The Riddle of the Aces

Abstract

1. In this note we shall try to unravel a puzzle. It is hoped that the triviality of the subject-matter will not persuade the reader that it is unworthy of his attention; our excuse for offering it is that the underlying principle turns out to be applicable to some very practical situations. In many of these we experience no problem and seem to be on firm ground, as may a man who on a winter's day strays from the common on to the pond, until the ice breaks.2. By repeatedly presenting theorems that appear intuitively to be unacceptable, a mathematical training is apt to engender distrust in the conclusions of common sense. When the mind has learned to convince itself of the truth of a host of astonishing facts, it hesitates to reject a fresh one however improbable it may seem. This reluctance has an unfortunate consequence, in that the more often the validity of such results is demonstrated, the less the urge is felt to scrutinize a new proof before entering yet another black mark upon the long crime sheet of intuition. Thus, paradoxically, the highly trained expert may sometimes be deceived into accepting a false statement that would be instantly denied by a layman incapable of following its supporting argument. ‘You can't fool me’, boasts the novice, ‘I'm too ignorant.’