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Kepler's allegory of containment, the making of modern astronomy, and the semiotics of mathematical thought.

That the German astronomer Johannes Kepler (1571-1630) is an allegorist comes as no surprise to anyone who has encountered his cryptic Somnium, a text largely finished by 1610, added to for some twenty years, and published posthumously in 1634 by Kepler's son Ludwig. A clever book which speculates about the astronomical perspective of dwellers on the moon, the Somnium seu Astronomia Lunari, as it was fully titled, summed up Kepler's lifelong interests in dream lore, symbolism, demonology, the mystical value of geometry, and medieval literature. A resolute believer, along with Galileo or Newton or Boyle, that the physical cosmos was the script of God, Kepler was steeped as well in classical literature even in his adult, professional life. When he arrived in the city of Graz in 1594 to teach arithmetic to the adolescent sons of the well-to-do at the Protestant Stiftsschule, he also taught rhetoric and the poetry of Vergil because his mathematical lectures were unpopular and undersubscribed (Caspar 56-57); likew ise, his letters seem peppered with intense, often overwrought mythographic metaphors. (1) As Graz school authorities in charge of Kepler's teaching assignments claimed and as Kepler biographer Max Caspar quotes, "mathematics is not everyone's meat" (56). But Kepler also had tropes, figures, and myth in his very bones.

My object in this essay is to place Kepler within the genealogy of the great European allegorists, the genealogy containing Martianus Capella, Bernardus Silvestris, Dante, and Chaucer. A consummate practitioner of literary allegory, the great astronomer crafted one of the most compelling otherworldly journeys in literature. But his allegory, I believe, captures far more than the hermeneutically encrypted laws of the new physics or the heliocentric astronomy. Kepler's Somnium encapsulates or incorporates topoi from a number of his theoretical writings, proffering what can be thought of as a narratology or "allegory of containment," an imagined geometry both figural and literal, textual and thematic. At the same time, it conducts a parable of cognitive or meta-discursive self-reflection. That is, the Somnium yields itself up best, first, when studied through contemporary narratological technique--with special focus on the narratological master trope of embedding or framing--and second, once it is understood as an allegory not only about speculations concerning space and heavenly bodies but also about the very psychic or cognitive mechanisms by which the mathematical or scientific investigator could be said to produce mathematics and science. The narrative structure of the Somnium is self-reflexive, furnishing a relentless "narrative of narrative" (Williams 24). With all of allegory's attendant tropes and figural mechanisms, the text reflects upon mathematical cognition itself; a cognition formalized eventually by the American semiotician C. S. Peirce and his twentieth-century successor, Brian Rotman--both of whom, I will argue, are prefigured in Kepler's many-leaved allegory of science and narrative.

My highlighting of Kepler's "allegory of containment" will thus bypass his major contributions (the Astronomia nova or the Harmonice mundi) and focus on some minor texts, such as the Somnium--perhaps the last of the cosmic allegories of the Latin Middle Ages--as well as the early Mysterium cosmographicum (1596) and Nova stereometria (1615). Whereas the Mysterium tends to be disparaged in the positivist sketches of philosophy or history of science because of its earliness and its unabashed abstract Platonism, (2) the Nova stereometria or Stereometria Doliorum Vinariorum ("The Measurement of Wine Casks") receives praise for its role in the advancement of seventeenth-century mathematics. Yet both brief texts thematize an allegory of geometry, a process that finds reciprocal representation in the geometry of allegory presented in Kepler's sensuous and concrete narrative of the Somnium.

Kepler's Somnium has often been thought of as the first work of modern science fiction. It is a juncture text, marking the start of diverging paths including speculative science fiction, modern astrophysics, and psychological fantasy. To say that it is a dream vision in which the narrator grasps radical scientific knowledge is to sell it short. It does stand in the tradition of Artemidorus' Oneirocritica, Cicero's dream of Scipio, Macrobius' commentary on that dream, and Chaucer's The House of Fame, a famous Middle English allegory which likewise involves the dream of an ascent into aetherial space punctuated by a visit to an insula, a planet-like "island," along with a lengthy lecture given by a personified magister (a talking, golden eagle) who imparts the secrets of physics, particularly what was known in the fourteenth century about acoustics and wave phenomena (Chaucer 348-73). But Kepler's Somnium is, literally and figuratively, wrapped in shells and layers of narratological complexity that render comme ntary on it, let alone mere summary of it, far more difficult.

Kepler begins the Somnium with the claim that he's been reading lots of Bohemian history because of a political dispute in local lands. He falls asleep one night while reading about a heroic woman from the Middle Ages, Libussa, and he recounts how, in a dream, he buys an old book at market which contains the personal account of an Icelandic astronomer named Duracotus. But Duracotus, after recounting his own early life with his mother Fiolxhilde, a sorceress, and a brief training period received in adolescence from the historical Tycho Brahe (Kepler's own true idol), repeats a long lecture told him by an unnamed Daemon who travels back and forth regularly to the moon, which is called Levania by its inhabitants. (Lebana is actually the Hebrew name for "moon," as Kepler notes.) In the lecture, the daemon ex Levania covers daemon-assisted transit by humans to Levania, lunar geography, the relative celestial mechanics and optical perspectives of Levania and Earth (called Volva by the Levanians), and the biologies of Levania's two zones, one lighted and one dark (Subvolva and Privolva, respectively). Finally, Kepler awakens suddenly, torn away from the Daemon's sermon and the book that contains it when wind and rain blast through his bedroom window. Add to this the most striking feature of the Somnium: from 1620 through 1630 Kepler added over 200 endnotes, some a couple of lines long and others a page or two in length, to his knotty narrative. These notes, layered with digressions, citations of learned and mythological texts, and adorned by geometrical diagrams and mathematical proofs and tables, purportedly consummate the self-confessed "allegory" of planetary and celestial mechanics furnished by the Daemon.

That this text would appeal to any student of allegory is evident from the start: Kepler again and again calls the text an allegory, since he's well aware that somnium doesn't just mean "dream" but an allegorical dream. (3) He also supplies, among his crowd of endnotes, frank and confident exegeses of many, many elements from among all the levels of narrative: Duracotus is "Science" at one point, while his mother Fiolxhilde is "Ignorance" and his unknown, unnamed, and long-dead father is "Reason" (endnote 10; GW11.2:334) (4); the nine chief daemons (novem praecipuum sapientissimi spiritus) with whom Fiolxhilde communicates are both the nine Muses and nine chief sciences, including mathematics, medicine, and astronomy (endnote 35; 336). Most important, Levania is an inverted image of Earth, but an Earth conceived according to the obsolescent Ptolemaic model of geocentrism. And thus is "the thesis of the entire allegorical dream" (hypothesin totius somnii), Kepler writes in endnote 96 (344), a matter he affirms more than once since it's his mission to dispel the outdated and erroneous view that the universe revolves around one's world because one seems to perceive circular trajectories in the sky of that world.

On the surface of things, Kepler's Somnium seems to be self-exhausting and supremely confident. The 223 endnotes that actually make up ninety-percent of the book spare no explanation--physical, geometrical, optical, geographical, mythographical, psychological, philological, or literary. Indeed, while reading this strange book one gets the feeling that the Kepler who authored the Somnium couldn't leave unspoken any thought; his was an imagination, as Max Caspar affirms with delight, richly synthetic and supremely playful and energetic. His scientific thinking lacks the austerity of eighteenth-century empiricism and conforms fully, as Gilles Deleuze would put it, to a cluttered or "folded" aesthetics of the baroque (3-13). This excessive, baroque quality has contributed, Gerald Holton laments, to the Somnium's soggy reputation among modern scientists and historians of science (69-90). It might also be that the book's ruthless self-commentary has helped to exclude it from interest among literary students of alle gory proper. Some early commentators had even deemed Kepler's Somnium a Swiftian satire (Rosen xxiii). What else could they make of Kepler's jovial boast, in note 56, that "I fire Satire's arrows indiscriminately at self-assured spectators" (vero spiculis Satyricis passim evibratis in spectators sui securos 339). One might think Kepler had allegorized himself as a personification of Satire, though memorable gibes of this sort helped condemn the text to being ignored as either serious science or literature.

But as Kepler pleads at the end of another long-winded, excursive endnote--number 70, a note about the coldness of the interplanetary aether and the necessary allegorizing of that coldness: "when you see me toiling thus, help me root out the causes" (et qua parte me vides laborare, adjuva ad causas eruendas), which we might prefer to interpret, "I could use help with the allegory here" (341). This frankness goes beyond satire and authorizes my critical toiling: I want to show how the Somnium spins off into other allegorical orbits, not the least of which seems to be a self-reflexive sense for allegory. It was Marjorie Nicolson who, some decades ago, sought to ally Kepler's "literary" method with the lyrical poetics of the English metaphysical poets (259-80), a critical move appreciative of the text's figural self-reflexivity and the author's enshrining of difficult tropes in all of his work (259-80). But I think it's more important to pursue a different though related critical vein by articulating some narrat ological propositions which would ally Kepler's tactics to the business of narrative embedding, thereby showing that the self-conscious matter of embedding evokes the ideas of geometry and mathematical cognition themselves.

Virtually alone among historians or philosophers of science who have treated the imaginative dimensions of Kepler's theoretical writings, Fernand Hallyn has conducted the first prolonged rhetorical and narratological analysis of the Somnium in an attempt to reconcile the text's large-scale narrative features with local rhetorical tropes of the sort close-readers like Nicolson had brought to the fore. In his fine book, The Poetic Structure of the World, Hallyn highlights the Somnium's summary structure of reversal or, more precisely, its reliance on what turns out to be early modern science's master trope, irony (35-52). After all, the perspectivalism of the levanicentric view held by the inhabitants of the Earth's "moon" corresponds inversely to the geocentric perspectivalism of the anti-Copernicans, the Ptolemaists, who still dominated seventeenth-century astronomy. Consequently, Hallyn goes so far as to diagram the narratological structure of embedded narrative or diegetic levels in the Somnium in order to demonstrate paired inversion or reversal: the text offers us, he elegantly summarizes, "a book containing a dream containing a book" (260-61). This chiasmus, dream in book and book in dream, Hallyn pictorializes as a series of concentric rectangles: four labeled frames constitute the text's narrative structure, wherein I. Explanatory notes: discontinuous contains II. Narrative 1: book about a dream which contains III. Narrative 2: dream about a book which contains IV. Didactic speech: continuous (Hallyn 262).

However, I think it is a crucial miscalculation for Hallyn to brand the Daemon's "didactic speech" as continuous, since that speech also contains a framing component and an adopted guise, and thus a further level or mode of performativity. That is, a fifth narrative level lies deep inside the already complex series of structural shells, occupying the bulk of the narrative proper itself. At one point, the Daemon, once it has supplied a description of how humans are telekinetically conveyed to Levania from the earth within eciptical shadows, claims that it will "subsequently speak of the form of [Levania's] provinces, commencing in the manner of the Geographers" ("Sequitur, vt de ipsius provinciac forma dicam, exorsus, more geographorum"; 324). That is, the Daemon, which had been speaking in its "blunt, hollow voice" (blaesae et obtusae vocis 323) steers away from the details of translunar transit (which seem to have bored it) and assumes the pronominal "we" as it goes on to sound, perhaps, like Mercator, Keple r's favorite contemporary geographer. But as a rule, or in accord with what might be identified as an implicit narrative code, the internal most framed discourse or endodiegesis of the Somnium seems to grow increasingly abstract and dense. In this final narrative voice, the voice of the professional geographers, the Somnium speaks the language of contemporary science, one given to empirical observations and lots of measurement and geometrical or trigonometrical rendering. It is a voice, curiously, that most closely mimes or echoes the "voice" of the outer most narrative frame in the text--the "discontinuous" though authoritative 200-plus endnotes sporadically furnished by Kepler through the years prior to the text's posthumous publication.

In contemporary French structural narratology, the effect of stories inside stories marks off the regressive embedding or containment of successive "metadiegeses," as Gerard Genette christens them (70-73). Despite Genette's rigorous formalization of this effect, diegetic embedding has received little attention to date (Nelles 79), although the most recent theorists of novelistic discourse maintain its absolute primacy as a formal trope in novelistic composition, especially in the English tradition (Nelles; Williams 2-4). The containment of one level of diegesis inside another (which I prefer to call endodiegesis rather than metadiegesis) closely resembles the traditional rhetorical name for a device called inclusio or emboitement ("emboxment") or, sometimes, "ring composition." (This is the intensive framing of thematic kernels, one within another, as part of a narratorially continuous diegesis.) Both diegetic embedding and emboxment suggest a phenomenology of rectilinear geometry, and in the case of either t rope, classical narratives, epics especially, seemed to thrive on the mysterious if playful impression of boxed inclusions. Such playfulness marks, perhaps significantly, the sort of maniacal framing characteristic of the monumental narratives constituting nothing less than the West's Great Books tradition. Vergil's Aeneid, Ovid's Metamorphoses, Boccaccio's Decameron, Chaucer's Canterbury Tales, de Sade's 120 Days of Sodom, Shelley's Frankenstein, Bronte's Wuthering Heights, or Conrad's Heart of Darkness, can contain four to five levels of embedded narration, (5) while they and other narratives (largely in the novella tradition) seem to want to lose the reader owing to their aggressive embedding or framing. Such narrative embedding, as Tzvetan Todorov has declared, may be the most decisive self-reflexive indicator of narrative's structural armature (70-71, 74). Narrative embedding itself underlines or thematizes, rather than its probity as a means of representing truth or reality, the artificial and tropologi cal nature of all narrative. It also signals the logical or phenomenological involvement of narrative in concatenated structures of elementary binarisms (inside/outside, contents/container, primary/secondary, anterior/posterior).

Moreover, the sort of correlation among narrative levels evident in the Somnium---concerning the replication of the prophetic yet geometrically invested discourse of the outermost frame of notes and the innermost frame of the Daemon's geographical sermon--bespeaks what Lucien Dallenbach calls the narrative misc en abyme, the embedded story-within-a-story that replicates or mirrors the larger narrative scheme in toto (8). Vergil's ekphrasis of Aeneas' shield in Book 8 of the Aeneid furnishes one of the earliest dynamic prototypes of the misc en abyme (all Roman myth and history get "narrated" in both ekphrastically presented shield and in Vergil's circumscribing, complete epic). A revered source for Kepler, Vergil could have instilled the preeminence of concentric geometry so essential to multi-degree narrative framing or embedding in general and to misc en abyme in particular.

Before I go further into the significance of nested or concentric geometries and Keplerian allegorical poetics, I should address another crucial tropological element descried by Hallyn in the conceptual scheme of the Somnium. It is of equal importance, Hallyn notes, that all of Kepler's major productions--the Somnium, the Astronomia nova, the Epitome astronomiae Copernicanae, the De cometis, and the Harmonice mundi especially--record some measure of perspectival anamorphosis, the distortion of a prior picture of earth and its related heavenly bodies in motion (101-3). Anamorphosis served as a master trope in Mannerist art through the sixteenth and seventeenth centuries; through the efforts of Stephen Greenblatt, it has even achieved notoriety in contemporary literary criticism and new-historicist theory as the conceptual aegis under which early modern ideology and aesthetics cooperated to construct the dominant model of the self (18-23). This emphasis on ellipses and anamorphosis opened Kepler to the well-kno wn negative charge from Galileo that the German astronomer was merely a Mannerist, as Alexandre Koyre had concluded (Caspar 137). Anamorphosis likewise stamps Kepler's work with the imprint of the baroque. But the rhetorical underpinnings beckon. If the enantiomorphic or "mirrored" visualization of a construct can be construed as the rhetorical trope irony--perhaps even as allegory itself, since inversio was in fact one of the classical rhetorical terms for allegoria (Quintilian 1.5.40; 8.6.44)--then anamorphic visualization might be taken as sheer troping (or "twisting" or crushing) itself. The ellipse, after all, is a troped or crushed circle; Kepler is the troping, or indeed the "othering," of Copernicus. His theoretical production allegorizes that of his great predecessor.

"Allegory" therefore appears throughout Kepler's works in more than one register or sense. Indeed, the Somnium, as Hallyn concludes, charts the anamorphic allegory of Copernicanism precisely as a nest of geometrically configured Chinese boxes which, it must be stressed, thematize another essence of what rhetoric called allegoria--allos agourein, "saying other." The nesting of discursive or narrative shells or boxes, one within another, recalls the Augustinian cliche of the occulted allegorical text, one which always demands that the initiated reader be able to penetrate beyond integumentum into substantia, beyond lettera into spiritus.

As I've just concluded, Kepler's "allegory" as a concept has more than one sense; it plays our in several registers. There is, to repeat and summarize, (1) the redactor-Kepler's notes, which resolve things and persons from the dream vision into reified or personified abstractions; (2) the ironic or inversional allegory of moon for earth and then sun for earth--that is, the ironizing of the Ptolemaic model; and (3) the anamorphic squeezing of the Coperican model. But the first two sorts of Keplerian allegory hold further promise for understanding a potential allegory of cognition. I therefore want to pursue Hallyn's geometricizing of the Somnium's narratology, a narratology of nested and rectilinearly framed levels or domains of discursivity. To sum up once again: we have a waking narrator; a sleeping narrator who reads a book presenting an autobiographical narrative by the Icelandic astronomer Duracotus; a lecture by a Daemon spurred by a necromantic ritual; that Daemon adopting a dramatic role or voice--a pr osopon, as classical rhetoric would have it (Quintilian 9.2.31); and the copious endnotes which are typically placed at a level beyond the whole series of concentricities. Kepler's text seems to narrate modes of thought as well as topics of thought.

Much more is implied in Hallyn's pictorially and diagrammatically constituted narratology. Plotting an ever more rarefied ontology and epistemology, the allegorical machinery presages, I believe, some modern theoretical descriptions of the increasingly rarefied constitution of not just literary and poetic but also scientific thought. The best direction regarding this hunch comes from the contemporary semiotics of mathematics. In a number of venues-notably his provocative essay "Thinking Dia-Grams: Mathematics, Writing, and Virtual Reality" and his book Ad Infinitum--semiologist Brian Rotman asserts that the history, philosophy, and pedagogy of mathematics elides an embarrassing or problematic component of the discipline's operations in the way it has customarily shown anxiety over the graphic diagram. Diagrams include all visual objects of convenience in the graphic discourse of the mathematician such as line drawings, labeled geometrical figures, analytical Cartesian plottings--in short, any pictogrammatic e nhancement or illustration that supports the mathematical scripts of equations or proofs. Rotman shows how mathematical "writing," and even writing in philosophy, abhors or at least shows anxiety in the face of textual diagrams. He focuses on the work of set-theoreticians as his prime example, with comparative gestures to Husserlian phenomenology ("Thinking" 20, 26-27). I have, so far, implied that Hallyn's signal reading of the "structure" of the Somnium advances Kepler's text as a self-reflexive advertisement of the isomorphism between mathematical systems with their geometric buttressing and literary systems such as the dream allegory, also buttressed by imaginary visual analogues. But in addition, the Somnium might be prioritizing diagrammaticality by incorporating a semiotics of mathematics translated into an imaginary narratological diagram of itself while it presents, at a topical level, geometries of celestial mechanics.

Rotman's theoretical picture of the workings of the mathematical imagination begins by employing Charles Saunders Peirce's dyadic structure of abstract cognition, a model of cognition specifically descriptive of the invisible or self-elided properties at work in mathematical thought. Pierce imagines two separate mathematical beings--the actual mathematician and a "skeleton diagram of the self" (Ad Infinitum 8). Rotman develops Peirce's picture into a triad, speculating how, in mathematical activity, there are actually three cognitive entities at work: there is a Person who functions under the auspices of a Metacode; there is a Subject who functions under the auspices of a formal Code; and there is an Agent who functions under a Virtual Code ("Thinking" 22-23; Ad Infinitum 7-10). Each entity or cognitive "self" is "embedded" inside the other, who exists at a prior ontological and epistemological level. In short, the Person is the real-life mathematician immersed in real time and "natural language." She brings to a mathematical exercise all sorts of additional associative paraphernalia, not the least of which is a phenomenological sense of writing out proofs on actual chalkboards, creating visual diagrams, logging actual errors or dead ends that never make it into journal articles or textbooks, and sweating out a personal life fraught by relations with other personal mathematicians or scientists. This is the scientist or mathematician made known to us by memoir or biography. The Subject, speaker of the Code, takes up "all rigorous sign practices--defining, proving, notating, and manipulating symbols--sanctioned by the mathematical community" ("Thinking" 23). In other words, the Subject is the idealized writer of this or that mathematical proof that gets published at a particular time. The Agent is an imaginary, indeed eidetic (as Husserl would say), mental "proxy" who "lives," as it were, among the pure abstractions of a mathematical or numerical or symbolic "landscape." One might say that the Agent is a prosopopei a of mathematical functionality itself. It is an imaginary embodiment of the semiotic or logical forces that dictate the seemingly programmed or "natural" relations among quantities: what "makes" two numbers add up to a third; what is actually causing summation once a summation operator has been scripted into an equation; what integrates the infinite number of infinitesimals in a calculus equation just because the mathematician (as Subject) had put at a certain point an integration symbol? This complex schema allows Peirce and then Rotman to render visible a structure of social and semiotic processes that mathematics actually ignores in its day-to-day functioning as a discipline, while the purpose of the schema is to reanimate the sense that mathematical production is an embodied, pro-temporal, immanent set of practices dependent upon rhetorical as well as purely logical functions. Although mathematical proof is ultimately a scriptive mode, it begins in or draws on visual, sensuous shapes; and although a fini shed proof may contain errors (made by the Subject), the working up of that proof has a hidden log of expunged errors. Mathematical work does not amount to a disembodied, atemporal, idealized set of graphic or scriptive recapitulations attempting to present, as do all Platonisms, universal truths or cosmic harmonies. But the conventional philosophy of mathematics, in fact, gives us only the Subject set forth in an idealized and completed workscape.

Although Rotman takes the pains to show how his (socially and institutionally recuperative) semiotic schema stems from Peirce's, some more discursive archeology demonstrates that it does not arrive on the semiological scene fully blown, as his argument (and as the procedures of many such semiotizations) might suggest. It is helpful, for instance, to see Rotman's scheme in dramatic terms: one might think of a one-character dramatic play--perhaps a Beckett play--which represents the conjoint work of an historical author, a stage manager, and an actor. The author or playwright stands in equivalence to the mathematical Person, speaker of the Metacode; the stage manager equates to the mathematical Subject, speaker of the Code; and the sole actor on stage is the Agent, speaker of the Virtual Code. Better, however, would be the more direct analogy between this semiotic system and its perfect parallel in the tripartite schema of Wayne Booth's narratology. In The Rhetoric of Fiction, Booth divides the composition and narration of a story among the work of a historical author, an "implied author," and a narrator (148-54). For example, Sir Arthur Conan Doyle authored consecutively all of the Sherlock Holmes tales; the Doyle Implied Author is the author of any one complete text completed and published at a certain moment (say, The Hound of the Baskervilles); the narrator of a Doyle novel or short story is always Dr. Watson (except for "The Musgrave Ritual"). Booth's paradigm bespeaks Chicago-Critical formalism--which, in such a scheme, invokes the text fetish of New Criticism, for the Implied Author embodies the temporal creation of any particular text at a specific time, the formalist's paramount object of interest.

I do not mean the analogy to mark a perfect correspondence. But Rotman's model and its lookalikes in literary theory have pragmatic or pedagogical import. Surely the historical and biographical mathematician, who labors over the span of a career, logging errors and dead ends, can feel at times severed from herself when she functions as the presenter of a finished paper at a conference. And these two entities are severed from the imaginary "being" who itself inhabits the paper's proofs, a fugitive being cut off semiotically from the communicational apparatus of the paper's personal (Person-al) or virtual presentation as well as from the needs and wants of the paper's audience or readership. Witness Roger Penrose's famous comments about how mathematicians or physicists don't really "follow" or "understand" the proofs of a mathematical paper when they're listening to that paper publicly. Nor do they fully understand it per se when they read it. Rather, they glean a general "sense" of the paper's argument (the Co de, spoken by the Subject) and then cobble together the argument's fuller significance over time, but never completely (Penrose 102). This unnerving confessional analysis reveals that Rotman's imaginary semiotic entities can frequently have a distant connection among themselves; their union requires an act of phenomenological or semiotic divination, dreaming together, visitation, perhaps even violence.

The link from here back to Kepler turns out to be frankly uncanny. Rotman's paradigm, I am convinced, recapitulates fantasized states of affairs and entities in Kepler's Somnium; or rather, the Somnium could be said to presage through the customary language of allegory the semiotics of mathematical cognition formalized first by Peirce and later by Rotman. It is astonishing that Rotman writes of the Metacode-bearing Person as a "dreamer awake," that is, the sociocultural real or virtual person whose life is steered towards "dreaming" mathematical fantasies, these numeric and algebraic imaginary worlds. The Subject, Rotman adds, is like a "dreamer asleep" while the agent, a truly phantom "proxy" or imago (both of these are actually Rotman's terms), dwells as the actant of mathematical signifiers and is the least corporeal of the three conjoint agencies in the triad ("Thinking" 23-24). More than any other narrative topos, the theme of dream texture evokes the language of medieval and early modern allegory, the d idactic literary mode most steeped in phantasmagoric, hallucinatory, surreal, or oneiric experience.

But to be precise, Kepler's text carefully decomposes his theme--the mathematical demonstration of Copernicus's heliocentric model--into the narratologically distributed actions of at least three ontologically separable agencies, each a version or proxy of the other: first, we have the "waking" Kepler who falls asleep and dreams of a book by an Icelandic astronomer named Duracotus. Although we don't quite see much of a dreaming Kepler other than the figure who, after reading about history and magic and after watching the stars for a bit, falls asleep to dream that he reads a book bought at market (atque mihi per somnum visas sum librum ex Nundinir, 321), we do have a second incarnation, a surrogate for the waking and sleeping narraror-Kepler--Duracorus, protege of Tycho Brahe, the interface figure between an external waking (or historical) reality and the twilight, sorciological world containing his mother Fiolxhilde and her daemonic familiars from Levania with whom Duracotus himself deals, but only in a cogn itively compromised, ritual state. (Incidentally, Max Caspar writes freely in his summary of the Somnium that Duracotus is merely Kepler "himself as a youth"; 352.) And third, we have the lecture-giving daemon ex Levania itself--a purely incorporeal, merely energetic entity (what else does Kepler say of it descriptively but that it has, or is, no more than a "hollow, indistinct voice," blaesa et obtusa vox) who takes on the elaborately mathematical voice of a Mercator or an Ortelius. The Daemon has a voice variably phanrasmatic and masculinely logical; the first voice is hollow, the second, that of rhetorical and scientific plenitude, just as the voice of the scientist or rhetorician should be full, plenus. (6) And yet the waking "Kepler" in the Somnium, the Boothean "narrator" proper who gets the shortest shrift in the text, calls up in the very scantiness of his representation the replete Kepler who writes those endnotes--the biographical Kepler who, in a case like few others in the history of science, is e nlivened for us by an unusually animated biography, Caspar's magisterial Kepler, another circumscribing shell or frame without which the Person Kepler would really be intangible to us (and without which this essay would have been compromised).

This state of affairs, both characterological and narrarological, seems to me an uncanny foretelling of the semiotic scheme endemic to mathematical cognition that Rotman has so fully expounded. Nor is this fit, to be sure, perfect, as Rotman himself admits regarding his own analogies. My homological or analogical "mapping" functions heuristically, while Rotman's fitting, for instance, of the structure Person/Subject/Agent to the structure "formal"! "vernacular"/ "set-theoretical" (three hierarchical levels or categories of mathematics) seems to function far more figurally ("Thinking" 24). Nevertheless, we would have to better account schematically for the Waking Kepler who furnished those 223 endnotyes over ten years; he, like the Geographer Daemon, shells out plenty of calculations and elaborate geometrical diagrams solely in his editorial excurses. Yet these endnotes, which seem to wish to occupy a place of epistemological authority, co-opt themselves precisely because of their resemblance to (and complicit y with) the most fictive narrative level of the Somnium--the lecture, site of the inrernalmost misc en abyme incorporating the scientific or scientistic raison d'etre of the whole project. To be sure, "endnotes" in general occupy a compromised or co-opted epistemological place--the consensual view taken by most contemporary textual theorists or editors working in the era ofposrmodern rextuality. As Ralph Hanna III has put it, commentational notes are as "fictive" as the texts they seek to illuminate; they open aporias as much as they seek to bestow closure; they disable further commentary or conversation as much as they claim to enable it (178-84). Notes mark nodes of strong desire once a text has been putatively "finished." They inscribe the desire of the authorial voice to skip freely among narrative or epistemological levels, though they also inscribe the difficulty, perhaps the impossibility, of really doing so. The truth of this is melodramatized in an experiment like Derrida's Glas, though it found half -realization even in "scientific" writing much earlier. Kepler the revolutionary scientist, who is as much bricoleur as he is rationalist, seems ever the mathematical Person wishing to be "fulfilled" as a mathematical Subject or Agent, as Rotman would put it. His Somnium records the narrative fantasy of the accords attempted among mathematical or scientific figurae or personae.

If the Somnium records the allegorical representation of layered cognitive entities that would one day be structurally constitutive of mathematical phenomenology, and if that representation, homologous to narrative embedding, contains too the desire to circumvent epistemological layering, then Kepler's text also literalizes such an allegory of containment in the virtual spaces of planets, their orbits, and in terrestrial objects themselves. After all, the Somnium is about transit between two ontically separate domains, the terrestrial and the lunar. The journeys of the traditional somnium coeleste (of the sorts Dante, Chaucer, and Kepler's Fiolxhilde or her Daemon undergo) can be seen to correspond to the transumptive collapsing of levels within any formalized structural narratology--either Booth's Aristotelian version, Gerard Genette's Saussurian one, or Rotman's Peircean one. (7)

But on another note, the self-reflexive narratology of the Somnium also holds a curious mediational function among the texts of Kepler's whole theoretical production. I have intimated already that the kind of cosmic allegory that powers up the Somnium refracts themes and structures that are ethical, personificational, astrophysical, poetic, rhetorical, epistemological, and even gendered and sexual. For instance, it is no coincidence that the Levanian word for Earth, Volva, which literally means "that which revolves about us" also means, via pun, vulva, in accord with more standard spellings of the anatomical term in seventeenth-century medical textbooks (OED, "vulva"). Latin vulva-volva (literally "wrapper" in English) figures as another sort of occulting container and as an object of desire in a narrative about separated mothers and sons, conjunction, and penetration. Such programmatic punning has long been seen as a constitutive formal component of allegory in contemporary literary theory--from the centrali ty of paranomasia in Maureen Quilligan's work to R. A. Shoaf's theory of "juxtology," a lexical process often energized by both cosmic and sexual faculties. Paramount for my purposes here, however, has been Kepler's unique geometrical allegory, the Pythagorean residuum that fueled his earliest theoretical writings and which has been taken, in conventional philosophy and history of science, as disjunct from his mature projects. The Somnium presents a geometrical-narratological allegory wherein mathematical cognition finds itself personified. But it also presents not just a personification allegory of mathematical cognition, but a reification allegory, if you will, which stages the phenomenology of geometrical containment--that is, the obsessive and systematic rendering of curvilinearities into/as rectilinearities and vice versa. Such ekphrastic exchanging of Archimedean simples--curves and lines (which constitute in themselves structural binary oppositions)--can be seen as the fundamentally Personal (or Virtua l) activity in the Keplerian imagination. If, as Caspar writes, the Somnium allowed Kepler "to bring into the open the child of his mind" (351-52), so Archimedean geometrical simples (recast by me as Saussurean or Greimasian contraries) enable that imaginative but cognitively therapeutic enshrinement of the basic, the fundamental, the primitive (not meant pejoratively), or the juvenile, which makes for a recuperation and invigoration of late Renaissance science.

Let me explain these generalizations by way of a look at two of Kepler's often diametrically opposed texts--the late Nova stereometria and the early Mysterium cosmographicum. The former receives praise as Kepler's mature mathematical opus endorsed by "modern" science, since it proffered a theory of rectilinear infinitesimals crucial to the calculation of areas and volumes under curved surfaces (GW9:121-31). The latter still gets scorned as an enthusiastic and misdirected opus redolent of Platonic or Pythagorean mysticism, the mysticism of nested polyhedra and their symbolic properties operating conjointly with a still circular (rather than elliptical) nest of planetary orbital patterns (GW 8: 31-128). So an imaginative way of reading the genealogy of Kepler's speculative or heuristic treatises would be to recognize the scientific "thema," as Gerald Holton would term it, of embedded or contained geometries--in particular polyhedral or rectilinear geometries embedded in or figurally exchanged for circumscribing or "translative" contours or curvilinear geometries. After all, the Mysterium cosmographicum presents a scheme whereby the five perfect or regular polyhedra--the tetrahedron, the cube, the icosahedron, the dodecahedron, and the octahedron--are each matched to a planet and its (Copernican) circular orbit. The scheme no doubt bespeaks the geometric phenomenology of tangential or inscribed sets, for as Kepler moves from the orbis magnum beyond Saturn inward, giving us the cube, then the tetrahedron, then the dodecahedron, then the icosahedron, then the octahedron (by which time he's taken us to the position of the sun in the center of things), he is guided, as one commentator has shown, by his favorite theme of harmony concerning the comparative radii between inscribed and circumscribing spheres and their respective polyhedra (Stephenson 80). Though the twelve-faced dodecahedron and the twenty-faced icosahedron lose out aesthetically (since they express imperfectae harmoniae owing to mathematically troublesome radius ratios; GW 8: 66-68), they do enjoy roles as the most sphere-like, and thus the most nearly perfect, of the solids.

This mathematical thema--the relationality of inscribed rectilinear to circumscribing curvilinear or spherical objects--also characterizes the indirect achievement of the Nova stereometria, which, along with a sustained annotation of Archimedes, touches on how Kepler observed wine merchants making accurate barrel content measurement by guesstimation (GW9: 10). He then works out a practical mathematics of this skill by breaking an interior curvilinear volume into an infinite number of imaginary rectilinear volumes that could be manipulated in mass calculation. The achievement would lead eventually to the forms of the calculus developed by Newton or Leibniz, while along the way it would be important, in its general spirit, to the development of Keplerian planetary trajectories and volumes. Modern history of science sees Platonic fantasy in the Mysterium cosmographicum, but the rudiments of differential calculus in the Nova stereometria. Yet both texts are the semiotic companion pieces of the Somnium, the obliqu e dream narrative all about--well, if we were to imagine a Monty Pythonesque Royal Society--"putting things inside of other things," a society for playfully putting geometries within one another. That is, of course, the classical function of allegory: to worry about concealment and containment, opening up and penetration, correspondence and fit-ness. But that allegorical impulse finds baroque and sensuous materialization or reification as a narrative romance of geometrical manipulations. Kepler's dream prefigures not just the great Victorian dream of Peirce, a dream of a real and living mathematician who dreams his "skeleton" or ghostly counterpart at the level of scriptive equations, but also the Victorian dream of Edwin Abbott, whose Flatland both reifies purely scriptive mathematics (the equations for curves or polygons) as geometrical images, and then personifies those figures as sentient agents in a narrative. Kepler's Somnium thus stands resolutely as the formal bridge between the oneiretic allegories o f Dante and Chaucer (which combine literal cosmographic description and celestial travel with ethical personification, always in a tropologically self-reflexive armature) and the modern allegories of Abbott and Peirce and, in turn, Rotman. (9)

On all levels, Kepler's Somnium proffers the fugitive desire to contain and be contained, and to open and disseminate, a set of cognitive and phenomenological protocols that come, then, not so much from "science" as from rhetoric. Kepler's baroque dream book, perhaps more than any other early modern text, celebrates science's dependence on rhetorical tropes and on the cognitive motor of allegory. This is because the master tropes, indeed their very names in Latin and Greek, kept reminding poets, rhetoricians, and scientists of the semiotic offices of figurality or rhetoricality themselves: the roots fingere, fictio, figura all connote the imaginary shaping or fitting of linguistic entities as if they were contoured objects. (And of course, term pairs such as ellipsis-ellipse, hyperbole-hyperbola, parable-parabola also indicate the lexical interdependence of geometry and rhetoric.) Figura was, all at once for an imagination like Kepler's, a physical human body, a geometrical shape, and a rhetorical device. One cannot escape the paranomastic or pun-driven energies among such rhetorical terms in early modern representation. Thus, Kepler's thematic obsession with the Earth's conical umbra or "shadow" on the neighboring island or planet Levania is only one of many encrypted rhetorical terms that point to the semiotics of allegory: umbra is the Hellenic rhetorical synonym, presciently rediscovered in the Renaissance, of a host of rhetorical concepts in Latin including imago, figura, forma (Auerbach 34), and by extension, allegoria. And like Rotman's or Peirce's proxy "Agent" in mathematical activity, Kepler's levanian Daemon (who should remind us of other nimble daemons that manipulate abstract or scientific objects, Maxwell's Demon preeminently) lurks in occulted, dark, closed places doing secret or elect things. Kepler's Daemon can move between Earth and Levania only inside ecliptical shadows, in umbrae, and it can manipulate travelling human bodies magically, telekinetically, but only inside or under cover of an act ual conic shadow (a phantom, abstract, virtual locus to be sure) since the moon-going Daemon, like its habitat of negation, stands always already as a metonym for allegory and mathematics (Clarke 10), both being dominant discursive formations built indispensably on the fantasization of contained geometries.

To the thesis that Kepler's dream book is a fully blown allegory of the phenomenology of space and time, I add that Rotman's Peircean semiotization of mathematical agency, which I've claimed finds prefiguration in the nested narratology of the Somnium, might be rewritten directly as a series of embedded geometries. Rather than in the way Rotman renders it--a diagram made up of a triangle which has at its three vertices the three entities (Person-Subject-Agent) and their respective codes (CIRTUAL CODECODE-METACODE)--we might instead be better served by a diagram of temporal emboitement or emboxment as a result of another latent allegorical structure constitutive of Brian Rotman's schema, temporality. Thus: [METACODE: Person [CODE: Subject [VIRTUAL CODE: Agent] ] ] Rotman himself emphasizes that the Person "foretells" the action of the Subject, who in turn foretells or is fulfilled by the action of the Agent ("Thinking" 24); like the endodiegetic insets of embedded narrative, the innermost narrative box or blot ter is the most chronologically prior--or, ironically, the most recent--in a receding architecture of imaginary polygons.

This temporal projection, incidentally, marks the language of traditional allegorical typology in medieval hermeneutics, by which I mean the temporally interactive but binary separation in traditional biblical exegesis, a major component of medieval literary allegory, of Old Testament and New Testament ontologies. In medieval typology, type prefigures antitype: Isaac prefigures Samson who prefigures Christ, just as Peircean Person prefigures Subject who prefigures Agent. (10) The narratological corollary would be to represent the temporal structures of typological allegory by nesting them, earlier events within newer events, among diegetic levels of discourse. That is why the personified Virtues of Prudentius' fifth-century allegory the Psychomachia can only narrate the deeds of, but not coexist with, their typological antecedents from the Old Testament--the heroic exempla Abraham, Samuel, David, Judith, Jonathan, and so forth (Paxson, Poetics 72-76). And of course, I have all along asserted that Kepler's nar rative prefigures or presages the semiologies expressed in Peirce and Rotman. I have mutually used Rotman's system to illuminate and formalize Keplerian cosmographics (from both early and later phases in his scientific career) while I have used Keplerian cosmographics to illuminate modern semiology.

Rhetoric beats both literature and science to the finish line at proclaiming itself to be the discourse or discipline of containers and contents. The kaleidoscopic structures and narrative codes of Kepler's allegorical Somnium foretell or "dream" the modern and postmodern discursive formations of semiotics and narratology. Those modern scientistic discursive formations, along with early modern astronomy and mathematics, also draw upon rhetoric and poetics. Neither Keplerian nor modern science has stepped out of the dream world of allegory, while modern art and criticism seeking to capture medieval allegory--witness a revolutionary film project such as Peter Greenaway's 1989 A TV Dante, replete with framed, pop-up boxes containing literary commentators overlaying the montage of Dante's continuous narrative of the first eight cantos of the Inferno--seem indebted more than ever to the structural interdependencies constituting Western thought's massive allegory of containment. Kepler, Peirce, Rotman, Abbott, and Greenaway are all the consanguine children of allegory's geometry, geometry's allegory.

James J. Paxson, associate professor of English at the University of Florida, is the author of The Poetics of Personification (Cambridge, 1994). He has co-edited Desiring Discourse: The Literature of Love, Ovid through Chaucer (Susquehanna, 1998), and The Performance of Middle English Culture: Essays on Chaucer and the Drama in Honor of Martin Stevens (D. S. Brewer, 1998). Paxson is also associate editor of Exemplaria.


(1.) In a letter of thanks treating the debt of the Mysterium to his teacher Michael Maestlin, Kepler writes, "In the origin of this work I was Semele, you Jupiter. Or, if you would rather compare the work with Minerva than with Bacchus, then I as Jupiter carried it in my head. But if you had not performed the midwife's task as Vulcan with the axe, I should never have given birth" (Caspar 66).

(2.) In an article I published a few years ago on allegory and Newton's calculus, I spurned Kepler's allegorical and Platonic writings in favor of his productions in pure analytical geometry, productions which helped make modern science by enabling the calculus which Newton and Leibniz would go on to invent in some few decades. That is, I dismissed the furnishings of traditional medieval dream allegory in order to articulate the proto-deconstructive features of temporality-allegory, a de Manian kind of allegory, in the structures of early modern calculus (Paxson, "Allegory of Temporality" 45,49). So in this essay, I wish to suture the various allegories constitutive of Keplerian metaphysical and mathematical poetics and to amend my hastiness of judgement in that article.

(3.) Whereas, in the Macrobean tradition, somnium means an allegorical dream conveying hidden truths via occulted, sensuous symbols, counterpart terms include visio, a direct, literal dream or premonition about current or future truths and events; and oracutum, a direct and literal relating of some hidden truth or future event by an authoritative dream narrator--usually the ghost of an ancestor or a spirit. Although Kepler's book is designated a somnium, it contains oracularist elements. For a handy introduction to the Macrobian system set in the Keplerian context, see Hallyn 256-57.

(4.) All citations are to the Latin text of the Somnium printed in the Gesammelte Werke 11.2:321-67. Translations are my own unless indicated. The two best available English translations are in Lear and Rosen. Hereafter, references to Kepler's texts printed in the Gesammelee Werke are indicated by GW, including volume followed by page numbers.

(5.) Ovid's Metamorphoses reaches this saturation point in Book 5, where we have the Ovidian narrator telling of the story exchange between Minerva and the Muses; the primary frame contains the story told by an unnamed Muse, which contains the story told by her sister Calliope, which contains the story of Arethusa, which contains little stories and declamations by Alpheus and others. The abusio of this trope of concentricity or embedding is thus reached early on (much earlier than, though prompting, John Barth's well-known parody of it in his short story, "The Menelaiad" [130-67]); out-pacing even Ovid is Chaucer's "The Nun's Priest's Tale," which contains a tale told by a murdered man in a tale told by the rooster Chaunteclere in a tale told by the Priest in the frame tale told by the Chaucerian narrator. But it is perhaps more important to recognize the Vergilian provenance of the effect: by Book 3 in the Aeneid, we see Vergil narrating the scene in which Aeneas narrates his long trials to the court of Dido in Carthage. Inside this narrative are the embedded narratives of ghosts, spirits, and monsters--such as Celaeno, leader of the Harpies. Significantly, such embedded tales to the third degree tend to contain the characteristic prophecies of the Aeneid voiced by rarefied beings--a narratological state of affairs that presages Kepler's poetics in the Somnium.

(6.) It is of further (aleatory) significance that such allegorical and scientistic plenitude was "erroneously" marked in the printing of Kepler's name in a 1597 catalogue that mentioned his first book, the Mysterium cosmographicum: instead of "Keplerus" we have "Reple[t]us" (Caspar 69).

(7.) I choose Gerard Genette's durable and rigorous narratological scheme, which separates diegetic from embedded or metadiegetic narrative levels, as representative of French structural narrative theory at its best. See Genette, passim, for descriptions of varieties of narrators (diegetic, metadiegetic, intradiegetic, paradiegetic, homodiegetic, heterodiegetic, extradiegetic, etc.) as they are keyed to narrative order, frequency, mood, duration, and level. For Genette, a transumption or metalepsis is the transit of a character from one diegetic level to another (234); this can be taken to correspond to the transit among sealed cosmic levels in an allegorical cosmography--Dante's ascent to paradiso or the Keplerian Daemon's descent to Earth.

(8.) See my Poetics of Personification for readings of tropological or figural self-reflexivity in the allegories of Prudentius, Chaucer, William Langland, Spenser, and Milton.

(9.) Typology itself was a structural system of signification: "typologues" (consisting of former type and latter antitype, and of ethical protype and counterrype) chart imaginary axes in what can be thought of as a proto-Greimasian structure of temporal relations. See my "Theory of Biblical Typology" 93-117. I mention parenthetically that the allegories of the high Middle Ages work in a "tropologically self-reflexive" way. In such allegorical narrative, the very structure and imagery of tropes and figures become thematically represented. If personification or prosopopeia, which comes from prosopon poein, "to make a face," serves as the master trope of much medieval allegory, various allegorical narratives plot the literalized demolition of faces (prosopa) at key dramatic junctures (Paxson,. Poetics 68-70). All tropes, as I have shown elsewhere, disclose their semiotic of insides/outsides (a structure well materialized in faciality), for tropes signal rhetoric's desire to alter, occult, or transform putativel y "literal," direct, proper, or substantial language.

(10.) In "Personification's Gender" 173-74, I develop the position in The Poetics of Personification regarding the ways in which the institutional or performative meta-rhetoric of Rhetoric itself transforms and compromises its own topics--the master tropes of literary discourse.

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