### Buridan’s Ass

Jean Buridan (c. 1300-1360).

“Buridan’s Ass” is a paradox in philosophy, in which a hungry donkey, located at the mid-point between two bales of hay, is frozen in indecision about which way to go and faces starvation — he is unable to move one way or the other.

Jean Buridan was a French philosopher who lived in the fourteenth century. He was not interested in donkeys, but in human morality. He wrote that if two courses of action are judged to be morally equal, we must suspend a decision until the right course of action becomes clear. The idea of the paradox can be found in the writings of the ancients, including Aristotle.

Dynamical Systems

We can illustrate Buridan’s argument in the context of simple dynamical systems. Suppose a test mass, constrained to move on a line, starts in equilibrium at the midpoint between two centres of attraction. If the force of attraction decreases with distance, as for a gravitational force, then any movement towards one of the centres will increase the force of attraction towards that centre so the test mass will continue to move away from the midpoint: the equilibrium is unstable.

If the force of attraction increases with distance, as when the test mass is attached by springs to the centres, then any movement away from a centre will increase the force of attraction towards that centre so the test mass will be drawn back to it: the equilibrium is stable.

It could be argued that a planet such as Earth, is “torn between” the gravitational attraction of the Sun and the outward centrifugal force. It is fortunate that Earth neither falls into the Sun or flies into deep space.

We can illustrate stability by considering a particle in a sinusoidal potential well. You might think of a ball-bearing on a corrugated steel roof. The highest points are unstable equilibria. The lowest points are stable equilibria.

Back to the Ass

Suppose the donkey is half-way between two haystacks. He can see the two stacks, which are equally distant and equally attractive (Fig. 1). However, once he moves away from the equilibrium point, he will be closer to one haystack than the other. It will appear larger, so he will continue towards it. He is at no risk of starvation.

Fig. 1. Once the donkey moves away from the equilibrium point, he will be closer to one haystack than the other and will continue towards that centre.

Now suppose the donkey is in a valley between two mountain ridges. At the top of each ridge is a large stack of hay. The two stacks are equally distant and equally attractive (Fig. 2). If the donkey moves towards the right-hand haystack, it will gradually disappear below the sky-line, while the left-hand one will remain visible — we assume here that the donkey “hunts by sight and not by smell!” He will head back down the hill towards the left-hand haystack. Alas, as he climbs towards it, it will disappear and the right-hand stack will re-appear. The unfortunate animal will oscillate interminably, never able to resist turning back towards the visible source of food.

Fig. 2.As the donkey moves away from the equilibrium point towards one haystack, that stack gradually disappears while the other one remains visible, so he turns back.

Mount Sin(a)

We represent the mountain ridges by

$\displaystyle h(x) = -\cos x \qquad\mbox{which has slope}\qquad m(x) = \sin x \,.$

The line-of-sight from the lowest point to the sky-line is the line from the point ${(0,-1)}$ that is tangent to ${h(x)}$ at some point ${a}$ in the range ${0 < a < \pi}$. A simple computation gives

$\displaystyle \frac{1-\cos a}{a} = \sin a \qquad\mbox{or}\qquad a \sin a + \cos a = 1 \,.$

This has an infinite sequence of positive solutions, the smallest being ${a = 2.33112\dots }$ which is about ${134^\circ}$. It is a simple matter to calculate the minimum height for a haystack to be visible from the lowest point and the maximum height before it remains visible from any point on the slope. Over to you.

Buridan’s Principle

The American computer scientist Leslie Lamport wrote the macro package that transformed Donald Knuth’s program TeX into the powerful typesetting system LaTeX widely used by mathematicians. In a paper Buridan’s Principle, he presents an argument that, subject to assumptions about continuity, there is always some starting condition under which the ass starves to death, no matter what strategy it takes. Lamport considers the dilemma facing a driver approaching a traffic light that changes to amber — whether to stop or to carry on, perhaps a matter of life and death.

Sources

${\bullet}$ Leslie Lamport, 1984: Buridan’s Principle. Foundations of Physics, 42(8): pp. 1056–1066. PDF here.

${\bullet}$ Wikipedia article Buridan’s Ass.