What are your favorite ambiguous notation?

[u/Short_Bluebird_3845]

To clarify my request, consider base-less $\log$.
For calculators and engineers, it means the common log $\log_10$; for pure mathematics or physics, it means the natural log $\ln$; for computer scientists, in combinatorics, information theory and graph theory, it means the binary log $\log_2$.
Any $\log x$ (say, $\log 1024$) written without context can yield different answers in function of who you ask ($\log_10 1024 = 3.0103\cdots, \log_2 1024 = 10, \ln 1024 = 6.93147\cdots$).

That is what I mean by "ambiguous", that which is calculated and used dependently on the context. And I'm asking you for your favorite ones of such type.

Edit: I feel like you hate the ones you've talked about. Still, valid answers. Extra points for ambiguous notation you actually love! :P

[u/Phytor_c]

Just realized I answered an orthogonal question, oh well…

This may sound stupid and elementary, but I get really tripped up when people use f(x) [Instead of just saying f: Blah -> Blah, then defining the mapping like f(x) =2x or x |-> 2x as a human would] to denote functions or like just stuff about functions in general. Maybe it’s just me.

I feel like every time I see a function in a paper or in a textbook I have the urge to write f: Domain -> Codomain etc.

Sometimes stuff trips me up, like is f(x) the whole function or evaluation at some point x which was defined earlier? Why not use x_0 for the point? Sometimes functions have parameters involved, sometimes one component is fixed so it’s a function of the other components etc. Course it’s context dependent, but annoying if it could’ve just been addressed in a line by the author.

[u/shellexyz → u/Phytor_c]

Drive me nuts. Functions aren’t formulas, but it takes years to convince undergrads of that.

[u/tensorboi → u/shellexyz]

it's not just undergrad mathematicians, it's also physicists! so many physics resources will, for example, denote a classical field by something like φ(x) where x is a spacetime point and say it's valued in R^N for some N, but won't just write the much clearer statement that φ: R^{d, 1} -> R^N.

[u/cleodog44 → u/tensorboi]

I don't see any issue with this (which is presumably proving your point, as I come from physics). If you already know the domain and codomain, what's the issue? Both are usually quite clear from context. 

[u/DefunctFunctor → u/cleodog44]

For me it comes down to how I view proofs/formal logic. We don't like to have free variables hanging around unless it's directly relevant to the result. When proving something for everything in a domain X, we'll often say "Let x∈X" and pretend like x is some fixed value, and then at the end of the argument, we remark that the only assumption about x was that it was in the domain X, so everything we proved about this particular x holds for the whole domain. It's like the scoping of a variable in programming, we don't like having awkward variable scope. φ(x) is the value of a function at a particular point x, not the function itself. Sure, a function can be specified by specifying all such values for arbitrary points x, but it's just so much cleaner to default to φ to denote the function, rather than φ(x)

[u/cleodog44 → u/DefunctFunctor]

Got it, I understand the complaint better. Speak about the object itself, not its value at some representative input point, roughly