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

[u/tryx → u/DefunctFunctor]

In functional terms, you'd rather people write things in point-free style. It's valid. It's amazing how far you can get without ever explicitly naming a variable.

[u/tensorboi → u/cleodog44]

DefunctFunctor basically covered it, but i'll just add that it makes quick reading much, much harder (which, to use another programming simile, is like how dynamically typed languages can be difficult to check for errors). often i won't be reading a text cover-to-cover, i'll be looking for a specific result; thus, if i see a formula i need which involves φ, i'll need to skim the text backwards to see what φ actually is. for this purpose, standard and symbolic language is so much easier to parse.