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Comment by nyeah | Hacker News Reader

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.

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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.

A bit like +0 and -0? It makes sense in some contexts, and none in others.

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.)

PP meant automorphisms, which is what the OP article is about.

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.

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'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.

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.

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