The Mathematical Cultures of Medieval Europe - Introduction

Victor J. Katz (University of the District of Columbia)

Mathematics in medieval Europe was not just the purview of scholars who wrote in Latin, although certainly the most familiar of the mathematicians of that period did write in that language, including Leonardo of Pisa, Thomas Bradwardine, and Nicole Oresme. These authors – and many others – were part of the Latin Catholic culture that was dominant in Western Europe during the Middle Ages. Yet there were two other European cultures that produced mathematics in that time period, the Hebrew culture found mostly in Spain, southern France, and parts of Italy, and the Islamic culture that predominated in Spain through the thirteenth century and, in a smaller geographic area, until its ultimate demise at the end of the fifteenth century. These two cultures had many relationships with the dominant Latin Catholic culture, but also had numerous distinct features. In fact, in many areas of mathematics, Hebrew and Arabic speaking mathematicians outshone their Latin counterparts. In what follows, we will consider several mathematicians from each of these three mathematical cultures and consider how the culture in which each lived influenced the mathematics they studied.

We begin by clarifying the words “medieval Europe”, because the dates for the activities of these three cultures vary considerably. Catholic Europe, from the fall of the Western Roman Empire up until the mid-twelfth century, had very little mathematical activity, in large measure because most of the heritage of ancient Greece had been lost. True, there was some education in mathematics in the monasteries and associated schools – as Charlemagne, first Holy Roman Emperor, had insisted – but the mathematical level was very low, consisting mainly of arithmetic and very elementary geometry. Even Euclid’s Elements were essentially unknown. About the only mathematics that was carried out was that necessary for the computation of the date of Easter.

Recall that Spain had been conquered by Islamic forces starting in 711, with their northward push being halted in southern France in 732. Beginning in 750, Spain (or al-Andalus) was ruled by an offshoot of the Umayyad Dynasty from Damascus. The most famous ruler of this transplanted Umayyad Dynasty, with its capital in Cordova, was ‘Abd al-Raḥmān III, who proclaimed himself Caliph early in the tenth century, cutting off all governmental ties with Islamic governments in North Africa. He ruled for a half century, from 912 to 961, and his reign was known as “the golden age” of al-Andalus. His son, and successor, al-Ḥakam II, who reigned from 961 to 977, was, like his father, a firm supporter of the sciences who brought to Spain the best scientific works from Baghdad, Egypt, and other eastern countries. And it is from this time that we first have mathematical works written in Spain that are still extant.

Al-Ḥakam’s son, Hishām, was very young when he inherited the throne on the death of his father. He was effectively deposed by a coup led by his chamberlain, who soon instituted a reign of intellectual terror that lasted until the end of the Umayyad Caliphate in 1031. At that point, al-Andalus broke up into many small Islamic kingdoms, several of which actively encouraged the study of sciences. In fact, Sā‘id al-Andalusī, writing in 1068, noted that “The present state, thanks to Allah, the Highest, is better than what al-Andalus has experienced in the past; there is freedom for acquiring and cultivating the ancient sciences and all past restrictions have been removed” [Sā‘id, 1991, p. 62].

Figure 1. Maps of Spain in 910 (upper left), 1037 (upper right), 1150 (lower left), and 1212-1492 (lower right)

Meanwhile, of course, the Catholic “Reconquista” was well underway, with a critical date being the reconquest of Toledo in 1085. Toledo had been one of the richest of the Islamic kingdoms, but was conquered in that year by Alfonso VI of Castile. Fortunately, Alfonso was happy to leave intact the intellectual riches that had accumulated in the city, and so in the following century, Toledo became the center of the massive transfer of intellectual property undertaken by the translators of Arabic material, including previously translated Greek material, into Latin. In fact, Archbishop Raymond of Toledo strongly encouraged this effort. It was only after this translation activity took place, that Latin Christendom began to develop its own scientific and mathematical capabilities.

But what of the Jews? There was a Jewish presence in Spain from antiquity, and certainly during the time of the Umayyad Caliphate, there was a strong Jewish community living in al-Andalus. During the eleventh century, however, with the breakup of al-Andalus and the return of Catholic rule in parts of the peninsula, Jews were often forced to make choices of where to live. Some of the small Islamic kingdoms welcomed Jews, while others were not so friendly. And once the Berber dynasties of the Almoravids (1086-1145) and the Almohads (1147-1238) from North Africa took over al-Andalus, Jews were frequently forced to leave parts of Muslim Spain. On the other hand, the Catholic monarchs at the time often welcomed them, because they provided a literate and numerate class – fluent in Arabic – who could help the emerging Spanish kingdoms prosper. By the middle of the twelfth century, most Jews in Spain lived under Catholic rule. However, once the Catholic kingdoms were well-established, the Jews were often persecuted, so that in the thirteenth century, Jews started to leave Spain, often moving to Provence. There, the Popes, in residence at Avignon, protected them. By the end of the fifteenth century, the Spanish Inquisition had forced all Jews to convert or leave Spain.

Figure 2. Papal territories in Provence

It was in Provence, and later in Italy, that Jews began to fully develop their interest in science and mathematics. They also began to write in Hebrew rather than in Arabic, their intellectual language back in Muslim Spain.