Posted by FillMaths 8 hours ago
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.
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.
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.
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.
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).
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!
That is why the "forgetful functor" seems at first sight stupid and when you think a bit, it is genius.
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.
This disagreement seems above the head of non mathematicians, including those (like me) with familiarity with complex numbers
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.
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.
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.
-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.
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.