Buridan’s Donkey: The Price of Indecision
Why do people die in traffic accidents? What if this article does not please you? Is it worth your time? On the other hand, what if this article pleases you ?
Why do people die in traffic accidents? What if this article you are about to read does not please you? Is it worth your time? On the other hand, what if this article pleases you? What should you do? These questions are reminiscent of the story of Buridan’s donkey, known as the paradox of indecision. In this article, we will discuss this paradox and consider the price of indecision from different perspectives. We will also try to explain this paradox mathematically.
Buridan’s donkey is a paradox attributed to the medieval philosopher Jean Buridan. According to this paradox, a donkey, caught between two equally attractive options, starves to death because it cannot make a decision. The donkey is caught between two piles of hay, both equidistant and of equal size. Unable to decide which one to choose, the donkey turns to neither and is left to die. This paradox serves to reflect on human free will, the ability to choose and the consequences of indecision.
To analyze the paradox in more detail, let us first look at its logical dimension. Bourdain’s donkey paradox may not actually be a paradox. Because, in real life, it is not possible to have a complete equality between two options. Each option has its own advantages and disadvantages. Therefore, the donkey can find a difference between two haystacks and decide accordingly. For example, a pile of hay might be fresher, greener, softer or bigger. Or the donkey can make a random choice or trust its instinct. In this case, instead of starving to death, the donkey heads for a pile of hay and survives. So how do we survive in this situation?
A mathematical expression of this situation may be a way out of the impasse.
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 assattime t>0 is a function of two arguments: the time t and the starting position x. Let At(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: At(0)=0,At(1)=1, and 0≤At(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 At(x) is a continuous function of x. Since At(0) = 0 and At(1) = 1, by continuity there must be a finite range of values of x for which 0 < At(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.
The emphasis in this representation may be on temporality. That is, the position of the donkey at a later time is a continuous function of its initial position. Since continuity covers a wide range of topics, from Modern physics to the Schrödinger equation, we will not dwell on it.
The underlying reason for starving Buridan’s donkey is actually based on a simple argument:
Buridan’s Principle. A discrete decision based upon an input having a continuous range of values cannot be made within a bounded length of time.
Buridan’s donkey starves to death because he cannot decide which of the piles of hay in front of him to choose. This applies to situations that require a discrete decision based on a starting position and made over a period of time. A continuous mechanism has to abandon intermittency, thus allowing a continuous range of decisions, or an unlimited amount of time to make a decision. Taking a modern example of this principle, the text argues that the discrete decision a driver must make when coming to a railroad crossing is impossible according to Buridan’s Principle. The position of the car is defined as a function of time and a parameter x.
Moreover, this principle does not decide how the decision is made, it only emphasizes the assumption of continuity. An often suggested method of escaping Buridan’s Principle is randomness. In theory, it is possible to balance a ball on a knife edge, but in practice, despite our best efforts, small random vibrations will cause the ball to fall. Furthermore, balancing the ball on a knife-edge requires very precise determination of both the position and momentum of the ball, which is prohibited by the Heisenberg Uncertainty Principle. Since a donkey, quadruped or human, must also have random noise and be subject to the Uncertainty Principle, it is impossible for it to hang forever in a state of perpetual instability.
Randomness can make it impossible for the donkey to starve to death consciously, but it cannot prevent the donkey from starving to death accidentally. Random vibrations make it impossible to balance the ball on the knife edge, but if the ball is positioned in a random way, random vibrations can cause it to fall as much as prevent it from falling. In classical physics, randomness is a manifestation of a lack of information — if we knew the positions and velocities of all atoms in the universe, even the smallest vibration would be predictable. Randomness due to insufficient information does not create discontinuity, so it does not invalidate the continuum hypothesis on which Buridan’s Principle is based. The status of continuity in quantum physics is not as clear as in classical physics. The laws of quantum mechanics (e.g. Schroedinger’s equation) are continuous, and the Uncertainty Principle, like random noise, seems to prevent the donkey from conscious starvation, but not accidental starvation.
Despite these arguments justifying Buridan’s Principle, its consequences may still seem absurd. How could an intelligent person knowingly step in front of a train? Buridan’s Principle manifests itself in human indecision. A driver caught in a state of “starvation” cannot decide whether or not to pass safely. He makes a decision and then hesitates, perhaps slamming the brakes on the accelerator several times. You have probably also encountered similar but less dramatic situations of indecision. For example, when you are waiting for the traffic light to change from green to red and you try to decide whether to stop and go.
Buridan’s Principle may seem less plausible when we realize that the outcome does not change even if there is a crossing gate. How could a crossing gate fail to prevent such an accident? Although the physical analysis of a car/gate interaction cannot be done in detail, if the driver hesitates as the gate falls, the gate could hit the top of the car, slowing its speed and causing it to enter the path of the train. In the most optimistic case, a crossing gate can reduce the effect of Buridan’s Principle, but not make it impossible. Despite the arguments justifying Buridan’s Principle, these conclusions may seem absurd. How could a driver deliberately cut in front of a train?
In fact, it does not invalidate Buridan’s Principle; it only shows that the range of starting positions in which a donkey starves to death is so small that the probability of placing it in that position is negligibly low, even if one tried. However, while it takes days for a donkey to starve to death, a few seconds of hesitation at a railroad crossing can be fatal. Are people getting hit by trains at railroad crossings because of Buridan’s Principle? Railroad crossing accidents really do happen. When the driver dies and no explanation can be found, it is natural to assume that he could not see the train or misjudged its speed or distance. Few people are aware of Buridan’s Principle, and the possibility that it could have caused such an accident has never been considered. Whether Buridan’s Principle is responsible for actual accidents seems to be an open question. All we can say, though, is that the probability of such an accident is very low, but no one knows whether “very low” means once in an era or once a year.
Thank you so much for taking the time to read this article. If you want to support me yo can follow me. 😊
References:
https://www.eurekalert.org/news-releases/573418
Buridan’s Principle- Leslie Lamport. 31 October 1984. To appear in Foundations of Physics.
