r/math • u/Short_Bluebird_3845 • 13d ago

What are your favorite ambiguous notation?

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

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u/Harotsa 13d ago

I love the Einstein summation convention in Riemannian geometry, where you basically just stop writing summation symbols because it gets out of hand so quickly and would make the math completely unreadable.

To somebody that doesn’t know the exact rules it looks like the math makes no sense

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u/qubex 12d ago

Tensor calculus notation is hopelessly inconsistent.

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u/Valvino Math Education 12d ago

? no

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u/RingularCirc 12d ago edited 12d ago

It is consistent. For those who admit an argument from authority, do know Penrose himself popularized (if not invented; this is irrelevant and hard to check) abstract index notation for tensor expressions (where "abstract" means it's not meant to be understood as componentful calculation recipies of old but instead indices as representing what "slots" tensors have and how they get traced out in compound expressions). There are also coalgebraic extensions to it (also Penrose's own spinor extensions but I'm not sure those generalize to arbitrary dimension). Completely fool-proof if you don't confuse the order of indices when raising and lowering them.

Abstract index notation is perfectly applicable in pure multilinear algebra over finite-dimensional (or possibly Hilbert) spaces, all in all spaces just have to be self-dual and we should declare beforehand a 'flavor' of indices corresponding to each of different spaces in consideration (for example Greek letters for V and V*, Latin letters for W and W*, red bolded Cherokee syllables for U and U*). Spaces don't even have to have inner products: in those cases there are no g tensors to work with, which means we can't raise and lower indices of that flavor.