James McLean wrote that the concept of the angle is special in
terms of being intrinsically dimentionless. I do not think so.
Unit of length was defined as a distance between two lines on a
chosen stick. What prevents me from giving the same status to a
unit for angles? Suppose the unit is defined as an angle between
two chosen stars, as seen from earth on January 1, 1900. Numerous
models (protractors) would be constructed from the gold-irridium
alloy to represent the unit. I am not saying that this is desirable, my
point is that the concept of the angle is not very different from
concepts which were historically expressed in units. Every physical
quantity, length, time, charge, etc. can be made dimentionless or
not, if we want.
It would not be very useful to make speed dimentionless in an
introductory physics course but theoretical physicists find this
approach very convenient. Angle is a physical concept, there is
nothing unique about it in this respect. I am perfectly happy with
dimentionless radians, but dimensional angles would not bother
me at all.