Does "to peer down on somebody" mean "to look at somebody with contempt, as if you think you are better"?
The context is this:
The sculptures peer down on visitors to America's Supreme Court.
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Sign up to join this communityNo, it does not. The meaning is literal (well, not truly literal—statues are inanimate—but literal given the anthropomorphizing of the statues). It means the statues are physically higher than the visitors, and they are looking ("peering") at the visitors, so by necessity they must be looking in a downward direction.
The phrases "to look down on someone" and especially "to look down one's nose at someone" can carry a connotation of self-imagined superiority. "Peer down" does not have that connotation.
I would actually guess that the author chose “peer down” instead of “look down” because “look down on” does have the figurative meaning “have contempt for.” The passage is probably using the more unusual word to avoid that.
That said, “peer” can also have the meaning, “to look searchingly, scrutinize,” so it could be read as implying that even the statues in the highest court in the land, who represent great lawgivers and Justice itself, are judging the people who pass beneath them.
I perceive some doublespeak in the provided phrase that others have not mentioned.
While "peer down" does not imply contempt, your example text "The sculptures peer down on visitors to America's Supreme Court", carries additional meaning.
Statues in a hallowed hall such as the Supreme Court "peering down" carries the connotation in the anthropomorphized sense that the statues are actively watching and at the least are forming opinions of their own of what they see.
This carries the implication that the visitors should be wary of their actions as they would similar to the "ghosts" of those in the same position were watching. It evokes something of the nature of being observed by a deity, though surely in this case the statues represent something of awe similar to demi-gods rather than actual deities: the supreme court holds sway over a great deal of personal liberty in the United States, though (importantly) by rendering an opinion of what is brought to the court, rather than by actively creating legislation.
This secondary meaning seems especially important here as this is literally the purpose of the court: to observe actions, opinions, and behaviors (peer) and then render an ruling (opinion) based on that observation.
The statues are statues, and if the author of your phrase intended to merely indicate they were positioned high up above visitors, there are other ways to convey this simply using positional semantics. That the author specifically pointed out that they are "peering down" indicates that the author wishes to convey that the visitors should consider themselves watched and evaluated.
For a contemptuous phrase, you would say: "look down on". Since the "viewers" are statues, and they cannot see or think, there is no contempt.
The use of "peer down" shows that they are partially or completely positioned above the heads of visitors, such as being positioned on pedestals or other elevated places.
Your question is a good one.
It could mean that, but it probably doesn't. To "peer down" on someone does literally mean to "look down" on them, but as others have pointed out, to "look down" on someone has gained the contemptuous meaning you suggest, so "peering down" is probably used as a way to say "look down" in its literal sense rather than its figurative sense. The author is probably purposefully avoiding "look down."
Having said that, "peer down" could simply be an artistic choice and is capable of being used to convey the meaning you are asking about, so additional context clues are necessary. In this case, your additional context suggests that the statues are physically above the visitors, using peer in its literal sense.