Why don't we see the violet in the rainbow as blue? I understand that we perceive the sky as blue and not violet because (1) sunlight has more blue than violet in it (see here), and (2) our eyes are more sensitive to blue than to violet.
However, I can't reconcile that with the reality that there's plenty of indigo and violet left in the rainbow, which is after all a breakdown of sunlight.
Whether I give more weight to (1) or to (2), I find a contradiction with the fact that we do in fact have plenty of violet:
If one were to argue point 1, "there isn't enough violet in sunlight", then one could counter with "but I can still see plenty violet in the rainbow".
If one were to argue point 2, "we're not sensitive enough to violet", then one could counter with a similar answer: we seem to be sensitive enough to violet as seen in the rainbow.
(If anything, the fact that the atmosphere prefers to scatter violet over blue should give extra points to violet in the color of the sky versus the rainbow, which is borne of less scattering—am I right here? The color of the sky is made from scattering from many, many rays that passed us overhead. The rainbow is made from much fewer rays that interact with water droplets. Less scattering.)
Yet, I can't deny the reality of what colors I'm perceiving. So, what is the physics that is eluding me in this regard?
 A: You are not comparing like with like. In a rainbow the blue light and the violet light are arriving at your eye from different directions and the colours can be clearly resolved. In scattered sunlight in the sky, the violet light and blue light arrive from the same direction and are superimposed.
There is less violet light in the solar spectrum and your eye is less sensitive to violet light than blue light. The net result is that the sky looks blue.
A: There is a difference between refraction of sunlight and scattering of sunlight. Different processes create different results.
The sky's color is caused by the scattering of sunlight by air molecules. This is a microscopic process (involving the interaction of a single photon with a single molecule at a time) that depends mainly on the absorption spectrum of air molecules.
On the other hand, a rainbow is caused by the refraction of sunlight through water droplets in the air. This is a macroscopic process (involving the interaction of a light ray with an entire droplet at once) that depends mainly on the refractive index and geometry of a water droplet.
As an aside, the reason that the light from a rainbow isn't really scattered too much by the atmosphere is because most of the liquid water droplets suspended in the atmosphere are quite close to the ground, so there simply isn't much atmosphere in the way.
