I was reading Raphael Finkel's book "Advanced Programming Language Design", and he mentions a language somebody named Finkel invented in the 1970s called AL. AL had dimensional analysis built in, with four base dimensions: 'time', 'distance', 'angle', and 'mass'.
(As Finkel says, this is the kind of "type safety" that's really needed in everyday calculations. His language did it statically, but you can do it dynamically too, as units(1) does.) He points out that 'angle' really shouldn't be included since radians are really dimensionless, which leaves us with 'time', 'distance', and 'mass'. But no 'speed', 'force' or 'energy'. However, they can be derived from the base units; speed is a ratio distance/time --- e.g. m/s. Acceleration is speed/time, or m/s², and force can be thought of as mass * acceleration, or kg*m/s². Finally, energy is force * distance, so you can express it in units of kg*m²/s². This is in fact how units(1) represents it. But do we really need all three of 'time', 'distance', and 'mass'? The speed of light provides a natural conversion factor between time and distance, and mass can be equivalently measured as energy, according to the well-known formula E = mc². So speed is really just a dimensionless quantity, and acceleration is the reciprocal of a time interval, namely the time to reach the speed of light at that constant acceleration; so its units are really 1/s. So force can really be measured merely with kg/s. Unfortunately this doesn't really help us get rid of mass: energy is now expressed in kg*s/s, or merely kg. So our two relativity-based equivalences (mass as energy, and time as distance) turned out to be merely two facets of the same equivalence. But I suspect that in everyday calculations, the equivalence of mass and energy is rarely useful; it's far more likely to hide errors than to reduce the amount of work necessary. With time, distance, and mass, the only incommensurable quantities I commonly run into with commensurable units are: - torque and energy; - various dimensionless quantities; - perhaps pressure and stress; - stress and young's modulus. Collapsing mass with energy and time with distance creates many more "units collisions".
