Mathematicians disagree on the essential structure of the complex numbers (2024)

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.

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.

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.

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.

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...

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 whole substack is great, I recommend reading all of it if you are interested in infinity

There's no disagreement, the algebraic one is the correct one, obviously. Anyone that says differently is wrong. :)

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.

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.

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?

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.

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.

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.

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.

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.

Is there agreement Gaussian integers?

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

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

:-)

What does Terry Tao think?

> 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.

Notably, neither `1 + i > 1 - i` or `1 + i < 1 - i` are correct statements, and obviously `1 + i = 1 - i` is absurd.

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.

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/

Whoever coined the terms ‘complex numbers’ with a ‘real part’ and ‘imaginary part’ really screwed a lot of people..

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.

Another "xyz" domain that doesn't resolve on my network.