[yau8edq12i]

As a mathematician, I have never, ever seen the equal sign used to denote proportionality. I cannot tell what you mean by "identity" that wouldn't be equality. And a definition in the sense you seem to mean is, indeed, also an equality. I guess this issue not mentioned in the paper because it doesn't exist.

[xanderlewis]

I agree it’s not really used symbolically, but isn’t it referring to the difference between, say:

(x + y)(x - y) = x^2 - y^2

(an identity, since it’s true for every x and every y) and something more arbitrary like

3x^2 + 2x - 7 = 0

(an equation certainly not valid for all x and whose solutions are sought).

Of course, really, the first one is a straightforward equality missing some universal quantification at the front… so maybe that’s just what the triple equals sign would be short for in this case.

[ducttapecrown]

[x:y] = [lx:ly]

and I suppose the model of localization discussed in the article above counts as well.

[yau8edq12i]

... What? That's just symbol soup. What are you trying to write? Do it in latex if it helps.

[laurent_du]

It's not symbol soup, it's projective coordinates. Did you claim to be a mathematician? I merely have a PhD in Mathematics and a few papers in peer-review journals, so, unlike you, not a real mathematician, and I immediately understood the point.

[yau8edq12i]

Uh... I have a PhD, I'm a tenured professor, and I even supervise a couple of PhD students. How about you? Given your snarky tone, it sounds like you'd have mentioned something like this. Not sure why you'd try to flex when you don't know much about me ;)

With my font, the letter l looked like a vertical bar |. Hence why this looked like symbol soup. But okay, let's say it's about projective spaces. Then surely, you realize that the equal sign still denotes equality. The point [1:2] is literally equal to [2:4] in P^2, not "proportional". Perhaps you need a refresher: https://en.wikipedia.org/wiki/Equivalence_class#quotient_set

[ducttapecrown]

I used l instead of \lambda, my bad I guess. My point was that the equality sign = is used to denote equality between projective points [x:y] = [ax:ay] and also proportionality between the ordered pairs [x:y] = [ax:ay], just depending on how you read and understand it.

I later remembered fractions a/b = ac/bc work this way too!