|By working ceaselessly on proving a new theorem, a successful math professor tries to avoid thinking about the fact that he has lost his wife who died in childbirth and about Paul, their "vegetable" of a son. In fact, he does seem to love the boy, but he is understandably troubled by his failure to speak or communicate, his lack of interest in people (today, we would say he is autistic), and the fact that he seems to do almost nothing. Still, his father lets the boy sit and watch him work on his theorem. At one point, the mathematician is losing hope, wondering if his idea was all wrong. When he returns to his desk, however, he finds that the boy has written the answer in the form of an elegant new type of integral transform.
A lot of mathematical terminology is thrown around in the story, quite correctly as well. When pondering different possible descriptions of his son's condition (e.g. "Mongoloid", the term used for people with Down Syndrome at the time) the father thinks about the definitions of different terms from abstract algebra. The story mentions that to prove his theorem he needs to show that a certain infinite series converges to an irrational number. But, the most interesting thing about the mathematical content of the story is not the words used but the very idea that the young child could learn math by watching his father and understand it well enough to come up with a key step in the proof.
I am very grateful to Fred Galvin for mentioning this story to me after someone posted an inquiry about it on a discussion board.
Originally published as pages 31-32 of Analog's April 1964 issue, it has since appeared in a few collections, most recently in The Rescuer and Other Science Fiction Stories (2014).