In this detailed and fascinating study David Albertson brings to light how Neopythagoreanism survived and prospered from Hellenic prehistory to the end of the fifteenth century, when mathematics began to split off from theology--for better and worse. Heroes of his story are the unlikely and almost submerged figures of Thierry of Chartres in the twelfth century and Nicholas of Cusa in the fifteenth.
Albertson not only makes sense of how mathematics figured in their theologies, he demonstrates that we cannot really grasp those theologies without understanding their mathematical themes. Moreover, his work is a tour de force of source criticism, letting us see just how understanding an author's sources (even those he did not totally understand) can make sense out of texts that were seldom considered as so dependent on previous sources. The result is a book worth careful consideration and study because of its proposals about Thierry and Nicholas. The review that follows can only touch on the highlights of Mathematical Theologies.
What is Christian Neopythagoreanism and where does it come from? Albertson returns to its Hellenic prehistory as universal mathematical science followed by its transformations in Plato and the early academy where mathematicals mediate between Forms and particulars. In the middle Platonists Eudorus and Moderatus he finds a transcendent henology, a theology in search of mediations of the Divine One. With Nicomachus of Gerasa there arrives a number ontology that integrates the four mathematical sciences of the quadrivium. Albertson describes this historical mathematical theology as: "(1) a species of Neopythagorean henology, (2) oriented around the coeval mediation of Logos and Arithmos, which assumes (3) a universal mathesis grounded in the quadrivium" (58). Such a summary cannot do justice to the insights the author brings to his survey of the fragmentary details of the various authors' ideas and proposals.
His detailed review of late antique Neopythagoreanism then moves to Iamblichus and Proclus among others and concludes with the early Augustine and Boethius. Both pagan and Christian thinkers ultimately strained out the Neopythagorean elements in their sources, focusing on Logos over Arithmos. Early on Augustine saw God as "Number without number"--a kind of supreme number, yet he finally turned to the Verbum as mediator. Most importantly, Boethius recast Nichomachus' views on number into the quadrivium, only to partition it off from his theological speculations. Instead of being the supreme Decad, God is the supreme mathematician.
In the twelfth century Thierry of Chartres marks the re-introduction of Neopythagorean thought into western theology. Albertson first singles out Bernard of Chartres' gloss on Plato's Timaeus that "rediscovered the autonomy of arithmetical mediation" (102) by combining Platonic mediation with the mathesis of the quadrivium. This set the immediate context for Thierry's commentary on Genesis (Tractatus de sex dierum operibus) where he uses the quadrivium to interpret the Christian doctrine of the Trinity. Using Augustine's triad of unitas, aequalitas, concordia Thierry proceeds to identify unity with divinity so that the arithmetical principles governing unity can be applied to divinity. Unity creates the numbers that participate in it, just as creatures depend on their Creator as source and sustenance. Aequalitas is simply unity multiplied by itself and thus the laws of mathematics apply to the unity of Father and Son.
Thierry then uses aequalitas existentiae to cover the self-identity of every being and connects this to Augustine's mathematical order of creation. In this way Augustine's triad and the Christian understanding of Trinity and creation and can be integrated with Boethian quadrivial mathematics. As the source of the quadrivium, God "as Creator is perpetually numerating" (115).
Thierry develops his mathematical theology in three further works that deal with Boethius'De Trinitate, building on his work in the Tractatus. To explain creation from mathematical principles he needed a way to organize physics, mathematics and theology in a single vision. Albertson takes the differences in Thierry's commentaries as a diachronic development with Thierry moving from the mathematical trinity to a full theology of the quadrivium, and moving from psychology to a proposed ontological hierarchy. Thierry proposes his signature conception of reciprocal "folding" (adapted from Boethius' Consolatio. He uses the concept of folding to mediate between Creator and creation and to pull together the disciplines of theology, mathematics and physics.
In Lectiones Thierry puts together four modes of being, hierarchically structured in terms of enfolding/unfolding from divine necessity to material possibility. God's oneness "enfolds" the whole universe and the cosmos marks unity's "unfolding" into plurality. There are two parallel orders that unfold the extremes: actuality and possibility. God's oneness is unfolded in the order of actual beings, while absolute possibility is unfolded into determinate possibility. Each of the four modes is a way of seeing the sciences: theology for God's unity, mathematics for that unity unfolded. Physics considers both modes of possibility.
The crucial second mode, necessitas complexionis, enables Thierry to define cosmic order as the unfolded enfolding of number in God: the creation of number is the creation of things since God is "Number without number." Mathematics stands to theology as the unfolded to the enfolded. Albertson argues persuasively that this necessity is not God's absolute necessity but the necessity within the seriality of number unfolded from God--retaining and expressing "the necessity of the original enfolding" (133). Albertson summarizes in four steps his detailed exposition: 1) Thierry looked to number as the ground of mediation between God and world, 2) he selected the quadrivium as a vehicle for henological theology, 3) he moves from the arithmetical Trinity to theorizing the quadrivium in the concept of necessitas complexionis. The question remaining is whether the eternal exemplary forms are to be conceived as numbers or as the divine Word.
Thierry finished his work at an inopportune time--just as Aristotle was recovered in Western Christendom. Those few who took up his ideas often separated their parts. Albertson discusses the anonymous De septem septenis, Clarembald of Arras' writing, and the recently discovered anonymous Fundamentum naturae (actually an attack on Thierry) as three works that take up his ideas as a whole. Yet the first treatise does not understand mediation, the second modified what was radical about Thierry to bring him closer to Augustine, and the third Aristotelian author rejects Thierry's theory of four modes as misguided Platonism. All three overlook Thierry's search for divine mathematical activity within creation and his stress that God is beyond creatures. None of the three understood entirely what Thierry was up to, but they all influenced Nicholas of Cusa.
Nicholas Cusanus is the most important later medieval thinker to use Thierry's ideas in key early and later works. His initial masterpiece, On Learned Ignorance, is here celebrated as authentically Neopythagorean since he used his Chartrian texts and counted the anonymous Fundamentum as one of them in order to experiment with Thierry's concepts of the four modes and reciprocal folding, adding in the Aristotelian contrast of absolute and contracted. (Albertson's use of the recently discovered Fundamentum is particularly helpful. He sees Nicholas as counting it among his Chartrian texts, even to the point of reproducing passages verbatim in book two of On Learned Ignorance. Cusanus scholars have mostly dismissed this copying or else cast doubt on Nicholas' originality in his masterpiece. Albertson's creative response manages to be conciliatory while casting more light on the book itself.) Albertson makes a persuasive case that the first two books of On Learned Ignorance have a parallel structure: prelude, exposition, discussion of source material, Cusan development, and postlude. In this way book one expounds on the maximum to introduce the arithmetical Trinity, and book two takes up contraction to examine Thierry's four modes of being.
This parallel order gives new meaning to aequalitas, with God as absolute, pure equality, while the measurements of the quadrivium fail to give precise results for all else that is contracted. Yet this very inexactness points to traces of God's identity and equality in God's creatures--illustrating learned ignorance. While Thierry used "enfolding/unfolding" to organize the four modes of being, Cusanus applies them to Creator and creature as well as to organizing the whole quadrivium. All folds are embraced by the one divine Enfolding that is at once unity, equality and maximum. The best way to understand "contraction" is via mathematics--as the infinite unfolding of unity into numerical singularities. Nicholas envisions the second mode of being, necessitas complexionis (the necessity of enfolding), as mediating contraction between divine aequalitas and creatures via the divine Word.
In book three of De docta ignorantia, Nicholas links this second mode with aequalitas and the divine Word. He uses two other concepts, explicatio and medium, to make Christ the axis of cosmic procession from and return to God. The eternal Word may be absolute, but the incarnate Word is the medium that connects absolute and contracted. Thus Christ unites the aequalitas essendi and the universal contractio, and reconciles the age-old tension between Logos and Arithmos. The Word enfolds the forms of all contracted things, connecting absolute and contracted (and thereby holding together the structure of the three books of On Learned Ignorance). Albertson emphasizes the originality of this Cusan Christology.
Nicholas' second lengthy treatise, De coniecturis, returns to his source materials and reworks them as a system of four unities: absolute divine unity is contracted to the universe of three lesser unities: the decadic root, its square, and its cube. What constitutes each unity is reciprocal enfolding and unfolding, so that the second unity (the primordial decade) is the mediator of divine unity to the rest. Albertson sees Cusanus as testing a different combination of Thierry and the Fundamentum treatise in "an integrated henological vision" even more Neopythagorean in character.
The 1450s and especially Idiota de mente find Nicholas displaying a deeper integration of Christian theology and Neopythagoreanism. Nicholas returns to Thierry's original account of the four modes and explores the necessitas complexionis. His account of created mens stands above all lesser unfoldings as an "image" of divine enfolding. As God's image and not an unfolding the human mind reflects divine enfolding as a unique mediator. Nicholas links mind as mensura with mind as second mode and thus combines the mathematical nature of mens with the imago Dei tradition. The mind's self-measurement is an image of God's mathematizing self-measure. In c.6 of De mente there occurs a theology of number that reflects the ascent of mind through the four modes. There is a radical intimacy of divine Number hidden within human number and once God is seen as innumerable Number, Arithmos is no less immediate to God than Logos.
Nicholas' lesser known treatise De theologicis complementis (1453) is a theological meditation on geometry. "God measures by measuring Godself and God creates by using Godself as a geometrical instrument" (247). The preeminent geometer is likewise the preeminent geometry. God is naturally number without multiplicity and magnitude without quantity. Cusan geometrical theology is visual and ecstatic in the sense that God's self-measuring is primarily self-seeing and God's self-seeing in the Word visualizes the world into being. And this theology is also ecstatic since Nicholas employs "rapture" to show how God intervenes to seize the human geometer with a gift of ecstasy.
In the 1460s Nicholas takes up the theological status of possibility as an equally valid divine name with actuality. Trialogus de possest introduces motion into geometrical space as Nicholas turns to "physics," matter and motion, the concrete and tangible, the fourth art of the quadrivium: solids in motion. Here Nicholas characterizes mathematical concepts as aenigmata; for Albertson these are "visual symbols that participate in the mathematizing divine reality that they signify" (260).
The final fruit of Albertson's investigations of Nicholas' sources appears in borne by his final chapter about Nicholas' lengthy and often puzzling De ludo globi. This chapter shows Albertson's interpretive powers at their strongest in service of the mathematical and Christological in Cusan thought. The game with the ball stands as a Christological figure, an image of varied human attempts to follow Christ's exemplary ball to the center, and as well a map of the movement in the universe that circles Christ as the sun, the coincident center of rest and motion. What Nicholas has done is make Christ a spatial medium, uniting the centers of Creator and cosmos.
Nicholas' curious discussion of coinage in book two of De ludo globi imagines a divine mint where the valuable coins are stamped with the face of Christ, the image of the Father, the coinmaker. Here Albertson sees Nicholas putting together divine form as image and divine form as value, using the metal's value and the image stamped upon it. Moreover, Nicholas collects and recalls in this work all the Chartrian doctrines discussed in earlier works, now linking more closely the quadrivium and the Incarnation.
For Albertson this late dialogue intensifies Nicholas' tendency toward geometrization by "coordinating multiple figures within an integrated spatial field" (267). Each of three worlds--God, cosmos, human soul--is repeated in each book of De ludo globi: book one shows the game in motion, book two the game at rest. Instead of geometrical figures that can be extracted, as in his earlier works, here Nicholas extends the geometrical image to encompass the whole work and makes Christ a geometrical center that integrates the co-sphericity of God, cosmos and soul. The dialogue also stands as a memoir of Nicholas' earlier encounter with Chartrian texts and Fundamentum's criticism. Both parts recall his borrowing of the four modes and present them anew in the passage about coinage.
Finally, Albertson finds in De ludo globi a reduction of quadrivial antinomies each mediated by Christ: from motion (spherics) to rest (geometry), from figure (magnitude) to number (multitude), from relative number (harmonics) to absolute unity (arithmetic). In this way Nicholas achieves a new harmony of components never united in his earlier works. He now makes the Incarnation the basis of the mathematical order of the cosmos rendered visible in the quadrivium. Logos and Arithmos are finally complementary.
I have left aside the splendid introduction and final coda of this thought-provoking book and focused on its excellent analysis of the history and the texts. Albertson proposes that his study shows how it is possible to put together mathematics and theology in a fashion where the walls we are familiar with between them never needed to be built. In that way, it reminds us of new possibilities for contemporary theology to explore. We may hope that Albertson's next project can show more concretely what this theology might amount to. In the meantime his gracefully written and telling study should help us return to Thierry of Chartres and Nicholas of Cusa with new understanding and deeper appreciation.