What determines the existence of a physical dimension, as in dimensional analysis? Dimensional analysis in physics is based on the idea that only commensurable quantities can be added, subtracted, equated or otherwise compared. The five dimensions which appear in dimensional analysis are: length, time, mass, temperature and electric charge (or current, if you prefer).
I am well acquainted with how to use dimensional analysis. However, it recently occurred to me that I am not quite certain of the origin of the above five physical dimensions and why they are incommensurate.
What, then, determines the existence of a particular physical dimension? Is it an empirical question? It must be to some extent, because the dimension of say electric charge was not apparent before this phenomenon was discovered.
Is it purely an empirical question? Could we somehow discover that two quantities, say length and mass, which are thought to be incommensurate could somehow be commensurable? It seems obvious that they are not commensurable, but I do not know how to prove it.
It should be noted that there are quantities, energy and torque for example, which do have the same physical dimension, but which might superficially appear incommensurate since they refer to seemingly different quantities. Thus, I think that just because two quantities "obviously" seem to refer to different phenomena, one cannot necessarily conclude that they are incommensurate.
 A: The number of physical dimensions is a matter of definition and personal choice in my opinion, and can be increased or diminished. Maxwell, following Gauss, believed that the number of dimensions of quantities was definitely three, (temperature not included) and Electromagnetism was defined to make this work: the force law between charges was $$F=\frac{q_1q_2}{r^2},$$ so unit charge was defined to be that charge which acts on an equal charge with unit of force at unit distance. Compare the situation in gravity, where Newton defined
$$F=\frac{Gm_1m_2}{r^2},$$ where the units of mass, length and time are already defined, and the cost is a new constant of nature, $G$. Later we did the same in electromagnetism, putting $4\pi\epsilon_0$ into the denominator, a new constant of nature, and defining charge independently. So you can increase the number of dimensions by introducing new constants. Consider for example modelling the atmosphere. The horizontal length scale is thousands of kilometres, and the vertical just a few kilometres. So you will use rectangular grid boxes with very small height compared to horizontal size, effectively defining different units of distance in the vertical and horizontal. All equations in the model must agree in both length dimensions. If you need the distance between two grid boxes it will be, in horizontal units, $\sqrt{\delta x^2+\delta y^2+\delta z^2/k^2}$, where $\delta x$, $\delta y$ and $\delta z$ are the number of grid boxes separating the points in each direction, and $k$ is a defined constant, horizontal distance per unit vertical distance.
Conversely, you could view the defined value of the speed of light as a reduction in the number of dimensions. The most recent change in the definition of SI base units has had a complex effect on how many dimensions there really are any more: you could view time as the only remaining dimension, because the speed of light, Boltzmann's constant, Planck's constant and the charge of the electron all have defined values.
A: To address a separate point in the question, units of angle do not have to balance in an equation so it doesn't work like a dimension for checking purposes. Take $\omega$ (radians per second). That doesn't balance in the case of SHM: $\omega^2=k/m$.
