In 1961, the great scientist Carl Sagan joined a semi-secret society called the Order of the Dolphin, which hoped to establish communication with intelligent extraterrestrials. Among the society's members was a neuroscientist named John Lilly, who had made a name for himself popularizing the idea that dolphins have their own language, as well as a kind of super-intelligence that rivals our own. Crack the code of dolphinese, argued Lilly, and we will be able to decipher any alien language we might encounter.

The order's members—including...