Comment by nyc111
1 year ago
When this subject comes up I always think that the equality sign is loaded. It can mean, definition, equality, identity, and proportionality. How come this state of the equality sign is not mentioned in the paper?
1 year ago
When this subject comes up I always think that the equality sign is loaded. It can mean, definition, equality, identity, and proportionality. How come this state of the equality sign is not mentioned in the paper?
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.
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.
[x:y] = [lx:ly]
and I suppose the model of localization discussed in the article above counts as well.
... What? That's just symbol soup. What are you trying to write? Do it in latex if it helps.
3 replies →
As far as I remember, that's not a problem in math:
:= definition ≡ identity = equality ∝ proportionality
But this is how it works:
= definition = identity = equality = proportionality
Big-O notation is a famous example where = is some flavor of (asymptotic) proportionality, and it catches a lot of people out when they first learn about it.
up to (non-)unique isomorphism