When I placed a lamp next to a mirror where I can see "two lamps", does that double the light output? I know all the light that would have been wasted on the wall is now being returned so is it a yes?
The lamp in the mirror is a reflection of light from the real lamp. The real lamp is producing light. The mirror lamp is a reflection of this light. The reflection doesnt add to the amount of total light
Yeah, we're not troll sciencing light out of nowhere, but if the lamp is next to a mirror instead of a wall, the room will have more illumination, because the mirror is reflecting the light emitted by the lamp better than a wall will do.
All true, but there's a reason for lamp shades as well, as they diffuse direct light and make the illumination of a room friendlier. Glare is a real issue and so while placing a mirror next to lamp increases the light in a certain direction it could be uncomfortable looking into the direction of the lamp. A white wall would reflect the light in a more diffuse and thus agreeable fashion, but the overall output measured in the room is gonna be smaller. Illumination depends on your needs in the end.
the mirror is reflecting the light emitted by the lamp better than a wall will do.
Will it? Maybe. You are making the assumption that specular reflection is better at reflecting than diffuse reflection, but that is a false assumption. It depends on the reflectivity of the wall and the mirror. A white wall can reflect more light than a mirror.