Teaching Math in the Middle Ages
Where is math in the Middle Ages? How are mathematical concepts articulated and taught? I attempt a partial answer to these questions in the video below, which is an example of my recent work and part of my second book project, The Mathematical Imagination in Medieval Literature.
In general, post-medieval accounts of the history of science in the Latin West have not been kind to the study of medieval mathematics. Writing in the 19th century, W. W. Rouse suggested that a masters’ student at Oxford or Cambridge would have very rarely studied the assigned mathematical texts for his course in the quadrivium; a century later Morris Kline was more explicit, saying that medieval mathematics was “totally ineffective” and that “no scientific, technical, or mathematical concept gained any foothold” until at least the sixteenth century. More recently, Judith Field has located the “real” beginnings of mathematics in the Renaissance, and, perhaps indicative of a systemic problem, history of science curricula, if they mention medieval mathematics at all, only briefly mention the Oxford Calculators on the way to accounts of the development of the calculus, skipping over the rest of the period in judgmental silence.
Even the most sympathetic historians of medieval science, attuned to the period as the font of European culture, tend to focus on medicine and astrology rather than number theory and geometric principles separate from astronomical investigations. While these studies certainly address arithmetical and geometrical concepts, they nonetheless stress the fundamental simplicity of both of these subjects on their way to discussions of the human body, the planets, and the stars. Contemporary accounts of medieval mathematics are negative in large part because they understand mathematics in the modern sense of the term – as a group of facts and postulates waiting to be discovered, always becoming more intricate and exact – rather than as a field of abstract knowledge which ponders the concepts of number, space, and change, complete within its own historical and cultural framework.
The Mathematical Imagination in Medieval Literature challenges these qualitative, teleological, and mechanical accounts of mathematics, arguing instead for an understanding of mathematics on the medieval period’s own – shifting – terms. In doing so, it follows the work of Jaqueline Stedall, the late historian of mathematics at Cambridge, who argued that the study of medieval mathematics ought to look not only at mathematical texts, but also at works of natural history, logic, and theology. However, I suggest that we go one step further, and look to devotional, poetic, and grammatical texts as well. After all, Isidore of Seville (560-636) began book three of the Etymologies – on the quadrivium – with a discussion about the function of numbers in the world, and his account is exhaustive in a way that is both very foreign to our concept of mathematics, and yet familiar if we consider the increasing trend towards the quantification of all manners of modern life. God, says Isidore, created the world using number, weight, and measure; humans use numbers to organize the seasons, to figure out when to pray and when to plant crops; numbers subtend musical and poetic meter, the arsis and thesis, that is, the rising and falling, of verse. “Take away number from all things,” Isidore says, “and everything perishes” (Tolle numerum in rebus omnibus, et omnia pereunt). Mathematical Theologies looks at the ways in which medieval mathematical thought found its way into devotional, theological, and philosophical texts from 1200-1500, and argues that we can better understand not only medieval mathematics, but also the origins of Renaissance and Enlightenment mathematics, if we recognize the importance of the medieval theological-mathematical ideas which influenced the work of Descartes, Newton, and Leibniz.
Walter William Rouse Ball (W.W. Rouse), A History of the Study of Mathematics at Cambridge, (Cambridge: CUP, 1889, p. 3; Morris Kline, Mathematics for Liberal Arts (Reading: Addison-Wesley, 1967) p. 199.
 Judith Veronica Field, The Invention of Infinity: Mathematics and Art in the Renaissance, (Oxford: OUP, 1997); Roger Cooke, The History of Mathematics: A Brief Course (New York: John Wiley and Sons, 1997).
 Isidore, Etymologies III.71.41 (p. 107); Eight hundred years later, John Trevisa would say the same thing in his fourteenth-century translation of Bartholomeus Anglicus’ De Proprietatibus Rerum: “The art of nombres and mesures serueth to diuinite, as doth the art of melody[…] For it is seide that the worlde is compowned and ymade in a certein acorde and proporcioun of armeny, as Isider seith libro iii.” Trevisa, Bk XXI. De musica, Capitulum cxxxi, p.1386.
Recorded by the Schoenberg Institute for Manuscript Studies at the Delaware Valley Medieval Association meeting, University of Pennsylvania, September 12, 2015.