Why don't nearby atmosphere look blue.

[FactChecker]

"The effect of secondary scattering (you seem to keep ignoring this) is absolutely minimal and not part of this explanation. (1/1,000,000 X 1/1,000,000 = Nothing)"

You contend that the blue scattering of direct sunlight by the air in the line of sight to distant mountains is many orders of magnitude greater than the scattering of blue-tinted light coming from the blue sky. This is even though the blue sky provides a great deal of light and has more blue in it to be scattered. On a cloudy day it is common for photographers to use a warming filter that is not necessary otherwise. That leads me to believe that the color of ambient light from the sky is a larger factor than you are assuming. Maybe you have provided more supporting calculations in earlier replies. If so, I missed it. (P.S. One of the factors of 1/1,000,000 above is common to both the direct and indirect light and is misleading in comparisons of the two)

 

[sophiecentaur]

@ FactChecker: It is nice to be getting a well though out response on this thread. (I like the handle, btw; it makes me stand to attention, mentally.)
That factor of 1/million was probably an overstatement (I'm not sure) But you can go on the evidence of photographic experience. The difference between the reflected light from an object in full sun and in shade (in the absence of light cloud and other reflecting objects) is 'several stops' or a factor of a factor of 1/16 or less. That figure does't actually compare the intensity of light in a 1' solid angle from the Sun and from the sky - which is what I was getting at. The 'stops' figure is based on an integral and so I contend it is an underestimate. But I take your final point, to some extent.
If we assume you are right, and that the overall effect of scattered light from the sky, compared with direct sunlight, on any arbitrary volume of air, the ratio would still be a factor of 1/16, which can hardly be a significant modification of the perceived colour (subjective) of an already very de-saturated Blue. Either way, to me it seems to be a distractor and it is not valid to attribute the ' blueness' to 'blue coming down from the sky' and being scattered.

I am not sure how directly the practice of using a warming filter relates to this. I often tinker with the colour balance in cloudy and shady shots but I think that is due to the absence of the red from the obscured Sun, rather than to any extra blue from the clouds. That's a bit of an open question, I think. The fact that warm means cold and cold means hot, subjectively, doesn't help here. (lol)