Kaitlin Moore
Prof. Kevin Brownlee
ENGL 323: Dante
8 May 2018
Navigating the "Mighty sea of being":
The Physics of Paradise and Dante’s Cosmology
Dante's ascent through Paradise, culminating in his gazing upon the Empyrean, presents a paradoxical model of the universe. While Dante employs Ptolemaic geocentrism –– with each of the heavenly spheres occupying distinct annuli in ever-increasing concentric orbits around the Earth –– he expands the paradigm to include Non-Euclidean spatial geometries and nonlinear temporal mechanics, where eternity is at once all of time and outside of time, and boundless space is bound by the singularity of the Empyrean. The cosmological mechanisms of the Tenth Heaven –– made evident through accounts of the kinetic motion of the Primum Mobile as well as the limitations of both human sense perception and the Poet’s linguistic capacities –– necessitate the subjugation of all events or actions to God, which in turn calls into question the capacity for human agency and free will. This essay will serve as a brief survey of Dante's cosmology, as well as examine the implications for God and the Divine in a model that transcends dimensions of space and time.
Like Hell and Mount Purgatory, Paradise is structured as a sequence of concentric orbits that share the same center. In the former two divisions of the cosmos, each annulus –– or region between two concentric circles of different radii –– corresponded to a circle of Hell or a tier of Purgatory. In Paradise, each annulus corresponds to an area swept out by a planet. Therefore, Dante’s universe is Ptolemaic in design, a geocentric model with Earth at the center, and the Sun, Moon, planets, and sphere of fixed stars all orbiting the Earth. In his Convivio, Dante suggests that there are nine concentric annuli of Heaven, each nested inside the other in a hierarchal organizational structure:
The order of the heavens is as follows. The first is that of Mars; the second Mercury; the third Venus; the fourth that of the Sun; the fifth Mars; the sixth Jupiter; the seventh Saturn; the eighth that of the fixed stars; the ninth being… the Crystalline sphere, that is one which is diaphanous or completely transparent. Moreover, beyond these… the Empyrean, which is to say the heaven of flame, or luminous heaven.
According to the Ptolemaic model, the further Dante travels from the Earth’s surface, the larger the radii of each spherical annulus will grow to accommodate the preceding orbit, and thus the more massive the area will be. By this logic, the Empyrean, the final Heaven, ought to be infinitely huge. However, instead of getting bigger, as Dante ascends, the fiery rings of the angelic orders grow smaller, approaching a singularity. The breakdown of the Ptolemaic model within the Empyrean demonstrates Dante’s poetical redaction of classical cosmology into a Christian framework –– in Dante’s conception, God circumscribes the world and yet occupies an infinitely small annulus –– "one individual light / The whole." The Empyrean, Dante suggests, is both a point and the outermost of the heavenly spheres, singular and infinite, which violates the laws of classical geometry.
According to Euclid, increasing the radius of a sphere increases its size and thus the volume of space the sphere occupies. This intuitive axiom is subverted, however, in Canto 28 when Dante writes:
Therefore the circle, whose swift course enwheels / The universal frame… Thus by the virtue, not the seeming breadth / Of substance, measuring, thou shalt see the Heavens, / Each to the intelligence that ruleth it, / Greater to more, and smaller unto less, / Suited in strict and wondrous harmony.
As Hell was stratified according to the direct correlative relationship between the gravity of a sin and a soul's proximity to Satan, in Heaven, the closer the blessed are to God, the greater the degree of divine perfection. Like a set of Matryoshka dolls, each heavenly sphere is designed to nest inside one another. In referring to the "enwheeling" of the Heavens as a function of "virtue" rather than "seeming breadth," Dante the Poet equates the hierarchal structuring of the spheres to degrees of divine perfection rather than an annulus's overall dimensions or magnitude. Dante writes, "Thus they to different havens are moved on / Through the vast sea of being, and each one / With instinct given, that bears it in its course," suggesting that the concentric circles of Heaven are structured such that everything and everyone in the universe has its place relative to God.
While the Empyrean violates the axiomatic principles of spatial geometry, the speed of the singularity nevertheless obeys the laws of circular motion. The law of the conservation of angular momentum holds that the smaller the radii of a rotating body, the faster the spin. As his narrative surrogate gazes upon the Empyrean, Dante writes: "there wheel’d about the point a circle of fire, / More rapid than the motion which surrounds, / Speediest, the world." Since the spin of Empyrean is infinitely fast, physics maintains that the circles must contract the closer they are to God. Furthermore, the conservation of angular momentum suggests that size and speed share an inverse ratio, therefore a point singularity would possess immense speed –– until a stroboscopic effect occurs whereupon continuous motion becomes indistinguishable from continuous rest. In the Convivio, Dante describes a stroboscopic Empyrean with a "speed… almost incomprehensible" where "stillness and peace are the qualities of that region of Supreme Deity, who alone beholds himself entire." Beyond the Primum Mobile, which imparts kinetic motion to all the subordinate Heavenly spheres, inertia breaks down, and motion and rest become one and the same. Dante’s Tenth Heaven, therefore, is a shape that does not abide by the classical geometry of Euclid, but obeys the angular momentum laws of mechanical physics.
The Non-Euclidean space of Empyrean finds an equivalent in contemporary studies of the expanding universe. This relationship between size, distance, and motion can be likened to Hubble's Law, which, in addition to maintaining the conservation of angular momentum in an inflationary universe, expresses velocity as a function of distance –– in other words, nearby objects move slowly, and more distant ones move rapidly. Hubbelian expansion carries a geometric corollary not dissimilar to Dante's, in that kinetic motion of both the Empyrean and an inflationary universe in a modern cosmological conception necessitates their being singularities at the beginning of time. Essentially, since the universe is getting bigger, Hubble's Law holds that the entire cosmos grew from an infinitesimally small point. If light-years can be likened to distance in a spacetime construct, the greater the distance measured, the further back in time the image, and therefore the smaller the size of the universe. As the radius of the universal sphere pushes closer and closer to the beginning of time, Hubble's Law suggests that the universe occupies smaller and smaller regions of space, despite the fact that those same regions circumscribe all of creation, in all directions of the sky. When Dante's guide, Beatrice, returns to her place in Heaven, she appears to Dante both infinitely far away and infinitely close –– "not from the centre of the sea so far / Unto the region of the highest thunder, / As was my ken from hers; and yet the form / Came through that medium down, unmix’d and pure." In that moment, boundless space is bound inside the spatial distance from Dante to Beatrice. This relationship does not necessitate a direct correlation between distance and size, unlike the Ptolemaic model of the universe.
The Non-Euclidean nature of Dante's Heaven carries significant spatial implications, which in turn inform the temporal mechanics of the Empyrean. Time, Dante suggests, is not an absolute measure uniformly applicable to all entities. Rather, its passage is relativistic, dependent upon one’s proximity of God. In Canto 29, as Dante the protagonist traverses the annulus of the Primum Mobile, Beatrice describes God as "Inhabiting His own eternity, / Beyond time’s limit or what bound soe’er / To circumscribe His being." Linear time –– that is, chronological time, or the isomorphic, unidirectional relationship between cause and effect –– does not exist for God, because He occupies a condition of ever-present eternity. Dante Scholar Alison Cornish cites Thomas Aquinas in suggesting that "an angel… can understand more or fewer things in a single glance, depending on its rank in the hierarchies." Therefore God, to whom all angels are subordinate, embraces infinite lengths of past and future, and views in direct comprehension everything as though it is taking place in the present. When Dante relates time and motion in the Convivio, he posits that the Primum Mobile determines the natural operation of the universe and serves as both the physical source of motion and the mechanical impetus for time. God, not bound by the Primum Mobile, occupies what Cornish refers to as a "single mathematical point, an instant of no duration." Therefore, the Empyrean, the point singularity circumscribing the Primum Mobile and all lower annuli of the Heavens, like God Himself, exists outside of time.
Due to its being bound by a single "mathematical instant," Cornish refers to a number of temporal and spatial "paradoxes" inherent to Dante's account of Heaven. It is an infinite point that nevertheless encompasses all of creation; a stroboscopic singularity with a speed so great it becomes synonymous with stillness; a realm that exists beyond time and yet serves as the progenitor for time. A type of Non-Euclidean topographical structure that reconciles these paradoxes, however, is a shape called a 3-sphere –– an enclosed, finite manifold that contains three spatial dimensions that form the boundary of a sphere in four dimensions, circumscribing height, width, and depth as well as time. Of note, however, is how the Non-Euclidean nature of the 3-sphere allows for any loop, or circular path, to be continuously shrunk to a point without leaving the surface. Any orbits carved out within a 3-sphere create simultaneously infinite regions in higher dimensions and finite singularities in three dimensions. The 3-sphere contains parallels and meridians similar to lines of latitude and longitude on a sphere, but the hypermerdians necessitated by mathematics would appear to human beings –– human beings like Dante –– as dots on a circle appearing and disappearing with variable degrees of separation. If one were to superimpose Empyrean over a 3-sphere, "all nature hangs upon" a series of single "point[s]."
Just as a two-dimensional shape cannot perceive a three-dimensional structure –– a stick figure on a piece of paper, for example, when viewing a sphere, would discern only a circular cross-section –– a fundamental disjunction of form and configuration exists between beings that inhabit a 3-sphere and mortal, three-dimensional souls like Dante's. This disparity is evidenced when Dante the Protagonist observes: "For Beatrice, she who passeth on / So suddenly from good to better, time / Counts not the act." Unlike Dante, who must progress sphere to sphere –– from the "good" annuli to the "better" annuli, or closer to God –– in sequence, the journey from one heaven to another takes no more than a moment's time for those like Beatrice. Just as depth, a three dimensional property, does not factor into two-dimensional space, so too does time not constitute a boundary of constraint in the Empyrean's 3-sphere. This gives the illusion of a fluctuating heaven, where the blessed can come and go as they please because they are not bound by the same spatial and temporal constraints as Dante. The timelessness and extra-dimensional space of the 3-sphere implies that the living and the dead are not separated by time or circumstance, but rather by perspectival limitations caused by a Non-Euclidean-realm superimposing itself over the finite dimensions Dante can perceive.
A fundamental issue concomitant with this Non-Euclidean geometry rests in the lack of visual correlatives –– in essence, humans cannot form a mental image of four-dimensional space, and thus the Empyrean defies codification by any illustrative or even linguistic means. While Dante the Protagonist, by virtue of Beatrice's intervention, herself being an inhabitant of the Empyrean, expands his capacity to see in all senses, Dante the Poet in retrospect finds himself wrestling with the inherent limitations of human language in describing God's Heaven. The sights of Paradise, the temporal and spatial antinomies, are too complex to be reconciled by the mortal mind. Dante writes that so "divine a song, that fancy's ear / records it not; and the pen passeth on," and admits that even the mind, the "inward shaping of the brain," does not have the "colours fine enough to trace such folds." The task of illustrating Heaven by any poetical means is beyond Dante's scope, and by describing the souls of the blessed as "folds," their physicality fugues into ambiguity. The vision is too strange, unfamiliar, and seemingly self-contradictory for Dante to arrange into a systemic survey by means of his poetry. The Poet admits his shortcomings in declarations of his limits of language.
As Dante the Poet struggles to transcribe Heaven, Dante the Protagonist struggles, with his limited three-dimensional perception, to reconcile God's eternal omniscience with free will. While Beatrice corrects for Dante's inability to perceive physically the Empyrean –– strengthening his eyes and ears –– Dante the Protagonist retains his inability to perceive causally God's relationship to time and destiny. From Dante's perspective, the divine foreordaining of all that will happen threatens the capacity of human agency, especially in regards to the concept of forgiveness and salvation. Beatrice summarizes Dante's doubts in reiterating that "each soul," in death, is "restored to its particular star" and in some regards the "influencing of these orbs revert / The honour and the blame in human acts," suggesting that humans cannot be held morally responsible for their actions. Like the locomotive force of the Primum Mobile imparting motion to the spheres of Heaven, the "influencing" of God shapes each man’s fated life. However, Beatrice goes on to argue that what can be understood as determinism to "the eye of man" is consequent of Dante's incapacity to behold the whole system of time and space within which God operates, where He dwells in eternity and is copresent with the entirety of time. What would appear to mortals as fixed fate, predestination, would to God in this external singularity be simple intelligences of observable phenomena. Therefore, His sovereignty can be equated to His presiding within the Empyrean, a place outside time, where past, present, and future –– and, by extension, the consequences of each human action –– are simultaneously beholden.
In Canto XXXIII, Dante begs God for a memory of the Tenth and Final Heaven so that he can convey its glory in his poetry. The Empyrean shifts to a vision of three interlocking circles, bounding within its limits not only the entire physical universe, but a spiritual domain otherwise inaccessible to the human senses. Dante's "wheel" circumscribes dimensions both small and large, still and stringent, human and Divine. In his journey, Dante the Protagonist sought truth in the enlightenment of God's love. In many ways, the climbs of modern astronomers and astrophysicists up the mountains housing their telescopes are single-minded treks towards places where their shutters, when left open to the sky, record the wheeling of the stars as a pattern much like Dante’s three concentric circles. In that regard, their searches, like Dante's, become searches for simplicity, a shedding, a sloughing off. Not just of light and atmosphere, but of distortions invisible to the eye. The Empyrean is not only a portrait of an astonishingly modern cosmology, but a means by which Dante the Poet can reconcile Divine Love with the complex forms and wonders of the universe. Works Cited
Alighieri, Dante. Paradiso. Translated by Henry Francis Cary. P.F. Collier & Son, 1967.
–––––––––. The Convivio (The Banquet). Translated by A. S. Kline, 2008.
The Bible: New International Version. Hodder & Stoughton, 2012.
Cornish, Alison. "Planets and Angels." Reading Dante's Stars. Yale University Press, 2000: 119–142.
Kirshner, Robert P. "Hubble’s Diagram and Cosmic Expansion." Proceedings of the National Academy of Sciences, vol. 101, no. 1, 6 January 2004: 8–13.
Weeks, Jeffrey R. The Shape of Space. CRC Press, 2001.