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.
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'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.
A bit like +0 and -0? It makes sense in some contexts, and none in others.
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)).
Agreed. To me it looks like the entire discussion is just bike-shedding.
It's math. Bikeshedding is the goal.
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.
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.