Why Are There Three Primary Colours?

One question that has long puzzled me (although not enough to motivate me to find an answer until now) is:

why are there three primary colours?

Another, possible statement of the problem might be: why is the space of colours three dimensional?

Once when I posed this question to a friend they suggested the reason was that the human eye has three cones. But that could be the result of rather than the reason for the three dimensional colour space.

One possible answer is that it isn't three dimensional, it's infinite dimensional but three gives a reasonably good approximation and the marginal utility of adding more dimensions drops off quickly. This reminds me of music where 12 notes gives a decent approximation to the harmonic series—much better than seven notes which is the next best under 12 and enough that few have been motivated to go to 19 notes where the next improvement happens.

After all, isn't white light a combination of all frequencies, not just three?

If you take a sine wave with the frequency of red and one with the frequency of green and add them, you get a wave whose periodicity resembles that of yellow. But it isn't the same as a pure sine wave of that frequency. This in itself suggests that additive colours are just approximations.

Newton recognized that there were colours that didn't appear in the spectrum but were achievable through combining spectral colours. What I'm not clear about is whether such combinations require three components. Will a combination of two spectral colours only give you an approximation of another spectral colour? Do you need a third component to get non-spectral colours? Does adding a fourth component give you better approximations but with a far reduced marginal utility?

Anyone care to shed some light? (pun intended)

UPDATE (2004/06/07): now see Update On The Primary Colours.