## The Quadrivium and the Stakes for Ordering the Mathematical Arts

Legend has it that Pythagoras sentenced the first person to discover irrational numbers, Hippasus of Metapontum (c.530-450 BC), to death. He was tossed overboard a ship to drown. Why? Pythagoras taught that number was the essence and cause of all things, and for Pythagoras and his followers, numbers meant integers. Hippasus’ discovery of irrational numbers appeared to undermine the very core of Pythagoras’ teachings about the numerical nature of the universe. The secret could not get out. Hippasus had to die.

The existence of irrational numbers became a Pythagorean secret. They were called “unutterables” because in Greek, the ratio between two integers was called logos, and so, irrational numbers were called, alogos, which can be translated as either “irrational” or “not spoken.” The worry caused by this secret knowledge was somewhat alleviated by Eudoxus of Cnidos (408-355 BC) when he argued that the basis of reality was a ratio of magnitudes. In effect, Eudoxus made geometry replace arithmetic as the highest mathematical discipline, the foundation of all others. Geometry and arithmetic were hardly even separate disciplines at the time. This change of emphasis allowed Pythagorean teachings about the numeric nature of the universe to continue.

The idea that the mathematical disciplines have some orderly relationship between each other is essential for understanding the medieval concept of “quadrivium.” While it is well known that the medieval liberal arts curriculum, at least in its ideal established by Boethius, taught that a student must study both the trivium and quadrivium before progressing to philosophy and theology, the exact nature and rationale for the quadrivium is often less understood. Lists of the arts comprising the quadrivium (arithmetic, geometry, astronomy, and music/harmony) are consistent, but the exact order for these lists can vary. While there is no doubt that sometimes there is truly no rationale for a given order of the mathematical arts, attention to the mathematical art considered the principle or highest can reveal at least three identifiable streams of quadrivial traditions coming from the ancient world (similar to Chenu’s identification of different kinds of Platonism): the Boethian, the Calcidean, and the Capellan. The mathematical art considered “principle” is the one closest to metaphysical reality of the universe and serves as the foundation for all other mathematical disciplines. While the problem of irrational numbers may not have been on the forefront of anyone’s mind in the Middle Ages…it was a closely guarded Pythagorean secret after all…the problem of the principle mathematical art, inherited from Pythagoreanism, was readily available in the source texts.

Boethius (c.480-525) not only established the seven liberal arts as the traditional curriculum for the Middle Ages, but he also wrote treatises on all of the trivium as well as arithmetic, music, and geometry (the latter work is now lost).  He, coined the term, “quadrivium” in his attempt to translate the tessares methodoi (four methods) of the Neopythagorean, Nicomachus of Gerasa (c.60-120). Boethius’ own De institutione arithmetica largely draws upon the work of Nicomachus. Modern day history of mathematics textbooks often observe that Nicomachus’ work is one of the first to distinguish arithmetic and geometry as separate disciplines but that the actual quality of the mathematics contains basic errors. Unlike Euclid, Nicomachus doesn’t always give his proofs. Nicomachus presents arithmetic as the principle mathematical art and as a result, so does Boethius. While Boethius was unlikely to have gotten the problem of irrational numbers from Nicomachus because Nicomachus presents arithmetic as the highest mathematical art, Boethius adopts his fourfold division of the mathematical arts along with the belief that arithmetic was the principle mathematical art (De institutio arithmetica 1,1,8).

In his work on arithmetic, Boethius explains that the order of the quadrivium he offers (music, astronomy, geometry, and arithmetic) both reflects the true nature of the universe and is the proper pedagogical order for the study of mathematics as a preparation for philosophy. Progression through each of the arts trains the mind to move from sense perception to intelligible reality.

This progression of the soul can be seen in the Consolation of Philosophy, where Boethius begins with music and is drawn to philosophy upward by means of astronomy, geometry, and finally arithmetic.

While Boethius’ highly influential order of the quadrivium was adopted by both Cassiodorus and Isidore, Calcidiuswrites very clearly in his commentary on Plato’s Timaeus that geometry is the foundation of all other mathematical arts (Commentum 2.32). His influence throughout the Middle Ages was also extensive. Calcidius’ translation and commentary of Plato’s Timaeus, was one of the only texts of Plato available throughout much of the Middle Ages. Although there were other translations of the Timaeus available, Calcidius’ commentary, as Reydams-Schils has demonstrated, was actually a very good introduction to Platonism as a whole because it was designed to introduce the reader to Platonic doctrine in a pedagogically sequenced way from mathematics to physics and then theology. Throughout the earlier Middle Ages, as Somfai has shown, the commentary was used to teach the quadrivium itself, and earlier versions contained numerous geometrical diagrams. While interest in his geometrical figures appears to fall out of favor in the twelfth century and in newer commentaries on the Timaeus, Nicholas of Cusa in the fourteenth century has both the old Calcidius’ commentaries and the newer commentaries, and geometry clearly plays a major role in his understanding of infinity and kinds of infinity.

The third line of quadrivial tradition can be found in Martianus Capella whose Marriage of Philology and Mercury, places music as the highest of the seven liberal arts, the culmination of his entire work. As Michael Masi has observed, this ordering can be found in many visual depictions of the quadrivium, including most famously, the Incarnation Portal at Chartres Cathedral, where arithmetic is paired with geometry as a mathematical study and music with astronomy as a study in harmony. While the complete reasons for this preference are too numerous to identify in a blog, there is a certain kind of Pythagorean logic even here. Music, for Pythagoras and his followers, was thought to be the best evidence for number being at the foundation of the universe. Even the movement of the stars and planets were considered to be one example of many kinds of music in the universe.

The stakes for getting the order of the quadrivium right in the Middle Ages may not have risen to the level of murder (although that might make a nice monastic murder mystery written by Umberto Eco, Murder Most Irrational….). And yet, three sources for the quadrivial tradition in the Middle Ages did present the idea that the order of the mathematical arts reflects the most fundamental nature of the universe itself. Furthermore, this fundamental order of the universe has implications for the order of education in the mathematical arts. These stakes, the metaphysical order of the universe and of education, would still have been considered pretty high for most thinkers throughout the Middle Ages.

Lesley-Anne Dyer Williams is a Professor for Memoria College’s Masters of Arts in Great Books program and graduated with her doctorate from the University of Notre Dame’s Medieval Institute in 2012. She was also the founding director Liberal Arts Guild at LeTourneau University. Her research focuses upon twelfth-century Platonism and poetry, especially Thierry of Chartres and Bernard Silvestris.

Lesley-Anne Dyer Williams
Public Humanities Postdoctoral Fellow
Medieval Institute
University of Notre Dame

Albertson, David. Mathematical Theologies: Nicholas of Cusa and the Legacy of Thierry of Chartres. Oxford University Press, 2014. https://doi.org/10.1093/acprof:oso/9780199989737.001.0001.

Boethius. Boethian Number Theory: A Translation of the “De Institutione Arithmetica” with Introduction and Notes. Translated by Michael Masi. Studies in Classical Antiquity; v. 6. Amsterdam: Rodopi, 1983.

Boethius. The Consolation of Philosophy. Translated by Victor Watts. London: Penguin, 1999.

Burton, David M. The History of Mathematics: An Introduction. Dubuque, Iowa: Wm. C. Brown Publishers, 1988.

Caiazzo, Irene. “Teaching the Quadrivium in the Twelfth-Century Schools.” In A Companion to Twelfth-Century Schools, edited by Cédric Giraud, translated by Ignacio Duran, 88:180–202. Brill’s Companions to the Christian Tradition. Brill, 2019. https://doi.org/10.1163/9789004410138_010.

Calcidius. On Plato’s Timaeus. Dumbarton Oaks Medieval Library 41. Cambridge, Massachusetts; London, England: Harvard University Press, 2016.

Chenu, M. D. Nature, Man, and Society in the Twelfth Century: Essays on New Theological Perspectives in the Latin West. Chicago and London: University of Chicago Press, 1957.

Eco, Umberto. The Name of The Rose. Reprint edition. Boston: HarperVia, 2014.

Evans, Gillian R. “The Influence of Quadrivium Studies in the Eleventh- and Twelfth-Century Schools.” Journal of Medieval History 1, no. 2 (July 1975): 151–64.

Fassler, Margot E. The Virgin of Chartres: Making History through Liturgy and the Arts. Yale University Press, 2010.

Fournier, Michael. “Boethius and the Consolation of the Quadrivium.” Medievalia et Humanistica, no. 34 (2008): 1–21.

Gersh, Stephen. Middle Platonism and Neoplatonism: The Latin Tradition. 2 vols. Notre Dame: University of Notre Dame Press, 1986.

Martianus Capella. Martianus Capella and the Seven Liberal Arts. Translated by William Harris Stahl, Richard Johnson, and E.L. Burge. Vol. II: The Marriage of Philology and Mercury. 2 vols. Records of Western Civilization 84. Columbia University Press, 1992.

Masi, Michael. “Boethius and the Iconography of the Liberal Arts.” Latomus 33, no. 1 (January 1, 1974): 57–75.

Nicholas of Cusa. Nicholas of Cusa on Learned Ignorance: A Translation and an Appraisal of De Docta Ignorantia. Edited by Jasper Hopkins. Minneapolis: The Arthur Banning Press, 1985.

Oosterhoff, Richard. Making Mathematical Culture: University and Print in the Circle of Lefèvre d’Étaples. Oxford-Warburg Studies. Oxford: University Press, 2018. https://doi.org/10.1093/oso/9780198823520.001.0001.

Reydam-Schils, Gretchen. “Meta-Discourse: Plato’s Timaeus According to Calcidius.” Phronesis 52 (2007): 301–27.

Somfai, Anna. “Calcidius’ Commentary on Plato’s Timaeus and Its Place in the Commentary Tradition: The Concept of Analogia in Text and Diagrams.” Bulletin of the Institute of Classical Studies 47, no. Supplement_83_Part_1 (January 1, 2004): 203–20. https://doi.org/10.1111/j.2041-5370.2004.tb02303.x.

Somfai, Anna. “The Eleventh-Century Shift in the Reception of Plato’s Timaeus and Calcidius’ Commentary.” Journal of the Warburg and Courtauld Institutes 65 (2002): 1–21.

Stahl, William H. “The Quadrivium of Martianus Capella: Its Place in the Intellectual History of Western Europe.” In Arts libéraux et philosophie au moyen âge, 959–67. Actes du IVe Congrès internationl de philosophie médiévale. Montreal Paris, 1969.

Stahl, William Harris, Richard Johnson, and E.L. Burge. Martianus Capella and the Seven Liberal Arts. Vol. I: The Quadrivium of Martianus Capella. 2 vols. Records of Civilization, Sources and Studies 84. New York: Columbia University Press, 1971.

## Moral Self-determination and the Byzantine Christian Tradition

Though diverging with regards to detail, most historians of intellectual history would readily acknowledge that the advent of Christian antiquity coincided with a new concept of moral self-governance and, consequently, individual culpability.[1] Antique and medieval Christian thinkers cultivated a universal notion of ethical self-determination, affirming that all possess an inherent and unnecessitated capacity for the recognition and pursuit of the good regardless of one’s social upbringing or physical circumstances. A prima facie examination of these late antique and medieval Christian notions might seem to suggest many common features with post-Enlightenment and contemporary conceptions of moral autonomy, which emphasize self-legislation and independently-derived moral criteria. Nevertheless, a closer reading of these sources discloses a mindset that grounds moral self-determination in an ethic of co-governance, establishing the heteronomous “other” as an indispensable aspect of the quest for the good.

A significant exemplar of this “ethic of co-governance” can be found in the corpus of the early Byzantine monk, Maximus the Confessor (c. 580–662 AD), a figure revered by both eastern and western Christian traditions. Imbued with the spirit of the eastern ascetic tradition, the Confessor draws upon both monastic literature and the Hellenic philosophy of the Alexandrian intellectual tradition in order to synthesize his theological vision. Prominent among the doctrines prized by the eastern monastic tradition is indeed the idea that every rational agent possesses a free will, a notion that Maximus himself would also ardently defend and develop. Equally prominent, however, is the practice of “obedience” (hypakoē) to a spiritual guide or superior. This practice became an indispensable aspect of spiritual life in the eastern monastic communities that coalesced in the fourth and fifth centuries, and it remained a venerated feature of eastern monasticism through the end of the Byzantine era. Though not a central motif in his spiritual writings, Evagrius of Pontus (345–399 AD), a pioneer of eastern monasticism, is careful to exhort both male and female monastics living in community to attend to the words of their spiritual guides.[2]

The most well-known literary source providing an exposition of obedience is The Ladder of Divine Ascent, authored by John of Sinai (c.579–659 AD).[3] In the fourth chapter or “step,” John addresses the practice, defining it thusly: “Obedience is absolute renunciation of our own life, clearly expressed in our bodily actions…Obedience is the tomb of the will and the resurrection of humility.”[4] His endorsement of the renunciation of “will” may sound odd to many readers, especially given the Christian emphasis upon moral self-governance. Nevertheless, John is not denying the concept of free will as such, nor is he suggesting that the volitional faculty must atrophy into non-existence. Scholarly evidence suggests that the term John uses here for “will,” thelēma or thelēsis, comes to be associated with the volitional faculty in a philosophical sense in the writings of Maximus the Confessor, whose engagement with the Christological controversies of the seventh century provided the impetus for the standardization of the expression.[5] Thus, when John speaks of “will” and its denial, he is arguably referring to what Maximus the Confessor and his theological progeny would call gnomē, which in the idiom of the time refers to a private or particular disposition of will, or even to a personal opinion.[6]  John’s monk is not so much denying his own intrinsic freedom of will as he is seeking the co-governance and insight of those who are more advanced in virtue, and, through them, struggling to direct his volitional disposition such that it harmonizes with the other members of the community.

Maximus discloses a similar approach to moral self-determination by establishing his ethical teaching on “love” or agapē, which figures prominently in his philosophical and dogmatic treatises as well as his ascetic writings.[7] Agapē is no mere private sentiment but constitutes the impetus and ground for moral practice as a whole, thereby suggesting that moral judgment and orientation presuppose an awareness of one’s community and the persistent presence of a real, tangible “other.” In this way, Maximus retools an older Aristotelian paradigm, exchanging justice for love as the central and all-defining virtue.[8] Insofar as agapē is the chief virtue, narcissistic self-love, or filautia, is its inverse and the progenitor of all vice. As he demonstrates in one of his earliest works, The Ascetic Life, ascetic discipline should not be considered a private enterprise intended primarily for the sake of internal moral perfection.[9] Rather, its purpose is the effacement of filautia and the diachronic restoration of temporal and eternal relationships with the creator and one’s fellow creatures. To quote the Confessor directly: “He who is unable to separate himself from the passionate yearning for material things shall neither love God nor his neighbor authentically.”[10] Defining this activity in ontological terms, Maximus argues that divine love shall eschatologically gather together the fragmented portions of human nature into a functional unity, existing as a single mode in solidarity of will and disposition.[11] If love is the metaphysical impetus for the pursuit of virtue and the ground of morals, mimēsis or “imitation” is the pedagogical means by which it is recognized and acquired. Creatively appropriating and redeploying principles of Neoplatonic philosophy, the Confessor establishes the imitatio Christi, the existential imitation of Christ and his virtues, as the epistemological core of his ethics.[12] True followers of Christ imitate his mode of existence, disclosing through their lives and examples divine virtue. The lives and modes of these “exact imitators” are in turn imitated and imparted unto the morally immature.[13]

When viewed through a contemporary lens, we might say that Maximus’ view and the tradition that informs him entail the recognition of “autonomy”—as we would construe it now—as the point of departure for human agency. However, the ideal of agapē calls for the voluntary sacrifice of autonomous moral space for the sake of moral co-governance and a reciprocal unity of wills, which depends upon the concrete example of Jesus Christ and his “exact imitators.”

Demetrios Harper
Byzantine Studies Post-doctoral Fellow

[1]This is strongly reaffirmed by Kyle Harper (From Shame to Sin: The Christian Transformation of Sexual Morality in Late Antiquity[Cambridge, Mass: Harvard University Press, 2013], 80-133), who objects to Michael Frede’s assertions that the concept of free will is not unique to the Christian tradition but can, in fact, be attributed to Epictetus. See Frede’s A Free Will: Origins of the Notion in Ancient Thought, Sather Classical Lectures 68, ed. A. A. Long(Berkeley: University of California Press, 2011), pp. 66-88.

[2]See The Two Treatises: To Monks in Monasteries, and Exhortation to a Virgin, in Evagrius of Pontus: The Greek Ascetic Corpus, trans. Robert Sinkewicz (Oxford: Oxford University Press, 2003), 127-28, 131.

[3]These dates are based on what still remains tentative conjecture. Cf. Alexis Torrance, Repentance in Late Antiquity: Eastern Christian Asceticism and the Framing of the Christian Life c. 400-650 CE (Oxford: Oxford University Press, 2013), 158-60.

[4]The Ladder of Divine Ascent 4.3, revised edition, trans. Lazarus Moore (Boston, MA: Holy Transfiguration Monastery, 1991), 21. For the original text, I consulted the Κλίμαξ, in Ἰωάννου τοῦ Σιναΐτου ἅπαντα τὰ ἔργα, Φιλοκαλία τῶν νηπτικῶν καὶ ἀσκητικῶν πατέρων 16, ΕΠΕ, Ἐλευθέριος Μερετάκη (Θεσσαλονίκη Πατερικαὶ Ἐκδόσεις Γρηγόριος ὁ Παλαμᾶς, 1996).

[5]John D. Madden is among the first to argue for the originality of Maximus’ contribution to the genealogy of the concept of will. Cf. his “The Authenticity of Early Definitions of Will (thelēsis)” in Maximus Confessor: Actes du Symposium sur Maxime le Confesseur, Fribourg (2-5 Septembre 1980), eds. Felix Heinzer and Christoph Schönborn (Fribourg: Éditions Universitaire Fribourg, 1982), 61-82. Madden’s “originality thesis” is defended by David Bradshaw, St Maximus the Confessor on the Will, in Knowing the Purpose of Creation Resurrection, Proceedings of the Symposium on St Maximus the Confessor, ed. Maxim Vasiljević (Alhambra: Sebastian Press, 2013), 143–58 For an up-to-date and comprehensive overview of Maximus’ view, see Ian McFarland, “The Theology of Will,” in The Oxford Handbook of Maximus the Confessor, eds. Pauline Allen and Bronwen Neil (Oxford: Oxford University Press, 2015), 516-32.

[6]Ian McFarland, “The Theology of Will,” 520-522. Cf. for the context and background of “will” and its correlative expressions in Maximus, cf. Paul Blowers, Maximus the Confessor: Jesus Christ and the Transfiguration of the World (Oxford: Oxford University Press, 2016), 156-65.

[7]Cf. Maximus’ Four Hundred Texts on Love, in The Philokalia, eds. and trans. Kallistos Ware et al., vol. 2 (London: Faber and Faber, 1981), 48-113; Letter 2: On Love,in Maximus the Confessor,The Early Church Fathers, trans. Andrew Louth (Abingdon: Routledge, 1996), 84-93.For a systematic account of Maximus’ aretology and its foundations, see Demetrios Harper, Chapter 4, The Analogy of Love: St. Maximus the Confessor and the Foundations of Ethics(Yonkers, NY: St. Vladimir’s Seminary Press, 2018).

[8]See Maximus’ Quaestiones ad Thalassium I 40.60-70, Corpus Christianorum, Series Graeca 7, eds. C. Laga and C. Steele (Turnhout: Brepols, 1980), 269-71.

[9]Liber asceticus 100-115, CorpusChristianorum, Series Graeca40, ed. P. Van Deun (Turnhout: Brepols, 2000), 17. Cf. also the introduction to the Quaestiones ad Thalassium I 380-390, 39-41.

[10]Liber asceticus 100-110, 17. The translation is mine.

[11]Letter 2: On Love, 88.

[12]Cf. St. Maximus the Confessor’s Questions and Doubts III, 1, trans. Despina Prassas (DeKalb: Northern Illinois Press, 2010),156-57;Ambiguum 48.6,in On Difficulties in the Church Fathers II, Dumbarton Oaks Medieval Library 29, ed. and trans. Nicholas Constas (Cambridge, MA: Harvard University Press, 2014), 218-20.

[13]Liber asceticus 635-665, 73-74.