That's my rabbit hole of the week.

> The current state of affairs leads inevitably to ghostly appearances and disappearances of the radian in the dimensional analysis of physical equations.

In "A spectral unit", Nature Physics (2020) - https://www.nature.com/articles/s41567-020-0997-3.pdf Giacomo Prando summarizes the troubled history of the radian, a unit with the odd property of appearing and disappearing seemingly at will in dimensional formulas

It led me to reading about "dimensionless quantity".

> There have been periodic proposals to "patch" the SI system to reduce confusion regarding physical dimensions. For example, a 2017 op-ed in Nature argued for formalizing the radian as a physical unit.

SI units need reform to avoid confusion (2017) - https://doi.org/10.1038%2F548135b

> The idea was rebutted on the grounds that such a change would raise inconsistencies for both established dimensionless groups, like the Strouhal number, and for mathematically distinct entities that happen to have the same units, like torque (a vector product) versus energy (a scalar product).

Don't tamper with SI-unit consistency (2017) - https://doi.org/10.1038%2F549160d

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What's surprising is that these discussions are so recent, considering the wide implications on so much practical work in mechanics and engineering. Maybe people got so used to working around the question, that any proposed solution would be too disruptive - so it's better to keep things backward compatible rather than conceptually simple/clear and explicit.

In another comment someone said, "types" (in programming and type theory, I'd guess) are a poor model of physics units. But I wonder about that, it seems "units" are something like types with associated quantities, like degrees with 360, or meters with the speed of light.

This line of inquiry also led me to "dimensionless physical constants". https://en.wikipedia.org/wiki/Dimensionless_physical_constan...

Certainly pi (and e) must be one of those fundamental constants regardless of any unit of measurement. A "turn", on the other hand - well, if we consider 1 as a dimensionless constant that means a "whole"..

How Many Fundamental Constants Are There? (2011) John Baez https://math.ucr.edu/home/baez/constants.html

Turns are Better than Radians (2022)