"Finding patterns where no patterns exist" simply refers to John Nash's tragic mental affliction of paranoid schizophrenia. Individuals with paranoid schizophrenia can have troubling symptoms ranging from mild to absolutely crippling. They can believe that strangers or friends are out to get them, people on television are talking about or to them, or even that a large group is hatching a conspiracy against them. The disease is very difficult to treat, owing in part to the fact that these delusions can be very convincing.

Nash began to show symptoms of schizophrenia around his early thirties. This is considered somewhat late for when the disease usually develops. He began with bouts of paranoia. For example, one particular delusion had him believing that any man in a red tie was part of a conspiracy against him. This is what is meant by seeing nonexistent patterns.

John Nash, a mathematician, saw patterns in numbers and in events that were not seen initially by others. This lead to his great strides in the field of game theory. In late 1958, his behavior became erratic and he became increasingly paranoid. This was the beginning of his battle with paranoid schizophrenia. 

Schizophrenia with paranoia often causes its sufferers to see connections that do not exist. For example, if a man leaves a newspaper on a bench, typically no one would think anything of it, however, someone suffering from schizophrenia might believe it to be some sort of message delivery. The unfortunate aspect of this is that, due to the paranoia, no one can convince the sufferer that what they believe is a pattern or has special meaning is an ordinary event. In Nash's case, although his initial pattern observations were valid, he began to see patterns in newspaper articles and occurrences that did not exist and were really separate, unrelated events.