Recreational Mathematics in the Middle Ages

Overview

Mathematics in the Middle Ages was largely concerned with commenting on traditional texts, most notably the Elements of Euclid (fl. 300 b.c.). This work supplied both the vocabulary in which much mathematics was done and the kind of problems to whose solution mathematical effort was devoted. After the fall of Rome in 476, it took many centuries even for a complete text of Euclid to be available to mathematicians in western Europe. As a result, much of the mathematics that was being done could be called "philosophy of mathematics" and involved an investigation of the properties of the numbers that were the building blocks of mathematics in Euclid and elsewhere. Amid all this work (which resembles a kind of number mysticism), there were some techniques that could be incorporated in mathematics when it was less involved with issues of theology.

Background

Geometry was the form of investigation that the most sophisticated mathematical work of the Greeks took. Arithmetic was more or less built into the system of geometry that Euclid developed, although he did not describe the system of calculation in detail. In the geometrical work of Euclid and his successors, the mathematical ideas were given a firm foundation by means of explicit definitions, axioms, and a sequence of theorems whose logical relations to one another were set out equally explicitly. Arithmetic was done by means of representing numbers of lengths, although this did have some peculiar consequences. Thus, when a number (represented by a length) was multiplied by a number (represented by a length), the product would be represented by an area. This would scarcely have made for an easy system of calculation.

By contrast, there were problems and speculations about arithmetic stretching back to Diophantus of Alexandria, about whose life little is known. He gave methods of solution to what are now called "word problems," but then there were no other sorts of problems available to students of arithmetic in Greek mathematics. There was an even longer tradition of number mysticism going back to the earliest commentators on the Bible. It seemed difficult to make sense of some of the numbers used in books of the Bible (from Genesis to Revelation), and theologians applied themselves to understanding how the numbers fit together. Issues ranged from seeking explanation for why human beings were created on the sixth day of creation to figuring out how the Israelites in Egypt could have increased so rapidly over four generations. The kind of mathematics being trotted out to solve such problems was not impressive by Greek standards, but in the absence of Greek mathematical texts it was the best available. It was not until the eleventh century that Greek mathematical texts began to be available in translation in western Europe, after they had been handed down through the mathematicians of Islamic civilization in the Middle East and Spain.

Typical of the sort of arithmetic investigation popular among the scholarly community during the Middle Ages was the work of Rabbi ben Ezra (1092-1167). A Jew working within Arabic civilization in Spain, he wrote about the positive whole numbers in a variety of settings and tended to take an attitude in which religion was mixed with observations of nature. In paragraphs about numbers like six or seven, he would bring in Biblical references as well as referring back to the kind of number speculation associated with the disciple of Pythagoras (c. 570-c. 500 b.c.) In this tradition the numbers had properties associated with them (in the same way that people did) like bravery or masculinity. Despite the rather far-fetched nature of some of these speculations, ben Ezra also worked in the area that would nowadays be called combinatorics. This involves the number of ways of arranging objects and making selections from them. It seemed as though his work was connected with astrological issues, keeping track of the number of ways that the planets could be grouped. Astrology had a more distinguished place in medieval science than it has had in more recent times.

Perhaps the best known of the medieval investigations of combinatorics was the work of Ramon Llull (c. 1232-1316). Just as with ben Ezra, Llull was not interested in the abstract study of mathematics and patterns for their own sake. Instead, he was strongly interested in trying to set out the truths of the Christian religion in a way that would make them convincing to the followers of Islam. Llull was from Catalonia in northeastern Spain, and the question of dealing with the Muslim enclaves still left in Spain at the time was not just a matter for idle speculation. Llull sought to put the attributes of God in a list and then to present the possible combinations of those attributes as facts of Christian belief. He wrote of manufacturing ways of enumerating these combinations and came up with different means of representing them throughout his life. Although the motivation for his efforts was religious, once he created the diagrams for representing the attributes of God, he subordinated theology to combinatorics.

Impact

The work on arithmetic and speculations about the whole numbers continued under the heading of religion and philosophy after they had been given up on as mathematics. Book VII of the Elements of Euclid had sounded something like a philosophical foundation for the whole numbers, and arithmetical reflections of religious ideas seemed not to fit in so badly. When Book V became available to western Europe, its advantages for doing mathematics enabled the arguments about Book VII to be relegated to a backwater. The mathematical traditions that were imported from the Arabic scientific world offered more substantial problems on which to work. Numerology did not long remain in the curriculum of mathematical studies at universities and earlier instruction.

The Pythagorean speculations about number did not entirely disappear from discussions in the philosophy of mathematics. The Indian mathematician Srinivasa Ramanujan (1887-1920) was said to have an intimate, firsthand acquaintance with many integers, which explained his ability to come up with generalizations about their property. There is a mystical strain to some of the attitudes of Kurt Gödel (1906-1978) in which one can hear an echo of medieval attitudes towards the integers. Within mathematics itself such speculation had vanished, as there were too many interesting problems that could be couched in more concrete terms.

The individual who followed most closely in the footsteps of Ramon Llull was the German polymath Gottfried Wilhelm Leibniz (1646-1716). Leibniz had a detailed acquaintance with the mathematics of his day and independently created the calculus (along with Isaac Newton), to name just one of the major contributions he made to mathematical progress. Combinatorics had been the study of some earlier work in the seventeenth century, most of it not concerned with the work of Llull.

Leibniz, by contrast, was trying to deal with the problem of communication between Christian denominations as well as between Christians and non-Christians. He harked back to the approach that Llull had used in search of a universal language in which all the truths of religion could be framed. His hope was to create a language in which statements could not only be given a comprehensible form, but their truth or falsity could be settled by a mechanical procedure. He wrote a good deal about the benefits of such a language but did not accomplish much by way of constructing it. Leibniz's view of logic and the form of sentences led him to the sort of enumeration of combinations that was built into Llull's work.

By the start of the twentieth century, mathematical logic had begun to supply a more concrete realization of Leibniz's vision. Technical developments in logic, however, had led to a more complex situation than anything found in Llull. It seems hard to believe the claims that Llull's work had anything to do with the development of computer science, since the structures for the computer scientist would be unrecognizable to Llull.

Recreational mathematics has frequently supplied topics for investigation outside the mainstream of contemporary mathematics and ended up providing new directions for mathematical research. In the Middle Ages the mathematical language and texts available were sufficiently impoverished that the recreational byways did not lead to anything near the boundaries of current developments. When too much of the mathematics being done is recreational, there is not enough mathematical technique to be called upon to be applied to the solution of recreational problems.

THOMAS DRUCKER