Would we be able to see anything else? Or would the air just block out the view of anything else?
Only if it were opaque. Actually, you are seeing air when objects appear to waver when viewed over a hot surface.
Actually, we are evolved not to see air. Mirages and shimmering over heat sources are not "seeing" air - they are effects of changes in the refractive properties of air.
What does it mean to be able to see an object? It means that we can detect the light that is reflected by that object. In the case of very sparse gasses like air, the amount of light reflected (scattered) by the object is very small but still detectable. For example, if you look up at the sky and see that it is blue, you are seeing the air; the blue light is scattered by the air to a greater extent than the shorter wavelengths, so the sky appears more blue than other objects around it.
Now if you were to concentrate the air, it would be much easier to see the light reflected by the air. This is actually possible; air is mostly nitrogen and if you liquify nitrogen, you can see the nitrogen as clearly as one would see any other liquid like water.
Perhaps you argue that we similarly can't see water or glass?
We can see these materials because they differ in refractive index from air, and exhibit some reflectivity at their surfaces. The way we "see" air is primarily from refractive effects due to temperature gradients. Go out at night and look at stars that twinkle. Are you "seeing" air, or are you seeing the effects of temperature gradients in air on the light-path from the star to your eye?
I'm seeing regions of differing refractive index. Those regions are composed chiefly of, well, air.
You can see air quite easily by looking at a daytime blue sky. Although it is apparently quite transparent the scattered blue light overwhelms the dimmer star light, at least to our eyes. So you can see air, but you need many miles of it and good lighting.
Maybe you'd have to consider it from another angle. The sun produces a certain frequency band of the electromagnetic spectrum, for which the mixture of gasses, which is called atmosphere, happens to be reasonable transparent. Evolution took advantage of that by develloping sensors aka eyes, which are the most sensitive in the center of that frequency band.
Should the atmosphere have been more opaque for the solar spectrum then we would have seen eternal fog.
We can 'see' the air indeed sometimes when simmering hot and cold air mix with have differenrent refraction indexes, causing 'simmering' distortions of the background and mirages.
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