Which sentence sounds correct to you:
They have opposing point of views.
They have opposing points of view.
They have opposing points of views.
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Only points of view is correct.
Point of views and points of views don’t make sense because of view only indicates which kind of point you’re talking about: a point from which the person takes their view. The word view does not function like houses in doors of houses, because the points are not possessed by views. Grammatically, of view works like of Paris in plaster of Paris. Plaster of Paris is a kind of plaster.
It might help to understand the literal meaning of “point of view”. Suppose that two ships are in the water at different places around an island. If you are standing on one of the ships, you can see a town on the island, but a person standing on the other ship cannot see the town. Standing on the other ship, you can see a beach, which can’t be seen from the first ship. The location of each ship is a geometrical point. And if you consider that point regarding what a person can or can’t see from that point, you are thinking of that point as a point of view. From one point of view, the town is visible; from the other point of view, the town cannot be seen.
People say point of view metaphorically to refer to a person’s preconceptions, previous experience, social position, and access to information, which influence the conclusions they draw when reasoning or making a judgement. When you describe a disagreement as “opposing points of view”, this suggests that the disagreement arises from differences in what each person can see, like the two ships. So, if each person could see the topic from the other person’s point of view, each person might learn from the other and they might agree, just as a person could see both the town and the beach by traveling to both points in the water near the island. Also, the disagreement might be irreconcilable if neither person will consider the matter from the other’s point of view.