Divination and Mathematics
These games and divination systems are remarkably old. Consider the die used in most games of chance: the reason it has pips instead of numbers on the faces is that the form of the die settled into its present form before the invention of Arabic numerals.
Divination drove the development of mathematics: much of Mayan, Egyptian, and Babylonian mathematics were used for astrological purposes. For example, our measurement of time and angles come from Babylonian astrologers’ division of the heavens in their base 60 system. The most advanced mechanical computers from Greek and from Arab inventors in the ancient world were complex representations of the heavens, used for navigation and astrology. The Antikythera mechanism (often called the first mechanical computer) is the best known of these, as few others have been preserved. Found in a shipwreck and dating from around 200 BC, it showed the position of all the known planets, the sun, and the moon, requiring over 30 gears to do so. Modern scientists, who find such a device fascinating for the level of mechanical sophistication it displays, seem reluctant to admit that the only practical use such a device could have had was casting horoscopes and determining auspicious days. Watching how the planets move back and forth around the wheel of the zodiac on a recreation of this device, it is not hard to see how such an irregular motion would give the impression of an intelligent and willful plan being acted out. Early attempts by archaeologists to understand the device focused on the words inscribed on it, and were unsuccessful. It was only when an attempt was made to understand the gearing system that the meaning of the device was recovered.
Later, it was the analysis of games of chance that led to the development of probability theory and statistics, which are key components of most modern AI systems, since absolute reasoning is often too brittle to deal with real-world situations.
The combinatoric principles of the I Ching and the geomantic divination (introduced at the beginning of this chapter) inspired the 17th century philosopher Gottfried Leibniz to develop binary notation. These binary codes are found in other divination systems around the world, such as the African Ifa or Sikidy systems of divination. In recent years, the fields of “ethno-mathematics” and “ethno-computation” have begun studying these cultural artifacts to explore the mathematical ideas of non-Western cultures.
Elements of recursion play a large role in these games and divination systems, where the state resulting from a series of actions is the beginning point for the same series of actions, performed again and again. In Mancala, for example, the game is played by choosing one pit, scooping up all the seeds from the pit, and planting one in each of the following pits. The object of the game is to be the first to get all of one’s seeds into the final pit. One strategy to do this is to find a pattern that persists over time, so that the seeds in multiple pits move together in a train. These patterns were discovered and used by experienced players across Africa. In the field of cellular automata, this is known as a “glider.” As a form that maintains itself as it moves through a space divided into discrete cells, it is an important component in the study of these computational systems, a study which only began in the 1940s as computers were invented.
These connections to mathematics are a natural extension of the representational nature of the tokens and spaces used in board games and divination. As a simpler system than the real world, it provided a fertile ground to begin development of mathematical ideas.
Divination and Ontology
The systems of classical elements (Earth, Air, Fire, and Water in Western cultures or Wood, Fire, Earth, Metal, and Water in China) used in divination rituals were attempts to find symmetries and order underlying reality, to find general systematic laws that applied to all aspects of nature and human life. These systems had appealing symmetries and provided a theoretical framework in which physics, anatomy, psychology, or any of dozens of other sciences could be understood. Most of the connections made were illusory, forced by overzealous application of symmetry, but the overall attempt to find such connections and symmetries is similar to much of modern science.
In his study of African divination methods, Wim van Binsbergen identified three features of geomantic divination:
Using an assortment of pre-created elements, rules to combine them, and a randomizer, these divination systems pioneered a way of getting seemingly original creations from a machine. Are machines necessarily limited to this kind of recombination of pre-created ideas, or is it possible for them to create new works of art, new ideas which we would judge as creative if they came from a person? This is a question we will return to periodically throughout the book, as other inventors and artists used these same methods to try to build creative machines.
Cicero on Divination
The Roman scholar and philosopher Cicero examined divination critically in 45 BC in the book On Divination. It’s hard to say exactly what Cicero believed about divination because he was careful to examine all the different possibilities in his work. One of the ideas he explored, however, was that divination might be accurate, even if it isn’t guided by the gods:
"For the presages which we deduce from an examination of a victim's entrails, from thunder and lightning, from prodigies, and from the stars, are founded on the accurate observation of many centuries. Now it is certain, that a long course of careful observation, thus carefully conducted for a series of ages, usually brings with it an incredible accuracy of knowledge; and this can exist even without the inspiration of the Gods, when it has been once ascertained by constant observation what follows after each omen, and what is indicated by each prodigy."
This is remarkably similar to how digital neural networks (a form of machine learning meant to imitate the structure of the brain) are trained. At first, the correlation between input and output is completely random, but as observations are made, the associations are strengthened or weakened until it comes to accurately reflect reality in some way. Cicero imagined a simple process that would lead a system of divination to evolve over time into something that could give intelligent and predictive answers without reflecting any hidden agent providing those answers. The serious question here (one Cicero himself raises) is whether the observations were accurate enough, the correlations strong enough, and the period of adjustment long enough that the system had developed to a point where it could be useful.
These games and divination systems are remarkably old. Consider the die used in most games of chance: the reason it has pips instead of numbers on the faces is that the form of the die settled into its present form before the invention of Arabic numerals.
Divination drove the development of mathematics: much of Mayan, Egyptian, and Babylonian mathematics were used for astrological purposes. For example, our measurement of time and angles come from Babylonian astrologers’ division of the heavens in their base 60 system. The most advanced mechanical computers from Greek and from Arab inventors in the ancient world were complex representations of the heavens, used for navigation and astrology. The Antikythera mechanism (often called the first mechanical computer) is the best known of these, as few others have been preserved. Found in a shipwreck and dating from around 200 BC, it showed the position of all the known planets, the sun, and the moon, requiring over 30 gears to do so. Modern scientists, who find such a device fascinating for the level of mechanical sophistication it displays, seem reluctant to admit that the only practical use such a device could have had was casting horoscopes and determining auspicious days. Watching how the planets move back and forth around the wheel of the zodiac on a recreation of this device, it is not hard to see how such an irregular motion would give the impression of an intelligent and willful plan being acted out. Early attempts by archaeologists to understand the device focused on the words inscribed on it, and were unsuccessful. It was only when an attempt was made to understand the gearing system that the meaning of the device was recovered.
Later, it was the analysis of games of chance that led to the development of probability theory and statistics, which are key components of most modern AI systems, since absolute reasoning is often too brittle to deal with real-world situations.
The combinatoric principles of the I Ching and the geomantic divination (introduced at the beginning of this chapter) inspired the 17th century philosopher Gottfried Leibniz to develop binary notation. These binary codes are found in other divination systems around the world, such as the African Ifa or Sikidy systems of divination. In recent years, the fields of “ethno-mathematics” and “ethno-computation” have begun studying these cultural artifacts to explore the mathematical ideas of non-Western cultures.
Elements of recursion play a large role in these games and divination systems, where the state resulting from a series of actions is the beginning point for the same series of actions, performed again and again. In Mancala, for example, the game is played by choosing one pit, scooping up all the seeds from the pit, and planting one in each of the following pits. The object of the game is to be the first to get all of one’s seeds into the final pit. One strategy to do this is to find a pattern that persists over time, so that the seeds in multiple pits move together in a train. These patterns were discovered and used by experienced players across Africa. In the field of cellular automata, this is known as a “glider.” As a form that maintains itself as it moves through a space divided into discrete cells, it is an important component in the study of these computational systems, a study which only began in the 1940s as computers were invented.
These connections to mathematics are a natural extension of the representational nature of the tokens and spaces used in board games and divination. As a simpler system than the real world, it provided a fertile ground to begin development of mathematical ideas.
Divination and Ontology
The systems of classical elements (Earth, Air, Fire, and Water in Western cultures or Wood, Fire, Earth, Metal, and Water in China) used in divination rituals were attempts to find symmetries and order underlying reality, to find general systematic laws that applied to all aspects of nature and human life. These systems had appealing symmetries and provided a theoretical framework in which physics, anatomy, psychology, or any of dozens of other sciences could be understood. Most of the connections made were illusory, forced by overzealous application of symmetry, but the overall attempt to find such connections and symmetries is similar to much of modern science.
In his study of African divination methods, Wim van Binsbergen identified three features of geomantic divination:
- a physical apparatus serving as a random generator
- a set of rules which allow for the translation or coding of the numerical outcome of the random generator in terms of culturally agreed specific values with a divinatory meaning
- an interpretative catalogue listing such divinatory meanings and accessing them through the assigned codes
Using an assortment of pre-created elements, rules to combine them, and a randomizer, these divination systems pioneered a way of getting seemingly original creations from a machine. Are machines necessarily limited to this kind of recombination of pre-created ideas, or is it possible for them to create new works of art, new ideas which we would judge as creative if they came from a person? This is a question we will return to periodically throughout the book, as other inventors and artists used these same methods to try to build creative machines.
Cicero on Divination
The Roman scholar and philosopher Cicero examined divination critically in 45 BC in the book On Divination. It’s hard to say exactly what Cicero believed about divination because he was careful to examine all the different possibilities in his work. One of the ideas he explored, however, was that divination might be accurate, even if it isn’t guided by the gods:
"For the presages which we deduce from an examination of a victim's entrails, from thunder and lightning, from prodigies, and from the stars, are founded on the accurate observation of many centuries. Now it is certain, that a long course of careful observation, thus carefully conducted for a series of ages, usually brings with it an incredible accuracy of knowledge; and this can exist even without the inspiration of the Gods, when it has been once ascertained by constant observation what follows after each omen, and what is indicated by each prodigy."
This is remarkably similar to how digital neural networks (a form of machine learning meant to imitate the structure of the brain) are trained. At first, the correlation between input and output is completely random, but as observations are made, the associations are strengthened or weakened until it comes to accurately reflect reality in some way. Cicero imagined a simple process that would lead a system of divination to evolve over time into something that could give intelligent and predictive answers without reflecting any hidden agent providing those answers. The serious question here (one Cicero himself raises) is whether the observations were accurate enough, the correlations strong enough, and the period of adjustment long enough that the system had developed to a point where it could be useful.
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