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.
One way to sharpen the question is to stop asking whether C is "fundamental" and instead ask whether it is forced by mild structural constraints. From that angle, its status looks closer to inevitability than convenience.
Take R as an ordered field with its usual topology and ask for a finite-dimensional, commutative, unital R-algebra that is algebraically closed and admits a compatible notion of differentiation with reasonable spectral behavior. You essentially land in C, up to isomorphism. This is not an accident, but a consequence of how algebraic closure, local analyticity, and linearization interact. Attempts to remain over R tend to externalize the complexity rather than eliminate it, for example by passing to real Jordan forms, doubling dimensions, or encoding rotations as special cases rather than generic elements.
More telling is the rigidity of holomorphicity. The Cauchy-Riemann equations are not a decorative constraint; they encode the compatibility between the algebra structure and the underlying real geometry. The result is that analyticity becomes a global condition rather than a local one, with consequences like identity theorems and strong maximum principles that have no honest analogue over R.
I’m also skeptical of treating the reals as categorically more natural. R is already a completion, already non-algebraic, already defined via exclusion of infinitesimals. In practice, many constructions over R that are taken to be primitive become functorial or even canonical only after base change to C.
So while one can certainly regard C as a technical device, it behaves like a fixed point: impose enough regularity, closure, and stability requirements, and the theory reconstructs it whether you intend to or not. That does not make it metaphysically fundamental, but it does make it mathematically hard to avoid without paying a real structural cost.
This is the way I think. C is "nice" because it is constructed to satisfy so many "nice" structural properties simultaneously; that's what makes it special. This gives rise to "nice" consequences that are physically convenient across a variety of applications.
I work in applied probability, so I'm forced to use many different tools depending on the application. My colleagues and I would consider ourselves lucky if what we're doing allows for an application of some properties of C, as the maths will tend to fall out so beautifully.
Not meaning to derail an interesting conversation, but I'm curious about your description of your work as "applied probability". Can you say any more about what that involves?
Pure probability focuses on developing fundamental tools to work with random elements. It's applied in the sense that it usually draws upon techniques found in other traditionally pure mathematical areas, but is less applied than "applied probability", which is the development and analysis of probabilistic models, typically for real-world phenomena. It's a bit like statistics, but with more focus on the consequences of modelling assumptions rather than relying on data (although allowing for data fitting is becoming important, so I'm not sure how useful this distinction is anymore).
At the moment, using probabilistic techniques to investigate the operation of stochastic optimisers and other random elements in the training and deployment of neural networks is pretty popular, and that gets funding. But business as usual is typically looking at ecological models involving the interaction of many species, epidemiological models investigating the spread of disease, social network models, climate models, telecommunication and financial models, etc. Branching processes, Markov models, stochastic differential equations, point processes, random matrices, random graph networks; these are all the common objects used. Actually figuring out their behaviour can require all kinds of assorted techniques though, you get to pull from just about anything in mathematics to "get the job done".
In my work in academia (which I’m considering leaving), I’m very familiar with the common mathematical objects you mentioned. Where could I look for a job similar to yours? It sounds very interesting
Sorry, I'm in academia too, but my ex-colleagues who left found themselves doing nearly identical work doing MFT research at hedge funds, climate modelling at our federal weather bureau, and SciML in big tech. I know of someone doing this kind of work in telecoms too, but I haven't spoken to them lately. Having said that, it's rough out there right now. A couple of people I know looking for another job right now (academia or otherwise) with this kind of training are not having much luck...
> Take R as an ordered field with its usual topology and ask for a finite-dimensional, commutative, unital R-algebra that is algebraically closed and admits a compatible notion of differentiation with reasonable spectral behavior.
No thank you, you can keep your R.
Damn... does this paragraph mean something in the real world?
Probably I've the brain of a gnat compared to you, but do all the things you just said have a clear meaning that you relate to the world around you?
> 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.
Math and reality are, in general completely distinct. Some math is originally developed to model reality, but nowadays (and for a long time) that's not the typical starting point, and mathematicians pushing boundaries in academia generally don't even think about how it relates to reality.
However, it is true (and an absolutely fascinating phenomenon) that we keep encountering phenomena in reality and then realize that an existing but previously purely academic branch of math is useful for modeling it.
To the best of our knowledge, such cases are basically coincidence.
Opposing view (that I happen to hold, at least if I had to choose one side or the other): not only is mathematics 'reality'; it is arguably the only thing that has a reasonable claim to being 'reality' itself.
After all, facts (whatever that means) about the physical world can only be obtained by proxy (through measurement), whereas mathematical facts are just... evident. They're nakedly apparent. Nothing is being modelled. What you call the 'model' is the object of study itself.
A denial of the 'reality' of pure mathematics would imply the claim that an alien civilisation given enough time would not discover the same facts or would even discover different – perhaps contradictory – facts. This seems implausible, excluding very technical foundational issues. And even then it's hard to believe.
> To the best of our knowledge, such cases are basically coincidence.
This couldn't be further from the truth. It's not coincidence at all. The reason that mathematics inevitably ends up being 'useful' (whatever that means; it heavily depends on who you ask!) is because it's very much real. It might be somewhat 'theoretical', but that doesn't mean it's made up. It really shouldn't surprise anyone that an understanding of the most basic principles of reality turns out to be somewhat useful.
I think you're not even disagreeing with me, we're just using different definitions of the word "reality". I meant it to use specifically "the physical world" - which you are treating as distinct from mathematics as well in your second paragraph.
Mathematics is an abstract game of symbols and rules invented by humans. It has nothing to do with reality. However it is quite useful for modelling our understanding of reality.
"that we keep encountering phenomena in reality and then realize that an existing but previously purely academic branch of math is useful for modeling it."
Would you have some examples?
(Only example that I know that might fit are quaternions, who were apparently not so useful when they were found/invented but nowdays are very useful for many 3D application/computergraphics)
Group theory entering quantum physics is a particularly funny example, because some established physicists at the time really hated the purely academic nature of group theory that made it difficult to learn.[1]
If you include practical applications inside computers and not just the physical reality, then Galois theory is the most often cited example. Galois himself was long dead when people figured out that his mathematical framework was useful for cryptography.
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.
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.
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.
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.
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.
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.
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?
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 have a real philosophical problem with complex numbers
> I believe real numbers to be completely natural
I have to say I find this perspective interesting but completely alien.
We need to have a way to find x such that x^2-2 = 0, and Q won’t cut it so we have R. (Or if you want, we need a complete ordered field so we have R)
We need to have a way to find x such that x^2+2 = 0, and R won’t cut it so we have C. (Or if you want, we need algebraic closure of R by the fundamental theorem of algebra so we need C)
I don’t really think any numbers (even “natural” numbers) are any more natural than any other kind of numbers. If you start to distinguish, where do you stop. Negative numbers are ok or not? What about zero? Is that “natural”? Mathematicians disagree about whether 0 is in N at least.
It reminds me of the famous quote from Gauss:
That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.
The 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>".
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.
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.
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.
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.
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.
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
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.
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).
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.
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.
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.
> 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.
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".
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.
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).
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)
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.
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.
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).
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.
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).
> 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.
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.
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.
> 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.
> 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.
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.
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).
> 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.
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.
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 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.
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.
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.
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.
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.
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..
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.
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.
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.
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.
The movement from R to C can be done rigorously. It gets hand-waved away in more application-oriented math courses, but it's done properly in higher level theoretically-focused courses. Lifting from a smaller field (or other algebraic structure) to a larger one is a very powerful idea because it often reveals more structure that is not visible in the smaller field. Some good examples are using complex eigenvalues to understand real matrices, or using complex analysis to evaluate integrals over R.
I hate when people casually move "between" Q and Z as if a rational number with unit denominator suddenly becomes an integer, and it's all because of this terrible "a/b" notation. It's more like (a, b). You can't ever discard that second component, it's always there. ;)
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!"
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.
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.
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!
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.
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.
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.
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.
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.
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)
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.
[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.
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.
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.
>
I began studying 3-manifolds after coming up with a novel way I preferred to draw their presentations. All approaches are formally equivalent, but they impose different cognitive loads in practice. My approach was trivially equivalent to triangulations, or spines, or Heegaard splittings, or ... but I found myself far more nimbly able to "see" 3-manifolds my way.
I showed various colleagues. Each one would ask me to demonstrate the equivalence to their preferred presentation, then assure me "nothing to see here, move along!" that I should instead stick to their convention.
Then I met with Bill Thurston, the most influential topologist of our lifetimes. He had me quickly describe the equivalence between my form and every other known form, effectively adding my node to a complete graph of equivalences he had in his muscle memory. He then suggested some generalizations, and proposed that circle packings would prove to be important to me.
Some mathematicians are smart enough to see no distinction between any of the ways to describe the essential structure of a mathematical object. They see the object.
I was interested in how it would make sense to define complex numbers without fixing the reals, but I'm not terribly convinced by the method here. It seemed kind of suspect that you'd reduce the complex numbers purely to its field properties of addition and multiplication when these aren't enough to get from the rationals to the reals (some limit-like construction is needed; the article uses Dedekind cuts later on). Anyway, the "algebraic conception" is defined as "up to isomorphism, the unique algebraically closed field of characteristic zero and size continuum", that is, you just declare it has the same size as the reals. And of course now you have no way to tell where π is, since it has no algebraic relation to the distinguished numbers 0 and 1. If I'm reading right, this can be done with any uncountable cardinality with uniqueness up to isomorphism. It's interesting that algebraic closure is enough to get you this far, but with the arbitrary choice of cardinality and all these "wild automorphisms", doesn't this construction just seem... defective?
It feels a bit like the article's trying to extend some legitimate debate about whether fixing i versus -i is natural to push this other definition as an equal contender, but there's hardly any support offered. I expect the last-place 28% poll showing, if it does reflect serious mathematicians at all, is those who treat the topological structure as a given or didn't think much about the implications of leaving it out.
More on not being able to find π, as I'm piecing it together: given only the field structure, you can't construct an equation identifying π or even narrowing it down, because if π is the only free variable then it will work out to finding roots of a polynomial (you only have field operations!) and π is transcendental so that polynomial can only be 0 (if you're allowed to use not-equals instead of equals, of course you can specify that π isn't in various sets of algebraic numbers). With other free variables, because the field's algebraically closed, you can fix π to whatever transcendental you like and still solve for the remaining variables. So it's something like, the rationals plus a continuum's worth of arbitrary field extensions? Not terribly surprising that all instances of this are isomorphic as fields but it's starting to feel about as useful as claiming the real numbers are "up to set isomorphism, the unique set whose cardinality matches the power set of the natural numbers", like, of course it's got automorphisms, you didn't finish defining it.
You need some notion of order or of metric structure if you want to talk about numbers being "close" enough to π. This is related to the property of completeness for the real numbers, which is rather important. Ultimately, the real numbers are also a rigorously defined abstraction for the common notion of approximating some extant but perhaps not fully known quantity.
There's a related idea in mathematics, the proof that the real numbers are a vector space over the rational numbers. If you scramble the basis vectors, you obtain an isomorphic vector space, but it is effectively a "permutation" of |R. Of course, vector spaces don't even have multiplication, but one interesting thing is that the proof requires the axiom of choice.
I think that actually constructing a "nontrivial" model of C using the field conception might require choosing a member from each of an infinite family of sets, i.e. it requires applying the axiom of choice, similar to the way you construct R as a vector space.
Most commenters are talking about the first part of the post, which lays out how you might construct the complex numbers if you're interested in different properties of them. I think the last bit is the real interesting substance, which is about how to think about things like this in general (namely through structuralism), and why the observations of the first half should not be taken as an argument against structuralism. Very interesting and well written.
It is very re-assuring to know, on a post where I can essentially not even speak the language (despite a masters in engineering) HN is still just discussing the first paragraph of the post.
To be clear, this "disagreement" is about arbitrary naming conventions which can be chosen as needed for the problem at hand. It doesn't make any difference to results.
The author is definitely claiming that it's not just about naming conventions: "These different perspectives ultimately amount, I argue, to mathematically inequivalent structural conceptions of the complex numbers". So you would need to argue against the substance of the article to have a basis for asserting that it is just about naming conventions.
Article: "They form the complex field, of course, with the corresponding algebraic structure, but do we think of the complex numbers necessarily also with their smooth topological structure? Is the real field necessarily distinguished as a fixed particular subfield of the complex numbers? Do we understand the complex numbers necessarily to come with their rigid coordinate structure of real and imaginary parts?"
So yes these are choices. If I care how the complex plane maps onto some real number somewhere, then I have to pick a mapping. "Real part" is only one conventional mapping. Ditto the other stuff: If I'm going to do contour integrals then I've implied some things about metric and handedness.
I still don't see how this really puts mathematicians in "disagreement." Let's pedestrian example:
I usually make an x,y plot with the x-axis pointing to the right and the y-axis pointing away from me. If I put a z-axis, personally I'll make it upwards out of the paper (sometimes this matters). Usually, but not always, my co-ordinates are meant to be smooth. But if somebody does some of this another way, are they really disagreeing with me? I think "no." If we're talking about the same problem, we'll eventually get the same answer (after we each fix 3 or 4 mistakes). If we're talking about different problems, then we need our answers to potentially "disagree."
Exactly. So I feel like all the article really says is that "Complex numbers" doesn't necessarily tell you everything you need to know. It depends on what you're doing with them.
In the article he says there is a model of ZFC in which the complex numbers have indistinguishable square roots of -1. Thus that model presumably does not allow for a rigid coordinate view of complex numbers.
Theorem. If ZFC is consistent, then there is a model of ZFC that has a definable complete ordered field ℝ with a definable algebraic closure ℂ, such that the two square roots of −1 in ℂ are set-theoretically indiscernible, even with ordinal parameters.
Haven’t thought it through so I’m quite possibly wrong but it seems to me this implies that in such a situation you can’t have a coordinate view. How can you have two indistinguishable views of something while being able to pick one view?
Mathematicians pick an arbitrary complex number by writing "Let c ∈ ℂ." There are an infinite number of possibilities, but it doesn't matter. They pick the imaginary unit by writing "Let i ∈ ℂ such that i² = −1." There are two possibilities, but it doesn't matter.
If two things are set theoretically indistinguishable then one can’t say “pick one and call it i and the other one -i”. The two sets are the same according to the background set theory.
They're not the same. i ≠
−i, no matter which square root of negative one i is. They're merely indiscernible in the sense that if φ(i) is a formula where i is the only free variable, ∀i ∈ ℂ. i² = −1 ⇒ (φ(i) ⇔ φ(−i)) is a true formula. But if you add another free variable j, φ(i, j) can be true while φ(−i, j) is false, i.e. it's not the case that ∀j ∈ ℂ. ∀i ∈ ℂ. i² = −1 ⇒ (φ(i, j) ⇔ φ(−i, j)).
This is a very interesting question, and a great motivator for Galois theory, kind of like a Zen koan. (e.g. "What is the sound of one hand clapping?")
But the question is inherently imprecise. As soon as you make a precise question out of it, that question can be answered trivially.
Generally, the nth roots of 1 form a cyclic group (with complex multiplication, i.e. rotation by multiples of 2pi/n).
One of the roots is 1, choosing either adjacent one as a privileged group generator means choosing whether to draw the same complex plane clockwise or counterclockwise.
The question is meaningless because isomorphic structures should be considered identical. A=A. Unless you happen to be studying the isomorphisms themselves in some broader context, in which case how the structures are identical matters. (For example, the fact that in any expression you can freely switch i with -i is a meaningful claim about how you might work with the complex numbers.)
Homotopy type theory was invented to address this notion of equivalence (eg, under isomorphism) being equivalent to identity; but there’s not a general consensus around the topic — and different formalisms address equivalence versus identity in varied ways.
Sure. Either that or the reverse. "They're not the same" in the sense that they can't both be clockwise. "They are the same" in the sense that we could make either one clockwise.
Names, language, and concepts are essential to and have powerful effects on our understanding of anything, and knowledge of mathematics is much more than the results. Arguably, the results are only tests of what's really important, our understanding.
In particular, the core disagreement seems to be about whether the automorphisms of C should keep R (as a subset) fixed, or not.
The easy solution here would be to just have two different names: (general) automorphisms (of which there might be many) and automorphisms-that-keep-R-fixed (of which there are just the two mentioned.
If you make this distinction, then the approach of construction of C should not matter, as they are all equivalent?
No the entire point is that it makes difference in the results. He even gave an example in which AI(and most humans imo) picked different interpretation of complex numbers giving different result.
I really know almost nothing about complex analysis, but this sure feels like what physicists call observational entropy applied to mathematics: what counts as "order" in ℂ depends on the resolution of your observational apparatus.
The algebraic conception, with its wild automorphisms, exhibits a kind of multiplicative chaos — small changes in perspective (which automorphism you apply) cascade into radically different views of the structure. Transcendental numbers are all automorphic with each other; the structure cannot distinguish e from π. Meanwhile, the analytic/smooth conception, by fixing the topology, tames this chaos into something with only two symmetries. The topology acts as a damping mechanism, converting multiplicative sensitivity into additive stability.
I'll just add to that that if transformers are implementing a renormalization group flow, than the models' failure on the automorphism question is predictable: systems trained on compressed representations of mathematical knowledge will default to the conception with the lowest "synchronization" cost — the one most commonly used in practice.
The way I think of complex numbers is as linear transformations. Not points but functions on points that rotate and scale. The complex numbers are a particular set of 2x2 matrices, where complex multiplication is matrix multiplication, i.e. function composition. Complex conjugation is matrix transposition. When you think of things this way all the complex matrices and hermitian matrices in physics make a lot more sense. Which group do I fall into?
Being an engineer by training, I never got exposed to much algebra in my courses (beyond the usual high school stuff in high school). In fact did not miss it much either. Tried to learn some algebraic geometry then... oh the horror. For whatever reason, my intuition is very geometric and analytic (in the calculus sense). Even things like counting and combinatorics, they feel weird, like dry flavorless pretzels made of dried husk. Combinatorics is good only when I can use Calculus. Calculus, oh that's different, it's rich savoury umami buttery briskets. Yum.
That's not the interesting part. The interesting part is that I thought everyone is the same, like me.
It was a big and surprising revelation that people love counting or algebra in just the same way I feel about geometry (not the finite kind) and feel awkward in the kind of mathematics that I like.
It's part of the reason I don't at all get the hate that school Calculus gets. It's so intuitive and beautifully geometric, what's not to like. .. that's usually my first reaction. Usually followed by disappointment and sadness -- oh no they are contemplating about throwing such a beautiful part away.
School calculus is hated because it's typically taught with epsilon delta proofs which is a formalism that happened later in the history of calculus. It's not that intuitive for beginners, especially students who haven't learn any logic to grok existential/universal quantifiers. Historically, mathematics is usually developed by people with little care for complete rigor, then they erase their tracks to make it look pristine. It's no wonder students are like "who the hell came up with all this". Mathematics definitely has an education problem.
IMO, the calculus is taught incorrectly. It should start with functions and completely avoid sequences initially. Once you understand how calculus exploits continuity (and sometimes smoothness), it becomes almost intuitive. That's also how it was historically developed, until Weierstrass invented his monster function and forced a bit more rigor.
But instead calculus is taught from fundamentals, building up from sequences. And a lot of complexity and hate comes from all those "technical" theorems that you need to make that jump from sequences to functions. E.g. things like "you can pick a converging subsequence from any bounded sequence".
In Maths classes, we started with functions. Functions as list of pairs, functions defined by algebraic expressions, functions plotted on graph papers and after that limits. Sequences were peripherally treated, just so that limits made sense.
Simultaneously, in Physics classes we were being taught using infinitesimals, with the a call back that "you will see this done more formally in your maths classes, but for intuition, infinitesimals will do for now".
Or you can hand wave a bit and trust intuition. Just like the titans who invented it all did!
The obsession with rigor that later developed -- while necessary -- is really an "advanced topic" that shouldn't displace learning the intuition and big picture concepts. I think math up through high school should concentrate on the latter, while still being honest about the hand-waving when it happens.
I broadly agree. But, the big risk here is that it's really easy for an adventurous student to stretch that handwaving beyond where it's actually valid. You at least have to warn them that the "intuitions" you give them are not general methods, just explanations for why the algorithms you teach them do something worthwhile (and for the ones inclined to explore, give them some fun edge cases to think about).
You can do it with synthetic differential geometry, but that introduces some fiddliness in the underlying logic in order to cope with the fact that eps^2 really "equals" zero for small enough eps, and yet eps is not equal to zero.
calculus works... because it was almost designed for Mechanics. If the machine it's getting input, you have output. When it finished getting input, all the output you get yields some value, yes, but limits are best understood not for the result, but for the process (what the functions do).
You are not sending 0 coins to a machine, do you? You sent X to 0 coins to a machine. The machine will work from 2 to 0, but 0 itself is not included because is not a part of a changing process, it's the end.
Limits are for ranges of quantities over something.
Of course everyone agrees that this is a nice way to construct the complex field. The question is what is the structure you are placing on this construction. Is it just a field? Do you intend to fix R as a distinguished subfield? After all, there are many different copies of R in C, if one has only the field structure. Is i named as a constant, as it seems to be in the construction when you form the polynomials in the symbol i. Do you intend to view this as a topological space? Those further questions is what the discussion is about.
I mean, yes of course i is an element in C, because it's a monic polynomial in i.
There's no "intend to". The complex numbers are what they are regardless of us; this isn't quantum mechanics where the presence of an observer somehow changes things.
It's not about observers, but about mathematical structure and meaning. Without answering the questions, you are being ambiguous as to what the structure of C is. For example, if a particular copy of R is fixed as a subfield, then there are only two automorphisms---the trivial automorphism and complex conjugation, since any automorphism fixing the copy of R would have to be the identity on those reals and thus the rest of it is determined by whether i is fixed or sent to -i. Meanwhile, if you don't fix a particular R subfield, then there is a vast space of further wild automorphisms. So this choice of structure---that is, the answer to the questions I posed---has huge consequences on the automorphism group of your conception. You can't just ignore it and refuse to say what the structure is.
Does anyone have any tips on how I would fundamentally understand this article without just going back to school and getting a degree in mathematics? This is the sort of article where my attempts to understand a term only ever increase the number of terms I don't understand.
PhD math guy here. Personally, I don't think this is a great article for a lay person to approach or appreciate. You might be able and interested to follow along with the various constructions of C, but without appreciation for formal logic and category theory it'll seem like distinctions without a difference (and to working mathematicians, they are). The model theory I found quite interesting, but was high effort even for me. And the philosophical points are also rather specific.
I thought i understood complex numbers and accepted them until I did countour integration for the first time.
Ever since then I have been deeply unsettled. I started questioning taking integrals to (+/-) infinity, and so I became unsettled with R too.
If C exists to fix R, then why does R even exist? Why does R need to be fixed? Why does the use of the upper or lower plane for counter integration not matter? I can do mathematically why, but why do we have a choice?
This blog post really articulated stuff formally that I have been bothered by for years.
In the case of quaternions, there is called double-covering, which turns out (rather than being an artefact), play fundamental role in particle physics.
As a non-mathematican, I found that trying to introduce C as a closure of R (i. e. analytically in author's terms) invariably triggers confusion and "hey, why do mathematicians keep changing rules on the fly, they just told me square of minus one doesn't exist". And in terms of practical applications it doesn't seem particularly useful on the first glance (who cares about solving cubics algebraically? The formula is too unwieldy anyway.) Most applications tends to start in the coordinate view and go from there. And it does introduce a nasty sharp edge to cut oneself on (i vs -i), but then for instance physics is full of such edges: direction of pseudo-vectors, sign of voltage on loads sources, holes in dimensional analysis (VA vs W, Ohm/square), the list could go on. And nobody really care.
> "hey, why do mathematicians keep changing rules on the fly, they just told me square of minus one doesn't exist
Mathematicians aren’t chasing numerical solutions, they’re chasing structure. ℂ isn’t just about solving cubics, it’s about eliminating holes in algebra so the theory behaves uniformly and is easier to build upon.
And as for "changing rules" they haven't changed, they have broadened the field (literally) over which the old rules applied in a clever way to remove a restriction.
For me personally, I prefer to perceive as more fundamental those constructions that straightforwardly generate the most with the least of novelty.
For example one may be introduced to the real numbers, and later to the complex numbers and then later perhaps the quaternions and the octonions, etc. in a haphazard disconnected way.
Given just the real numbers and "geometric algebra" (i.e. Grassmann algebras), this generates basically all these structures for different "settings".
I hence view geometric algebra as a lower level and more fundamental than complex numbers specifically.
A more interesting question (if we had to limit discussion to complex numbers) would be the following: which representations of complex numbers are known, what are their advantages and disadvantages, and can novel representation of complex numbers be devised that display less of the disadvantages?
For example, we have -just to list a few- the following representations:
1. cartesian representation of complex numbers: a + bi : the complex numbers permit a straightforward single-valued addition and single-valued multiplication, but only multivalued integer-powered-roots. addition and multiplication change smoothly with smooth changes in the input values, taking N-th roots do not, unless you use multivalued roots but then you don't have single valued roots.
2. polar representation of complex numbers: r(cos(theta)+i sin(theta)): this representatin admits smooth and single valued multiplication and N-th roots, but no longer permits smooth and single valued additions!
Please follow up with your favourite representations for complex numbers, and expound the pro's and con's of that representation.
For example can you generate a representation where addition and N-th roots are smooth and single valued, but multiplications are not?
Can you prove that any representation for complex numbers must suffer this dilemma in representation, or can you devise a representation where all 3 addition, multiplication, roots are smooth and single valued?
The field C is a tool, just as with any other abstraction in math. Accountants don't need it, so ignore it. In abstract algebra they arise - and so do many other abstractions, such as quaternions. Quaternions can be regarded as a further development of the concept of number. Note that every such development gives up some property its predecessors had. In the case of quaternions, commutativity. If a field of application finds a mathematical invention useful, it gets used. Often, too, it's a matter of simplicity. Use mathematical invention X in applied field Y and it simplifies the work. Abstain from using invention X, and then the work could still be done in a less simple way. That's all there is to it.
Real men know that infinite sets are just a tool for proving statements in Peano arithmetic, and complex numbers must be endowed with the standard metric structure, as God intended, since otherwise we cannot use them to approximate IEEE 754 floats.
Knowledge is the output of a person and their expertise and perspective, irreducibly. In this case, they seem to know something of what they're talking about:
> Starting 2022, I am now the John Cardinal O’Hara Professor of Logic at the University of Notre Dame.
> From 2018 to 2022, I was Professor of Logic at Oxford University and the Sir Peter Strawson Fellow at University College Oxford.
Also interesting:
> I am active on MathOverflow, and my contributions there (see my profile) have earned the top-rated reputation score.
There is perfect agreement on the Gaussian integers.
The disagreement is on how much detail of the fine structure we care about. It is roughly analogous to asking whether we should care more about how an ellipse is like a circle, or how they are different. One person might care about the rigid definition and declare them to be different. Another notices that if you look at a circle at an angle, you get an ellipse. And then concludes that they are basically the same thing.
This seems like a silly thing to argue about. And it is.
However in different branches of mathematics, people care about different kinds of mathematical structure. And if you view the complex numbers through the lens of the kind of structure that you pay attention to, then ignore the parts that you aren't paying attention to, your notion of what is "basically the same as the complex numbers" changes. Just like how one of the two people previously viewed an ellipse as basically the same as a circle, because you get one from the other just by looking from an angle.
Note that each mathematician here can see the points that the other mathematicians are making. It is just that some points seem more important to you than others. And that importance is tied to what branch of mathematics you are studying.
The Gaussian integers usually aren't considered interesting enough to have disagreements about. They're in a weird spot because the integer restriction is almost contradictory with considering complex numbers: complex numbers are usually considered as how to express solutions to more types of polynomials, which is the opposite direction of excluding fractions from consideration. They're things that can solve (a restricted subset of) square-roots but not division.
This is really a disagreement about how to construct the complex numbers from more-fundamental objects. And the question is whether those constructions are equivalent. The author argues that two of those constructions are equivalent to each other, but others are not. A big crux of the issue, which is approachable to non-mathematicians, is whether it i and -i are fundamentally different, because arithmetically you can swap i with -i in all your equations and get the same result.
I don't think anyone thinks is "i and -i are fundamentally different". What they care more about is whether the 5 5th roots of 2 have an natural ordering or not.
Basically C comes up in the chain R \subset C \subset H (quaternions) \subset O (octonions) by the so-called Cayley-Dickson construction. There is a lot of structure.
> But in fact, I claim, the smooth conception and the analytic conception are equivalent—they arise from the same underlying structure.
Conjugation isn’t complex-analytic, so the symmetry of i -> -i is broken at that level. Complex manifolds have to explicitly carry around their almost-complex structure largely for this reason.
We could've called the imaginaries "orthogonals", "perpendiculars", "complications", "atypicals", there's a million other options. I like the idea that a number is complex because it has a "complicated component".
Usually they explain it something like: oh, at first people didn't know what 2-5 added up to, but then we invented negative numbers. Well, complex numbers are that but for square roots of negative numbers.
But that's a completely misleading way to explain these things. Complex numbers aren't numbers aren't numbers really.
It entirely depends on what we mean for something to be a number. Humans over the passage of time have been recognizing that their earlier conception had been too restrictive, narrow minded.
As one broadens the idea of what it means for anything to be a number, we acquire/invite new members to the family and with great benefit.
I mean, yeah, they aren't real numbers, they are composed of a real number and another one that is the multiplicative of the square root of -1. Hence they're called complex, i.e. composed of parts.
If the square root of -1 is not a number, what is it? How come you can do arithmetic with it?
For what it's worth, Errett Bishop, the famous constructivist did not have this kind of existential issue with the complex numbers, commenting that the Reals were inadequate for some things. I really liked the trig cos isin connection in High School
the title is a bit clickbait - mathematicians don't disagree, all the "conceptions" the article proposes agree with each other. It also seems to conflate the algebraic closure of Q (which would contain the sqrt of -1) and all of the complex numbers by insisting that the former has "size continuum". Once you have "size continuum" then you need some completion to the reals.
anyhow. I'm a bit of an odd one in that I have no problems with imaginary numbers but the reals always seemed a bit unreal to me. that's the real controversy, actually. you can start looking up definable numbers and constructivist mathematics, but that gets to be more philosophy than maths imho.
The famous constructivist Errett Bishop did not have this sort of existential issue with the Complex Numbers, only saying the Reals were inadequate for some things.
To the ones objecting to "choosing a value of i" I might argue that no such choice is made. i is the square root of -1 and there is only one value of i. When we write -i that is shorthand for (-1)i. Remember the complex numbers are represented by a+bi where a and b are real numbers and i is the square root of -1. We don't bifurcate i into two distinct numbers because the minus sign is associated with b which is one of the real numbers. There is a one-to-one mapping between the complex numbers and these ordered pairs of reals.
You say that i is "the square root of -1", but which one is it? There are two. This is the point in the essay---we cannot tell the difference between i and -i unless we have already agreed on a choice of which square root of -1 we are going to call i. Only then does the other one become -i. How do we know that my i is the same as your i rather than your -i?
To fix the coordinate structure of the complex numbers (a,b) is in effect to have made a choice of a particular i, and this is one of the perspectives discussed in the essay. But it is not the only perspective, since with that perspective complex conjugation should not count as an automorphism, as it doesn't respect the choice of i.
One perspective of the complex numbers is that they are the even subalgebra of the 2D geometric algebra. The "i" is the pseudoscalar of that 2D GA, which is an oriented area.
If you flip the plane and look at it from the bottom, then any formula written using GA operations is identical, but because you're seeing the oriented area of the pseudoscalar from behind, its as if it gains a minus sign in front.
This is equivalent to using a right-handed versus left-handed coordinate systems in 3D. The "rules of physics" remain the same either way, the labels we assign to the coordinate systems are just a convention.
Is it two, or is it infinite? The quaternions have three imaginary units, i, j, and k. They're distinct, and yet each of them could be used for the complex numbers and they'd work the same way. How would I know that "my" imaginary unit i is the same as some other person's i? Maybe theirs is j, or k, or something else entirely.
There are 2 square roots of 9, they are 3 and -3. Likewise there are two square roots of -1 which are i and -i. How are people trying to argue that there are two different things called i? We don't ask which 3 right? My argument is that there is only 1 value of i, and the distinction between -i and i is the same as (-1)i and (1)i, which is the same as -3 vs 3. There is only one i. If there are in fact two i's then there are 4 square roots of -1.
Notably, the real numbers are not symmetrical in this way: there are two square roots of 1, but one of them is equal to it and the other is not. (positive) 1 is special because it's the multiplicative identity, whereas i (and -i) have no distinguishing features: it doesn't matter which one you call i and which one you call -i: if you define j = -i, you'll find that anything you can say about i can also be shown to be true about j. That doesn't mean they're equal, just that they don't have any mathematical properties that let you say which one is which.
Your view of the complex numbers is the rigid one. Now suppose you are given a set with two binary operations defined in such a way that the operations behave well with each other. That is you have a ring. Suppose that by some process you are able to conclude that your ring is algebraically equivalent to the complex numbers. How do you know which of your elements in your ring is “i”? There will be two elements that behave like “i” in all algebraic aspects. So you can’t say that this one is “i” and this one is “-i” in a non arbitrary fashion.
There is no way to distinguish between "i" and "-i" unless you choose a representation of C. That is what Galois Theory is about: can you distinguish the roots of a polynomial in a simple algebraic way?
For instance: if you forget the order in Q (which you can do without it stopping being a field), there is no algebraic (no order-dependent) way to distinguish between the two algebraic solutions of x^2 = 2. You can swap each other and you will not notice anything (again, assuming you "forget" the order structure).
Building off of this point, consider the polynomial x^4 + 2x^2 + 2. Over the rationals Q, this is an irreducible polynomial. There is no way to distinguish the roots from each other. There is also no way to distinguish any pair of roots from any other pair.
But over the reals R, this polynomial is not irreducible. There we find that some pairs of roots have the same real value, and others don't. This leads to the idea of a "complex conjugate pair". And so some pairs of roots of the original polynomial are now different than other pairs.
That notion of a "complex conjugate pair of roots" is therefore not a purely algebraic concept. If you're trying to understand Galois theory, you have to forget about it. Because it will trip up your intuition and mislead you. But in other contexts that is a very meaningful and important idea.
And so we find that we don't just care about what concepts could be understood. We also care about what concepts we're currently choosing to ignore!
"God made the integers, all else is the work of man", Kronecker, 1886.
Complex numbers are just a fancy way to represent 2-dimensional numbers that are convenient in geometry through the sq(-1)=π/2 rotation.
They are nothing special and they could just be a matrix. Out of my head I think about complex integers. Also there are higher-order complex numbers which are defined by sq(sq(-1)) etc. In Greek complex numbers are called "mongrel". Both names are bad, I would just use 2-dimensional numbers.
My biggest pet peeve in complex analysis is the concept of multi-value functions.
Functions are defined as relations on two sets such that each element in the first set is in relation to at most one element in the second set. And suddenly we abandon that very definitions without ever changing the notation! Complex logarithms suddenly have infinitely many values! And yet we say complex expressions are equal to something.
> You can think of it as returning an equivalence class if you like. Then it's single-valued.
I can, but I still have to be very mindful of how I use the "=" sign. Sometimes it's an (in)equality, sometimes it's... the equivalence class thing. The ambiguity doesn't seem very elegant.
> Also note the very cool and fun topology connection here. The keyword to search for is Riemann surface.
Idk, to me it feels much much better than just picking one root when defining the inverse function.
This desire to absolutely pick one when from the purely mathematical perspective they're all equal is both ugly and harmful (as in complicates things down the line).
But couldn't we just switch the nomenclature? Instead of an oxymoronic concept of "multivalue function", we could just call it "relation of complex equivalence" or something of sorts.
I am confused by both the article and the discussion and I don't mean confused by the discussion on the complex which is all fairly clear but by this very weird idea of essential structure whatever that's supposed to mean. I'm wondering if there is something cultural here (I'm French) and linked to the central place algebra has in our curriculum or if I'm just reading a lot of IA generated convoluted discussion.
To me, the question doesn't even make sense. There is no disagreement here and I don't understand what is an essential structure. All the properties presented on the complex are consistent. They all exist. There is nothing ever essential about mathematics.
I mean, it's just mathematics. As long as you are internally consistent with your axioms, things just are. It's like asking if water is a liquid or a collection of molecules. There is nothing to disagree about here.
The square root of any number x is ±y, where +y = (+1)*y = y, and -y = (-1)*y.
So we define i as conforming to ±i = sqrt(-1). The element i itself has no need for a sign, so no sign needs to be chosen. Yet having defined i, we know that that i = (+1)*i = +i, by multiplicative identity.
We now have an unsigned base element for complex numbers i, derived uniquely from the expansion of <R,0,1,+,*> into its own natural closure.
We don't have to ask if i = +i, because it does by definition of the multiplicative identity.
TLDR: Any square root of -1 reduced to a single value, involves a choice, but the definition of unsigned i does not require a choice. It is a unique, unsigned element. And as a result, there is only a unique automorphism, the identity automorphism.
Honestly, the rigid conception is the correct one. Im of the view that i as an attribute on a number rather than a number itself, in the same way a negative sign is an attribute. Its basically exists to generalize rotations through multiplication. Instead of taking an x,y vector and multiplying it by a matrix to get rotations, you can use a complex number representation, and multiply it by another complex number to rotate/scale it. If the cartesian magnitude of the second complex number is 1, then you don't get any scaling. So the idea of x/y coordinates is very much baked in to the "imaginary attribute".
I feel like the problem is that we just assume that e^(pi*i) = -1 as a given, which makes i "feel" like number, which gives some validity to other interpretations. But I would argue that that equation is not actually valid. It arises from Taylor series equivalence between e, sin and cos, but taylor series is simply an approximation of a function by matching its derivatives around a certain point, namely x=0. And just because you take 2 functions and see that their approximations around a certain point are equal, doesn't mean that the functions are equal. Even more so, that definition completely bypasses what it means to taking derivatives into the imaginary plane.
If you try to prove this any other way besides Taylor series expansion, you really cant, because the concept of taking something to the power of "imaginary value" doesn't really have any ties into other definitions.
As such, there is nothing really special about e itself either. The only reason its in there is because of a pattern artifact in math - e^x derivative is itself, while cos and sin follow cyclic patterns. If you were to replace e with any other number, note that anything you ever want to do with complex numbers would work out identically - you don't really use the value of e anywhere, all you really care about is r and theta.
So if you drop the assumption that i is a number and just treat i as an attribute of a number like a negative sign, complex numbers are basically just 2d numbers written in a special way. And of course, the rotations are easily extended into 3d space through quaternions, which use i j an k much in the same way.
> As such, there is nothing really special about e itself either. The only reason its in there is because of a pattern artifact in math - e^x derivative is itself
Not sure I follow you here... The special thing about e is that it's self-derivative. The other exponential bases, while essentially the same in their "growth", have derivatives with an extra factor. I assume you know e is special in that sense, so I'm unclear what you're arguing?
Im saying that the definition of polar coordinates for complex numbers using e instead of any other number is irrelevant to the use of complex numbers, but its inclusion in Eulers identity makes it seem like a i is a number rather than an attribute. And if you assume i is a number, it leads to one thinking that that you can define the complex field C. But my argument is that Eulers identity is not really relevant in the sense of what the complex numbers are used for, so i is not a number but rather a tool.
We as humans had a similar argument regarding 0. The thought was that zero is not a number, just a notational trick to denote that nothing is there (in the place value system of the Mesopotamians)
But then in India we discovered that it can really participate with the the other bonafide numbers as a first class citizen of numbers.
It is not longer a place holder but can be the argument of the binary functions, PLUS, MINUS, MULTIPLY and can also be the result of these functions.
With i we have a similar observation, that it can indeed be allowed as a first class citizen as a number. Addition and multiplication can accept them as their arguments as well as their RHS. It's a number, just a different kind.
But you can define the complex field C. And it has many benefits, like making the fundamental theorem of algebra work out. I'm not seeing the issue?
On a similar note, why insist that "i" (or a negative, for that matter) is an "attribute" on a number rather than an extension of the concept of number? In one sense, this is a just a definitional choice, so I don't think either conception is right or wrong. But I'm still not getting your preference for the attribute perspective. If anything, especially in the case of negative numbers, it seems less elegant than just allowing the negatives to be numbers?
Sure, you can define any field to make your math work out. None of the interpretations are wrong per say, the question is whether or not they are useful.
The point of contention that leads to 3 interpretations is whether you assume i acts like a number. My argument is that people generally answer yes, because of Eulers identity (which is often stated as example of mathematical beauty).
My argument is that i does not act like a number, it acts more like an operator. And with i being an operator, C is not really a thing.
Rotations fell out of the structure of complex numbers. They weren't placed there on purpose. If you want to rotate things there are usually better ways.
> If you want to rotate things there are usually better ways.
Can you elaborate? If you want a representation of 2D rotations for pen-and-paper or computer calculations, unit complex numbers are to my knowledge the most common and convenient one.
For pen and paper you can hold tracing paper at an angle. Use a protractor to measure the angle. That's easier than any calculation. Or get a transparent coordinate grid, literally rotate the coordinate system and read off your new coordinates.
For computers, you could use a complex number since it's effectively a cache of sin(a) and cos(a), but you often want general affine transformations and not just rotations, so you use a matrix instead.
> For computers, you could use a complex number since it's effectively a cache of sin(a) and cos(a), but you often want general affine transformations and not just rotations, so you use a matrix instead.
That makes sense in some contexts but in, say, 2D physics simulations, you don't want general homogeneous matrices or affine transformations to represent the position/orientation of a rigid body, because you want to be able to easily update it over time without breaking the orthogonality constraint.
I guess you could say that your tuple (c, s) is a matrix [ c -s ; s c ] instead of a complex number c + si, or that it's some abstract element of SO(2), or indeed that it's "a cache of sin(a) and cos(a)", but it's simplest to just say it's a unit complex number.
Why use a unit complex number (2 numbers) instead of an angle (1 number)? Maybe it optimizes out the sins and cosses better — I don't know — but a cache is not a new type of number.
There's a significant advantage in using a tuple over a scalar to represent angles.
For many operations you can get rid of calls to trigonometric functions, or reduce the number of calls necessary. These calls may not be supported by standard libraries in minimalistic hardware. Even if it were, avoiding calls to transcendental can be useful.
Because rotations with complex numbers is not just rotations, its rotations+scaling.
The advantage of complex numbers is to rotate+scale something (or more generally move somewhere in a complex plane), is a one step multiplication operation.
If you need to support zoom, scaling shows up very frequently.
I can give an example from real life. A piece of code one of my colleagues was working on required finding a point on the angular bisector. The code became a tangle of trigonometry calls both the forward and inverse functions. The code base was Python, so there was native support of complex numbers.
So you need angular bisector of two points p and q ? just take their geometric mean and you are done. At the Python code base level you only have a call to sqrt. That simplifies things.
The whole idea of imaginary number is its basically an extension of negative numbers in concept.
When you have a negative number, you essentially have scaling + attribute which defines direction. When you encounter two negative attributes and multiply them, you get a positive number, which is a rotation by 180 degrees. Imaginary numbers extend this concept to continuous rotation that is not limited to 180 degrees.
With just i, you get rotations in the x/y plane. When you multiply by 1i you get 90 degree rotation to 1i. Multiply by i again, you get another 90 degree rotation to -1 . And so on. You can do this in xyz with i and j, and you can do this in 4dimentions with i j and k, like quaternions do, using the extra dimension to get rid of gimbal lock computation for vehicle control (where pointed straight up, yaw and roll are identicall)
The fact that i maps to sqrt of -1 is basically just part of this definition - you are using multiplication to express rotations, so when you ask what is the sqrt of -1 you are asking which 2 identical number create a rotation of 180 degrees, and the answer is 1i and 1i.
Note that the definition also very much assumes that you are only using i, i.e analogous to having the x/y plane. If you are working within x y z plane and have i and j, to get to -1 you can rotate through x/y plane or x/z plane. So sqrt of -1 can either mean "sqrt for i" or "sqrt for j" and the answer would be either i or j, both would be valid. So you pretty much have to specify the rotation aspect when you ask for a square root.
Note also that you can you can define i to be <90 degree rotation, like say 60 degrees and everything would still be consistent. In which case cube root of -1 would be i, but square root of -1 would not be i, it would be a complex number with real and imaginary parts.
The thing to understand about math is under the hood, its pretty much objects and operations. A lot of times you will have conflicts where doing an operation on a particular object is undefined - for example there are functions that assymptotically approach zero but are never equal to it. So instead, you have to form other rules or append other systems to existing systems, which all just means you start with a definition. Anything that arises from that definition is not a universal truth of the world, but simply tools that help you deal with the inconsistencies.
The whole idea of an imaginary number is that it squares to a negative number. Everything else is accidental. Nobody expected that exp(i*a)=cos(a)+i*sin(a). Totally wacky discovery.
Imaginary numbers don't work in 3D, by the way. The most natural representation of a 3D rotation is a normalized 4D quaternion, and it's still pretty weird.
There's more to it than rotation by 180 degrees. More pedagogically ...
Define a tuple (a,b) and define addition as pointwise addition. (a, b) + (c, d) = (a+c, b+d). Apples to apples, oranges to oranges. Fair enough.
How shall I define multiplication, so that multiplication so defined is a group by itself and interacts with the addition defined earlier in a distributive way. Just the way addition and multiplication behave for reals.
Ah! I have to define it this way. OK that's interesting.
But wait, then the algebra works out as if
(0, 1) * (0, 1) = (-1, 0)
but right hand side is isomorphic to -1. The (x, 0)s behave with each other just the way the real numbers behave with each other.
All this writing of tuples is cumbersome, so let me write (0,1) as i.
Addition looks like the all too familiar vector addition. What does this multiplication look like? Let me plot in the coordinate axes.
Ah! It's just scaled rotation, These numbers are just the 2x2 scaled rotation matrices that are parameterized not by 4 real numbers but just by two. One controls degree of rotation the other the amount of scaling.
If I multiply two such matrices together I get back a scaled rotation matrix. OK, understandable and expected, rotation composed is a rotation after all. But if I add two of them I get back another scaled rotation matrix, wow neato!
Because there are really only two independent parameters one isomorphic to the reals, let's call the other one "imaginary" and the tupled one "complex".
What if I negate the i in a tuple? Oh! it's reflection along the x axis. I got translation, rotation and reflection using these tuples.
What more can I do? I can surely do polynomials because I can add and multiply. Can I do calculus by falling back to Taylor expansions ? Hmm let me define a metric and see ...
You made it seem like rotations are an emergent property of complex numbers, where the original definition relies on defining the sqrt of -1.
Im saying that the origin of complex numbers is the ability to do arbitrary rotations and scaling through multiplication, and that i being the sqrt of -1 is the emergent property.
> Im saying that the origin of complex numbers is the ability to do arbitrary rotations and scaling through multiplication, and that i being the sqrt of -1 is the emergent property.
Not true historically -- the origin goes back to Cardano solving cubic equations.
But that point aside, it seems like you are trying to find something like "the true meaning of complex numbers," basing your judgement on some mix of practical application and what seems most intuitive to you. I think that's fruitless. The essence lies precisely in the equivalence of the various conceptions by means of proof. "i" as a way "to do arbitrary rotations and scaling through multiplication", or as a way give the solution space of polynomials closure, or as the equivalence of Taylor series, etc -- these are all structurally the same mathematical "i".
So "i" is all of these things, and all of these things are useful depending on what you're doing. Again, by what principle do you give priority to some uses over others?
>he origin goes back to Cardano solving cubic equations.
Whether or not mathematicians realized this at the time, there is no functional difference in assuming some imaginary number that when multiplied with another imaginary number gives a negative number, and essentially moving in more than 1 dimension on the number line.
Because it was the same way with negative numbers. By creating the "space" of negative numbers allows you do operations like 3-5+6 which has an answer in positive numbers, but if you are restricted to positive only, you can't compute that.
In the same way like I mentioned, Quaternions allow movement through 4 dimentions to arrive at a solution that is not possible to achieve with operations in 3 when you have gimbal lock.
So my argument is that complex numbers are fundamental to this, and any field or topological construction on that is secondary.
"Rotations fell out of the structure of complex numbers. They weren't placed there on purpose. If you want to rotate things there are usually better ways."
I see Complex numbers in the light of doing addition and multiplication on pairs. If one does that, rotation naturally falls out of that. So I would agree with the parent comment especially if we follow the historical development. The structure is identical to that of scaled rotation matrices parameterized by two real numbers, although historically they were discovered through a different route.
I think all of us agree with the properties of complex numbers, it's just that we may be splitting hairs differently.
>"Rotations fell out of the structure of complex numbers. They weren't placed there on purpose. If you want to rotate things there are usually better ways."
I mean, the derivation to rotate things with complex numbers is pretty simple to prove.
If you convert to cartesian, the rotation is a scaling operation by a matrix, which you have to compute from r and theta. And Im sure you know that for x and y, the rotation matrix to the new vector x' and y' is
x' = cos(theta)*x - sin(theta)*y
y' = sin(theta)*x + cos(theta)*y
However, like you said, say you want to have some representation of rotation using only 2 parameters instead of 4, and simplify the math. You can define (xr,yr) in the same coordinates as the original vector. To compute theta, you would need ArcTan(yr/xr), which then plugged back into Sin and Cos in original rotation matrix give you back xr and yr. Assuming unit vectors:
x'= xr*x - yr*y
y'= yr*x + xr*y
the only trick you need is to take care negative sign on the upper right corner term. So you notice that if you just mark the y components as i, and when you see i*i you take that to be -1, everything works out.
So overall, all of this is just construction, not emergence.
Yes it's simple and I agree with almost everything except that arctan bit (it loses information, but that's aside story).
But all that you said is not about the point that I was trying to convey.
What I showed was you if you define addition of tuples a certain, fairly natural way. And then define multiplication on the same tuples in such a way that multiplication and addition follow the distributive law (so that you can do polynomials with them). Then your hands are forced to define multiplication in very specific way, just to ensure distributivity. [To be honest their is another sneaky way to do it if the rules are changed a bit, by using reflection matrices]
Rotation so far is nowhere in the picture in our desiderata, we just want the distributive law to apply to the multiplication of tuples. That's it.
But once I do that, lo and behold this multiplication has exactly the same structure as multiplication by rotation matrices (emergence? or equivalently, recognition of the consequences of our desire)
In other words, these tuples have secretly been the (scaled) cos theta, sin theta tuples all along, although when I had invited them to my party I had not put a restriction on them that they have to be related to theta via these trig functions.
Or in other words, the only tuples that have distributive addition and multiplication are the (scaled) cos theta sin theta tuples, but when we were constructing them there was no notion of theta just the desire to satisfy few algebraic relations (distributivity of add and multiply).
> "How shall I define multiplication, so that multiplication so defined is a group by itself and interacts with the addition defined earlier in a distributive way. Just the way addition and multiplication behave for reals."
which eventually becomes
> "Ah! It's just scaled rotation"
and the implication is that emergent.
Its like you have a set of objects, and defining operations on those objects that have properties of rotations baked in ( because that is the the only way that (0, 1) * (0, 1) = (-1, 0) ever works out in your definition), and then you are surprised that you get something that behaves like rotation.
Meanwhile, when you define other "multiplicative" like operations on tuples, namely dot and cross product, you don't get rotations.
However, the fact remains that rotations can "emerge" just from the desire to do additions and multiplications on tuples to be able to do polynomials with them ... which is more directly tied to its historical path of discovery, to solve polynomial equations, starting with cubic.
>historical path of discovery, to solve polynomial equations, starting with cubic.
Even with polynomial equations that have complex roots, the idea of a rotation is baked in in solving them. Rotation+scaling with complex numbers is basically an arbitrary translation through the complex plane. So when you are faced with a*x*x + b*x + c = 0, where a b and c all lie on the real number line, and you are trying to basically get to 0, often you can't do it by having x on a number line, so you have to start with more dimentions and then rotate+scale so you end up at zero.
Its the same reason for negative numbers existing. When you have positive numbers only, and you define addition and subtraction, things like 5-6+10 become impossible to compute, even though all the values are positive. But when you introduce the space of negative numbers, even though they don't represent anything in reality, that operation becomes possible.
Yes but it was a fundamental mathematical achievement to see this equivalence. That knowledge had to emerge, be discovered. This eventually led to the theory of Galois fields.
The connection with rotation emerged naturally from a line of thought that initially had nothing to do with rotations. It was a consequence of a desire to satisfy distributive laws and maintain vector addition.
Connection between seemingly unrelated mathematical fields happen from time to time and those are considered events of surprise, understanding and celebration.
You can define that, but (if you don't already know about complex numbers) it's not obvious that it does anything mathematically interesting. It's just a cache for sin and cos, not a new type of anything. I could say that when evaluating 4th degree polynomials it's useful to have x, x^2 and x^3 immediately at hand, but the combination of those three isn't a new type of number, just a cache.
You are right - its not interesting. You already know that rotation can be done through multiplication (i.e rotation matrix), and you are just simplifying it further.
After all, the only application of imaginary numbers outside their definition is roots of a polynomial. And if you think of rotation+scaling as simple movement through the complex plane to get back to the real one, it makes perfect sense.
You can apply this principle generically as well. Say you have an operation on some ordered set S that produces elements in a smaller subset of S called S' It then follows that the inverse operation of elements of the complement of S' with respect to the original set S is undefined.
But you can create a system where you enhance the dimension of the original set with another set, giving the definition of that inverse operation for compliment of S'. And if that extra set also has ordering, then you are by definition doing something analogous to rotation+scaling.
It seems obvious now only because of significant mathematical discoveries of prominent mathematicians.
If one is taught what those discoveries revealed then of course they would seem obvious.
Arguing as you are, it would appear one can call all and every theorem in mathematics that connects to different fields as something obvious. They weren't, till someone proved the connection and that knowledge percolated down to how maths is taught, to text books.
That the integral of
exp(-x*x)
over the entire real line is sqrt pi can be surprising or obvious depending on how you were taught. At face value it has nothing to do with circles, unless you are taught the connection or you are a mathematician of high calibre who can see it without being taught the background information.
This completely misses the point of why the complex numbers were even invented. i is a number: it is one of the 2 solutions to the equation x^2 = -1 (the other being -i, of course). The whole point of inventing the complex numbers was to have a set of numbers for which any polynomial has a root. And sure, you can call this number (0,1) if you want to, but it's important to remember that C is not the same as R².
Your whole point about Taylor series is also wrong, as Taylor series are not approximations, they are actually equal to the original function if you take their infinite limit for the relevant functions here (e^x, sin x, cos x). So there is no approximation to be talked about, and no problem in identifying these functions with their Taylor series expansions.
I'd also note that there is no need to use Taylor series to prove Euler's formula. Other series that converge to e^x,cos x, sin x can also get you there.
>The whole point of inventing the complex numbers was to have a set of numbers for which any polynomial has a root
Think about what this implies.
You have an operation, like exponentiation, that has limits. Something squared can never be negative if you are talking about any real number.
In terms of Sets, you essentially have an operation that produces results only in a finite subset of the overall set. And so the inverse of that operation, when applied to the complement of that finite subset, is undefined.
However you can introduce another (ordered) set in complement to your original set and combine them to form a new set, with operations that define how you move around the values of those sets. So in the case of imaginary numbers, you basically redefine all your reals as "real number + 0 i". And now you have a way to apply that inverse operation to the complement of the finite subset, which means you can get answers to the roots of the polynomial.
And in defining the operation of multiplication, you essentially define a way to move around the 2 dimensional set now. And moving around 2 dimensions is exactly the same thing as rotation+scaling.
And note that when you say sqrt(-1) = i, you basically assume that the complex plane is 2d. There is nothing that is stopping you from making a complex plane 3d or 4d or nd. So sqrt(-1) can also be j, or it can be k. To know what it is, you have to specify the axis of the plane when you specify the sqrt operation, which again, brings it back to the concept of rotations.
And thats my whole point, there is nothing special about i, its simply just a construct that bakes in rotations through any way you wanna define it.
>our whole point about Taylor series is also wrong, as Taylor series are not approximations, they are actually equal to the original function if you take their infinite limit
Looking back at what I wrote, I worded it very poorly.
I don't have a problem with any math involved, not trying to say that Eulers identity is not valid.
What Im trying to say is that all the definitions sort of assume that if you have some operation that you can do on real numbers, and have a result, if you just plug in a complex value, it all works out, so people think that i behaves like a number. I personally don't think that this is the case, specifically about i behaving like a number just because those results work out.
For example, even without Taylor series, you can prove Eulers identity using the limit formula for e(x). The idea is that you have (1+xi/n)^n as n goes to infinity, but because you baked in the rotation as a multiplication in the definition, all you are doing is starting at 1+0i and doing smaller and smaller rotations to get to some value, and the limit of that value is essentially the unit vector rotated by a certain angle. So naturally the cos and sin equivalence arises.
My issue is that the limit equation for e, in the case of the reals, take e x times in multiplication and then compute the limit equation, and you get equivalence. But in the case of the complex, you don't really have any idea what it takes something to ith power, but you can compute the limit equation, and so you end up with a definition of what it means to take something to the ith power.
My argument is that its not really applicable - not that its wrong, but the fact that its not defining exponentiation to the ith power in the sense that i has "number like" qualities like real numbers do. You would have to prove that an equivalence
What is really happening is that you never really escape the real numbers, and your complex numbers are just simplified operations that rotate/scale a number, like rotation matricies do through multiplication, and that in the nature of the definition of those rotations, you get stuff like Eulers identity, which is somewhat pointless because like I mentioned - the value of e (i.e 2.7) is never really used to compute anything in regards to complex numbers in polar form of re^ix, all you care about is r and the x which is the angle.
And for this reason, I don't consider i a number, so the analytic/smooth interpretations to me are meaningless.
> And in defining the operation of multiplication, you essentially define a way to move around the 2 dimensional set now. And moving around 2 dimensions is exactly the same thing as rotation+scaling.
Again, this is not how complex numbers were defined. The only original goal was to come up with a number that can solve the equation x^2 + 1 = 0, so that (R + {this number}, + , *), becomes an algebraically closed field. Once you've set this goal, there is really a single simple choice for how the operations will work, because everything else is already constrained. If x^2 + 1 = 0, we already know that:
So the formula for complex number multiplication comes out of the arithmetic of real numbers, extended with this extra entity defined simply by being a root of x^2 + 1. The fact that this operation happens to represent a rotation in the RxR plane is "an accident" (I'm sure there are deep ties that make this necessary, probably related to the structure of polynomials themselves).
And while you can define other algebraically closed fields that include the reals as a subfield, the complex numbers are the simplest such set. R^n for n>2 is clearly more complex, for example. So there is a clear reason to prefer sqrt(-1) = i, and thus ending up with a 2d vector space.
> What Im trying to say is that all the definitions sort of assume that if you have some operation that you can do on real numbers, and have a result, if you just plug in a complex value, it all works out, so people think that i behaves like a number. I personally don't think that this is the case, specifically about i behaving like a number just because those results work out.
Again, complex multiplication is not some intentional construction, it is baked into how we defined the complex numbers the first time we did. Again, our goal was to find (well, define) the solution(s) of the equation x^2 + 1 = 0. The fact that we can plug in this x to any formula that involves other numbers just falls out of this goal, it's not an additional assumption. In the case of e(x), this is simpler to see with the power series formula:
e(nx) = sum(1 + nx + (nx)^2/2! + (nx)^3/3! + (nx)^4/4! + ...)
but, by definition, x^2 + 1 = 0, so x^2 = -1, so the formula becomes:
e(nx) = sum(1 + nx - n^2/2! - xn^3/3! + n^4/4! + ...) =
= sum(1 - n^2/2! + n^4/4! ...) + sum(xn - xn^3/3! + xn^5/5! ...) =
= sum(1 - n^2/2! + n^4/4! ...) + x sum(n - n^3/3! + n^5/5! ...) =
= cos n + x sin n
Note that this falls out of the properties of e(x), cos(x), and sin(x) for real numbers, and the single property of i that it is a solution of x^2 + 1 = 0.
I also think that the definition that e(x) is "take e x times in multiplication and then compute the limit" is any more intuitive. I certainly don't think that `e^2 = e(2) = lim (1 + 2/n) ^ n, with n-> infinity` is any more intuitive than the definition of `e(2i) = lim (1 + 2i/n)^n, with n -> infinity`.
> which is somewhat pointless because like I mentioned - the value of e (i.e 2.7) is never really used to compute anything in regards to complex numbers in polar form of re^ix, all you care about is r and the x which is the angle
This is also not really true, because the value of e is deeply tied to the values of cos x and sin x. This also becomes visible if you want to compute 2^(ix). 2^(ix) = e ^ ix (log_e 2) = cos (x log_e 2) + i sin (x log_e 2), using log_e to denote the natural logarithm to make it clearer that it is related to e. So the value of e itself is still there in the formula, even if we discount the relationship between e and cos and sin.
The things is, there are no accidents in math. If you end up with formulas that look like something else, that something else was a defining part of the original definition, whether it was obvious or not.
You would agree that in the quest to solve x^2+1=0, we had to introduce another dimension, with a multiply operator that lets us move through that dimension. All im saying is that introducing another dimension and ability to move in that dimension is the same thing as rotation and scaling.
As for e, the point im trying to make is that if you look at what it means to take something to the power of something when it comes to reals, there is a clear definition. But taking something to imaginary power is meaningless it iself. For reals, the power operator has a strict definition with multiplication and division for rationals, and generically extended to reals through limits. And to compute a limit means that you have to have continuity and smoothness. So by extension, to compute exponentiation, you have to have continuity and smoothness, and vice versa.
My argument is that there is no guarantee of continuity/smoothness on the complex plane - one can define it as such (i.e the analytic view) but in my stricter philosophical view, to make something with similar properties like real numbers, you have to have all the analogous operations work within the numbers itself. I.e you have to be able to define what 2i^3i is without ever referencing anything from the real numbers.
This is not done for complex plane - to define something to the i power you need to borrow definitions from the real plane.
As such, you can't define exponentiation to the i, because you can't compute limits. Computing the taylor series expansion equivalence or limit formulas in the way me and you presented them are "holograms" - i.e meaningless results.
And it seems that way. Say that you ignore taylor series formulation, skip it completely, and define the polar form to be r(10)^ix = cos(x)+isin(x), where r is radius and x is angle. JUST BECAUSE YOU CAN.
Would you lose anything? Not really. You would still have a way to compute 2^i - it would just be 10^(ilog(2)) -> r = 1, angle=log(2). I.e the map still exists in quite a good shape.
Basically it doesn't matter if r(10)^ix = cos(x)+isin(x) or r(e)^ix = cos(x)+isin(x), as counterintuitive as it may seem, which furthers my point that exponentiation operation to imaginary numbers is not defined.
So without exponentiation, all you are left with is basically a more strict rigid construction of another set with an ordering property, a multiplication operation definition, (i^2 = -1), and the resultant orthogonality that represents a cartesian plane.
And if you think thats silly, consider the fact that the algebraic view of complex numbers doesn't even consider calculus on them to be valid.
> And it seems that way. Say that you ignore taylor series formulation, skip it completely, and define the polar form to be r(10)^ix = cos(x)+isin(x), where r is radius and x is angle. JUST BECAUSE YOU CAN.
The thing is, you can't do this. Your numbers will not work out correctly. For example, (e^(pi*i))^i = 1/e^pi is a direct consequence of how exponentiation works and the definition of i. If you define 10^(xi) = cos x + i sin x, you will get:
(e^(pi\*i))^i =
= (10^(log_10 e \* pi \* i))^i
= (cos (log_10 e \* pi) + i sin (log_10 e \* pi)) ^ i
= 10^(log_10 $exp)i
= cos $exp + i sin $exp
I very much doubt that sin(cos (log_10 e * pi) + i sin (log_10 e * pi)) is 0, so this number can't possibly be equal to the real number 1/e^pi. So, with your definition, the property (a^x)^y = a^(x*y) doesn't hold for all a, x, y. So, your definition doesn't represent the exponential function at all.
I think the point is to look at it the other way: we invented/chose i so that exponentiation (and all other functions) maintain the same properties for the complex numbers as they have for the reals. The whole goal of defining the complex numbers is to extend R in a way that maintains all of the properties of functions on R.
> You would agree that in the quest to solve x^2+1=0, we had to introduce another dimension, with a multiply operator that lets us move through that dimension.
I don't really agree, no. In the analytic definition, there is no second dimension. Sure, C is isomorphic to R×R, but that is not how you construct it. C is not R×R, it's R+{a + b * i | a,b in R, b≠0} in this view. Just like Z is N+{a * -1 | a in N, a≠0}. You introduce one new number that you need to solve the equation, and then all of the numbers needed to make the new construction a field again. You don't introduce a new notion of multiplication, C uses the exact same multiplication operation as R does, or at least as polynomials over R do, as I showed.
The fact that C is isomorphic to R×R, and that multiplication of numbers in C is isomorphic to scaling+rotation of vectors in R×R, is not part of the construction.
I do agree that there are no accidents in math, so I imagine this isomorphism is related to some more fundamental relationship between polynomials, 2d vectors, and rotation matrices - because our construction of C is strictly motivated by polynomials.
> For reals, the power operator has a strict definition with multiplication and division.
I don't agree with this statement. It's true for the integers, but already for the the rationals it loses any direct relationship to multiplication and division (how do you get 4^(1/2) = 2 by repeated multiplication?). And for the reals, it's completely gone. We can't even define many properties of real-valued exponentials - we don't even know if e^pi is an irrational number, for example.
> When you search for something like taylor series or limit form of e and you see a way to compute e^i what you are doing is basically using operators designed for real numbers and extending them to the complex numbers (when you substitute i for where normally a real number would be in the power term.
We've already agreed that there are no accidents in math. So just as the multiplication of polynomials is fundamentally linked with rotations of 2d real-valued vectors, we must accept that real exponentiation is fundamentally linked to complex exponentiation, otherwise the formulas wouldn't work the way they do.
Note also that calculus is not limited to operations on the number line - you can take the derivatives or integrate or calculate limits of n-dimensional curves, and even differentiate over surfaces and n-dimensional manifolds more generally. Smoothness and continuity are anyway part of the structure of C, regardless what definition of it you use.
> And if you think thats silly, consider the fact that the algebraic view of complex numbers doesn't even consider calculus on them to be valid.
I don't understand what you mean by this. Calculus is well defined for functions over any field of size continuum, and that is exactly what <C,+,*,0,1> is in the algebraic view.
>C uses the exact same multiplication operation as R does
Not quite. If it did, the -i*-i would be i^2, not -1. And yes, I totally agree that C is R+, not RxR. The point is that you still introducing something extra with some rules, where you introduce the concept of geometric orthogonality into i^2 = -1, whether that is your intention or not.
>For reals, the power operator has a strict definition with multiplication and division for rationals, and generically extended to reals through limits.
I accidentally swapped reals and rationals there. The whole point was to highlight that exponentiation for real exponents relies on limits which relies on continutity.
>So just as the multiplication of polynomials is fundamentally linked with rotations of 2d real-valued vectors, we must accept that real exponentiation is fundamentally linked to complex exponentiation, otherwise the formulas wouldn't work the way they do.
Don't agree.
Multiplication of polynomials involves operations that are clearly defined. When you do (a+bi)^2, you have defined what it means to multiply complex numbers in their construction, without needing to use any such formula from real numbers.
Exponentiation where you have i exponent however, is not defined solely in the complex field.
>I don't understand what you mean by this. Calculus is well defined for functions over any field of size continuum, and that is exactly what <C,+,*,0,1> is in the algebraic view.
Algebraic view is basically the idea of that numbers are only defined if there is some relation that expresses their definition.
For example, 1+2=3, locks all 3 number down. 1 is 3-2, 2 is 3-1 and 3 is 1+2.
pi or e on the other hand, are "something else", because there is no algebraic formula that defines them. To do so you have to invoke the computation of limits, which is an analytic view, not algebraic.*
> Not quite. If it did, the -i-i would be i^2, not -1.
It is i², though, but that is equal to -1. Just like 22 = 2² = 4. I also maintain that the historical view is that this something extra comes from the properties of polynomials and their roots, not from geometric orthogonality.
> Exponentiation where you have i exponent however, is not defined solely in the complex field.
We can take another tack for defining the complex exponential function, if you'd prefer. One of the definitions of the exponential function is that e^x is the only function that respects f'(x) = f(x) (well, up to constant multiplication).
So, we need to look for a function f(z) such that f'(z) = f(z). There are various ways to do this (for example, using the Taylor series expansion and noting that all of the f derived n times factors are equal to f(z), which yields the power series definition). You don't need to appeal to limits of e^x to get there this way.
> Algebraic view is basically the idea of that numbers are only defined if there is some relation that expresses their definition.
Understood, you are using a different sense of "algebraic" than what I was - I was thinking more of the abstract algebraic definition of C.
Still, the sense you are using seems to be the concept of algebraic numbers, which is more formalized - the algebraic numbers are all those that represent the root of a polynomial with integer or rational coefficients. Interestingly, while pi and e are not algebraic numbers, i or i+7 are still algebraic.
However, I'm not sure what the point of bringing this up is. Exponentiation is simply not defined over the algebraic numbers, especially not e^x where x is algebraic - so if we restrict ourselves to algebraic numbers, e^i is not defined, true, but neither is e^1. And while 2^2 is defined, of course, I'm not even sure you can define 2^sqrt(2), so I wouldn't be surprised if 2^i doesn't make sense either.
Either way, the algebraic numbers are not "a way of thinking about numbers", they are a restricted subset of what is generally meant by "number", and many famous and useful results from many branches of math do not work over this subset (for example, you can't even use the same set of algebraic numbers to refer to the length of a circle and the length of a square).
I still think you misunderstanding what im saying. i don't have a problem with the math working out, its that i have a problem with the math being able to be applied in the first place.
Let me try a different way.
Suppose you start with just the i number line with the rules that exist. You have zero, integer i's, rational is, and even irrational i's. All seems good. Then you start to define operations. i*i is undefined (because it goes to -1, and the reals are outside of your domain that you are working with currently). And this means you can't effectively do any sort of futher work in defining exponentiation, or limits, because you can have purely complex polynomials with just undefined terms.
So like you said, complex is R+, not something like RxR. The definition of complex numbers is intrinsic to real numbers - its an enhancement on real numbers. And by extension, all the math works out when you do taylor series with e^i and such.
But this pretty much means its a rigid definition, i.e you are defining something in a certain construction to supplement reals.
And as for geometric claim, my argument with that is that just like when you have x, and then you add <x,y> in some form and way, you are defining geometry. So in defining a+bi, you are defining geometry.*
i is not a "trick" or a conceit to shortcut certain calculations like, say, the small angle approximation. i is a number and this must be true because of the fundamental theorem of algebra. Disbelieving in the imaginary numbers is no different from disbelieving in negative numbers.
"Imaginary" is an unfortunate name which gives makes this misunderstanding intuitive.
However, what's true about what you and GP have suggested is that both i and -1 are used as units. Writing -10 or 10i is similar to writing 10kg (more clearly, 10 × i, 10 × -1, 10 × 1kg). Units are not normally numbers, but they are for certain dimensionless quantities like % (1/100) or moles (6.02214076 × 10^23) and i and -1. That is another wrinkle which is genuinely confusing.
i is a complex number, complex numbers are of the form real + i*real... Don't you see the recursive definition ? Same with 0 and 1 they are not numbers until you can actually define numbers, using 0 and 1
i*i=-1 makes perfect sense
This is one definition of i. Or you could geometrically say i is the orthogonal unit vector in the (real,real) plane where you define multiplication as multiplying length and adding angles
i is also a quaternion. So by this logic we could say complex numbers are made up of quaternions. But we don’t say such things because they wouldn’t be a good mental model of what we want to talk about.
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.
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?
> 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.
Math and reality are, in general completely distinct. Some math is originally developed to model reality, but nowadays (and for a long time) that's not the typical starting point, and mathematicians pushing boundaries in academia generally don't even think about how it relates to reality.
However, it is true (and an absolutely fascinating phenomenon) that we keep encountering phenomena in reality and then realize that an existing but previously purely academic branch of math is useful for modeling it.
To the best of our knowledge, such cases are basically coincidence.
Opposing view (that I happen to hold, at least if I had to choose one side or the other): not only is mathematics 'reality'; it is arguably the only thing that has a reasonable claim to being 'reality' itself.
After all, facts (whatever that means) about the physical world can only be obtained by proxy (through measurement), whereas mathematical facts are just... evident. They're nakedly apparent. Nothing is being modelled. What you call the 'model' is the object of study itself.
A denial of the 'reality' of pure mathematics would imply the claim that an alien civilisation given enough time would not discover the same facts or would even discover different – perhaps contradictory – facts. This seems implausible, excluding very technical foundational issues. And even then it's hard to believe.
> To the best of our knowledge, such cases are basically coincidence.
This couldn't be further from the truth. It's not coincidence at all. The reason that mathematics inevitably ends up being 'useful' (whatever that means; it heavily depends on who you ask!) is because it's very much real. It might be somewhat 'theoretical', but that doesn't mean it's made up. It really shouldn't surprise anyone that an understanding of the most basic principles of reality turns out to be somewhat useful.
I think you're not even disagreeing with me, we're just using different definitions of the word "reality". I meant it to use specifically "the physical world" - which you are treating as distinct from mathematics as well in your second paragraph.
But then you must agree that it's not a coincidence.
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.
"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...
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.
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.
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.
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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:
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.
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...
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.
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.
> 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.
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.
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.
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.
> 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.
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?
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.
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).
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.
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).
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.
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).
> but I believe that all of mathematics arises from an underlying fundamental structure to the universe and that this results in it both being "discoverable" (rather than invented) and "useful" (as in helpful for describing, expressing and calculating things).
That is an interesting idea. Can you elaborate? As in, us, that is our brains live in this physical universe so we’re sort of guided towards discovering certain mathematical properties and not others. Like we intuitively visualize 1d, 2d, 3d spaces but not higher ones? But we do operate on higher dimensional objects nevertheless?
Anyway, my immediate reaction is to disagree, since in theory I can imagine replacing the universe with another with different rules and still maintaining the same mathematical structures from this universe.
Why do you believe that the same mathematical properties hold everywhere in the universe?
Not 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.
> 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.
> 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.
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).
> 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.
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.
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.
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:
Now algebraic closure, but better: Next, general theory of fields: I think maybe differential geometry can provide some help here.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.
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.
The movement from R to C can be done rigorously. It gets hand-waved away in more application-oriented math courses, but it's done properly in higher level theoretically-focused courses. Lifting from a smaller field (or other algebraic structure) to a larger one is a very powerful idea because it often reveals more structure that is not visible in the smaller field. Some good examples are using complex eigenvalues to understand real matrices, or using complex analysis to evaluate integrals over R.
I hate when people casually move "between" Q and Z as if a rational number with unit denominator suddenly becomes an integer, and it's all because of this terrible "a/b" notation. It's more like (a, b). You can't ever discard that second component, it's always there. ;)
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!"
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...
Related: https://news.ycombinator.com/item?id=18310788
I've always been satisfied with the explanation "Just as you need signed numbers for translation, you need complex numbers to express rotation." Nobody asks if negative numbers are really a natural thing, so it doesn't make sense to ask if complex ones are, IMO.
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.)
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.
The good news is that Q is not really entitled to a closure either.
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. >
I began studying 3-manifolds after coming up with a novel way I preferred to draw their presentations. All approaches are formally equivalent, but they impose different cognitive loads in practice. My approach was trivially equivalent to triangulations, or spines, or Heegaard splittings, or ... but I found myself far more nimbly able to "see" 3-manifolds my way.
I showed various colleagues. Each one would ask me to demonstrate the equivalence to their preferred presentation, then assure me "nothing to see here, move along!" that I should instead stick to their convention.
Then I met with Bill Thurston, the most influential topologist of our lifetimes. He had me quickly describe the equivalence between my form and every other known form, effectively adding my node to a complete graph of equivalences he had in his muscle memory. He then suggested some generalizations, and proposed that circle packings would prove to be important to me.
Some mathematicians are smart enough to see no distinction between any of the ways to describe the essential structure of a mathematical object. They see the object.
Would you mind sharing your representation? :-)
I was interested in how it would make sense to define complex numbers without fixing the reals, but I'm not terribly convinced by the method here. It seemed kind of suspect that you'd reduce the complex numbers purely to its field properties of addition and multiplication when these aren't enough to get from the rationals to the reals (some limit-like construction is needed; the article uses Dedekind cuts later on). Anyway, the "algebraic conception" is defined as "up to isomorphism, the unique algebraically closed field of characteristic zero and size continuum", that is, you just declare it has the same size as the reals. And of course now you have no way to tell where π is, since it has no algebraic relation to the distinguished numbers 0 and 1. If I'm reading right, this can be done with any uncountable cardinality with uniqueness up to isomorphism. It's interesting that algebraic closure is enough to get you this far, but with the arbitrary choice of cardinality and all these "wild automorphisms", doesn't this construction just seem... defective?
It feels a bit like the article's trying to extend some legitimate debate about whether fixing i versus -i is natural to push this other definition as an equal contender, but there's hardly any support offered. I expect the last-place 28% poll showing, if it does reflect serious mathematicians at all, is those who treat the topological structure as a given or didn't think much about the implications of leaving it out.
More on not being able to find π, as I'm piecing it together: given only the field structure, you can't construct an equation identifying π or even narrowing it down, because if π is the only free variable then it will work out to finding roots of a polynomial (you only have field operations!) and π is transcendental so that polynomial can only be 0 (if you're allowed to use not-equals instead of equals, of course you can specify that π isn't in various sets of algebraic numbers). With other free variables, because the field's algebraically closed, you can fix π to whatever transcendental you like and still solve for the remaining variables. So it's something like, the rationals plus a continuum's worth of arbitrary field extensions? Not terribly surprising that all instances of this are isomorphic as fields but it's starting to feel about as useful as claiming the real numbers are "up to set isomorphism, the unique set whose cardinality matches the power set of the natural numbers", like, of course it's got automorphisms, you didn't finish defining it.
You need some notion of order or of metric structure if you want to talk about numbers being "close" enough to π. This is related to the property of completeness for the real numbers, which is rather important. Ultimately, the real numbers are also a rigorously defined abstraction for the common notion of approximating some extant but perhaps not fully known quantity.
The common metric on the reals one of only two that exist as completions of Q!
The other is the p-padics (basically, low-order bits matter more than high-order bits), which have distance but not ordering.
There are infinitely many different p-adic completions of the rationals for each prime p, so we have 2-adics, 3-adics, 5-adics etc, all different.
There's a related idea in mathematics, the proof that the real numbers are a vector space over the rational numbers. If you scramble the basis vectors, you obtain an isomorphic vector space, but it is effectively a "permutation" of |R. Of course, vector spaces don't even have multiplication, but one interesting thing is that the proof requires the axiom of choice.
I think that actually constructing a "nontrivial" model of C using the field conception might require choosing a member from each of an infinite family of sets, i.e. it requires applying the axiom of choice, similar to the way you construct R as a vector space.
Vector space over which field ? The reals ? In that case you have got a chicken and egg problem.
Most commenters are talking about the first part of the post, which lays out how you might construct the complex numbers if you're interested in different properties of them. I think the last bit is the real interesting substance, which is about how to think about things like this in general (namely through structuralism), and why the observations of the first half should not be taken as an argument against structuralism. Very interesting and well written.
It is very re-assuring to know, on a post where I can essentially not even speak the language (despite a masters in engineering) HN is still just discussing the first paragraph of the post.
To be clear, this "disagreement" is about arbitrary naming conventions which can be chosen as needed for the problem at hand. It doesn't make any difference to results.
The author is definitely claiming that it's not just about naming conventions: "These different perspectives ultimately amount, I argue, to mathematically inequivalent structural conceptions of the complex numbers". So you would need to argue against the substance of the article to have a basis for asserting that it is just about naming conventions.
I mean, fine, I guess? But on the other hand Meh.
Article: "They form the complex field, of course, with the corresponding algebraic structure, but do we think of the complex numbers necessarily also with their smooth topological structure? Is the real field necessarily distinguished as a fixed particular subfield of the complex numbers? Do we understand the complex numbers necessarily to come with their rigid coordinate structure of real and imaginary parts?"
So yes these are choices. If I care how the complex plane maps onto some real number somewhere, then I have to pick a mapping. "Real part" is only one conventional mapping. Ditto the other stuff: If I'm going to do contour integrals then I've implied some things about metric and handedness.
I still don't see how this really puts mathematicians in "disagreement." Let's pedestrian example:
I usually make an x,y plot with the x-axis pointing to the right and the y-axis pointing away from me. If I put a z-axis, personally I'll make it upwards out of the paper (sometimes this matters). Usually, but not always, my co-ordinates are meant to be smooth. But if somebody does some of this another way, are they really disagreeing with me? I think "no." If we're talking about the same problem, we'll eventually get the same answer (after we each fix 3 or 4 mistakes). If we're talking about different problems, then we need our answers to potentially "disagree."
It depends on what that "other way" is, and if it has a different structure, making it a different thing.
What's special about C is that it's almost a completely unified and uniform view of algebra of analysis, but not quite.
Exactly. So I feel like all the article really says is that "Complex numbers" doesn't necessarily tell you everything you need to know. It depends on what you're doing with them.
In the article he says there is a model of ZFC in which the complex numbers have indistinguishable square roots of -1. Thus that model presumably does not allow for a rigid coordinate view of complex numbers.
It just means that there are two indistinguishable coordinate views a + bi and a - bi, and you can pick whichever you prefer.
Theorem. If ZFC is consistent, then there is a model of ZFC that has a definable complete ordered field ℝ with a definable algebraic closure ℂ, such that the two square roots of −1 in ℂ are set-theoretically indiscernible, even with ordinal parameters.
Haven’t thought it through so I’m quite possibly wrong but it seems to me this implies that in such a situation you can’t have a coordinate view. How can you have two indistinguishable views of something while being able to pick one view?
Mathematicians pick an arbitrary complex number by writing "Let c ∈ ℂ." There are an infinite number of possibilities, but it doesn't matter. They pick the imaginary unit by writing "Let i ∈ ℂ such that i² = −1." There are two possibilities, but it doesn't matter.
If two things are set theoretically indistinguishable then one can’t say “pick one and call it i and the other one -i”. The two sets are the same according to the background set theory.
They're not the same. i ≠ −i, no matter which square root of negative one i is. They're merely indiscernible in the sense that if φ(i) is a formula where i is the only free variable, ∀i ∈ ℂ. i² = −1 ⇒ (φ(i) ⇔ φ(−i)) is a true formula. But if you add another free variable j, φ(i, j) can be true while φ(−i, j) is false, i.e. it's not the case that ∀j ∈ ℂ. ∀i ∈ ℂ. i² = −1 ⇒ (φ(i, j) ⇔ φ(−i, j)).
I studied commutative algebra. I’m not set theorist. I wasn’t sure exactly what “set theoretically indiscernible” meant.
I'm not a professional, but to me it's clear that whether i and -i are "the same" or "different" is actually quite important.
I'm a professional mathematician and professor.
This is a very interesting question, and a great motivator for Galois theory, kind of like a Zen koan. (e.g. "What is the sound of one hand clapping?")
But the question is inherently imprecise. As soon as you make a precise question out of it, that question can be answered trivially.
Generally, the nth roots of 1 form a cyclic group (with complex multiplication, i.e. rotation by multiples of 2pi/n).
One of the roots is 1, choosing either adjacent one as a privileged group generator means choosing whether to draw the same complex plane clockwise or counterclockwise.
They would never be the same. It's just that everything still works the same if you switch out every i with -i (and thus every -i with i).
There are ways to build C that result in:
1) Exactly one C
2) Exactly two isomorphic Cs
3) Infinitely many isomorphic Cs
It's not really the question of whether i and -i are the same or not. It's the question of whether this question arises at all and in which form.
The question is meaningless because isomorphic structures should be considered identical. A=A. Unless you happen to be studying the isomorphisms themselves in some broader context, in which case how the structures are identical matters. (For example, the fact that in any expression you can freely switch i with -i is a meaningful claim about how you might work with the complex numbers.)
Homotopy type theory was invented to address this notion of equivalence (eg, under isomorphism) being equivalent to identity; but there’s not a general consensus around the topic — and different formalisms address equivalence versus identity in varied ways.
PP meant automorphisms, which is what the OP article is about.
They're different. Multiplication by i gives a quarter turn counterclockwise. -i gives a quarter turn clockwise.
Opposite quarter turns cancel: (-i)(i) = (-1)(i^2) = +1
Quarter turn twice counterclockwise gives a half turn: (i)(i) = -1
Quarter turn twice clockwise also gives a half turn: (-i)(-i) = -1
Sure. Either that or the reverse. "They're not the same" in the sense that they can't both be clockwise. "They are the same" in the sense that we could make either one clockwise.
A bit like +0 and -0? It makes sense in some contexts, and none in others.
Names, language, and concepts are essential to and have powerful effects on our understanding of anything, and knowledge of mathematics is much more than the results. Arguably, the results are only tests of what's really important, our understanding.
Agreed. To me it looks like the entire discussion is just bike-shedding.
It's math. Bikeshedding is the goal.
In particular, the core disagreement seems to be about whether the automorphisms of C should keep R (as a subset) fixed, or not.
The easy solution here would be to just have two different names: (general) automorphisms (of which there might be many) and automorphisms-that-keep-R-fixed (of which there are just the two mentioned.
If you make this distinction, then the approach of construction of C should not matter, as they are all equivalent?
No the entire point is that it makes difference in the results. He even gave an example in which AI(and most humans imo) picked different interpretation of complex numbers giving different result.
The whole substack is great, I recommend reading all of it if you are interested in infinity
He's also very active at Stack Exchange
– https://stackexchange.com/users/510234/joel-david-hamkins
– https://mathoverflow.net/users/1946/joel-david-hamkins
I really know almost nothing about complex analysis, but this sure feels like what physicists call observational entropy applied to mathematics: what counts as "order" in ℂ depends on the resolution of your observational apparatus.
The algebraic conception, with its wild automorphisms, exhibits a kind of multiplicative chaos — small changes in perspective (which automorphism you apply) cascade into radically different views of the structure. Transcendental numbers are all automorphic with each other; the structure cannot distinguish e from π. Meanwhile, the analytic/smooth conception, by fixing the topology, tames this chaos into something with only two symmetries. The topology acts as a damping mechanism, converting multiplicative sensitivity into additive stability.
I'll just add to that that if transformers are implementing a renormalization group flow, than the models' failure on the automorphism question is predictable: systems trained on compressed representations of mathematical knowledge will default to the conception with the lowest "synchronization" cost — the one most commonly used in practice.
https://www.symmetrybroken.com/transformer-as-renormalizatio...
The way I think of complex numbers is as linear transformations. Not points but functions on points that rotate and scale. The complex numbers are a particular set of 2x2 matrices, where complex multiplication is matrix multiplication, i.e. function composition. Complex conjugation is matrix transposition. When you think of things this way all the complex matrices and hermitian matrices in physics make a lot more sense. Which group do I fall into?
This would be the rigid interpretation since i and -i are concrete distinguishable elements with Im and Re defined.
if only matrices would've been invented before i..
There's no disagreement, the algebraic one is the correct one, obviously. Anyone that says differently is wrong. :)
Being an engineer by training, I never got exposed to much algebra in my courses (beyond the usual high school stuff in high school). In fact did not miss it much either. Tried to learn some algebraic geometry then... oh the horror. For whatever reason, my intuition is very geometric and analytic (in the calculus sense). Even things like counting and combinatorics, they feel weird, like dry flavorless pretzels made of dried husk. Combinatorics is good only when I can use Calculus. Calculus, oh that's different, it's rich savoury umami buttery briskets. Yum.
That's not the interesting part. The interesting part is that I thought everyone is the same, like me.
It was a big and surprising revelation that people love counting or algebra in just the same way I feel about geometry (not the finite kind) and feel awkward in the kind of mathematics that I like.
It's part of the reason I don't at all get the hate that school Calculus gets. It's so intuitive and beautifully geometric, what's not to like. .. that's usually my first reaction. Usually followed by disappointment and sadness -- oh no they are contemplating about throwing such a beautiful part away.
School calculus is hated because it's typically taught with epsilon delta proofs which is a formalism that happened later in the history of calculus. It's not that intuitive for beginners, especially students who haven't learn any logic to grok existential/universal quantifiers. Historically, mathematics is usually developed by people with little care for complete rigor, then they erase their tracks to make it look pristine. It's no wonder students are like "who the hell came up with all this". Mathematics definitely has an education problem.
IMO, the calculus is taught incorrectly. It should start with functions and completely avoid sequences initially. Once you understand how calculus exploits continuity (and sometimes smoothness), it becomes almost intuitive. That's also how it was historically developed, until Weierstrass invented his monster function and forced a bit more rigor.
But instead calculus is taught from fundamentals, building up from sequences. And a lot of complexity and hate comes from all those "technical" theorems that you need to make that jump from sequences to functions. E.g. things like "you can pick a converging subsequence from any bounded sequence".
Interesting.
In Maths classes, we started with functions. Functions as list of pairs, functions defined by algebraic expressions, functions plotted on graph papers and after that limits. Sequences were peripherally treated, just so that limits made sense.
Simultaneously, in Physics classes we were being taught using infinitesimals, with the a call back that "you will see this done more formally in your maths classes, but for intuition, infinitesimals will do for now".
You can do it with infinitesimals if you like, but the required course in nonstandard analysis to justify it is a bastard.
Or you can hand wave a bit and trust intuition. Just like the titans who invented it all did!
The obsession with rigor that later developed -- while necessary -- is really an "advanced topic" that shouldn't displace learning the intuition and big picture concepts. I think math up through high school should concentrate on the latter, while still being honest about the hand-waving when it happens.
I broadly agree. But, the big risk here is that it's really easy for an adventurous student to stretch that handwaving beyond where it's actually valid. You at least have to warn them that the "intuitions" you give them are not general methods, just explanations for why the algorithms you teach them do something worthwhile (and for the ones inclined to explore, give them some fun edge cases to think about).
You can do it with synthetic differential geometry, but that introduces some fiddliness in the underlying logic in order to cope with the fact that eps^2 really "equals" zero for small enough eps, and yet eps is not equal to zero.
while (i > 0) { operate_over_time }
calculus works... because it was almost designed for Mechanics. If the machine it's getting input, you have output. When it finished getting input, all the output you get yields some value, yes, but limits are best understood not for the result, but for the process (what the functions do).
You are not sending 0 coins to a machine, do you? You sent X to 0 coins to a machine. The machine will work from 2 to 0, but 0 itself is not included because is not a part of a changing process, it's the end.
Limits are for ranges of quantities over something.
"The Axiom of Choice is obviously true, the Well-ordering theorem obviously false, and who can tell about Zorn's lemma?"
(attributed to Jerry Bona)
Hah. This perspective is how you get an embedding of booleans into the reals in which False is 1 and True is -1 :-)
(Yes, mathematicians really use it. It makes parity a simpler polynomial than the normal assignment).
It works if you don't care about magnitudes, distances, or angles of complex numbers. Those properties aren't algebraic.
The complex numbers are just elements of R[i]/(i^2+1). I don't even understand how people are able to get this wrong.
Of course everyone agrees that this is a nice way to construct the complex field. The question is what is the structure you are placing on this construction. Is it just a field? Do you intend to fix R as a distinguished subfield? After all, there are many different copies of R in C, if one has only the field structure. Is i named as a constant, as it seems to be in the construction when you form the polynomials in the symbol i. Do you intend to view this as a topological space? Those further questions is what the discussion is about.
I mean, yes of course i is an element in C, because it's a monic polynomial in i.
There's no "intend to". The complex numbers are what they are regardless of us; this isn't quantum mechanics where the presence of an observer somehow changes things.
It's not about observers, but about mathematical structure and meaning. Without answering the questions, you are being ambiguous as to what the structure of C is. For example, if a particular copy of R is fixed as a subfield, then there are only two automorphisms---the trivial automorphism and complex conjugation, since any automorphism fixing the copy of R would have to be the identity on those reals and thus the rest of it is determined by whether i is fixed or sent to -i. Meanwhile, if you don't fix a particular R subfield, then there is a vast space of further wild automorphisms. So this choice of structure---that is, the answer to the questions I posed---has huge consequences on the automorphism group of your conception. You can't just ignore it and refuse to say what the structure is.
You're assuming there has to be a "meaning". There isn't. We're just manipulating meaningless symbols.
Obviously.
Does anyone have any tips on how I would fundamentally understand this article without just going back to school and getting a degree in mathematics? This is the sort of article where my attempts to understand a term only ever increase the number of terms I don't understand.
PhD math guy here. Personally, I don't think this is a great article for a lay person to approach or appreciate. You might be able and interested to follow along with the various constructions of C, but without appreciation for formal logic and category theory it'll seem like distinctions without a difference (and to working mathematicians, they are). The model theory I found quite interesting, but was high effort even for me. And the philosophical points are also rather specific.
Study analysis and algebra from books and videos on your own?
I thought i understood complex numbers and accepted them until I did countour integration for the first time.
Ever since then I have been deeply unsettled. I started questioning taking integrals to (+/-) infinity, and so I became unsettled with R too.
If C exists to fix R, then why does R even exist? Why does R need to be fixed? Why does the use of the upper or lower plane for counter integration not matter? I can do mathematically why, but why do we have a choice?
This blog post really articulated stuff formally that I have been bothered by for years.
In short: Reposting my comment from https://news.ycombinator.com/item?id=46775758 thread:
scaling -> real numbers
2d rotations and scaling -> complex numbers
3d rotations and scaling -> quaternions
In the case of quaternions, there is called double-covering, which turns out (rather than being an artefact), play fundamental role in particle physics.
As a non-mathematican, I found that trying to introduce C as a closure of R (i. e. analytically in author's terms) invariably triggers confusion and "hey, why do mathematicians keep changing rules on the fly, they just told me square of minus one doesn't exist". And in terms of practical applications it doesn't seem particularly useful on the first glance (who cares about solving cubics algebraically? The formula is too unwieldy anyway.) Most applications tends to start in the coordinate view and go from there. And it does introduce a nasty sharp edge to cut oneself on (i vs -i), but then for instance physics is full of such edges: direction of pseudo-vectors, sign of voltage on loads sources, holes in dimensional analysis (VA vs W, Ohm/square), the list could go on. And nobody really care.
> "hey, why do mathematicians keep changing rules on the fly, they just told me square of minus one doesn't exist
Mathematicians aren’t chasing numerical solutions, they’re chasing structure. ℂ isn’t just about solving cubics, it’s about eliminating holes in algebra so the theory behaves uniformly and is easier to build upon.
And as for "changing rules" they haven't changed, they have broadened the field (literally) over which the old rules applied in a clever way to remove a restriction.
For me personally, I prefer to perceive as more fundamental those constructions that straightforwardly generate the most with the least of novelty.
For example one may be introduced to the real numbers, and later to the complex numbers and then later perhaps the quaternions and the octonions, etc. in a haphazard disconnected way.
Given just the real numbers and "geometric algebra" (i.e. Grassmann algebras), this generates basically all these structures for different "settings".
I hence view geometric algebra as a lower level and more fundamental than complex numbers specifically.
A more interesting question (if we had to limit discussion to complex numbers) would be the following: which representations of complex numbers are known, what are their advantages and disadvantages, and can novel representation of complex numbers be devised that display less of the disadvantages?
For example, we have -just to list a few- the following representations:
1. cartesian representation of complex numbers: a + bi : the complex numbers permit a straightforward single-valued addition and single-valued multiplication, but only multivalued integer-powered-roots. addition and multiplication change smoothly with smooth changes in the input values, taking N-th roots do not, unless you use multivalued roots but then you don't have single valued roots.
2. polar representation of complex numbers: r(cos(theta)+i sin(theta)): this representatin admits smooth and single valued multiplication and N-th roots, but no longer permits smooth and single valued additions!
Please follow up with your favourite representations for complex numbers, and expound the pro's and con's of that representation.
For example can you generate a representation where addition and N-th roots are smooth and single valued, but multiplications are not?
Can you prove that any representation for complex numbers must suffer this dilemma in representation, or can you devise a representation where all 3 addition, multiplication, roots are smooth and single valued?
The field C is a tool, just as with any other abstraction in math. Accountants don't need it, so ignore it. In abstract algebra they arise - and so do many other abstractions, such as quaternions. Quaternions can be regarded as a further development of the concept of number. Note that every such development gives up some property its predecessors had. In the case of quaternions, commutativity. If a field of application finds a mathematical invention useful, it gets used. Often, too, it's a matter of simplicity. Use mathematical invention X in applied field Y and it simplifies the work. Abstain from using invention X, and then the work could still be done in a less simple way. That's all there is to it.
Real men know that infinite sets are just a tool for proving statements in Peano arithmetic, and complex numbers must be endowed with the standard metric structure, as God intended, since otherwise we cannot use them to approximate IEEE 754 floats.
Knowledge is the output of a person and their expertise and perspective, irreducibly. In this case, they seem to know something of what they're talking about:
> Starting 2022, I am now the John Cardinal O’Hara Professor of Logic at the University of Notre Dame.
> From 2018 to 2022, I was Professor of Logic at Oxford University and the Sir Peter Strawson Fellow at University College Oxford.
Also interesting:
> I am active on MathOverflow, and my contributions there (see my profile) have earned the top-rated reputation score.
https://jdh.hamkins.org/about/
Is there agreement Gaussian integers?
This disagreement seems above the head of non mathematicians, including those (like me) with familiarity with complex numbers
There is perfect agreement on the Gaussian integers.
The disagreement is on how much detail of the fine structure we care about. It is roughly analogous to asking whether we should care more about how an ellipse is like a circle, or how they are different. One person might care about the rigid definition and declare them to be different. Another notices that if you look at a circle at an angle, you get an ellipse. And then concludes that they are basically the same thing.
This seems like a silly thing to argue about. And it is.
However in different branches of mathematics, people care about different kinds of mathematical structure. And if you view the complex numbers through the lens of the kind of structure that you pay attention to, then ignore the parts that you aren't paying attention to, your notion of what is "basically the same as the complex numbers" changes. Just like how one of the two people previously viewed an ellipse as basically the same as a circle, because you get one from the other just by looking from an angle.
Note that each mathematician here can see the points that the other mathematicians are making. It is just that some points seem more important to you than others. And that importance is tied to what branch of mathematics you are studying.
The Gaussian integers usually aren't considered interesting enough to have disagreements about. They're in a weird spot because the integer restriction is almost contradictory with considering complex numbers: complex numbers are usually considered as how to express solutions to more types of polynomials, which is the opposite direction of excluding fractions from consideration. They're things that can solve (a restricted subset of) square-roots but not division.
This is really a disagreement about how to construct the complex numbers from more-fundamental objects. And the question is whether those constructions are equivalent. The author argues that two of those constructions are equivalent to each other, but others are not. A big crux of the issue, which is approachable to non-mathematicians, is whether it i and -i are fundamentally different, because arithmetically you can swap i with -i in all your equations and get the same result.
I don't think anyone thinks is "i and -i are fundamentally different". What they care more about is whether the 5 5th roots of 2 have an natural ordering or not.
One agreement could be that Eisenstein integers are more beautiful...
The link is about set theory, but others may find this interesting which discusses division algebras https://nigelvr.github.io/post-4.html
Basically C comes up in the chain R \subset C \subset H (quaternions) \subset O (octonions) by the so-called Cayley-Dickson construction. There is a lot of structure.
> But in fact, I claim, the smooth conception and the analytic conception are equivalent—they arise from the same underlying structure.
Conjugation isn’t complex-analytic, so the symmetry of i -> -i is broken at that level. Complex manifolds have to explicitly carry around their almost-complex structure largely for this reason.
Whoever coined the terms ‘complex numbers’ with a ‘real part’ and ‘imaginary part’ really screwed a lot of people..
How come? They are part real numbers, what would you call the other part?
We could've called the imaginaries "orthogonals", "perpendiculars", "complications", "atypicals", there's a million other options. I like the idea that a number is complex because it has a "complicated component".
Complex means that it is composed of parts, contrary to simplex, i.e. single or simple.
Iirc Gauss suggested "lateral numbers". Not the worst idea, but it's too late now.
I mean that they're not really numbers.
Usually they explain it something like: oh, at first people didn't know what 2-5 added up to, but then we invented negative numbers. Well, complex numbers are that but for square roots of negative numbers.
But that's a completely misleading way to explain these things. Complex numbers aren't numbers aren't numbers really.
It entirely depends on what we mean for something to be a number. Humans over the passage of time have been recognizing that their earlier conception had been too restrictive, narrow minded.
As one broadens the idea of what it means for anything to be a number, we acquire/invite new members to the family and with great benefit.
I mean, yeah, they aren't real numbers, they are composed of a real number and another one that is the multiplicative of the square root of -1. Hence they're called complex, i.e. composed of parts.
If the square root of -1 is not a number, what is it? How come you can do arithmetic with it?
Twisted ? Rotated ?
For what it's worth, Errett Bishop, the famous constructivist did not have this kind of existential issue with the complex numbers, commenting that the Reals were inadequate for some things. I really liked the trig cos isin connection in High School
the title is a bit clickbait - mathematicians don't disagree, all the "conceptions" the article proposes agree with each other. It also seems to conflate the algebraic closure of Q (which would contain the sqrt of -1) and all of the complex numbers by insisting that the former has "size continuum". Once you have "size continuum" then you need some completion to the reals.
anyhow. I'm a bit of an odd one in that I have no problems with imaginary numbers but the reals always seemed a bit unreal to me. that's the real controversy, actually. you can start looking up definable numbers and constructivist mathematics, but that gets to be more philosophy than maths imho.
The famous constructivist Errett Bishop did not have this sort of existential issue with the Complex Numbers, only saying the Reals were inadequate for some things.
To the ones objecting to "choosing a value of i" I might argue that no such choice is made. i is the square root of -1 and there is only one value of i. When we write -i that is shorthand for (-1)i. Remember the complex numbers are represented by a+bi where a and b are real numbers and i is the square root of -1. We don't bifurcate i into two distinct numbers because the minus sign is associated with b which is one of the real numbers. There is a one-to-one mapping between the complex numbers and these ordered pairs of reals.
You say that i is "the square root of -1", but which one is it? There are two. This is the point in the essay---we cannot tell the difference between i and -i unless we have already agreed on a choice of which square root of -1 we are going to call i. Only then does the other one become -i. How do we know that my i is the same as your i rather than your -i?
To fix the coordinate structure of the complex numbers (a,b) is in effect to have made a choice of a particular i, and this is one of the perspectives discussed in the essay. But it is not the only perspective, since with that perspective complex conjugation should not count as an automorphism, as it doesn't respect the choice of i.
One perspective of the complex numbers is that they are the even subalgebra of the 2D geometric algebra. The "i" is the pseudoscalar of that 2D GA, which is an oriented area.
If you flip the plane and look at it from the bottom, then any formula written using GA operations is identical, but because you're seeing the oriented area of the pseudoscalar from behind, its as if it gains a minus sign in front.
This is equivalent to using a right-handed versus left-handed coordinate systems in 3D. The "rules of physics" remain the same either way, the labels we assign to the coordinate systems are just a convention.
Is it two, or is it infinite? The quaternions have three imaginary units, i, j, and k. They're distinct, and yet each of them could be used for the complex numbers and they'd work the same way. How would I know that "my" imaginary unit i is the same as some other person's i? Maybe theirs is j, or k, or something else entirely.
It's 2 because there cannot be just one. There must be 2. Same as with magnetic poles and Siths. (and not same as with Highlanders.)
It's not infinite, because if you add more, you get a different space of numbers.
j and k are not complex numbers, in the same way that i is not a real number.
Changing the name of a number doesn't change the number. "Two" is not a different number from "2".
There are 2 square roots of 9, they are 3 and -3. Likewise there are two square roots of -1 which are i and -i. How are people trying to argue that there are two different things called i? We don't ask which 3 right? My argument is that there is only 1 value of i, and the distinction between -i and i is the same as (-1)i and (1)i, which is the same as -3 vs 3. There is only one i. If there are in fact two i's then there are 4 square roots of -1.
Notably, the real numbers are not symmetrical in this way: there are two square roots of 1, but one of them is equal to it and the other is not. (positive) 1 is special because it's the multiplicative identity, whereas i (and -i) have no distinguishing features: it doesn't matter which one you call i and which one you call -i: if you define j = -i, you'll find that anything you can say about i can also be shown to be true about j. That doesn't mean they're equal, just that they don't have any mathematical properties that let you say which one is which.
Your view of the complex numbers is the rigid one. Now suppose you are given a set with two binary operations defined in such a way that the operations behave well with each other. That is you have a ring. Suppose that by some process you are able to conclude that your ring is algebraically equivalent to the complex numbers. How do you know which of your elements in your ring is “i”? There will be two elements that behave like “i” in all algebraic aspects. So you can’t say that this one is “i” and this one is “-i” in a non arbitrary fashion.
There is no way to distinguish between "i" and "-i" unless you choose a representation of C. That is what Galois Theory is about: can you distinguish the roots of a polynomial in a simple algebraic way?
For instance: if you forget the order in Q (which you can do without it stopping being a field), there is no algebraic (no order-dependent) way to distinguish between the two algebraic solutions of x^2 = 2. You can swap each other and you will not notice anything (again, assuming you "forget" the order structure).
Building off of this point, consider the polynomial x^4 + 2x^2 + 2. Over the rationals Q, this is an irreducible polynomial. There is no way to distinguish the roots from each other. There is also no way to distinguish any pair of roots from any other pair.
But over the reals R, this polynomial is not irreducible. There we find that some pairs of roots have the same real value, and others don't. This leads to the idea of a "complex conjugate pair". And so some pairs of roots of the original polynomial are now different than other pairs.
That notion of a "complex conjugate pair of roots" is therefore not a purely algebraic concept. If you're trying to understand Galois theory, you have to forget about it. Because it will trip up your intuition and mislead you. But in other contexts that is a very meaningful and important idea.
And so we find that we don't just care about what concepts could be understood. We also care about what concepts we're currently choosing to ignore!
Exactly.
That is why the "forgetful functor" seems at first sight stupid and when you think a bit, it is genius.
When you think about it, creating a structure modulo some relation or kind of symmetry, is also a kind of targeted forgetting.
Exactly.
"God made the integers, all else is the work of man", Kronecker, 1886.
Complex numbers are just a fancy way to represent 2-dimensional numbers that are convenient in geometry through the sq(-1)=π/2 rotation.
They are nothing special and they could just be a matrix. Out of my head I think about complex integers. Also there are higher-order complex numbers which are defined by sq(sq(-1)) etc. In Greek complex numbers are called "mongrel". Both names are bad, I would just use 2-dimensional numbers.
Idk if this perspective is naive, but complex numbers to me are most motivated by spinning things.
Tired: De Moivre’s Theorem Wired: The complex number-line Expired: The complex plane
My biggest pet peeve in complex analysis is the concept of multi-value functions.
Functions are defined as relations on two sets such that each element in the first set is in relation to at most one element in the second set. And suddenly we abandon that very definitions without ever changing the notation! Complex logarithms suddenly have infinitely many values! And yet we say complex expressions are equal to something.
Madness.
You can think of it as returning an equivalence class if you like. Then it's single-valued.
More explicitly, it returns an equivalence class whose members are complex numbers that differ by integer multiples of 2*pi*i.
When it's important to distinguish members of the class, we speak of branches of the logarithm.
Also note the very cool and fun topology connection here. The keyword to search for is Riemann surface.
> You can think of it as returning an equivalence class if you like. Then it's single-valued.
I can, but I still have to be very mindful of how I use the "=" sign. Sometimes it's an (in)equality, sometimes it's... the equivalence class thing. The ambiguity doesn't seem very elegant.
> Also note the very cool and fun topology connection here. The keyword to search for is Riemann surface.
I'll check that out, thanks.
Idk, to me it feels much much better than just picking one root when defining the inverse function.
This desire to absolutely pick one when from the purely mathematical perspective they're all equal is both ugly and harmful (as in complicates things down the line).
Well, yeah, the alternative is also bad.
But couldn't we just switch the nomenclature? Instead of an oxymoronic concept of "multivalue function", we could just call it "relation of complex equivalence" or something of sorts.
Just think of it as a function that returns an array or a set: it still one value in a sense
I am confused by both the article and the discussion and I don't mean confused by the discussion on the complex which is all fairly clear but by this very weird idea of essential structure whatever that's supposed to mean. I'm wondering if there is something cultural here (I'm French) and linked to the central place algebra has in our curriculum or if I'm just reading a lot of IA generated convoluted discussion.
To me, the question doesn't even make sense. There is no disagreement here and I don't understand what is an essential structure. All the properties presented on the complex are consistent. They all exist. There is nothing ever essential about mathematics.
I mean, it's just mathematics. As long as you are internally consistent with your axioms, things just are. It's like asking if water is a liquid or a collection of molecules. There is nothing to disagree about here.
The square root of any number x is ±y, where +y = (+1)*y = y, and -y = (-1)*y.
So we define i as conforming to ±i = sqrt(-1). The element i itself has no need for a sign, so no sign needs to be chosen. Yet having defined i, we know that that i = (+1)*i = +i, by multiplicative identity.
We now have an unsigned base element for complex numbers i, derived uniquely from the expansion of <R,0,1,+,*> into its own natural closure.
We don't have to ask if i = +i, because it does by definition of the multiplicative identity.
TLDR: Any square root of -1 reduced to a single value, involves a choice, but the definition of unsigned i does not require a choice. It is a unique, unsigned element. And as a result, there is only a unique automorphism, the identity automorphism.
Notably, neither `1 + i > 1 - i` or `1 + i < 1 - i` are correct statements, and obviously `1 + i = 1 - i` is absurd.
What do > and < mean in the context of an infinite 2D plane?
Typically, the order of complex numbers is done by projecting C onto R, i.e. by taking the absolute value.
Yes I’m aware. It’s a work around but doesn’t give you a sensible ordering the way most people expect, i.e:
-2 > 1 (in C)
Which is why I prefer to leave <,> undefined in C and just take the magnitude if I want to compare complex numbers.
One is above the plane and the other is below it. ;)
In a word - "true".
In more words - it's interesting, but messy:
https://en.wikipedia.org/wiki/Partial_order
https://en.wikipedia.org/wiki/Ordered_field
> The complex numbers also cannot be turned into an ordered field, as −1 is a square of the imaginary unit i.
I found the article mildly interesting "light reading", until I got to this part:
> I was astounded to see that the Google AI overview in effect takes a stand amongst three conceptions
Uh oh. Hype alert. Should we continue reading?
... [a few moments later] ...
Oh, ok, the answer is yes. That was a bit of pandering but the author goes on to discuss how mathematicians think if this issue.
Also, don't miss this gem of a pun :
> Choosing the square root of -1 is a mathematical sin
:-)
Honestly, the rigid conception is the correct one. Im of the view that i as an attribute on a number rather than a number itself, in the same way a negative sign is an attribute. Its basically exists to generalize rotations through multiplication. Instead of taking an x,y vector and multiplying it by a matrix to get rotations, you can use a complex number representation, and multiply it by another complex number to rotate/scale it. If the cartesian magnitude of the second complex number is 1, then you don't get any scaling. So the idea of x/y coordinates is very much baked in to the "imaginary attribute".
I feel like the problem is that we just assume that e^(pi*i) = -1 as a given, which makes i "feel" like number, which gives some validity to other interpretations. But I would argue that that equation is not actually valid. It arises from Taylor series equivalence between e, sin and cos, but taylor series is simply an approximation of a function by matching its derivatives around a certain point, namely x=0. And just because you take 2 functions and see that their approximations around a certain point are equal, doesn't mean that the functions are equal. Even more so, that definition completely bypasses what it means to taking derivatives into the imaginary plane.
If you try to prove this any other way besides Taylor series expansion, you really cant, because the concept of taking something to the power of "imaginary value" doesn't really have any ties into other definitions.
As such, there is nothing really special about e itself either. The only reason its in there is because of a pattern artifact in math - e^x derivative is itself, while cos and sin follow cyclic patterns. If you were to replace e with any other number, note that anything you ever want to do with complex numbers would work out identically - you don't really use the value of e anywhere, all you really care about is r and theta.
So if you drop the assumption that i is a number and just treat i as an attribute of a number like a negative sign, complex numbers are basically just 2d numbers written in a special way. And of course, the rotations are easily extended into 3d space through quaternions, which use i j an k much in the same way.
> As such, there is nothing really special about e itself either. The only reason its in there is because of a pattern artifact in math - e^x derivative is itself
Not sure I follow you here... The special thing about e is that it's self-derivative. The other exponential bases, while essentially the same in their "growth", have derivatives with an extra factor. I assume you know e is special in that sense, so I'm unclear what you're arguing?
Im saying that the definition of polar coordinates for complex numbers using e instead of any other number is irrelevant to the use of complex numbers, but its inclusion in Eulers identity makes it seem like a i is a number rather than an attribute. And if you assume i is a number, it leads to one thinking that that you can define the complex field C. But my argument is that Eulers identity is not really relevant in the sense of what the complex numbers are used for, so i is not a number but rather a tool.
We as humans had a similar argument regarding 0. The thought was that zero is not a number, just a notational trick to denote that nothing is there (in the place value system of the Mesopotamians)
But then in India we discovered that it can really participate with the the other bonafide numbers as a first class citizen of numbers.
It is not longer a place holder but can be the argument of the binary functions, PLUS, MINUS, MULTIPLY and can also be the result of these functions.
With i we have a similar observation, that it can indeed be allowed as a first class citizen as a number. Addition and multiplication can accept them as their arguments as well as their RHS. It's a number, just a different kind.
But you can define the complex field C. And it has many benefits, like making the fundamental theorem of algebra work out. I'm not seeing the issue?
On a similar note, why insist that "i" (or a negative, for that matter) is an "attribute" on a number rather than an extension of the concept of number? In one sense, this is a just a definitional choice, so I don't think either conception is right or wrong. But I'm still not getting your preference for the attribute perspective. If anything, especially in the case of negative numbers, it seems less elegant than just allowing the negatives to be numbers?
Sure, you can define any field to make your math work out. None of the interpretations are wrong per say, the question is whether or not they are useful.
The point of contention that leads to 3 interpretations is whether you assume i acts like a number. My argument is that people generally answer yes, because of Eulers identity (which is often stated as example of mathematical beauty).
My argument is that i does not act like a number, it acts more like an operator. And with i being an operator, C is not really a thing.
Rotations fell out of the structure of complex numbers. They weren't placed there on purpose. If you want to rotate things there are usually better ways.
> If you want to rotate things there are usually better ways.
Can you elaborate? If you want a representation of 2D rotations for pen-and-paper or computer calculations, unit complex numbers are to my knowledge the most common and convenient one.
For pen and paper you can hold tracing paper at an angle. Use a protractor to measure the angle. That's easier than any calculation. Or get a transparent coordinate grid, literally rotate the coordinate system and read off your new coordinates.
For computers, you could use a complex number since it's effectively a cache of sin(a) and cos(a), but you often want general affine transformations and not just rotations, so you use a matrix instead.
> For computers, you could use a complex number since it's effectively a cache of sin(a) and cos(a), but you often want general affine transformations and not just rotations, so you use a matrix instead.
That makes sense in some contexts but in, say, 2D physics simulations, you don't want general homogeneous matrices or affine transformations to represent the position/orientation of a rigid body, because you want to be able to easily update it over time without breaking the orthogonality constraint.
I guess you could say that your tuple (c, s) is a matrix [ c -s ; s c ] instead of a complex number c + si, or that it's some abstract element of SO(2), or indeed that it's "a cache of sin(a) and cos(a)", but it's simplest to just say it's a unit complex number.
Why use a unit complex number (2 numbers) instead of an angle (1 number)? Maybe it optimizes out the sins and cosses better — I don't know — but a cache is not a new type of number.
There's a significant advantage in using a tuple over a scalar to represent angles.
For many operations you can get rid of calls to trigonometric functions, or reduce the number of calls necessary. These calls may not be supported by standard libraries in minimalistic hardware. Even if it were, avoiding calls to transcendental can be useful.
Because rotations with complex numbers is not just rotations, its rotations+scaling.
The advantage of complex numbers is to rotate+scale something (or more generally move somewhere in a complex plane), is a one step multiplication operation.
Do you find yourself scaling things a lot in 2D physics, but not translating them? I'd think translation and rotation are more common than scaling.
If you need to support zoom, scaling shows up very frequently.
I can give an example from real life. A piece of code one of my colleagues was working on required finding a point on the angular bisector. The code became a tangle of trigonometry calls both the forward and inverse functions. The code base was Python, so there was native support of complex numbers.
So you need angular bisector of two points p and q ? just take their geometric mean and you are done. At the Python code base level you only have a call to sqrt. That simplifies things.
No.
The whole idea of imaginary number is its basically an extension of negative numbers in concept. When you have a negative number, you essentially have scaling + attribute which defines direction. When you encounter two negative attributes and multiply them, you get a positive number, which is a rotation by 180 degrees. Imaginary numbers extend this concept to continuous rotation that is not limited to 180 degrees.
With just i, you get rotations in the x/y plane. When you multiply by 1i you get 90 degree rotation to 1i. Multiply by i again, you get another 90 degree rotation to -1 . And so on. You can do this in xyz with i and j, and you can do this in 4dimentions with i j and k, like quaternions do, using the extra dimension to get rid of gimbal lock computation for vehicle control (where pointed straight up, yaw and roll are identicall)
The fact that i maps to sqrt of -1 is basically just part of this definition - you are using multiplication to express rotations, so when you ask what is the sqrt of -1 you are asking which 2 identical number create a rotation of 180 degrees, and the answer is 1i and 1i.
Note that the definition also very much assumes that you are only using i, i.e analogous to having the x/y plane. If you are working within x y z plane and have i and j, to get to -1 you can rotate through x/y plane or x/z plane. So sqrt of -1 can either mean "sqrt for i" or "sqrt for j" and the answer would be either i or j, both would be valid. So you pretty much have to specify the rotation aspect when you ask for a square root.
Note also that you can you can define i to be <90 degree rotation, like say 60 degrees and everything would still be consistent. In which case cube root of -1 would be i, but square root of -1 would not be i, it would be a complex number with real and imaginary parts.
The thing to understand about math is under the hood, its pretty much objects and operations. A lot of times you will have conflicts where doing an operation on a particular object is undefined - for example there are functions that assymptotically approach zero but are never equal to it. So instead, you have to form other rules or append other systems to existing systems, which all just means you start with a definition. Anything that arises from that definition is not a universal truth of the world, but simply tools that help you deal with the inconsistencies.
The whole idea of an imaginary number is that it squares to a negative number. Everything else is accidental. Nobody expected that exp(i*a)=cos(a)+i*sin(a). Totally wacky discovery.
Imaginary numbers don't work in 3D, by the way. The most natural representation of a 3D rotation is a normalized 4D quaternion, and it's still pretty weird.
This is the most confident Ive seen someone in being so very wrong lol.
Nope. (Just to imitate your style)
There's more to it than rotation by 180 degrees. More pedagogically ...
Define a tuple (a,b) and define addition as pointwise addition. (a, b) + (c, d) = (a+c, b+d). Apples to apples, oranges to oranges. Fair enough.
How shall I define multiplication, so that multiplication so defined is a group by itself and interacts with the addition defined earlier in a distributive way. Just the way addition and multiplication behave for reals.
Ah! I have to define it this way. OK that's interesting.
But wait, then the algebra works out as if (0, 1) * (0, 1) = (-1, 0) but right hand side is isomorphic to -1. The (x, 0)s behave with each other just the way the real numbers behave with each other.
All this writing of tuples is cumbersome, so let me write (0,1) as i.
Addition looks like the all too familiar vector addition. What does this multiplication look like? Let me plot in the coordinate axes.
Ah! It's just scaled rotation, These numbers are just the 2x2 scaled rotation matrices that are parameterized not by 4 real numbers but just by two. One controls degree of rotation the other the amount of scaling.
If I multiply two such matrices together I get back a scaled rotation matrix. OK, understandable and expected, rotation composed is a rotation after all. But if I add two of them I get back another scaled rotation matrix, wow neato!
Because there are really only two independent parameters one isomorphic to the reals, let's call the other one "imaginary" and the tupled one "complex".
What if I negate the i in a tuple? Oh! it's reflection along the x axis. I got translation, rotation and reflection using these tuples.
What more can I do? I can surely do polynomials because I can add and multiply. Can I do calculus by falling back to Taylor expansions ? Hmm let me define a metric and see ...
I think we are agreeing.
You made it seem like rotations are an emergent property of complex numbers, where the original definition relies on defining the sqrt of -1.
Im saying that the origin of complex numbers is the ability to do arbitrary rotations and scaling through multiplication, and that i being the sqrt of -1 is the emergent property.
> Im saying that the origin of complex numbers is the ability to do arbitrary rotations and scaling through multiplication, and that i being the sqrt of -1 is the emergent property.
Not true historically -- the origin goes back to Cardano solving cubic equations.
But that point aside, it seems like you are trying to find something like "the true meaning of complex numbers," basing your judgement on some mix of practical application and what seems most intuitive to you. I think that's fruitless. The essence lies precisely in the equivalence of the various conceptions by means of proof. "i" as a way "to do arbitrary rotations and scaling through multiplication", or as a way give the solution space of polynomials closure, or as the equivalence of Taylor series, etc -- these are all structurally the same mathematical "i".
So "i" is all of these things, and all of these things are useful depending on what you're doing. Again, by what principle do you give priority to some uses over others?
>he origin goes back to Cardano solving cubic equations.
Whether or not mathematicians realized this at the time, there is no functional difference in assuming some imaginary number that when multiplied with another imaginary number gives a negative number, and essentially moving in more than 1 dimension on the number line.
Because it was the same way with negative numbers. By creating the "space" of negative numbers allows you do operations like 3-5+6 which has an answer in positive numbers, but if you are restricted to positive only, you can't compute that.
In the same way like I mentioned, Quaternions allow movement through 4 dimentions to arrive at a solution that is not possible to achieve with operations in 3 when you have gimbal lock.
So my argument is that complex numbers are fundamental to this, and any field or topological construction on that is secondary.
Maybe.
You disagreed with the parent comment that said
"Rotations fell out of the structure of complex numbers. They weren't placed there on purpose. If you want to rotate things there are usually better ways."
I see Complex numbers in the light of doing addition and multiplication on pairs. If one does that, rotation naturally falls out of that. So I would agree with the parent comment especially if we follow the historical development. The structure is identical to that of scaled rotation matrices parameterized by two real numbers, although historically they were discovered through a different route.
I think all of us agree with the properties of complex numbers, it's just that we may be splitting hairs differently.
>"Rotations fell out of the structure of complex numbers. They weren't placed there on purpose. If you want to rotate things there are usually better ways."
I mean, the derivation to rotate things with complex numbers is pretty simple to prove.
If you convert to cartesian, the rotation is a scaling operation by a matrix, which you have to compute from r and theta. And Im sure you know that for x and y, the rotation matrix to the new vector x' and y' is
x' = cos(theta)*x - sin(theta)*y
y' = sin(theta)*x + cos(theta)*y
However, like you said, say you want to have some representation of rotation using only 2 parameters instead of 4, and simplify the math. You can define (xr,yr) in the same coordinates as the original vector. To compute theta, you would need ArcTan(yr/xr), which then plugged back into Sin and Cos in original rotation matrix give you back xr and yr. Assuming unit vectors:
x'= xr*x - yr*y
y'= yr*x + xr*y
the only trick you need is to take care negative sign on the upper right corner term. So you notice that if you just mark the y components as i, and when you see i*i you take that to be -1, everything works out.
So overall, all of this is just construction, not emergence.
Yes it's simple and I agree with almost everything except that arctan bit (it loses information, but that's aside story).
But all that you said is not about the point that I was trying to convey.
What I showed was you if you define addition of tuples a certain, fairly natural way. And then define multiplication on the same tuples in such a way that multiplication and addition follow the distributive law (so that you can do polynomials with them). Then your hands are forced to define multiplication in very specific way, just to ensure distributivity. [To be honest their is another sneaky way to do it if the rules are changed a bit, by using reflection matrices]
Rotation so far is nowhere in the picture in our desiderata, we just want the distributive law to apply to the multiplication of tuples. That's it.
But once I do that, lo and behold this multiplication has exactly the same structure as multiplication by rotation matrices (emergence? or equivalently, recognition of the consequences of our desire)
In other words, these tuples have secretly been the (scaled) cos theta, sin theta tuples all along, although when I had invited them to my party I had not put a restriction on them that they have to be related to theta via these trig functions.
Or in other words, the only tuples that have distributive addition and multiplication are the (scaled) cos theta sin theta tuples, but when we were constructing them there was no notion of theta just the desire to satisfy few algebraic relations (distributivity of add and multiply).
I just don't like this characterization of
> "How shall I define multiplication, so that multiplication so defined is a group by itself and interacts with the addition defined earlier in a distributive way. Just the way addition and multiplication behave for reals."
which eventually becomes
> "Ah! It's just scaled rotation"
and the implication is that emergent.
Its like you have a set of objects, and defining operations on those objects that have properties of rotations baked in ( because that is the the only way that (0, 1) * (0, 1) = (-1, 0) ever works out in your definition), and then you are surprised that you get something that behaves like rotation.
Meanwhile, when you define other "multiplicative" like operations on tuples, namely dot and cross product, you don't get rotations.
> I just don't like this characterization
That's ok. It's a personal value judgement.
However, the fact remains that rotations can "emerge" just from the desire to do additions and multiplications on tuples to be able to do polynomials with them ... which is more directly tied to its historical path of discovery, to solve polynomial equations, starting with cubic.
>historical path of discovery, to solve polynomial equations, starting with cubic.
Even with polynomial equations that have complex roots, the idea of a rotation is baked in in solving them. Rotation+scaling with complex numbers is basically an arbitrary translation through the complex plane. So when you are faced with a*x*x + b*x + c = 0, where a b and c all lie on the real number line, and you are trying to basically get to 0, often you can't do it by having x on a number line, so you have to start with more dimentions and then rotate+scale so you end up at zero.
Its the same reason for negative numbers existing. When you have positive numbers only, and you define addition and subtraction, things like 5-6+10 become impossible to compute, even though all the values are positive. But when you introduce the space of negative numbers, even though they don't represent anything in reality, that operation becomes possible.
Yes but it was a fundamental mathematical achievement to see this equivalence. That knowledge had to emerge, be discovered. This eventually led to the theory of Galois fields.
The connection with rotation emerged naturally from a line of thought that initially had nothing to do with rotations. It was a consequence of a desire to satisfy distributive laws and maintain vector addition.
Connection between seemingly unrelated mathematical fields happen from time to time and those are considered events of surprise, understanding and celebration.
You can define that, but (if you don't already know about complex numbers) it's not obvious that it does anything mathematically interesting. It's just a cache for sin and cos, not a new type of anything. I could say that when evaluating 4th degree polynomials it's useful to have x, x^2 and x^3 immediately at hand, but the combination of those three isn't a new type of number, just a cache.
>it does anything mathematically interesting
You are right - its not interesting. You already know that rotation can be done through multiplication (i.e rotation matrix), and you are just simplifying it further.
After all, the only application of imaginary numbers outside their definition is roots of a polynomial. And if you think of rotation+scaling as simple movement through the complex plane to get back to the real one, it makes perfect sense.
You can apply this principle generically as well. Say you have an operation on some ordered set S that produces elements in a smaller subset of S called S' It then follows that the inverse operation of elements of the complement of S' with respect to the original set S is undefined.
But you can create a system where you enhance the dimension of the original set with another set, giving the definition of that inverse operation for compliment of S'. And if that extra set also has ordering, then you are by definition doing something analogous to rotation+scaling.
It seems obvious now only because of significant mathematical discoveries of prominent mathematicians.
If one is taught what those discoveries revealed then of course they would seem obvious.
Arguing as you are, it would appear one can call all and every theorem in mathematics that connects to different fields as something obvious. They weren't, till someone proved the connection and that knowledge percolated down to how maths is taught, to text books.
That the integral of
over the entire real line is sqrt pi can be surprising or obvious depending on how you were taught. At face value it has nothing to do with circles, unless you are taught the connection or you are a mathematician of high calibre who can see it without being taught the background information.This completely misses the point of why the complex numbers were even invented. i is a number: it is one of the 2 solutions to the equation x^2 = -1 (the other being -i, of course). The whole point of inventing the complex numbers was to have a set of numbers for which any polynomial has a root. And sure, you can call this number (0,1) if you want to, but it's important to remember that C is not the same as R².
Your whole point about Taylor series is also wrong, as Taylor series are not approximations, they are actually equal to the original function if you take their infinite limit for the relevant functions here (e^x, sin x, cos x). So there is no approximation to be talked about, and no problem in identifying these functions with their Taylor series expansions.
I'd also note that there is no need to use Taylor series to prove Euler's formula. Other series that converge to e^x,cos x, sin x can also get you there.
>The whole point of inventing the complex numbers was to have a set of numbers for which any polynomial has a root
Think about what this implies.
You have an operation, like exponentiation, that has limits. Something squared can never be negative if you are talking about any real number.
In terms of Sets, you essentially have an operation that produces results only in a finite subset of the overall set. And so the inverse of that operation, when applied to the complement of that finite subset, is undefined.
However you can introduce another (ordered) set in complement to your original set and combine them to form a new set, with operations that define how you move around the values of those sets. So in the case of imaginary numbers, you basically redefine all your reals as "real number + 0 i". And now you have a way to apply that inverse operation to the complement of the finite subset, which means you can get answers to the roots of the polynomial.
And in defining the operation of multiplication, you essentially define a way to move around the 2 dimensional set now. And moving around 2 dimensions is exactly the same thing as rotation+scaling. And note that when you say sqrt(-1) = i, you basically assume that the complex plane is 2d. There is nothing that is stopping you from making a complex plane 3d or 4d or nd. So sqrt(-1) can also be j, or it can be k. To know what it is, you have to specify the axis of the plane when you specify the sqrt operation, which again, brings it back to the concept of rotations.
And thats my whole point, there is nothing special about i, its simply just a construct that bakes in rotations through any way you wanna define it.
>our whole point about Taylor series is also wrong, as Taylor series are not approximations, they are actually equal to the original function if you take their infinite limit
Looking back at what I wrote, I worded it very poorly.
I don't have a problem with any math involved, not trying to say that Eulers identity is not valid.
What Im trying to say is that all the definitions sort of assume that if you have some operation that you can do on real numbers, and have a result, if you just plug in a complex value, it all works out, so people think that i behaves like a number. I personally don't think that this is the case, specifically about i behaving like a number just because those results work out.
For example, even without Taylor series, you can prove Eulers identity using the limit formula for e(x). The idea is that you have (1+xi/n)^n as n goes to infinity, but because you baked in the rotation as a multiplication in the definition, all you are doing is starting at 1+0i and doing smaller and smaller rotations to get to some value, and the limit of that value is essentially the unit vector rotated by a certain angle. So naturally the cos and sin equivalence arises.
My issue is that the limit equation for e, in the case of the reals, take e x times in multiplication and then compute the limit equation, and you get equivalence. But in the case of the complex, you don't really have any idea what it takes something to ith power, but you can compute the limit equation, and so you end up with a definition of what it means to take something to the ith power.
My argument is that its not really applicable - not that its wrong, but the fact that its not defining exponentiation to the ith power in the sense that i has "number like" qualities like real numbers do. You would have to prove that an equivalence
What is really happening is that you never really escape the real numbers, and your complex numbers are just simplified operations that rotate/scale a number, like rotation matricies do through multiplication, and that in the nature of the definition of those rotations, you get stuff like Eulers identity, which is somewhat pointless because like I mentioned - the value of e (i.e 2.7) is never really used to compute anything in regards to complex numbers in polar form of re^ix, all you care about is r and the x which is the angle.
And for this reason, I don't consider i a number, so the analytic/smooth interpretations to me are meaningless.
> And in defining the operation of multiplication, you essentially define a way to move around the 2 dimensional set now. And moving around 2 dimensions is exactly the same thing as rotation+scaling.
Again, this is not how complex numbers were defined. The only original goal was to come up with a number that can solve the equation x^2 + 1 = 0, so that (R + {this number}, + , *), becomes an algebraically closed field. Once you've set this goal, there is really a single simple choice for how the operations will work, because everything else is already constrained. If x^2 + 1 = 0, we already know that:
So the formula for complex number multiplication comes out of the arithmetic of real numbers, extended with this extra entity defined simply by being a root of x^2 + 1. The fact that this operation happens to represent a rotation in the RxR plane is "an accident" (I'm sure there are deep ties that make this necessary, probably related to the structure of polynomials themselves).And while you can define other algebraically closed fields that include the reals as a subfield, the complex numbers are the simplest such set. R^n for n>2 is clearly more complex, for example. So there is a clear reason to prefer sqrt(-1) = i, and thus ending up with a 2d vector space.
> What Im trying to say is that all the definitions sort of assume that if you have some operation that you can do on real numbers, and have a result, if you just plug in a complex value, it all works out, so people think that i behaves like a number. I personally don't think that this is the case, specifically about i behaving like a number just because those results work out.
Again, complex multiplication is not some intentional construction, it is baked into how we defined the complex numbers the first time we did. Again, our goal was to find (well, define) the solution(s) of the equation x^2 + 1 = 0. The fact that we can plug in this x to any formula that involves other numbers just falls out of this goal, it's not an additional assumption. In the case of e(x), this is simpler to see with the power series formula:
Note that this falls out of the properties of e(x), cos(x), and sin(x) for real numbers, and the single property of i that it is a solution of x^2 + 1 = 0.I also think that the definition that e(x) is "take e x times in multiplication and then compute the limit" is any more intuitive. I certainly don't think that `e^2 = e(2) = lim (1 + 2/n) ^ n, with n-> infinity` is any more intuitive than the definition of `e(2i) = lim (1 + 2i/n)^n, with n -> infinity`.
> which is somewhat pointless because like I mentioned - the value of e (i.e 2.7) is never really used to compute anything in regards to complex numbers in polar form of re^ix, all you care about is r and the x which is the angle
This is also not really true, because the value of e is deeply tied to the values of cos x and sin x. This also becomes visible if you want to compute 2^(ix). 2^(ix) = e ^ ix (log_e 2) = cos (x log_e 2) + i sin (x log_e 2), using log_e to denote the natural logarithm to make it clearer that it is related to e. So the value of e itself is still there in the formula, even if we discount the relationship between e and cos and sin.
The things is, there are no accidents in math. If you end up with formulas that look like something else, that something else was a defining part of the original definition, whether it was obvious or not.
You would agree that in the quest to solve x^2+1=0, we had to introduce another dimension, with a multiply operator that lets us move through that dimension. All im saying is that introducing another dimension and ability to move in that dimension is the same thing as rotation and scaling.
As for e, the point im trying to make is that if you look at what it means to take something to the power of something when it comes to reals, there is a clear definition. But taking something to imaginary power is meaningless it iself. For reals, the power operator has a strict definition with multiplication and division for rationals, and generically extended to reals through limits. And to compute a limit means that you have to have continuity and smoothness. So by extension, to compute exponentiation, you have to have continuity and smoothness, and vice versa.
My argument is that there is no guarantee of continuity/smoothness on the complex plane - one can define it as such (i.e the analytic view) but in my stricter philosophical view, to make something with similar properties like real numbers, you have to have all the analogous operations work within the numbers itself. I.e you have to be able to define what 2i^3i is without ever referencing anything from the real numbers.
This is not done for complex plane - to define something to the i power you need to borrow definitions from the real plane.
As such, you can't define exponentiation to the i, because you can't compute limits. Computing the taylor series expansion equivalence or limit formulas in the way me and you presented them are "holograms" - i.e meaningless results.
And it seems that way. Say that you ignore taylor series formulation, skip it completely, and define the polar form to be r(10)^ix = cos(x)+isin(x), where r is radius and x is angle. JUST BECAUSE YOU CAN.
Would you lose anything? Not really. You would still have a way to compute 2^i - it would just be 10^(ilog(2)) -> r = 1, angle=log(2). I.e the map still exists in quite a good shape.
Basically it doesn't matter if r(10)^ix = cos(x)+isin(x) or r(e)^ix = cos(x)+isin(x), as counterintuitive as it may seem, which furthers my point that exponentiation operation to imaginary numbers is not defined.
So without exponentiation, all you are left with is basically a more strict rigid construction of another set with an ordering property, a multiplication operation definition, (i^2 = -1), and the resultant orthogonality that represents a cartesian plane.
And if you think thats silly, consider the fact that the algebraic view of complex numbers doesn't even consider calculus on them to be valid.
I forgot to respond to this:
> And it seems that way. Say that you ignore taylor series formulation, skip it completely, and define the polar form to be r(10)^ix = cos(x)+isin(x), where r is radius and x is angle. JUST BECAUSE YOU CAN.
The thing is, you can't do this. Your numbers will not work out correctly. For example, (e^(pi*i))^i = 1/e^pi is a direct consequence of how exponentiation works and the definition of i. If you define 10^(xi) = cos x + i sin x, you will get:
I very much doubt that sin(cos (log_10 e * pi) + i sin (log_10 e * pi)) is 0, so this number can't possibly be equal to the real number 1/e^pi. So, with your definition, the property (a^x)^y = a^(x*y) doesn't hold for all a, x, y. So, your definition doesn't represent the exponential function at all.> a direct consequence of how exponentiation works and the definition of i
I dunno if Im just not coming across clearly or my thinking is too abstract, but when you say
"how exponentiation works"
you are referring to how exponentiation works for real exponents
You would have to prove that it works the same way for i exponents before you can go further.
I think the point is to look at it the other way: we invented/chose i so that exponentiation (and all other functions) maintain the same properties for the complex numbers as they have for the reals. The whole goal of defining the complex numbers is to extend R in a way that maintains all of the properties of functions on R.
> You would agree that in the quest to solve x^2+1=0, we had to introduce another dimension, with a multiply operator that lets us move through that dimension.
I don't really agree, no. In the analytic definition, there is no second dimension. Sure, C is isomorphic to R×R, but that is not how you construct it. C is not R×R, it's R+{a + b * i | a,b in R, b≠0} in this view. Just like Z is N+{a * -1 | a in N, a≠0}. You introduce one new number that you need to solve the equation, and then all of the numbers needed to make the new construction a field again. You don't introduce a new notion of multiplication, C uses the exact same multiplication operation as R does, or at least as polynomials over R do, as I showed.
The fact that C is isomorphic to R×R, and that multiplication of numbers in C is isomorphic to scaling+rotation of vectors in R×R, is not part of the construction.
I do agree that there are no accidents in math, so I imagine this isomorphism is related to some more fundamental relationship between polynomials, 2d vectors, and rotation matrices - because our construction of C is strictly motivated by polynomials.
> For reals, the power operator has a strict definition with multiplication and division.
I don't agree with this statement. It's true for the integers, but already for the the rationals it loses any direct relationship to multiplication and division (how do you get 4^(1/2) = 2 by repeated multiplication?). And for the reals, it's completely gone. We can't even define many properties of real-valued exponentials - we don't even know if e^pi is an irrational number, for example.
> When you search for something like taylor series or limit form of e and you see a way to compute e^i what you are doing is basically using operators designed for real numbers and extending them to the complex numbers (when you substitute i for where normally a real number would be in the power term.
We've already agreed that there are no accidents in math. So just as the multiplication of polynomials is fundamentally linked with rotations of 2d real-valued vectors, we must accept that real exponentiation is fundamentally linked to complex exponentiation, otherwise the formulas wouldn't work the way they do.
Note also that calculus is not limited to operations on the number line - you can take the derivatives or integrate or calculate limits of n-dimensional curves, and even differentiate over surfaces and n-dimensional manifolds more generally. Smoothness and continuity are anyway part of the structure of C, regardless what definition of it you use.
> And if you think thats silly, consider the fact that the algebraic view of complex numbers doesn't even consider calculus on them to be valid.
I don't understand what you mean by this. Calculus is well defined for functions over any field of size continuum, and that is exactly what <C,+,*,0,1> is in the algebraic view.
>C uses the exact same multiplication operation as R does
Not quite. If it did, the -i*-i would be i^2, not -1. And yes, I totally agree that C is R+, not RxR. The point is that you still introducing something extra with some rules, where you introduce the concept of geometric orthogonality into i^2 = -1, whether that is your intention or not.
>For reals, the power operator has a strict definition with multiplication and division for rationals, and generically extended to reals through limits.
I accidentally swapped reals and rationals there. The whole point was to highlight that exponentiation for real exponents relies on limits which relies on continutity.
>So just as the multiplication of polynomials is fundamentally linked with rotations of 2d real-valued vectors, we must accept that real exponentiation is fundamentally linked to complex exponentiation, otherwise the formulas wouldn't work the way they do.
Don't agree.
Multiplication of polynomials involves operations that are clearly defined. When you do (a+bi)^2, you have defined what it means to multiply complex numbers in their construction, without needing to use any such formula from real numbers.
Exponentiation where you have i exponent however, is not defined solely in the complex field.
>I don't understand what you mean by this. Calculus is well defined for functions over any field of size continuum, and that is exactly what <C,+,*,0,1> is in the algebraic view.
Algebraic view is basically the idea of that numbers are only defined if there is some relation that expresses their definition.
For example, 1+2=3, locks all 3 number down. 1 is 3-2, 2 is 3-1 and 3 is 1+2.
pi or e on the other hand, are "something else", because there is no algebraic formula that defines them. To do so you have to invoke the computation of limits, which is an analytic view, not algebraic.*
> Not quite. If it did, the -i-i would be i^2, not -1.
It is i², though, but that is equal to -1. Just like 22 = 2² = 4. I also maintain that the historical view is that this something extra comes from the properties of polynomials and their roots, not from geometric orthogonality.
> Exponentiation where you have i exponent however, is not defined solely in the complex field.
We can take another tack for defining the complex exponential function, if you'd prefer. One of the definitions of the exponential function is that e^x is the only function that respects f'(x) = f(x) (well, up to constant multiplication).
So, we need to look for a function f(z) such that f'(z) = f(z). There are various ways to do this (for example, using the Taylor series expansion and noting that all of the f derived n times factors are equal to f(z), which yields the power series definition). You don't need to appeal to limits of e^x to get there this way.
> Algebraic view is basically the idea of that numbers are only defined if there is some relation that expresses their definition.
Understood, you are using a different sense of "algebraic" than what I was - I was thinking more of the abstract algebraic definition of C.
Still, the sense you are using seems to be the concept of algebraic numbers, which is more formalized - the algebraic numbers are all those that represent the root of a polynomial with integer or rational coefficients. Interestingly, while pi and e are not algebraic numbers, i or i+7 are still algebraic.
However, I'm not sure what the point of bringing this up is. Exponentiation is simply not defined over the algebraic numbers, especially not e^x where x is algebraic - so if we restrict ourselves to algebraic numbers, e^i is not defined, true, but neither is e^1. And while 2^2 is defined, of course, I'm not even sure you can define 2^sqrt(2), so I wouldn't be surprised if 2^i doesn't make sense either.
Either way, the algebraic numbers are not "a way of thinking about numbers", they are a restricted subset of what is generally meant by "number", and many famous and useful results from many branches of math do not work over this subset (for example, you can't even use the same set of algebraic numbers to refer to the length of a circle and the length of a square).
I still think you misunderstanding what im saying. i don't have a problem with the math working out, its that i have a problem with the math being able to be applied in the first place.
Let me try a different way.
Suppose you start with just the i number line with the rules that exist. You have zero, integer i's, rational is, and even irrational i's. All seems good. Then you start to define operations. i*i is undefined (because it goes to -1, and the reals are outside of your domain that you are working with currently). And this means you can't effectively do any sort of futher work in defining exponentiation, or limits, because you can have purely complex polynomials with just undefined terms.
So like you said, complex is R+, not something like RxR. The definition of complex numbers is intrinsic to real numbers - its an enhancement on real numbers. And by extension, all the math works out when you do taylor series with e^i and such.
But this pretty much means its a rigid definition, i.e you are defining something in a certain construction to supplement reals.
And as for geometric claim, my argument with that is that just like when you have x, and then you add <x,y> in some form and way, you are defining geometry. So in defining a+bi, you are defining geometry.*
Yeah i is not a number. Once you define complex numbers from reals and i, i becomes a complex numbers but that's a trick
i is not a "trick" or a conceit to shortcut certain calculations like, say, the small angle approximation. i is a number and this must be true because of the fundamental theorem of algebra. Disbelieving in the imaginary numbers is no different from disbelieving in negative numbers.
"Imaginary" is an unfortunate name which gives makes this misunderstanding intuitive.
https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJ...
However, what's true about what you and GP have suggested is that both i and -1 are used as units. Writing -10 or 10i is similar to writing 10kg (more clearly, 10 × i, 10 × -1, 10 × 1kg). Units are not normally numbers, but they are for certain dimensionless quantities like % (1/100) or moles (6.02214076 × 10^23) and i and -1. That is another wrinkle which is genuinely confusing.
https://en.wikipedia.org/wiki/Dimensionless_quantity
Yes I agree, I was just arguing against "i=0+1*i so it's by definition a complex number" which is a self referential definition
If you take this tack, then 0 and 1 are not numbers either.
i is not a real number, is not an integer, is not a rational etc.
You need a base to define complex numbers, in that new space i=0+1*i and you could call that a complex number
0 and 1 help define integers, without {Empty, Something} (or empty, set of the empty, or whatever else base axioms you are using) there is no integers
The simple fact you wanted to write this:
i=0+1*i
Makes i a number. Since * is a binary operator in your space, i needs to be a number for 1*i to make any sense.
Similarly, if = is to be a binary relation in your space, i needs to be a number for i={anything} to make sense.
Comparing i with a unary operator like - shows the difference:
i*i=-1 makes perfect sense
-*-=???? does not make sense
i is a complex number, complex numbers are of the form real + i*real... Don't you see the recursive definition ? Same with 0 and 1 they are not numbers until you can actually define numbers, using 0 and 1
This is one definition of i. Or you could geometrically say i is the orthogonal unit vector in the (real,real) plane where you define multiplication as multiplying length and adding anglesThere's no issue with recursive definitions. That's how arithmetic was original formalized by Peano's axioms [1].
[1] https://en.wikipedia.org/wiki/Peano_axioms
i is also a quaternion. So by this logic we could say complex numbers are made up of quaternions. But we don’t say such things because they wouldn’t be a good mental model of what we want to talk about.
> i is also a quaternion
Yeah, so ?
The fraction 1/2 is a rational and also a real and also a complex numbers and also a quaternion also an octonion ....
We use a number along with the minimal abstraction that is sufficient for our purpose.
i=0+1*i =0+1*(0+1*(0+1*(0+1*(0+1*(0+1*( ...*
... = -2 - (-3) = -1 - (-2) = 0 - (-1) = 1 - 0 = 1 = 2 - 1 = 3 - 2 = 4 - 3 = 5 - 4 = ...
What does Terry Tao think?
Another "xyz" domain that doesn't resolve on my network.
Yep- there’s some issues representing complex numbers in 3D space. You may want to check out quaternions.
Instructions unclear, gimbal locked
https://web.archive.org/web/20251015174117/https://www.infin...
This one has the paywall, but the main site has no paywall currently.
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