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Comment by clintonc | Hacker News Reader

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.

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

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

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

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

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

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

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

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

Absolutely, thanks for asking!

Pure probability focuses on developing fundamental tools to work with random elements. It's applied in the sense that it usually draws upon techniques found in other traditionally pure mathematical areas, but is less applied than "applied probability", which is the development and analysis of probabilistic models, typically for real-world phenomena. It's a bit like statistics, but with more focus on the consequences of modelling assumptions rather than relying on data (although allowing for data fitting is becoming important, so I'm not sure how useful this distinction is anymore).

At the moment, using probabilistic techniques to investigate the operation of stochastic optimisers and other random elements in the training and deployment of neural networks is pretty popular, and that gets funding. But business as usual is typically looking at ecological models involving the interaction of many species, epidemiological models investigating the spread of disease, social network models, climate models, telecommunication and financial models, etc. Branching processes, Markov models, stochastic differential equations, point processes, random matrices, random graph networks; these are all the common objects used. Actually figuring out their behaviour can require all kinds of assorted techniques though, you get to pull from just about anything in mathematics to "get the job done".

In my work in academia (which I’m considering leaving), I’m very familiar with the common mathematical objects you mentioned. Where could I look for a job similar to yours? It sounds very interesting

Sorry, I'm in academia too, but my ex-colleagues who left found themselves doing nearly identical work doing MFT research at hedge funds, climate modelling at our federal weather bureau, and SciML in big tech. I know of someone doing this kind of work in telecoms too, but I haven't spoken to them lately. Having said that, it's rough out there right now. A couple of people I know looking for another job right now (academia or otherwise) with this kind of training are not having much luck...

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

No thank you, you can keep your R.

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

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

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

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

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

"that we keep encountering phenomena in reality and then realize that an existing but previously purely academic branch of math is useful for modeling it."

Would you have some examples?

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

Group theory entering quantum physics is a particularly funny example, because some established physicists at the time really hated the purely academic nature of group theory that made it difficult to learn.[1]

If you include practical applications inside computers and not just the physical reality, then Galois theory is the most often cited example. Galois himself was long dead when people figured out that his mathematical framework was useful for cryptography.

[1] https://hsm.stackexchange.com/questions/170/how-did-group-th...

Opposing view (that I happen to hold, at least if I had to choose one side or the other): not only is mathematics 'reality'; it is arguably the only thing that has a reasonable claim to being 'reality' itself.

After all, facts (whatever that means) about the physical world can only be obtained by proxy (through measurement), whereas mathematical facts are just... evident. They're nakedly apparent. Nothing is being modelled. What you call the 'model' is the object of study itself.

A denial of the 'reality' of pure mathematics would imply the claim that an alien civilisation given enough time would not discover the same facts or would even discover different – perhaps contradictory – facts. This seems implausible, excluding very technical foundational issues. And even then it's hard to believe.

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

This couldn't be further from the truth. It's not coincidence at all. The reason that mathematics inevitably ends up being 'useful' (whatever that means; it heavily depends on who you ask!) is because it's very much real. It might be somewhat 'theoretical', but that doesn't mean it's made up. It really shouldn't surprise anyone that an understanding of the most basic principles of reality turns out to be somewhat useful.

And yet another view:

Mathematics is an abstract game of symbols and rules invented by humans. It has nothing to do with reality. However it is quite useful for modelling our understanding of reality.

I think you're not even disagreeing with me, we're just using different definitions of the word "reality". I meant it to use specifically "the physical world" - which you are treating as distinct from mathematics as well in your second paragraph.

But then you must agree that it's not a coincidence.

> does this paragraph mean something in the real world?

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

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

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

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

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

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

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

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

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

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

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

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

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

And so on.

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

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

Mathematics is a game of symbols and rules.

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

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

I used to feel the same way. I now consider complex numbers just as real as any other number.

The key to seeing the light is not to try convincing yourself that complex number are "real", but to truly understand how ALL numbers are abstractions. This has indeed been a perspective that has broadened my understanding of math as a whole.

Reflect on the fact that negative numbers, fractions, even zero, were once controversial and non-intuitive, the same as complex are to some now.

Even the "natural" numbers are only abstractions: they allow us to categorize by quantity. No one ever saw "two", for example.

Another thing to think about is the very nature of mathematical existence. In a certain perspective, no objects cannot exist in math. If you can think if an object with certain rules constraining it, voila, it exists, independent of whether a certain rule system prohibit its. All that matters is that we adhere to the rule system we have imagined into being. It does not exist in a certain mathematical axiomatic system, but then again axioms are by their very nature chosen.

Now in that vein here is a deep thought: I think free will exists just because we can imagine a math object into being that is neither caused nor random. No need to know how it exists, the important thing is, assuming it exists, what are its properties?

Correct. And this is the key distinction between the mathematical approach and the everyday / business / SE approach that dominates on hacker news.

Numbers are not "real", they just happen to be isomorphic to all things that are infinite in nature. That falls out from the isomorphism between countable sets and the natural numbers.

You'll often hear novices referencing the 'reals' as being "real" numbers and what we measure with and such. And yet we categorically do not ever measure or observe the reals at all. Such thing is honestly silly. Where on earth is pi on my ruler? It would be impossible to pinpoint... This is a result of the isomorphism of the real numbers to cauchy sequences of rational numbers and the definition of supremum and infinum. How on earth can any person possibly identify a physical least upper bound of an infinite set? The only things we measure with are rational numbers.

People use terms sloppily and get themselves confused. These structures are fundamental because they encode something to do with relationships between things

The natural numbers encode things which always have something right after them. All things that satisfy this property are isomorphic to the natural numbers.

Similarly complex numbers relate by rotation and things satisfying particular rotational symmetries will behave the same way as the complex numbers. Thus we use C to describe them.

As a Zen Koan:

A novice asks "are the complex numbers real?"

The master turns right and walks away.

Very similar arguments date back to at least Plato. Ancient Greek math was based in geometry and Plato argued one could never demonstrate incommensurable lengths of rope due to physical constraints. And yet incommensurable lengths exist in math. So he said the two realms are forever divided.

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

Mathematics (and computer science) is often taught independent of philosophy, which is a loss for both fields.

Philiosophers aren't aware but Science itself and math curb-stomped most of the bullshit from philosophy and for the good.

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.

I get that philosophy was outmoded in lots of ways by modern science, but Aristotle is “the father of biology”, basically starting the discipline, so the last point is hard to understand.

I think free will exists just because we can imagine a math object into being that is neither caused nor random.

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.

> But neither being deterministic nor being random qualifies as free will for me

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.

Not sure what you mean here, but non-random + non-caused is the very definition of free will.

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?

> 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).

That's my point. The fail to exist only in a certain axiomatic system that is familiar to us. But in a certain mathematical/platonic sense there is nothing essential about that axiomatic system.

Well, what does random mean? Unpredictable, right? Why is it unpredictable? Because the outcome is not determined by anything else. [1] So random just means not determined. And instead of caused I would say determined, because caused is a pretty problematic term, but for this discussions the two should be pretty much interchangeable. And this is probably the best place to attack my argument, to point out something wrong with that. Once you agree to this, it will be a real uphill battle.

So your non-random + not-caused just says non-(non-determined) and non-determined. Now you have to pick a fight with the law of excluded middle [2]. You are saying that there exists a thing that has some property but also does not have that property. Do you see the problem? Nothing makes sense anymore, having a property no longer means having a property, everything starts falling apart.

Maybe you can resolve that problem in a clever way, but you will have to do a lot more work than saying there is some axiomatic system where this is not an issue. Which one? Or at least a proof of existence? And even if you have one, does it apply to our universe?

[1] Things may also seem random because you do not have access to the necessary state, for example a coin flip is not truly random, you just do not have detailed enough information about the initial state to predict the outcome. Or you may not know the laws or have the computing power to use the laws and that bares you from seeing the deterministic truth behind something seemingly random. But all those cases are not true randomness, they are just ignorance making things look random.

[2] https://en.wikipedia.org/wiki/Law_of_excluded_middle

Yep, the law of the excluded middle is one place to start attacking your argument, I assume you know not all philosophers accept it.

Then, you are also right that semantics intertwine with logic in a way that needs careful interrogation and is open to different perspectives. I'd be very careful making the leap you make from:

> non-random + not-caused

to:

> non-(non-determined) and non-determined.

Your arguments also contain an interesting thing to think about: True randomness. If you really think about it, true randomness should not exist. And yet we think radioactive decay at the quantum level is truly, fundamentally, irreducibly random. If that is so, here is an example of things happening that we, by definition, cannot explain in any more fundamental way.

Which is to say, the universe is not bound by the logic of our experience. In the same way we had to break out of our basic intuition about numbers to create new ones that gave us more power, in the same way we could never have logically reasoned our way into quantum mechanics and needed experimental evidence to accept something so radical, yes in the same way math does not care that our minds/logic is currently too weak to conceive of a mechanism for free will.

Here is mind twister for you: Imagine a chain of antecedents for an action. In our intuition, the chain stretches backwards infinitely. But what is it could somehow wrap around to form a ring at infinity? Analogous to the way cosmologists think the universe is not infinite in all dimensions

I was wrong about the law of excluded middle, that is not an issue. Intuitionistic logic rejects it, because it says P or not P is definitely true, whether or not we have any proof for P or not P. But that is not really relevant here, the real question is whether there are things that are neither determined nor random. If random means not determined, then no such thing can exist, unless you accept a violation of the law of noncontradiction [1]. So are random things and determined things complementary sets with respect to some universe of things under consideration?

[1] https://en.wikipedia.org/wiki/Law_of_noncontradiction

I like this approach. I especially agree with the comparison of complex numbers to negative numbers. Remember that historically, not every civilization even had a number for zero. Likewise, mathematicians struggled with a generalized solution to the Quadratic. The problem was that there were at least 6 possible equations to solve a quadratic without using negative numbers. Back then, its application was limited to area and negative numbers seemed irrelevant based on the absolute value nature of distance. It was only by abandoning our simplistic application rooted in reality that we could develop a single Quadratic Equation and with it open a new world of possibilities.

> I think free will exists just because we can imagine a math object into being that is neither caused nor random.

You can absolutely be deterministic and still believe you have free will.

> 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.

This is exactly my thoughts too !

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/...

I like to think of complex numbers as “just” the even subset of the two dimensional geometric algebra.

Almost every other intuition, application, and quirk of them just pops right out of that statement. The extensions to the quarternions, etc… all end up described by a single consistent algebra.

It’s as if computer graphics was the first and only application of vector and matrix algebra and people kept writing articles about “what makes vectors of three real numbers so special?” while being blithely unaware of the vast space that they’re a tiny subspace of.

Clifford algebras are harder to philosophically motivate than complex numbers, so you've reduced a hard problem to a harder problem.

There is such a thing of using overly simple abstractions, which can be especially tempting when there's special cases at "low `n`". This is common in the 1D, 2D and 3D cases and then falls apart as soon as something like 4D Special Relativity comes along.

This phenomenon is not precisely named, but "low-dimensional accidents", "exceptional isomorphisms", or "dimensional exceptionalism" are close.

Something that drives me up the wall -- as someone who has studied both computer science and physics -- is that the latter has endless violations of strong typing. I.e.: rotations or vibrations are invariably "swept under the rug" of complex numbers, losing clarity and generality in the process.

They're not objectively harder to motivate, just preferentially harder for people who aren't interested in them. But they're extremely interesting. They offer a surface for modelling all kinds of geometrical relationships very succinctly, semantically anyway.

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...

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.

For people who read this parent comment and are tempted to say “well of course complex numbers can be ordered, I could just define an ordering like if I have two complex numbers z_1 and z_2 I just sort them by their modulus[1].”

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.

im not very good at all this, having just a basic engineers education in maths. But the sentence

> 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

> Can't a textual description be infinitely long?

That's a good question. The usual answer is no.

The idea is that every book/description that has ever been written can be seen as string of finite length over a finite alphabet. For example, the PDF of the book is a file, i.e., a string of finite length over the byte-alphabet.

Another way to think about it, that every book/description has to have been written. Writing started some time in the past. Since then a finite amount of time has passed. Assuming one writes one character per second at most, one obtains an upper length on the number of characters in the book. This implies that it is finite in length.

That being said, one way to define the real numbers is to start with infinite sequences of rational numbers. Next, one defines when they converge against the same number. A real number x is then defined as the classes of sequences that converges against x. The set of infinite sequences of rational numbers is uncountable infinite. That's where the cardinality of the real numbers at the end of the day comes from.

The reason I bring this up is because one can view an "infinite sequence of rational numbers" as "infinitely long textual description". So your question really scratches the core of the problem.

> The number of persons that have and will ever live is countably infinite

I don't think you can say that their number is infinite. Countable, yes. But there is no rule that new people will keep spawning.

If you say that humanity will end at some point, then yes, there is only a finite number of people. This, however, does not go against my overall argument. In that case, you end up with a finite number of thoughts. That's even smaller than countable infinite and makes the real numbers even weirder.

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.

Yep. Humans (and other animals) have an inbuilt ability to count small numbers of objects, so whole numbers seem more natural to us, but it's just a bias.

For me, the complex numbers arise as the quotients of 2-dimensional vectors (which arise as translations of the 2-dimensional affine space). This means that complex numbers are equivalence classes of pairs of vectors is a 2-dimesional vector space, like 2-dimensional vectors are equivalence classes of pairs of points in a 2-dimensional affine space or rational numbers are equivalence classes of pairs of integers, or integers are equivalence classes of pairs of natural numbers, which are equivalence classes of equipotent sets.

When you divide 2 collinear 2-dimensional vectors, their quotient is a real number a.k.a. scalar. When the vectors are not collinear, then the quotient is a complex number.

Multiplying a 2-dimensional vector with a complex number changes both its magnitude and its direction. Multiplying by +i rotates a vector by a right angle. Multiplying by -i does the same thing but in the opposite sense of rotation, hence the difference between them, which is the difference between clockwise and counterclockwise. Rotating twice by a right angle arrives in the opposite direction, regardless of the sense of rotation, therefore i*i = (-i))*(-i) = -1.

Both 2-dimensional vectors and complex numbers are included in the 2-dimensional geometric algebra, whose members have 2^2 = 4 components, which are the 2 components of a 2-dimensional vector together with the 2 components of a complex number. Unlike the complex numbers, the 2-dimensional vectors are not a field, because if you multiply 2 vectors the result is not a vector. All the properties of complex numbers can be deduced from those of the 2-dimensional vectors, if the complex numbers are defined as quotients, much in the same way how the properties of rational numbers are deduced from the properties of integers.

A similar relationship like that between 2-dimensional vectors and complex numbers exists between 3-dimensional vectors and quaternions. Unfortunately the discoverer of the quaternions, Hamilton, has been confused by the fact that both vectors and quaternions have multiple components and he believed that vectors and quaternions are the same thing. In reality, vectors and quaternions are distinct things and the operations that can be done with them are very different. This confusion has prevented for many years during the 19th century the correct use of quaternions and vectors in physics (like also the confusion between "polar" vectors and "axial" vectors a.k.a. pseudovectors).

Also, with elementary math: y+ as positive exponential numbers, y- as negative. Try rotating 90 deg the axis, into the -x part. What happens?

Problem is: you have chosen an orientation (x rightwards, y upwards). That makes your choice of i/-i not canonical: as is natural, because it cannot be canonical.

It is an interesting question whether it would be possible to distinguish the 2 senses of rotation in a plane that is not embedded in a 3-dimensional space where right and left are easily distinguished. The answer seems to be no.

While in a plane, if you choose 2 orthogonal vectors, from that moment on you can distinguish clockwise from counterclockwise and -i from +i, based on the order of the 2 chosen vectors.

However, from the point of view of a 3-dimensional observer that would watch this choice, it will probably look random, i.e. the senses of rotation would either match those that the 3-dimensional observer thinks as correct, or be the opposite, and within the plane there would be no way to recognize what choice has been made.

This is no big deal. Similarly, in an affine plane there is no origin, but after you choose a particular point then you have an origin to which you can bind a vector space with a system of coordinates, where the senses of rotation are established after the choice of 2 non-collinear vectors.

In an affine plane, the choice of 1 point eliminates the symmetry of translation, then the choice of 1 vector eliminates the symmetry of rotation, and then the choice of a 2nd non-collinear vector eliminates the symmetry between the 2 senses of rotation, allowing the complete determination of a system of coordinates for the 2-dimensional vector space and also the complete determination of the associated field of complex numbers.

A question I enjoy asking myself when I'm wondering about this stuff is "if there are alien mathematicians in a distant galaxy somewhere, do they know about this?"

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

This is precisely why I’ve always lived physics, as used to something like “geography or history”.

For the reason you just stated.

In my view nonnegative real numbers have good physical representations: amount, size, distance, position. Even negative integers don't have this types of models for them. Negative numbers arise mostly as a tool for accounting, position on a directed axis, things that cancel out each other (charge). But in each case it is the structure of <R,+> and not <R,+,*> and the positive and negative values are just a convention. Money could be negative, and debt could be positive, everything would be the same. Same for electrons and protons.

So in our everyday reality I think -1 and i exist the same way. I also think that complex numbers are fundamental/central in math, and in our world. They just have so many properties and connections to everything.

> In my view nonnegative real numbers have good physical representations

In my view, that isn’t even true for nonnegative integers. What’s the physical representation of the relatively tiny (compared to ‘most integers’) Graham’s number (https://en.wikipedia.org/wiki/Graham's_number)?

Back to the reals: in your view, do reals that cannot be computed have good physical representations?

Good catch. Some big numbers are way too big to mean anything physical, or exist in any sense. (Up to our everyday experiences at least. Maybe in a few years, after the singularity, AI proves that there are infinite many small discrete structures and proves ultrafinitist mathematics false.)

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

> In my view nonnegative real numbers have good physical representations: amount, size, distance, position

I'm not a physicist, but do we actually know if distance and time can vary continuously or is there a smallest unit of distance or time? A physics equation might tell you a particle moves Pi meters in sqrt(2) seconds but are those even possible physical quantities? I'm not sure if we even know for sure whether the universe's size is infinite or finite?

I am not a physicist either but isn't the smallest unit of distance planck's length?

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.

Planck's length has absolutely no known physical significance. It is just a combination of fundamental constants that happens to have the dimension of a length.

The so called Planck system of units, proposed by him in 1899, when he computed what is now called Planck's constant, is just an example of how a system of fundamental units must not be defined. To explain exactly the mistakes done by Planck then requires a longer space than here.

Unfortunately, probably because most textbooks of physics do an extremely poor job in explaining the foundation of physics, which is the theory of the measurement of the physical quantities, most people are not aware that the Planck system of units is completely bogus, like also a few other similar attempts, like the Stoney system of units.

Thus far too often one can see on Internet people talking about the "Planck units" as if they would mean something.

Unlike with the "Planck units", there are fundamental constants that really mean something. For instance, the so called "constant of fine structure", a.k.a. Sommerfeld's constant, is the ratio between the speed of an electron and the speed of light, when the electron moves on the orbit corresponding to the lowest total energy around a nucleus of infinite mass.

This "constant of fine structure" is a measure of the strength of the electromagnetic interaction, like the Newtonian constant of gravitation is a measure of the strength of the gravitational interaction. The Planck length and time are derived from the Newtonian constant of gravitation, and they are so small because the gravitational interaction is much weaker, but they do not correspond to any quantities that could characterize a physical system.

For now, there exists no evidence whatsoever of some minimum value for length or time, i.e. there exists no evidence that time and length are not indefinitely divisible.

> In my view nonnegative real numbers have good physical representations: amount, size, distance, position.

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 should've mentioned nonnegative integers, as they correspond to the amount of discrete things.

I don't see any difference between rational numbers and reals. Their decimal expansion has nothing to do if they correspond anything physically existing or not, nor do any other difference between rationals and reals seem relevant.

That physical representation argument never made any sense to me. Like say I have a rock. I split it in two. Do I now have 2 rocks? So 2=1? Or maybe 1/2 =1 and 1+1=1.

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.

I have MS in math and came to a conclusion that C is not any more "imaginary" than R. Both are convenient abstractions, neither is particularly "natural".

How do you feel about N?

It’s only “natural” up to a point. I’ve never seen 10^100 of something so there’s that.

The number of ways to shuffle a deck with 70 cards, approximately?

"The number of x" is not "natural".

This is caveman logic and I support it.

Why? The number of ways to shuffle 5 cards is a relevant thing to talk about, and is a perfectly fine number we use all the time.

What about 6 cards? 7 cards? At what point does it become "not natural"?

Not op, but the number of cards doesn't matter. Only one shuffle can exist at a time, the "number of shuffles" is not a number of natural objects but rather a cardinality of a set. And as we know sets and cardinalities open the gates of hell.

This doesn't mean it's not a "relevant thing to talk about". It just means that these mathematical constructs while useful don't maintain a direct connection to reality, kind of like complex numbers.

> Only one shuffle can exist at a time, the "number of shuffles" is not a number of natural objects but rather a cardinality of a set.

I really don't understand what this means in practice. If there are exactly 50 rocks in front of me right now, I can't talk about 51? It doesn't maintain a direct connection to reality to talk about what would happen if I threw another rock on the pile? Or if that's connected cause another rock exists, what about if I have exactly 20 chickens, and I want to talk about what would happen when another is born? Is this "connected to reality" and "a number of natural objects"? Or "the cardinality of a set" instead?

I'm not the person you're asking, but I also have an MS in math and the same opinions.

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.

N pretty much is "arbitrary-length information theory". As soon as you leave the realm of the finite, you end up with N. I'm not convinced that any alien civilization could get very far mathematically or computationally without reinventing N somewhere, even if unintentionally (e.g, how does one state the halting problem).

Natural numbers are "natural" enough but N as the "set of all natural numbers" not so much. It only takes N to build the Hilbert's hotel. Uncountable set of all subsets of N is probably even worse.

All that, of course, doesn't make N bad or useless. It just shows that mathematical objects don't have to follow the laws or intuition of the real world to be useful in the real world.

We have too much mental baggage about what a "number" is.

Real numbers function as magnitudes or objects, while complex numbers function as coordinatizations - a way of packaging structure that exists independently of them, e.g. rotations in SO(2) together with scaling). Complex numbers are a choice of coordinates on structure that exists independently of them. They are bookkeeping (a la double‑entry accounting) not money

> We have too much mental baggage about what a "number" is.

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.

> I believe real numbers to be completely natural

You can teach middle school children how to define complex numbers, given real numbers as a starting point. You can't necessarily even teach college students or adults how to define real numbers, given rational numbers as a starting point.

well it's hard to formally define them, but it's not hard to say "imagine that all these decimals go on forever" and not worry about the technicalities.

An infinite decimal expansion isn't enough. It has to be an infinite expansion that does not contain a repeating pattern. Naively, this would require an infinite amount of information to specify a single real number in that manner, and so it's not obvious that this is a meaningful or well-founded concept at all.

The way I think of it is this:

Imagine you have a ruler. You want to cut it exactly at 10 cm mark.

Maybe you were able to cut at 10.000, but if you go more precise you'll start seeing other digits, and they will not be repeating. You just picked a real number.

Also, my intuition for why almost all numbers are irrational: if you break a ruler at any random part, and then measure it, the probability is zero that as you look at the decimal digits they are all zero or have a repeating pattern. They will basically be random digits.

> 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.

A reasonably defensible inference would be that adding a finite amount of precision adds a finite number of additional digits. That is a physically realizable operation. There's no obvious physical meaning to the idea of repeating that operation infinitely many times, so this is not clearly a meaningful way of defining or constructing real numbers. If you were trying to use this construction to convince a skeptic that irrational real numbers exist, you would fail -- they would simply retort that arbitrary finite precision exists and that you have failed to demonstrate infinite non-repeating, non-terminating precision.

I don't quite get what you mean here. While you need to allow infinite expansions without repeating patterns, you also need to expansions with these pattern to get all reals. Maybe the most difficult part is to explain why 0.(9) and 1 should be the same, though, while no such identification happens for repeating patterns that are not (9).

What are you talking about? Infinite decimals give reals, do they not? Repeating decimals give rational which are a subset of the reals.

The colloquial phrase 'infinite decimal' is perfectly intelligible without reference to whether it's an infinite amount of data or rigorously defined or whatever else.

There's a lot of trickery involved din dealing with the reals formally but they're still easy to conceptualize intuitively.

“What I’m taking about” is that they are not easy to conceptualize intuitively.

If I were a skeptic of real numbers, I’d tell you that talking about an infinite decimal expansion that never terminated and contains no repeating pattern is nonsense. I’d say such a thing doesn’t exist, because you can’t specify a single example by writing down its decimal expansion — by definition. So if that’s the only idea you have to convince a skeptic, you’ve already failed and are out of the game. To convince the skeptic, you’d have to develop a more sophisticated method to show indirectly an example of a real number that is not rational (for instance, perhaps by proving that, should sqrt(2) exist, it cannot be rational).

I guess we are talking about different things. It seems to me that it's trivial to imagine then conceptually. They go on forever and most of them never repeat? Sounds good to me. Sqrt(2) never repeats? sure, whatever. I never found the proofs of this stuff very interesting.

Now, I am a skeptic of their use in physics / science. But that's a different question, and more about pedagogy than the raw content of the theories.

With that approach, all anyone has to say is that you'd have to provide infinite information to specify an example and that the way these objects interact is completely undefined; therefore you haven't defined or done anything at all. You are indeed simply imagining something -- and nothing more. You can imagine whatever you want, but nobody else is inclined to believe that what you imagine exists or behaves in the intended manner.

Beyond that, if a skeptic were inclined to accept the existence of objects with "infinite information content" by definition, they could then ask you to simply add two of them together. That would most likely be the end of it -- trying to add infinite non-repeating decimal expansions does not act intuitively. To answer this type of question in general, you would have to prove that the set of all infinite decimal expansions, if we grant its existence, has a property called completeness, as you would eventually discover that you would have to define addition x+y of these numbers as a limit: x+y = lim_{k -> infinity} (x_k+y_k) where {x,y}_k = the rational number obtained by truncating {x,y} after k digits. You must prove this limit always exists and is unique and well-defined. And even having done all that work, you still couldn't give a single example of one of these numbers without additional nontrivial work, so a skeptic could still easily reject all of this.

This is far beyond what you could reasonably expect the typical middle school student or even general member of the adult population to follow and far more difficult than simply defining complex numbers as having the form x+iy.

yes, I am describing imagining something. Imagine taking decimals and letting them go on without ending. That is conceptualizing them intuitively. It is easy.

I don't really know what you're arguing about. You are describing the sorts of things that have to be solved to construct them rigorously. But I don't know why. No one is talking about that.

I was talking about that, specifically, the relative difficulty of defining reals from rationals vs complex numbers from reals. You replied to me. :)

Moreover, I disagree that you have imagined real numbers. I don’t think you’ve imagined a single real number at all in the manner you describe. Why should I believe you've even described anything that isn't rational to begin with? For instance, 0.999... is the same as 1. Why should I not think that whatever decimal expansion you're imagining is, similarly, equivalent to a rational number we already know about? Occam's razor would reasonably suggest you're just imagining different representations of objects already accounted for in the rationals. After all, an infinite amount of precision captured by an infinite nonrepreating string of digits could easily just converge back to a number we already know.

I am very confused why you are continually talking about rationals as if they are not real. every real number is also a rational number, in the usual conception of things, are they not? Perhaps you are distinguishing the two? like regarding 1.000 as an equivalence classes of cauchy sequences is not the same as 1.000 as the equivalence class of a/a?

because when I picture 1.000 I am clearly imagining a real number. Likewise if I imagine pi, as defined any way you like.

My language was sloppy, but I'll admit I thought it was pretty obvious that we were talking about defining the rest of the reals starting from the rationals -- obvious enough that it didn't need clarification. I can't edit my prior comment, but you may imagine it has been amended in the obvious way with that clarity made explicit rather than implicit and reply to it again if you're interested in continuing the conversation.

I don't know if this will help, but I believe that all of mathematics arises from an underlying fundamental structure to the universe and that this results in it both being "discoverable" (rather than invented) and "useful" (as in helpful for describing, expressing and calculating things).

Why do you believe that the same mathematical properties hold everywhere in the universe?

Not the person you're replying too, but ... because it would be weird if they didn't.

There are legitimate questions if physical constants are constant everywhere in the universe, and also whether they are constant over time. Just because we conceive something "should" be a certain way doesn't make it true. The zero and negative numbers were also weird yet valid. How is the structure of mathematics different from fundamental constants, which we also cannot prove are invariant.

The constants don't have to be the same everywhere. It is sufficient that everywhere in the universe follows some structure and rules, that's all.

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

> It is sufficient that everywhere in the universe follows some structure and rules, that's all.

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?

What the heck, how is "undiscovered laws" different from "structure and rules"?

Not OP but I think they are making a slightly different claim — that the universe sort of dictates or guides the mathematical structure we discover. Not whether they hold everywhere or not.

> but I believe that all of mathematics arises from an underlying fundamental structure to the universe and that this results in it both being "discoverable" (rather than invented) and "useful" (as in helpful for describing, expressing and calculating things).

That is an interesting idea. Can you elaborate? As in, us, that is our brains live in this physical universe so we’re sort of guided towards discovering certain mathematical properties and not others. Like we intuitively visualize 1d, 2d, 3d spaces but not higher ones? But we do operate on higher dimensional objects nevertheless?

Anyway, my immediate reaction is to disagree, since in theory I can imagine replacing the universe with another with different rules and still maintaining the same mathematical structures from this universe.

> I doubt anyone could make a reply to this comment that would make me feel any better about it.

I am also a complex number skeptic. The position I've landed on is this.

1) complex numbers are probably used for far more purposes across math than they "ought" to be, because people don't have the toolbox to talk about geometry on R^2 but they do know C so they just use C. In particular, many of the interesting things about complex analysis are probably just the n=2 case of more general constructions that can be done by locating R inside of larger-dimensional algebras.

2) The C that shows up in quantum mechanics is likely an example of this--it's a case of physics having a a circular symmetry embedded in it (the phase of the wave functions) and everyone getting attached to their favorite way of writing it. (Ish. I'm not sure how the square the fact that wave functions add in superposition. but anyway it's not going to be like "physics NEEDS C", but rather, physics uses C because C models the algebra of the thing physics is describing.

3) C is definitely intrinsic in a certain sense: once you have polynomials in R, a natural thing to do is to add a sqrt(-1). This is not all that different conceptually from adding sqrt(2), and likely any aliens we ever run into will also have done the same thing.

I can't entirely follow the details, but apparently quantum mechanics actually doesn't work for fields other than C, including quaternions. https://scottaaronson.blog/?p=4021

That makes sense, but it assumes that the thing you would replace C with is a field. If physics' C is sitting inside a larger space I imagine that that space will not be a field (probably a lie group or something instead).

> 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.

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 object X models object Y, then I’m going to say that X is Y

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.

> 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.

I would love to hear an example… but before you do, I’m going to clarify that my statement was expressing a notion of what “is” sometimes means to a mathematician, and caution that

1. This notion is contextual, that sometimes we use the word “is” differently, and

2. It requires an understanding of “forgetfulness”.

So if I say that “Cauchy sequences in Q is R” and “Dedekind cuts is R”, you have to forget the structure not implied by R. In a set-theoretic sense, the two constructions are unequal, because you use constructed different sets.

I think this weird notion of “is” is the only sane way to talk about math. YMMV.

I think the problem with insisting on using "is" that way is that you then can't distinguish between two things you might reasonably want to express, i.e. "is isomorphic to"/"has the same structure as" and "refers to the same object". I totally agree that math is all about forgetting about the features of your objects that are not relevant to your problem (and in particular as the post argues things like R and C do not refer to any concrete construction but rather to their common structure), but if you want to describe that position you have to be able to distinguish between equality and isomorphism.

(Of course using "is" that way in informal discussion among mathematicians is fine -- in that case everyone is on the same page about what you mean by it usually)

> I think the problem with insisting on using "is" that way is that you then can't distinguish between two things you might reasonably want to express, i.e. "is isomorphic to"/"has the same structure as" and "refers to the same object".

It’s reasonable to want to express that difference in specific circumstances, but it would be completely unreasonable to make this the default.

For example, I can say that Z is a subset of Q, and Q is a subset of R. I can do this, but maybe you cannot—you’ve expressed a preference for a more rigid and inflexible terminology, and I don’t think you’re prepared to deal with the consequences.

Tensor are much less unequivocal to me. They seem to follow naturally from basic geometric considerations. C on the other hand is definitely i there but I'm not sure it's the best way to write or conceptualize what it's doing.

*much more, oops

I think the issue with "modeling" is really a human one, not a mathematical one.

It's helpful sometimes to think of our collective body of mathematical knowledge as like a "codebase", and our notations and concepts as the "interface" to the abstractions at play within. Any software engineer would immediately acknowledge that some interfaces are FAR better than others.

The complex numbers numbers are one interface to the thing they model, and as you say, in a certain sense, it may be the case that the thing is C. But other interfaces exist: 2x2 antisymmetric traceless matrices, or a certain bivector in the geometric-algebra sense.

Different interfaces: a) suggest different extensions, b) interface with other abstractions more or less naturally, c) lend themselves to different physical interpretations d) compress the "real" information of the abstraction to different degrees.

An example of (a): when we first learn about electric and magnetic fields we treat them both as the "same kind of thing"—vector fields—only to later find they are not (B is better thought of as bivector field, or better still, both are certain components of dA). The first hint is their different properties under reflections and rotations. "E and B are both vector fields" is certainly an abstraction you CAN use, but it is poorly-matched to the underlying abstraction and winds up with a bunch of extra epicycles.

Of (d): you could of course write all of quantum mechanics with `i` replaced by a 2x2 rotation matrix. (This might be "matrix mechanics", I'm not sure?) This gives you many more d.o.f. than you need, and a SWE-minded person would come in and say: ah, see, you should make invalid states unrepresentable. Here, use this: `i = (0 -1; 1 0)`. An improvement!

Of (b): the Pauli matrices, used for spin-1/2 two-state systems, represent the quaternions. Yet here we don't limit ourselves to `{1, i, j, k}`; we prefer a 2-state representation—why? Because (IIRC) the 2 states emerge intuitively from the physical problems which lead to 2-state systems; because the 2 states mix in other reference frames; things like that (I can't really remember). Who's to say something similar doesn't happen with the 2 states of the phase `i`, but that it's obscured by our abstraction? (Probably it isn't, but, prove it!)

I have not given it much more thought than this, but, I find that this line of thinking places the "discontent with the complex numbers in physics" a number of people in this thread attest to in a productive light. That dissatisfaction is with the interface of the abstraction: why? Where was the friction? In what way does it feel unnecessarily mystifying, or unparsimonious?

Of course, the hope is that something physical is obscured by the abstraction: that we learn something new by viewing the problem in another frame, and might realize, say, that the interface we supposed to be universally applicable actually ceases to work in some interesting case, and turns out to explain something new.

> 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

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.

Yes I know and agree with that. But still I think physics can be described with either. There will, I expect be a physical meaning to that quotient. Maybe the larger space without the quotient is also physically meaningful too.

> I believe real numbers to be completely natural,

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

Why would we expect most real numbers to be computable? It's an idealized continuum. It makes perfect sense that there are way too many points in it for us to be able to compute them all.

The idea is we can't actually prove a non-computable real number exists without purposefully having axioms that allow for deriving non-computable things. (We can't prove they don't exist either, without making some strong assumptions).

> The idea is we can't actually prove a non-computable real number exists without purposefully having axioms that allow for deriving non-computable things.

Sorry, what do you mean?

The real numbers are uncountable. (If you're talking about constructivism, I guess it's more complicated. There's some discussion at https://mathoverflow.net/questions/30643/are-real-numbers-co... . But that is very niche.)

The set of things we can compute is, for any reasonable definition of computability, countable.

I am talking about constructivism, but that's not entirely the same as saying the reals are not uncountable. One of the harder things to grasp one's head around in logic is that there is a difference between, so to speak, what a theory thinks is true vs. what is actually true in a model of that theory. It is entirely possible to have a countable model of a theory that thinks it is uncountable. (In fact, there is a theorem that countable models of first order theories always exist, though it requires the Axiom of Choice).

I think that what matters here (and what I think is the natural interpretation of "not every real number is computable") is what the theory thinks is true. That is, we're working with internal notions of everything.

I'd agree with that for practical purposes, but sometimes the external perspective can be enlightening philosophically.

In this case, to actually prove the statement internally that "not every real number is computable", you'd need some non-constructive principle (usually added to the logical system rather than the theory itself). But, the absence of that proof doesn't make its negation provable either ("every real number is computable"). While some schools of constructivism want the negation, others prefer to live in the ambiguity.

I hold that the discovery of computation was as significant as the set theory paradoxes and should have produced a similar shift in practice. No one does naive set theory anymore. The same should have happened with classical mathematics but no one wanted to give up excluded middle, leading to the current situation. Computable reals are the ones that actually exist. Non-computable reals (or any other non-computable mathematical object) exist in the same way Russel’s paradoxical set exists, as a string of formal symbols.

Formal reasoning is so powerful you can pretend these things actually exist, but they don’t!

I see you are already familiar with subcountability so you know the rest.

What do you really mean exists - maybe you mean has something to do with a calculation in physics, or like we can possibly map it into some physical experience?

Doesn't that formal string of symbols exist?

Seems like allowing formal string of symbols that don't necessarily "exist" (or well useful for physics) can still lead you to something computable at the end of the day?

Like a meta version of what happens in programming - people often start with "infinite" objects eg `cycle [0,1] = [0,1,0,1...]` but then extract something finite out of it.

They don’t exist as concepts. A rational number whose square is 2 is (convenient prose for) a formal symbol describing some object. It happens that it does not describe any object. I am claiming that many objects described after the explosion of mathematics while putting calculus on a firmer foundation to resolve infinitesimals do not exist.

List functions like that need to be handled carefully to ensure termination. Summations of infinite series deal are a better example, consider adding up a geometric series. You need to add “all” the terms to get the correct result.

Of course you don’t actually add all the terms, you use algebra to determine a value.

You can go farther and say that you can't even construct real numbers without strong enough axioms. Theories of first order arithmetic, like Peano arithmetic, can talk about computable reals but not reals in general.

Maybe I'm getting hung up on words, but my beef is with the parent saying they find real numbers "completely natural".

It's a reasonable assumption that the universe is computable. Most reals aren't, which essentially puts them out of reach - not just in physical terms, but conceptually. If so, I struggle to see the concept as particularly "natural".

We could argue that computable numbers are natural, and that the rest of reals is just some sort of a fever dream.

> It's a reasonable assumption that the universe is computable

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

It feels like less of an expectation and more of a: the "leap" from the rationals to the reals is a far larger one than the leap from the reals to the complex numbers. The complex numbers aren't even a different cardinality.

> for us to be able to compute them all

It's that if you pick a real at random, the odds are vanishingly small that you can compute that one particular number. That large of a barrier to human knowledge is the huge leap.

One nice way of seeing the inevitability of the complex numbers is to view them as a metric completion of an algebraic closure rather than a closure of a completion.

Taking the algebraic closure of Q gives us algebraic numbers, which are a very natural object to consider. If we lived in an alternative timeline where analysis was never invented and we only thought about polynomials with rational coefficients, you’d still end up inventing them.

If you then take the metric completion of algebraic numbers, you get the complex numbers.

This is sort of a surprising fact if you think about it! the usual construction of complex numbers adds in a bunch of limit points and then solutions to polynomial equations involving those limit points, which at first glance seems like it could give a different result then adding those limit points after solutions.

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.

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...

> In particular, they arose historically as a tool for solving polynomial equations.

That is how they started, but mathematics becomes remarkable "better" and more consistent with complex numbers.

As you say, The Fundamental Theorem of Algebra relies on complex numbers.

Cauchy's Integral Theorem (and Residue Theorem) is a beautiful complex-only result.

As is the Maximum Modulus Principle.

The Open Mapping Theorem is true for complex functions, not real functions.

---

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.

As you say, "the fundamental theorem of algebra relies on complex numbers" gets to the heart of the view that complex numbers are the algebraic closure of R.

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!

The author mentioned that the theory of the complex field is categorical, but I didn't see them directly mention that the theory of the real field isn't - for every cardinal there are many models of the real field of that size. My own, far less qualified, interpretation, is that even if the complex field is just a convenient tool for organizing information, for algebraic purposes it is as safe an abstraction as we could really hope for - and actually much more so than the real field.

The real field is categorically characterized (in second-order logic) as the unique complete ordered field, proved by Huntington in 1903. The complex field is categorically characterized as the unique algebraic closure of the real field, and also as the unique algebraically closed field of characteristic 0 and size continuum. I believe that you are speaking of the model-theoretic first-order notion of categoricity-in-a-cardinal, which is different than the categoricity remarks made in the essay.

I believe the author does talk about the first-order model theoretic perspective at one point, but yes, I was referring to that notion.

A long time ago on HN, I said that I didn't like complex numbers, and people jumped all over my case. Today I don't think that there's anything wrong with them, I just get a code smell from them because I don't know if there's a more fundamental way of handling placeholder variables.

I get the same feeling when I think about monads, futures/promises, reactive programming that doesn't seem to actually watch variables (React.. cough), Rust's borrow checker existing when we have copy-on-write, that there's no realtime garbage collection algorithm that's been proven to be fundamental (like Paxos and Raft were for distributed consensus), having so many types of interprocess communication instead of just optimizing streams and state transfer, having a myriad of GPU frameworks like Vulkan/Metal/DirectX without MIMD multicore processors to provide bare-metal access to the underlying SIMD matrix math, I could go on forever.

I can talk about why tau is superior to pi (and what a tragedy it is that it's too late to rewrite textbooks) but I have nothing to offer in place of i. I can, and have, said a lot about the unfortunate state of computer science though: that internet lottery winners pulled up the ladder behind them rather than fixing fundamental problems to alleviate struggle.

I wonder if any of this is at play in mathematics. It sure seems like a lot of innovation comes from people effectively living in their parents' basements, while institutions have seemingly unlimited budgets to reinforce the status quo..

A decent substitute for i is R, an explicit rotation operator. Just a change of symbol but it clears a lot of things up.

As a math enjoyer who got burnt out on higher math relatively young, I have over time wondered if complex numbers aren’t just a way to represent an n-dimensional concept in n-1 dimensions.

Which makes me wonder if complex numbers that show up in physics are a sign there are dimensions we can’t or haven’t detected.

I saw a demo one time of a projection of a kind of fractal into an additional dimension, as well as projections of Sierpinski cubes into two dimensions. Both blew my mind.

Might you mean an n-dimensional concept in n/2 dimensions?

The complex numbers is just the ring such that there is an element where the element multiplied by itself is the inverse of the multiplicative identity. There are many such structures in the universe.

For example, reflections and chiral chemical structures. Rotations as well.

It turns out all things that rotate behave the same, which is what the complex numbers can describe.

Polynomial equations happen to be something where a rotation in an orthogonal dimension leaves new answers.

I think you would enjoy (and possibly have your mind blown) this series of videos by the “Rebel Mathematician” Prof Norman Wildburger. https://youtu.be/XoTeTHSQSMU

He constructs “true” complex numbers, generalises them over finite and unbounded fields, and demonstrates how they somewhat naturally arise from 2x2 matrices in linear algebra.

I wonder off and on if in good fiction of "when we meet aliens and start communicating using math"- should the aliens be okay with complex residue theorems? I used to feel the same about "would they have analytic functions as a separate class" until I realized how many properties of polynomials analytic functions imitate (such as no nontrivial bounded ones).

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.

> Is this the shadow of something natural that we just couldn't see, or just a convenience?

They originally arose as tool, but complex numbers are fundamental to quantum physics. The wave function is complex, the Schrödinger equation does not make sense without them. They are the best description of reality we have.

The schroedinger equation could be rewritten as two coupled equations without the need for complex numbers. Complex numbers just simplify things and "beautify it", but there is nothing "fundamental" about it, its just representation.

But if you rewrite it as "two coupled equations", you are still using complex numbers, just in another guise.

Complex numbers are just two dimensional numbers, lol

I've been thinking about this myself.

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.

Now algebraic closure, but better:

  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.

Next, general theory of fields:

  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.

I think maybe differential geometry can provide some help here.

  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.

Complex numbers are just a field over 2D vectors, no? When you find "complex solutions to an equation", you're not working with a real equation anymore, you're working in C. I hate when people talk about complex zeroes like they're a "secret solution", because you're literally not talking about the same equation anymore.

There's this lack of rigor where people casually move "between" R and C as if a complex number without an imaginary component suddenly becomes a real number, and it's all because of this terrible "a + bi" notation. It's more like (a, b). You can't ever discard that second component, it's always there.

We identify the real number 2 with the rational number 2 with the integer 2 with the natural number 2. It does not seem so strange to also identify the complex number 2 with those.

If you say "this function f operates on the integers", you can't turn around and then go "ooh but it has solutions in the rationals!" No it doesn't, it doesn't exist in that space.

You can't do this for general functions, but it's fine to do in cases where the definition of f naturally embeds into the rationals. For example, a polynomial over Z is also a polynomial over Q or C.

I hate when people casually move "between" Q and Z as if a rational number with unit denominator suddenly becomes an integer, and it's all because of this terrible "a/b" notation. It's more like (a, b). You can't ever discard that second component, it's always there. ;)

Yes, you're right. You can't say your function operates in Z "but has solutions in Q". That's what people are doing when they take a real function and go "ooh look, secret complex solutions!"

The movement from R to C can be done rigorously. It gets hand-waved away in more application-oriented math courses, but it's done properly in higher level theoretically-focused courses. Lifting from a smaller field (or other algebraic structure) to a larger one is a very powerful idea because it often reveals more structure that is not visible in the smaller field. Some good examples are using complex eigenvalues to understand real matrices, or using complex analysis to evaluate integrals over R.

Sure, you can say that complex numbers are 2-tuples with a special rule for multiplication (a, b) * (c, d) = (ac - bd, ad + bc). Same way you can define rationals as 2-tuples with a special rule for addition (a, b) + (c, d) = ((ad + cb) / gcd(b, d), bd / gcd(b, d)).

But I think this way you'd lose insight as to where these rules really come from. The rule for complex multiplication is the way it is, precisely because it gives you an algebra that works as if you were manipulating a quantity that squared to -1.

Unsure this would help, but maybe thinking in English Prime could be an interesting exercice. https://en.wikipedia.org/wiki/E-Prime

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.

1. Algebra: Let's say we have a linear operator T on a real vector space V. When trying to analyze a linear operator, a key technique is to determine the T-invariant subspaces (these are subspaces W such that TW is a subset of W). The smallest non-trivial T-invariant subspaces are always 1- or 2-dimensional(!). The first case corresponds to eigenvectors, and T acts by scaling by a real number. In the second case, there's always a basis where T acts by scaling and rotation. The set of all such 2D scaling/rotation transformations are closed under addition, multiplication, and the nonzero ones are invertible. This is the complex numbers! (Correspondence: use C with 1 and i as the basis vectors, then T:C->C is determined by the value of T(1).)

2. Topology: The fact the complex numbers are 2D is essential to their fundamentality. One way I think about it is that, from the perspective of the real numbers, multiplication by -1 is a reflection through 0. But, from an "outside" perspective, you can rotate the real line by 180 degrees, through some ambient space. Having a 2D ambient space is sufficient. (And rotating through an ambient space feels more physically "real" than reflecting through 0.) Adding or multiplying by nonzero complex numbers can always be performed as a continuous transformation inside the complex numbers. And, given a number system that's 2D, you get a key topological invariant of closed paths that avoid the origin: winding number. This gives a 2D version of the Intermediate Value Theorem: If you have a continuous path between two closed loops with different winding numbers, then one of the intermediate closed loops must pass through 0. A consequence to this is the fundamental theorem of algebra, since for a degree-n polynomial f, when r is large enough then f(r*e^(i*t)) traces out for 0<=t<=2*pi a loop with winding number n, and when r=0 either f(0)=0 or f(r*e^(i*t)) traces out a loop with winding number 0, so if n>0 there's some intermediate r for which there's some t such that f(r*e^(i*t))=0.

So, I think the point is that 2D rotations and going around things are natural concepts, and very physical. Going around things lets you ensnare them. A side effect is that (complex) polynomials have (complex) roots.

> I believe real numbers to be completely natural

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.

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 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.

It's not like I have a real answer, of course, but something flipped inside of me after hearing the following story by Aaronson. He is asking[0], why quantum amplitudes would have to be complex. I.e., can we imagine a universe, where it's not the case?

> Why did God go with the complex numbers and not the real numbers?

> Years ago, at Berkeley, I was hanging out with some math grad students -- I fell in with the wrong crowd -- and I asked them that exact question. The mathematicians just snickered. "Give us a break -- the complex numbers are algebraically closed!" To them it wasn't a mystery at all.

Apparently, you weren't one of these math grad students, and, to be fair, Aaronson is starting with the question that is somewhat opposite to yours, but still, doesn't it intuitively make sense somehow? We are modeling something. In the process of modeling something we discover functions, and algebra, and find out that we'd like to use square roots all over the place. And just that alone leads us naturally to complex numbers! We didn't start with them, we only imagined an algebra that allows us to describe some process we'd like to describe, and suddenly there's no way around complex numbers! To me, thinking this way makes it almost obvious that ℂ-numbers are "real" somehow, they are indeed the fundamental building block of some complex-enough model, while ℝ are not.

Now, I must admit, that of course it doesn't reveal to me what the fuck they actually are, how to "imagine" them in the real world. I suppose, it's the same with you. But at least it makes me quite sure that indeed this is "the shadow of something natural that we just couldn't see", and I just don't know what. I believe it to be the consequence of us currently representing all numbers somehow "wrong". Similarly to how ancient Babylonian fraction representations were preventing ancient Babylonians from asking the right questions about them.

P.S. I think I must admit, that I do NOT believe real numbers to be natural in any sense whatsoever. But this is completely besides the point.

[0] https://www.scottaaronson.com/democritus/lec9.html

I 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.

How does your question differ from the classic question more normally applied to maths in general - does it exist outside the mind (eg platonism) or no (eg. nominalism)?

If it doesn't differ, you are in the good company of great minds who have been unable to settle this over thousands of years and should therefore feel better!

More at SEP:

https://plato.stanford.edu/entries/philosophy-mathematics/

Given that you have a Ph.D. in mathematics, this might seem hopelessly elementary, but who knows--I found it intuitive and insightful: https://betterexplained.com/articles/a-visual-intuitive-guid...

I've always been satisfied with the explanation "Just as you need signed numbers for translation, you need complex numbers to express rotation." Nobody asks if negative numbers are really a natural thing, so it doesn't make sense to ask if complex ones are, IMO.

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?

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

> 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.

Oh -- also --

e^(i Ⲡ) + 1 = 0

Nevertheless, you may be right.

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.

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.

Does this seemingly-amazing 2swap math exposition video offer any extra perspective? https://youtu.be/9HIy5dJE-zQ

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

Maybe the bottom ~1/3, starting at "The complex field as a problem for singular terms", would be helpful to you. It gives a philosophical view of what we mean when we talk about things like the complex numbers, grounded in mathematical practice.

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.

HN once recommended a book

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

> Is this the shadow of something natural that we just couldn't see

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

> I believe real numbers to be completely natural, but far greater mathematicians than I found them objectionable only a hundred years ago

I believe even negative numbers had their detractors

I'm presuming this is old news to you, but what helped me get comfortable with ℂ was learning that it's just the algebraic closure of ℝ.

And why would R be "entitled" to an algebraic closure?

(I have a math degree, so I don't have any issues with C, but this is the kind of question that would have troubled me in high school.)

The good news is that Q is not really entitled to a closure either.

When it doesn't, we yearn for something that will fill the void so that it does. It's like that note you yearn for in a musical piece that the composer seems to avoid. One yearns for a resolution of the tension.

Complex numbers offers that resolution.

> And why would R be "entitled" to an algebraic closure?

It's the birthright of every field.

Why would N be entitled to it? We made up negative numbers and more just to have a closure. You just learn about them at an age when you don't question it yet.

Even the counting numbers arose historically as a tool, right?

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

Maybe it is a notation issue.

What is a negative number? What is multiplication? What is a complex "number"? Complex are not even orderable. Is complex addition an overloading of the addition operator. Same with multiplication?

What i squared is -1 ? What does -1 even mean? Is the sign, a kind of operator?

The geometric interpretation help. These are transformations. Instead of 1 + i, we could/should write (1,i)

The AI might be clearer: https://gemini.google.com/share/6e00fab74749

A lot of math is not very clear because it is not very well taught. The notations are unclear. For instance, another example is: what is the difference between a matrix and a tensor? But that is another debate for anyone who wants to think about it. The definition found in books is often kind of wrong making a distinction that shouldn't really exist more often than not.

erratum: (1,1) rather

[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 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.

Thanks for your comment.

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.

Personally, no number is natural. They are probably a human construct. Mathematics does not come naturally to a human. Nowadays, it seems like every child should be able to do addition, but it was not the case in the past. The integers, rationals, and real numbers are a convenience, just like the complex numbers.

A better way to understand my point is: we need mental gymnastics to convert problems into equations. The imaginary unit, just like numbers, are a by-product of trying to fit problems onto paper. A notable example is Schrodinger's equation.

That's how I see complex numbers:

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, vibrate, or oscillate (like radio waves, electricity, or quantum particles). > The Peer-to-Peer Truth: Think of a complex number as a vector with an attitude. It lives in the same 2D house, but it knows how to spin and transform itself algebraically in ways a simple (x, y) coordinate cannot. >

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