This book extends the remarkable studies of medieval practices of thinking that Mary Carruthers began with The Book of Memory: A Study of Memory in Medieval Culture (1990) and has continued in subsequent books and articles. What the current volume adds to her project is less clear from the title than from the image on its cover, one of the many medieval images of God creating the world that puts in God’s hand the kind of drawing and measuring compass used in geometry (British Library Royal ms. 1 E vii, fol. 1v, visible online at the British Library website). For one thing, this book is about what Carruthers calls beginnings, that is, what we most often call creativity but was taught in rhetoric under the heading of invention and developed furthest, perhaps, in practices of meditation. Further, while it is deeply grounded, like her previous books, in the verbal arts, this one’s principal contribution is its focus on the arts of number, mostly geometry but with some reference to the other arts of the quadrivium (arithmetic, music, and astronomy).
World-changing modern advances such as mathematized science, precise measurement, and calculus have led us to see a sharp separation between literacy and numeracy, but in medieval learning, the arts taught in the trivium and the quadrivium were much more interwoven. For us, the primary academic purpose of math is its scientific applications. If we try to imagine a premodern purpose other than something practical like building, what comes to mind might be number symbolism, such as in Dante’s Commedia or the Middle English Pearl, or the mathematical aspect of medieval aesthetics as analyzed by Edgar de Bruyne and Umberto Eco. But Carruthers here explores uses of the arts of number as tools for thinking allied to those she has previously studied more through the arts of the trivium, grammar, rhetoric, and logic.
With this emphasis, she both revisits authors and works she has discussed before, such as her deservedly favorite Hugh of St. Victor, and explores works that expressly teach these mathematical tools and put them to most vigorous use, such as the one that opens her first chapter, Nicholas of Cusa’s fifteenth-century De ludo globi, a dialogue in the tradition of texts for educating princes but conducted through imagining a complex bowling game. This mathematical scheme turns traditional doctrine and philosophical topics into a playful process of discovery. While Cusa’s text may be an extreme case, Carruthers uses it to introduce a general medieval approach that she states in terms that also invite consideration of how the human mind works in general: “complex Invention is always also a matter of Geometry” (27, capitalization original). “Invention” here indicates the first step of rhetorical teaching but extends far beyond the purpose of producing a text to anything that might be seen as the “designing thought” of her title, including religious contemplation.
A more familiar example might be Bonaventure’s thirteenth-century Itinerarium mentis in Deum (“The Mind’s Way toward God”), discussed later in the same chapter. The mathematical basis of Bonaventure’s own compositional invention is apparent in many of his treatises. Here, however, he turns his own meditation on St. Francis’s vision of a six-winged seraph in the form of Christ crucified into a mathematically structured practice that can be followed by readers. Seen in this way, the text “translates Francis’s singular miracle into an instrument for shared community use” (38). Bonaventure is drawing on medieval training in the arts of number and developing techniques taught by earlier authors also discussed here, especially Hugh and his fellow twelfth-century Victorines.
As Carruthers explains in her acknowledgements, this book began in three series of invited lectures. Its loosely organized essays seem to aim at being suggestive rather than exhaustive. Readers may wish for more detail in the discussion of some examples, but the notes and bibliography give thorough guidance to scholarship on texts discussed as well as many related topics (though the author-date system of citations leads to some confusion and leaves at least one footnote without a corresponding bibliographic entry). The titles of the four main chapters identify topics that are in fact woven throughout: “Syntax and Geometry,” “Investigation and Invention,” “homo sapiens as homo mensurans,” and “The Instruments of Thinking.” A better indication of each chapter’s contents is its subheadings (and readers would have been well served if these were included on the table of contents). Each chapter’s range is such that one would be hard pressed to give a briefer summary than Carruthers does in the last three pages of her introduction. Chapter 4 gathers the insights of previous chapters about the use of geometric schemes of thinking around the “basic shapes”: squares and cubes, circles and wheels, lines and trees, ladders and towers, and combinations of them as exemplified in diagrams of the liberal arts themselves. A coda shows how a little-known text attributed to Richard of St. Victor, Some Allegories of the Tabernacle of the Covenant, makes much more sense as a guide for applying the techniques the book has discussed (the translation cited as Zinn 1979 is to be found not under Zinn in the list of secondary sources but under Richard of St. Victor in the list of primary sources). A gallery of 26 black-and-white figures is given at the beginning of the book, most of which are given full pages and discussed specifically at one or more points in the text.
Because the works treated here range across the European Middle Ages in both time and geography, it is challenging to know whose ways of thinking the book is meant to be reconstructing. Carruthers is careful to convey her reasons for choosing works to focus on. Some were influential as basic instructional texts, some synthesize the techniques she is tracing, some, such as De ludo globi, exemplify a complexity that especially rewards this approach, and many combine more than one of these factors. References to scholarship on what happened in medieval classrooms, though not highly localized, serves as a reminder not to imagine some kind of ghost medieval reader with access to everything available to modern scholars. The book’s penetration to a pervasive common inheritance of thinking is most persuasive in its frequent analysis of the semantic nuances and interrelationships of key terms in medieval Latin. Carruthers considers, for instance, why it is significant that many verbs for mental activities, often late Latin and medieval coinages, are deponents, a Latin way of combining the senses of active and passive voice (like the middle voice in Greek). Occasionally she also draws on recent scientific studies of mental processes that echo or illuminate her medieval picture.
One section speculates provocatively about how practices that combine verbal and mathematical tools seem to have shaped medieval conceptions of time. Though it would go against the grain of this blending of tools, I wondered how they might also give distinct and different access to our experience of time, the verbal as more sequential and the geometric as more simultaneous. To some extent medieval practices seem to use mathematical figures in order to engage the capacities of the visual for grasping a complex whole at once, but Carruthers also uncovers more process-oriented meanings in medieval handling of geometry.
At several points, Carruthers invokes 1 Corinthians 13:12, the verse known from its translation in the Geneva Bible (and adopted in the Authorized or King James Version) as “For now we see through a glass darkly.” She usually refers to it in the Vulgate, where the key phrase, per speculum in aenigmate (“through a mirror in a riddle”) does not obscure the combination of visual and verbal metaphors present in the Greek original. (On page 84, the reference to 2 Corinthians seems to be an error for 1 Corinthians, though speculantes in 2 Cor. 3:18 is also relevant.) Whereas the Geneva translation reflects the Calvinist and broadly modern sense of a gulf separating the human from the divine, medieval readers understood this verse to convey the possibility of movement from the apparent to the mysterious, what Carruthers captures in the use of both verbal and mathematical figures of varying complexity to move from basic memory to contemplative heights.
Carruthers mentions at one point her long perplexity at Hugh of St. Victor’s statement in the Didascalicon (2.15) that “geometry is ‘the well-spring of our sensory cognitions and the originator of things we say’” (27) because it repurposes a phrase previously applied by Cassiodorus and Isidore to the idea of “topics” as taught in the trivium, rather than to geometry. I can imagine that this might be one of many texts concerning the medieval mathematical arts that she had stored up during her studies until she was ready to make sense of them for us. This volume, both in its central argument and in its numerous asides on related topics—such as the place of affect in invention—adds to her treasury of resources for understanding the medieval “craft of thought” (to cite the title of another of her books) as a collection of practices rather than settled conceptions. “Conceive,” as she observes, reflects a characteristically gestational rather than mechanical model of thinking. In a time preoccupied with the claims of artificial intelligence, this book gives access to a long view of what is human about thinking, of how we make the most of our embodied capacity for intelligence.