Strauss’s criticism of the existentialists, especially Heidegger, is that they tried to elicit an ethic out of the abyss. This was the ethic of resoluteness – choose whatever you like and be loyal to it to the death; its content does not matter."Night of the world" ("Nacht der Welt") is one of Hegel's. Search Google books for: "night of the world" or "Nacht der Welt" inauthor:Heidegger, and it'll returns zero hits.
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But Strauss’s worries about America’s global aspirations are entirely different. Like Heidegger, Schmitt, and Kojève, Strauss would be more concerned that America would succeed in this enterprise than that it would fail. In that case, the “last man” would extinguish all hope for humanity (Nietzsche); the “night of the world” would be at hand (Heidegger); the animalisation of man would be complete (Kojève); and the trivialisation of life would be accomplished (Schmitt). That is what the success of America’s global aspirations meant to them.