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> 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 o

by ajkjk 9mo ago

> 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.

andrewflnr 9mo ago
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 https://scottaaronson.blog/?p=4021
- ajkjk 9mo ago
  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).
klodolph 9mo ago
> 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.
- topaz0 9mo ago
  > 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.
  - klodolph 9mo ago
    > 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.
    - topaz0 9mo ago
      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)
nyssos 9mo ago
> 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.
- ajkjk 9mo ago
  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.

by ajkjk 9mo ago

andrewflnr 9mo ago

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 https://scottaaronson.blog/?p=4021

ajkjk 9mo ago
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).

klodolph 9mo ago

> 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.

topaz0 9mo ago
> 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.
- klodolph 9mo ago
  > 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.
  - topaz0 9mo ago
    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)

nyssos 9mo ago

> 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.

ajkjk 9mo ago
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.