget some of the axioms of complex numbers in order to get there.

Likewise, we can “forget” that vectors exist and write Maxwell’s equations in terms of separate x, y, and z variables. You end up with a lot more equations—20 equations instead of 4. Or you can go in the opposite direction and discover a new formalism, geometric algebra, and rewrite Maxwell’s equation as a single equation over multivectors. (Fewer equations doesn’t mean better, I just want to describe the concept of forgetting structure in mathematics.)

You can play similar games with tensors. Does physics really use tensors, or just things that happen to transform like tensors? Well, it doesn’t matter. Anything that transforms like a tensor is actually a tensor. And anything that has the algebraic properties of C is, itself, C.

If you haven't read to the end of the post, you might be interested in the philosophical discussion it builds to. The idea there, which I ascribe to, is not quite the same as what you are saying, but related in a way, namely, that in the case that X models Y, the mathematician is only concerned with the structure that is isomorphic between them. But on the other hand, I think following "therefore X is Y" to its logical conclusion will lead you to commit to things you don't really believe.

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.

Otherwise we have a random universe, which does not seem to be the case.

What is that sufficient for?

>Otherwise we have a random universe, which does not seem to be the case.

Why jump to randomness, rather than to the possibility of undiscovered laws?

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.

Most of real numbers are not even computable. Doesn't that give you a pause?

> 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.

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.

Literally every elementary particle enters the chat to disagree. Also every cloud of smoke and each whisp of dissipated heat.

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.

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.

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.

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 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..

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.

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.

> 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 ask as someone who doesn't understand as much as you, but who is charmed by such visual explanations :)

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.

Complex numbers are just two dimensional numbers, lol

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.

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.

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.

https://press.princeton.edu/books/paperback/9780691133911/ne...

Related: https://news.ycombinator.com/item?id=18310788

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/

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.

This is also super interesting and I don't know why anyone would be uninterested in it philosophically: https://en.wikipedia.org/wiki/Classification_of_Clifford_alg...

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.

In special relativity there are solutions that allow FTL if you use imaginary numbers. But evidence suggests that this doesn’t happen.

I believe even negative numbers had their detractors

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.

Even negative numbers and zero were objected to until a few hundred years ago, no?

(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.)

Complex numbers offers that resolution.

It's the birthright of every field.

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.

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.

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.

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.

---

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.

But also, the most slick, sexy proof I know for the fundamental theorem of algebra is via complex analysis, where it's an easy consequence of Liouville's Theorem, which states that any function which is complex-differentiable and bounded on all of C must in fact be constant.

Like many other theorems in complex analysis, this is extremely surprising and has no analogue in real analysis!

First, let's try differential equations, which are also the point of calculus:

  Idea 1: The general study of PDEs uses Newton(-Kantorovich)'s method, which leads to solving only the linear PDEs,
  which can be held to have constant coefficients over small regions, which can be made into homogeneous PDEs,
  which are often of order 2, which are either equivalent to Laplace's equation, the heat equation,
  or the wave equation. Solutions to Laplace's equation in 2D are the same as holomorphic functions.
  So complex numbers again.

  Idea 2: Infinitary algebraic closure. Algebraic closure can be interpeted as saying that any rational functions can be factorised into monomials.
  We can think of the Mittag-Leffler Theorem and Weierstrass Factorisation Theorem as asserting that this is true also for meromorphic functions,
  which behave like rational functions in some infinitary sense. So the algebraic closure property of C holds in an infinitary sense as well.
  This makes sense since C has a natural metric and a nice topology.

  Idea 3: Fields of characteristic 0. Every algebraically closed field of characteristic 0 is isomorphic to R[√-1] for some real-closed field R.
  The Tarski-Seidenberg Theorem says that every FOL statement featuring only the functions {+, -, ×, ÷} which is true over the reals is
  also true over every real-closed field.

  Idea 4: Conformal geometry in 2D. A conformal manifold in 2D is locally biholomorphic to the unit disk in the complex numbers.

  Idea 5: This one I'm not 100% sure about. Take a smooth manifold M with a smoothly varying bilinear form B \in T\*M ⊗ T\*M.
  When B is broken into its symmetric part and skew-symmetric part, if we assume that both parts are never zero, B can then be seen as an almost
  complex structure, which in turn naturally identifies the manifold M as one over C.

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.

I think that actually constructing a "nontrivial" model of C using the field conception might require choosing a member from each of an infinite family of sets, i.e. it requires applying the axiom of choice, similar to the way you construct R as a vector space.

scaling -> real numbers

2d rotations and scaling -> complex numbers

3d rotations and scaling -> quaternions

In the case of quaternions, there is called double-covering, which turns out (rather than being an artefact), play fundamental role in particle physics.

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?

This is a very interesting question, and a great motivator for Galois theory, kind of like a Zen koan. (e.g. "What is the sound of one hand clapping?")

But the question is inherently imprecise. As soon as you make a precise question out of it, that question can be answered trivially.

One of the roots is 1, choosing either adjacent one as a privileged group generator means choosing whether to draw the same complex plane clockwise or counterclockwise.

1) Exactly one C

2) Exactly two isomorphic Cs

3) Infinitely many isomorphic Cs

It's not really the question of whether i and -i are the same or not. It's the question of whether this question arises at all and in which form.

Haven’t thought it through so I’m quite possibly wrong but it seems to me this implies that in such a situation you can’t have a coordinate view. How can you have two indistinguishable views of something while being able to pick one view?

– https://stackexchange.com/users/510234/joel-david-hamkins

– https://mathoverflow.net/users/1946/joel-david-hamkins

That's not the interesting part. The interesting part is that I thought everyone is the same, like me.

It was a big and surprising revelation that people love counting or algebra in just the same way I feel about geometry (not the finite kind) and feel awkward in the kind of mathematics that I like.

It's part of the reason I don't at all get the hate that school Calculus gets. It's so intuitive and beautifully geometric, what's not to like. .. that's usually my first reaction. Usually followed by disappointment and sadness -- oh no they are contemplating about throwing such a beautiful part away.

The obsession with rigor that later developed -- while necessary -- is really an "advanced topic" that shouldn't displace learning the intuition and big picture concepts. I think math up through high school should concentrate on the latter, while still being honest about the hand-waving when it happens.

calculus works... because it was almost designed for Mechanics. If the machine it's getting input, you have output. When it finished getting input, all the output you get yields some value, yes, but limits are best understood not for the result, but for the process (what the functions do).

You are not sending 0 coins to a machine, do you? You sent X to 0 coins to a machine. The machine will work from 2 to 0, but 0 itself is not included because is not a part of a changing process, it's the end.

Limits are for ranges of quantities over something.

But instead calculus is taught from fundamentals, building up from sequences. And a lot of complexity and hate comes from all those "technical" theorems that you need to make that jump from sequences to functions. E.g. things like "you can pick a converging subsequence from any bounded sequence".

In Maths classes, we started with functions. Functions as list of pairs, functions defined by algebraic expressions, functions plotted on graph papers and after that limits. Sequences were peripherally treated, just so that limits made sense.

Simultaneously, in Physics classes we were being taught using infinitesimals, with the a call back that "you will see this done more formally in your maths classes, but for intuition, infinitesimals will do for now".

(attributed to Jerry Bona)

(Yes, mathematicians really use it. It makes parity a simpler polynomial than the normal assignment).

There's no "intend to". The complex numbers are what they are regardless of us; this isn't quantum mechanics where the presence of an observer somehow changes things.

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...

Mathematicians aren’t chasing numerical solutions, they’re chasing structure. ℂ isn’t just about solving cubics, it’s about eliminating holes in algebra so the theory behaves uniformly and is easier to build upon.

And as for "changing rules" they haven't changed, they have broadened the field (literally) over which the old rules applied in a clever way to remove a restriction.

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.

This disagreement seems above the head of non mathematicians, including those (like me) with familiarity with complex numbers

The disagreement is on how much detail of the fine structure we care about. It is roughly analogous to asking whether we should care more about how an ellipse is like a circle, or how they are different. One person might care about the rigid definition and declare them to be different. Another notices that if you look at a circle at an angle, you get an ellipse. And then concludes that they are basically the same thing.

This seems like a silly thing to argue about. And it is.

However in different branches of mathematics, people care about different kinds of mathematical structure. And if you view the complex numbers through the lens of the kind of structure that you pay attention to, then ignore the parts that you aren't paying attention to, your notion of what is "basically the same as the complex numbers" changes. Just like how one of the two people previously viewed an ellipse as basically the same as a circle, because you get one from the other just by looking from an angle.

Note that each mathematician here can see the points that the other mathematicians are making. It is just that some points seem more important to you than others. And that importance is tied to what branch of mathematics you are studying.

This is really a disagreement about how to construct the complex numbers from more-fundamental objects. And the question is whether those constructions are equivalent. The author argues that two of those constructions are equivalent to each other, but others are not. A big crux of the issue, which is approachable to non-mathematicians, is whether it i and -i are fundamentally different, because arithmetically you can swap i with -i in all your equations and get the same result.

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.

To fix the coordinate structure of the complex numbers (a,b) is in effect to have made a choice of a particular i, and this is one of the perspectives discussed in the essay. But it is not the only perspective, since with that perspective complex conjugation should not count as an automorphism, as it doesn't respect the choice of i.

If you flip the plane and look at it from the bottom, then any formula written using GA operations is identical, but because you're seeing the oriented area of the pseudoscalar from behind, its as if it gains a minus sign in front.

This is equivalent to using a right-handed versus left-handed coordinate systems in 3D. The "rules of physics" remain the same either way, the labels we assign to the coordinate systems are just a convention.

For instance: if you forget the order in Q (which you can do without it stopping being a field), there is no algebraic (no order-dependent) way to distinguish between the two algebraic solutions of x^2 = 2. You can swap each other and you will not notice anything (again, assuming you "forget" the order structure).

But over the reals R, this polynomial is not irreducible. There we find that some pairs of roots have the same real value, and others don't. This leads to the idea of a "complex conjugate pair". And so some pairs of roots of the original polynomial are now different than other pairs.

That notion of a "complex conjugate pair of roots" is therefore not a purely algebraic concept. If you're trying to understand Galois theory, you have to forget about it. Because it will trip up your intuition and mislead you. But in other contexts that is a very meaningful and important idea.

And so we find that we don't just care about what concepts could be understood. We also care about what concepts we're currently choosing to ignore!

That is why the "forgetful functor" seems at first sight stupid and when you think a bit, it is genius.

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.

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.

This desire to absolutely pick one when from the purely mathematical perspective they're all equal is both ugly and harmful (as in complicates things down the line).

But couldn't we just switch the nomenclature? Instead of an oxymoronic concept of "multivalue function", we could just call it "relation of complex equivalence" or something of sorts.

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.

-2 > 1 (in C)

Which is why I prefer to leave <,> undefined in C and just take the magnitude if I want to compare complex numbers.

In more words - it's interesting, but messy:

https://en.wikipedia.org/wiki/Partial_order

https://en.wikipedia.org/wiki/Ordered_field

> The complex numbers also cannot be turned into an ordered field, as −1 is a square of the imaginary unit i.