alt Hacker NewsI 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.
More on not being able to find π, as I'm piecing it together: given only the field structure, you can't construct an equation identifying π or even narrowing it down, because if π is the only free variable then it will work out to finding roots of a polynomial (you only have field operations!) and π is transcendental so that polynomial can only be 0 (if you're allowed to use not-equals instead of equals, of course you can specify that π isn't in various sets of algebraic numbers). With other free variables, because the field's algebraically closed, you can fix π to whatever transcendental you like and still solve for the remaining variables. So it's something like, the rationals plus a continuum's worth of arbitrary field extensions? Not terribly surprising that all instances of this are isomorphic as fields but it's starting to feel about as useful as claiming the real numbers are "up to set isomorphism, the unique set whose cardinality matches the power set of the natural numbers", like, of course it's got automorphisms, you didn't finish defining it.