Medieval mathematics
Medieval mathematics encompasses the mathematical developments and contributions made during the European Middle Ages, a period that spanned from the fall of the Roman Empire to the onset of the Renaissance. Contrary to the perception of this era as one of stagnation, significant advancements were achieved in fields such as algebra and astronomy, with many scholars preserving and commenting on classical Greek works. The integration of knowledge from various cultures—including Arabic, Persian, and Indian—played a crucial role in the evolution of mathematical thought during this time.
Key figures such as Boethius and later mathematicians like Fibonacci were instrumental in bridging ancient and medieval scholarship, translating vital texts from Greek to Latin and introducing new numerical systems to Europe. The era also saw a rise in educational institutions, with monasteries and early universities fostering a culture of learning that included the study of mathematics as part of the liberal arts. This period experienced a flourishing of intellectual exchange, particularly in regions influenced by the Islamic Golden Age, where advancements in various sciences significantly impacted European thought. Overall, medieval mathematics reflects a rich tapestry of cultural interactions and intellectual development that laid foundational stones for future discoveries.
Medieval mathematics
Summary: Medieval mathematics developments included Scholasticism and the emergence of secular universities.
The European Middle Ages, or the “medieval period,” lasted from the fall of Rome to the Renaissance and was identified by Renaissance thinkers as separating their own period from that of classical civilization. The Middle Ages were construed as a time of backwardness, but in fact progressed in spite of economic, medical, and political difficulties. Mathematicians made original contributions to such areas as algebra and astronomy and commentaries on historic texts preserved Greek works. Mathematics historians have studied Arabic, Persian, Turkish, Indian, Islamic, and European contributions during the Middle Ages. For example, Adolf Yushkevich wrote a seminal work on the history of mathematics in the Middle Ages. He highlighted similar features of medieval mathematics based on the cultures in Europe and Asia and, along with Boris Rozenfeld, studied Arabic contributions.
Early Middle Ages
The transfer of western Europe from the Roman Empire to the Goths occurred gradually through the fourth and fifth centuries, partly by conquest and partly by migration and assimilation. The old travel and trade network decayed and scholarship retreated mostly into monasteries. The philosopher Boethius straddled the Roman and Goth eras. He valued mathematics highly, endeavoring to translate several important mathematical works from Greek to Latin and dividing the seven liberal arts into two tiers: a lower tier, the trivium—containing logic, grammar and rhetoric—and an upper tier, the quadrivium—containing the four mathematical arts of arithmetic, geometry, astronomy, and music theory. Boethius is remembered primarily for his work Consolation of Philosophy, written while he was imprisoned before execution. Christianity became a primary supporter of higher learning, music, and art in Europe, and also a strong participant in government owing to the high levels of literacy among Church officials. Monasticism also gained momentum during the early middle ages, inspired by the isolated communities in Syria and Egypt. Owing to the importance of study in religious life, many monasteries functioned also as schools and libraries.
Carolingian Renascence
Around the ninth century, Charlemagne and his successor, Louis the Pious, enacted various reforms to effect uniform standards in a renascence of the Roman Empire. Charlemagne had schools created to restore education across Europe, reunifying the dialectized Latin and creating a script for it, the Carolingian minuscule. The standard curriculum saw Boethius’s trivium and quadrivium become the foundations for the bachelor and master of arts degrees. A standard currency facilitated reformation of the economy and long-distance trade and taxation. The Roman influence is evident in monumental architecture, which incorporates elements from classical styles in clear, relatively simple arrangements. Circles, squares, cubes, and cones feature prominently, as does symmetry. Carolingian architecture and painting became the basis for the more ornate Romanesque style and, ultimately, the Gothic.
Byzantium and the East
The Greek-speaking part of the Roman Empire, also called Byzantium, survived the Latin half’s decline. In the sixth century, Byzantium extended around the eastern Mediterranean from Egypt to Greece, expanded across all of north Africa, and even took Carthage and Italy from the Goths. Then, severely weakened by epidemics thought to be the Black Death, the Byzantine Empire shrank to what is now Turkey and Greece, plus Carthage and some parts of Italy. Even after this decline, Byzantine culture stood as the standard for both western Europe and the Near East. Owing to increasing influence from Christianity, art and monumental architecture tended to manifest in churches (such as Hagia Sophia), and philosophy intertwined with Christianity on many topics, including ethics, existence, governance, and death.
Hellenistic knowledge percolated gradually eastward from Byzantium, first in translation into Syriac and then into Arabic, which fueled a philosophical community in Damascus. By the seventh century, Neoplatonism, which had been Christianized in late antiquity, had been accommodated into the Islamic framework. This set the backdrop against which Aristotelianism, and all of its disagreements with Platonism, had to be accommodated next.
In the eighth century, Baghdad became the cultural focus of the East. The scholarly community there attracted scholars of diverse races and religions. The Islamic Golden Age continued into the eleventh century, with many advances of significance to western Europe, including those by al-Khwarizmi in algebra, by Ibn al-Haytham (Alhazen) in optics and scientific method, by al-Battani (Albategnius) in astronomy, by Jabir ibn Hayyan (Geber) in alchemy, and by Ibn Sina (Avicenna) in medicine. A rich tradition of poetry and calligraphy also emerged.
Al-Andalus
In the eighth century, the Moors of north Africa took most of the Iberian Peninsula that ultimately became the Umayyad caliphate based at Córdoba after the Abbasids came to power in Baghdad. While the Abbasid caliphate suffered from political fragmentation, the Umayyad territories in the Iberian Peninsula thrived.
Astronomy and botany were especially active in al-Andalus, both for intellectual interest and for applications in timekeeping, astrology, and medicine. While the societal framework was predominantly Islamic, numerous Jews and Christians participated in high culture during extended periods of cosmopolitanism. Al Zarqali (Arzarchel) discovered the ellipticity of planetary orbits in the eleventh century, and ibn Baija (Avempace) deduced that the Milky Way was not a continuous cloud but numerous stars. Studies of Aristotle by ibn Rushd (Averroës) shaped philosophy and religion for centuries later.
High Middle Ages
From the eleventh to the thirteenth centuries, western Europe was peaceful enough to entertain a high degree of cultural development. Windmill- and waterwheel-powered industries developed, economies flourished, and urban populations grew quickly, spreading into formerly Moorish Iberia, into southern Italy, and even into the Baltic and the Near East. The Arabic heritage was absorbed and then reacted against in a philosophical movement called “Scholasticism.”
Scholasticism emerged from the works of Aristotle. They were translated from Arabic into Latin and provided a basis for a worldview based on empiricism and logic. Although the philosophy was secular, it was pursued largely for its power to support Christian doctrine. The Arabic writers had already weighed Platonist versus Aristotelian views and largely harmonized the philosophy with religious givens. Much of the result was hence incompatible with new movements in Christianity, and the Scholastics sought to rebuild it by returning to the original sources. The scientific content was developed notably by Robert Grosseteste and Roger Bacon in England and Albertus Magnus, Thomas Aquinas, and Duns Scotus in France. These five also ranked highly in the Church, illustrating the continuing need that religion had for higher education and the support for intellectuals that the Church provided.
Early in the Middle Ages, higher learning had been concentrated in monasteries and Church schools. With the new secular engagement, universities appeared, beginning with Bologna in 1088, then Paris in 1150, then Oxford in 1167, then others. Learning emerged from the monasteries into urban surroundings and engaged more with secular needs, such as commerce and industry. Gothic architecture replaced the hefty, solid Romanesque, with height and lightness built from thin stone ribs reaching up and out to become the ribs of vaulted ceilings. Acute arches and vaults replaced the Romanesque semicircle, and walls gave way to large glass windows. Gothic designs manifest Euclidean geometry problems, including constructing regular polygons, dividing arbitrary angles into equal parts, dividing lines into equal parts, fitting circles through points, tangent to lines or tangent to other circles.
In the fourteenth century, frequent plagues and crop failures decimated the population, undermining social structure, industry, and economies. From the turmoil sprang new outlooks on all fronts. Among the more famous literary achievements, Dante wrote his Commedia and other tracts (including some scientific ones), Chaucer wrote his Canterbury Tales, Bocaccio wrote the Decameron. Such fresh thoughts ultimately gave rise to the Renaissance in fifteenth-century Italy.
A number of European mathematicians were important in helping to introduce eastern mathematics into Europe. Many Greek works were unknown in Europe and were found only in Arabic. Adelard de Bada translated the Arabic texts of Arabic and Greek mathematicians into Latin. Leonardo Pisano Fibonacci was educated in north Africa and traveled extensively. In Pisa he introduced the Hindu–Arabic place-valued decimal system and the use of Arabic numerals into Europe, while also making fundamental contributions of his own.
Bibliography
Cantor, Norman F. The Civilization of the Middle Ages: A Completely Revised and Expanded Edition of Medieval History, the Life and Death of a Civilization. New York: HarperCollins, 1993.
Crosbie, Alfred F. The Measure of Reality: Quantification and Western Society, 1250–1600. Cambridge, England: Cambridge University Press, 1997.
Lindberg, David C. The Beginnings of Western Science: The European Scientific Tradition in Philosophical, Religious, and Institutional Context, Prehistory to A.D. 1450. 2nd ed. Chicago: University of Chicago Press, 2007.
Menocal, M. R. The Ornament of the World: How Muslims, Jews and Christians Created a Culture of Tolerance in Medieval Spain. Boston: Little, Brown, 2002.