Buridan’s Principle

The problem of Buridan's Ass, named after the fourteenth century French philosopher Jean Buridan, states that an ass placed equidistant between two bales of hay must starve to death because it has no reason to choose one bale over the other. With the benefit of modern mathematics, the argument can be expressed as follows. Assume that, at time 0, the ass is placed at position x along the line joining the bales of hay, where one bale is at position 0 and the other at position 1, so 0 < x < 1. The position of the ass at time t > 0 is a function of two arguments: the time t and the starting position x. Let A t (x) denote that position. For simplicity, assume that when the ass reaches a bale of hay it stays there forever, so for all t ≥ 0: A t (0) = 0, A t (1) = 1, and 0 ≤ A t (x) ≤ 1 for any x with 0 < x < 1. The ass is a physical mechanism subject to the laws of physics. Any such mechanism is continuous, so A t (x) is a continuous function of x. Since A t (0) = 0 and A t (1) = 1, by continuity there must be a finite range of values of x for which 0 < A t (x) < 1. These values represent initial positions of the ass for which it does not reach either bale of hay within t seconds. Such a range of values of x exists for any time t, including times large enough to insure that the ass has starved to death by then. Thus, there exists a finite range of starting positions for which the ass starves to death. The key assumption in this argument is continuity: the ass's position at a later time is a continuous function of its initial position. Continuity has been a guiding principle in the development of modern physics. Phenomena that appear discontinuous, such as discrete atomic spectral lines, are explained in terms of continuous physical laws, such as Schroedinger's equation. The assumption of continuity is discussed at length in Section 6. For now, let us accept it and investigate its consequences. The general principle underlying the starvation of Buridan's ass can be stated as follows: Buridan's Principle. A discrete decision based upon an input having a …