FillMaths

The real field is categorically characterized (in second-order logic) as the unique complete ordered field, proved by Huntington in 1903. The complex field is categorically characterized as the unique algebraic closure of the real field, and also as the unique algebraically closed field of characteristic 0 and size continuum. I believe that you are speaking of the model-theoretic first-order notion of categoricity-in-a-cardinal, which is different than the categoricity remarks made in the essay.

It's not about observers, but about mathematical structure and meaning. Without answering the questions, you are being ambiguous as to what the structure of C is. For example, if a particular copy of R is fixed as a subfield, then there are only two automorphisms---the trivial automorphism and complex conjugation, since any automorphism fixing the copy of R would have to be the identity on those reals and thus the rest of it is determined by whether i is fixed or sent to -i. Meanwhile, if you don't fix a particular R subfield, then there is a vast space of further wild automorphisms. So this choice of structure---that is, the answer to the questions I posed---has huge consequences on the automorphism group of your conception. You can't just ignore it and refuse to say what the structure is.

Of course everyone agrees that this is a nice way to construct the complex field. The question is what is the structure you are placing on this construction. Is it just a field? Do you intend to fix R as a distinguished subfield? After all, there are many different copies of R in C, if one has only the field structure. Is i named as a constant, as it seems to be in the construction when you form the polynomials in the symbol i. Do you intend to view this as a topological space? Those further questions is what the discussion is about.

You say that i is "the square root of -1", but which one is it? There are two. This is the point in the essay---we cannot tell the difference between i and -i unless we have already agreed on a choice of which square root of -1 we are going to call i. Only then does the other one become -i. How do we know that my i is the same as your i rather than your -i?

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.

This one has the paywall, but the main site has no paywall currently.

Subsequent posts developing the paradox further into the transfinite ordinals and uncountable ordinals: https://www.infinitelymore.xyz/t/infinite-subway-paradox

Sorry, I tried my best. I wanted to mention the thought experiment part, since that is the most interesting bit. (But I'm not sure why it was misleading?)

The pdf file is available at: https://arxiv.org/pdf/2407.02463

That's similar to what the author says in the second paragraph. But he goes on to consider many other subtle notions arising from the fact that the complex field is not rigid. How can we tell i from -i? They have all the same properties with respect to the field structure.