|God & Nature Magazine||
God, Chance, and Buridan's ox
By John Hall
The puzzle is an old one. A man takes his ox to be fed and, finding two identical piles of hay, tethers the animal exactly between them. The lead on the animal is long enough that it can reach either pile. Now this particular ox is very intelligent and, because this is the 14th century, knows the latest in Aristotelian logic. However, the piles are identical, and so are the distances to them, so he cannot find any logical reason to prefer going to one rather than to the other. Consequently, the ox starves to death. As a rational person and animal lover, could you save the ox from starvation?
Usually referred to as Buridan’s ox (or donkey, or ass), this puzzle is associated with the name of Jean Buridan, a French scholar of the 14th century. It did not originate with him, though. Its precursors can be traced back to Aristotle, who put a man in the central role. The puzzle has been discussed ever since and is even the subject of YouTube videos. Few people recognize, however, that this is not merely a thought experiment: the Bible gives us a real example. In Leviticus, God placed Aaron in the position of the ox. Fortunately for Aaron, He didn’t leave him there.
In Leviticus 16, God gave Moses detailed instructions for the ritual that Aaron would perform on the Day of Atonement. This ritual is important for us because it anticipates the atonement of Christ on the cross. Here, however, we will not focus on this central meaning of the ritual but on a bit of Bible trivia on the periphery. As part of the ritual, Aaron used two goats, one to be sacrificed and the other to be sent out into the wilderness as the scapegoat. Both goats were without blemish and thus acceptable as offerings. They were not physically identical, but God in his sovereignty could ensure that their behavior would always serve his purpose. Either goat could fulfill either role.
This left Aaron in the position of Buridan’s ox. Whichever choice he made would serve, but neither choice was preferable to the other. So how did God rescue Aaron? He told him to flip a coin. Well, not quite. Coins did not come into use until centuries later, about the time the Assyrian empire collapsed. Instead, God told Aaron to cast lots, which, with two alternatives, is equivalent to a coin toss. God used chance to achieve his purpose and to solve the puzzle of Buridan’s Ox. God also controlled which lot was chosen. As Proverbs states, “The lot falls into the lap but every decision is from the Lord.” To avoid a speculative morass, we will not consider how God would make his choice when the alternatives were equally acceptable. It is sufficient to recognize that God’s purpose could be achieved in more ways than one, but that only one way would become a reality.
The Bible gives us other examples of the use of lots in connection with tabernacle and temple worship. Lots were used administratively to schedule the duties of priests and Levites (1 Chr. 24: 5 – 31), musicians (1 Chr. 25: 8 – 31), and to assign gatekeepers to gates (1 Chr. 26: 12 – 16). When they returned from Babylon, the priests, Levites, and the people pledged to God that they would follow His laws, regulations, and decrees (Neh. 10: 28 – 39). As part of their pledge they stated that they had cast lots to determine when each family would provide firewood for the altar (Neh. 10: 34).
On these occasions God’s purposes could be achieved in more than one way. This implies some flexibility in the ongoing workings of the creation. Sometimes though, only one outcome would do, and then God could certainly ensure that it would occur. Before his son John was born, Zechariah was chosen by lot to go into the temple and burn incense (Luke 1: 8 – 10). What is clear from the account is that of the available priests, God specifically chose Zechariah so that he could receive some private one-on-one counselling from the angel of the Lord.
A random or stochastic process like those we have been examining is a process that, like a coin flip, has more than one possible outcome, although only one actually occurs. The goats faced two possibilities. Twenty-four distinct groups of musicians could be ordered in many ways. There are many kinds of random processes. Each has its own mathematical representation, dynamics, and uses. Because God used random processes in tabernacle and temple worship, it should come as no surprise to find them in the natural processes of his creation too. In fact, they have a role in both biology and physics. We use them ourselves.
The biological world is only possible because of the richness that God built into carbon chemistry, which takes place within the constraints of chemical thermodynamics. In the last few decades computational biology has revealed more of this wealth in biology and biochemistry. A protein with a specific biochemical function can have many variants. These variants can be thought of as being inter-related in a web-like pattern, the nearest neighbors differing by a single amino acid substitution. Functional metabolic systems of chemical reactions and circuits that control gene expression form similar networks. Not all of the variants that have been discovered by computational biology are actually seen in the biological world. Those that are seen are produced during reproduction.
Reproduction does not occur with perfect accuracy— daughter organisms can include some changes or mutations. In one reproductive event, only a few of the many changes that are possible actually occur, and which ones do is not determined by their consequences for the organism. For these reasons the process of mutation is largely a random one. At first these small deviations from perfect accuracy might seem to be a peripheral feature of biology. Yet they allow the expression of many variants with the same function, as well as the discovery of previously unseen functions. Without change, biological history would be static. With change, the history is dynamic: biological systems can adjust to gradually altering environmental conditions. Because they are constrained by the features that God built into organic chemistry and biological processes, this history cannot be said to be “just random” or contrary to design. Random processes are part of God’s overall design.
The rise of science in Europe can be attributed, in part, to Christianity. Medieval scholars concluded that a just God who is not arbitrary or capricious would create a rationally coherent world. Europe’s earlier polytheism provided no such assurance. Scholars later recognized that God was not limited by his rationality. He had many choices in creating the world, so learning which choices He did make requires observation and experiment, not merely rational thought. Consequently, an empirical study of the universe that assumes it is real and has a coherent order reflects a Christian worldview.
In physics, Newton’s studies exemplified this anticipated coherence. Scholars had long known that the heavenly bodies have orderly motions. Contrary to Aristotle’s earlier belief, though, Newton assumed that the same equations that described these motions would also apply on Earth. Einstein improved on Newton’s work, but special and general relativity reflect the same underlying unity.
At smaller, quantum scales, it is Schrödinger’s equation that describes movements rather than Newton’s or Einstein’s. This reflects an additional feature of the physical world. The motions of particles like electrons cannot be calculated with certainty. Schrödinger’s equation produces a probability density function rather than an exact answer. Random processes again have a role in the creation.
We, God’s creatures, also use random processes for our own purposes. Here are three examples. In medicine, the gold standard in clinical research is the randomized controlled trial. The randomized complete block design has been the work horse in agricultural field trials for over eighty years. Computer scientists are currently developing probabilistic programming and quantum computing methods to address problems that would not be tractable otherwise
People sometimes say, “God does nothing by chance,” but this statement is ambiguous. The word ‘chance’ can mean either opportunity, or a lack of intention and purpose, or a random process. Finding the latter two meanings in the same word can trap us into believing that random processes are always purposeless. They are not. Random vibrations in a piece of machinery can frustrate our goals, but random processes can also be used purposefully. Both the book of scripture and the book of nature provide examples. God used random processes to achieve his purposes in the sacred context of tabernacle and temple worship. He also used them in his creation to express his glory.
John W. Hall is a retired biostatistician who earned his Ph.D. in mathematical statistics. His undergraduate degree was in honors physics. During his working career, John was involved in both medical and agricultural research. His favorite version of Buridan’s puzzle requires a large glass of cold milk, two plates of oreos, and some pocket change.