2.2 we used to think “we used to think”

—but now we thought we didn’t. Or, as Bruno Latour put it to what may be the all-time champion of nonpostmodern book titles, We Have Never Been Modern—a sentiment equally skeptical of pretense to post-critical apprehension and critical of pretentious postmodern purport. That is to say, the party line sung en choeur by the bulk of contemporary scientific and philosophical theorists has it that the definitive shift took place along a n auspicious arc from Galileo to Kant, in result of which the absolutist precepts of premodern metaphysics were singularly displaced by modern relativistic critical skepticism. Shake and fake, or wake and bake? Back to the frying board, or into the fire? Lest you mistake our dealing from the bottom of the deck for nonsensical nostalgia or some other sentimental disinclination, we should reiterate our revocation of premodern, modern, and postmodern demarcations alike, being wishfully revisionist historical brackets, one and all.

Moreover, we would not purport ‘anti-modernism’—but we may well be punting ‘amodernism’ by way of our imperfect anterior auger that “we will never have been postmodern.” In short, our historical interests point to the overlooked, erased, suppressed, repressed, co-opted, distorted, hijacked, and otherwise unseemly apprehensions that have yet to be processed, comprehended, integrated, instituted, or accepted sufficient to warrant as “historical” rather than presently pressing. Conversely, we find names and dates slightly less compelling than post-colonial cricket statistics. Latour, not surprisingly, has adopted the ‘actant’ from Greimas (not having read enough to warrant, but so it seems,) to designate causative entities amid relational vectors irrespective of subjectivized dispositions and objectivized presuppositions. This would (again, not surprisingly) put (one Bruno,) Latour in company with (Giordano, the other) Bruno, so, all’s well that ends

well, if we cannot continue to evade Eco’s echoes of “Giordano Bruno’s space of an infinity of worlds,” what with its inky deep and black horizon, perhaps we can preempt it by recourse to his progenitors, namely “the divine Cusanus” (as Bruno tirelessly testifies,) with whom the displacement of the Aristotelian cosmos began in earnest (never mind Copernicus)—but in less ambitious theoretical terms than Bruno, if not more ambitious practical aims—and with greater rigor, if not lesser repercussion. To revert Mein Herr’s query (from Lewis Carroll’s last gasp, Sylvie and Bruno, thus satisfying the comedic ‘rule of threes’), what is the largest world you would care to inhabit? Its extrema are sure to exceed even the most ambitious 1:1 map—that is, so long as we axiomatically decree that one fold cannot stand for another—but to so proscribe would unleash a transfinite concatenation of recursive pre-sup-positions through which seemingly invariant links fail irregularly under different strains given discrete variables under continuous variance. “. . . no synonyms . . . everything differs . . .”

are we T/Here yet?

In redressing the partially-aborted birth of modernity, we shan’t fail to cite the usual suspects elsewhere and otherwise; however, what may get lost in Kuhn, Cassirer, or Koyre we may regain in recent reassessments by Karsten Harries (2001) and Dalibor Vesely (2004)—both of whom re-center the topic of de-centered topics (reorient the disorientation, if you will) from an architectural perspective on perspectives (cue harrumphs and nutcracks). Regarding our present topic, both expound the exchange whereby (1) the esoteric reformulation of epistemological perspective by Cusanus reciprocally mirrored in (2) the exoteric formalization of visuospatial perspective by (his confrère) Leon Battista Alberti.

In De Coniecturis (1443), Cusanus explicates his Figura Paradigmatica as a matched set of interpenetrating pyramids, the antipodes of which are analogous to light and shadow (Lat. lucis/tenebrae), unity and alterity, sameness and otherness, and so on—an exhaustive litany by which of each term pair, the non-infinite maximum of one basis coincides with a non-zero minimum of the other. (Cf. Yeats’s 1907 A Vision, in which selfsame “gyres” unspool from his better-read wife in a fashionable fit of mesmerism.) As Vesely notes, this “diagram matches Leonardo [da Vinci]’s description of the two pyramids and their role in perspectival vision [‘one of which has its apex in the eye and the base as distant as the horizon. The other has the base towards the eye and the apex on the horizon.’].” From here, Vesely follows the esoteric/exoteric reciprocation through Cusanus’s onto-epistemic metaphysics:

For Cusanus, the intersection of the pyramids represents the dialectics of human and divine vision, described in De visione Dei (1453) as reciprocity of seeing and being, as being seen by the divine eye. Communication with the divine is possible only through the likeness of human and divine mind. The human mind represents in one sense the unity of vision, in another sense a reference to measure (Lat. mensura), which, as “the essence of number is the first exemplar of the mind.” Because measure is the main characteristic of proportion, the association of human mind and measure also speaks about the proportional structure of mind; and because proportion, as we have seen, is also the essence of perspectivity, the structure of human mind is in Cusanus’s understanding perspectival.

—Dalibor Vesely, 2004

The perspectival shifts undertaken by sophisticated proto-moderns resemble the counter-motions of a double pendulum—a deceptively simple machine whose behavior nonetheless appears chaotic or contradictory to all but the mathematical physicist. All the same, we shall attempt to retrace its path in brief. (Not the pendulum, the other. Well, so much for that analogy.) Working backwards (as it were), what we have in (2) is a shift from (2A) the representation of a theocentric perspective (‘God’s eye view’) to (2B) the representation of a human perspective (in guise of painterly ‘realism’). What we have in (1) is nothing short of the realignment of (1B) anthropocentric subjectivity to reciprocal status with (1A) theocentric subjectivity on the basis of a (topological) homeomorphism—and with it, a host of problematics on cause, knowledge, freedom, and possibility. Harries sketches out the vanishing points:

Alberti’s perspective invites us to look through the material as if it were transparent, a window through which we can see what the painter has chosen to represent. But this is very much a human perspective, which has its center in the observer: what we see is appearance for us. The spiritual perspective of [earlier] medieval art would have us look through the painting in a very different sense: through the material to its spiritual significance. The mundane is transformed into a divine sign. Alberi’s art is incompatible with this spiritual perspective. A God-centered art gives way to a human centered art.

—Karsten Harries, 2001

First, to address (1), with extreme brevity: As Harries notes, by 1277 (but three years after the deaths of two key Scholastics, Aquinas and Bonaventure), the Aristotelian geocentric cosmos was sufficiently entrenched that it took a Condemnation to revoke a litany of precepts on the grounds of placing artificial (finite) human restrictions on (infinite) divine prerogative (providence, will, vicissitude, whathaveyou). In consequence of the high institutional regard for ratiocination (to wit, see proto-Rumsfeldian torture loophole of 1252), the Church couldn’t very well purport God’s omnipotence while at the same time insisting that He orbit this or any other cosmos around this or any other planet.

Condemned propositions included #27, “That the first cause cannot make more than one world [Quod prima causa nonpotest mundos facere.]” and #16, “That the first cause is the most remote cause of all things.—This is erroneous if it is so understood as to mean that it is not the most proximate. [Quod prima causa est causa omnium remotissima.—Error, si intelligatur ita, quod non propinquissima.]” Such revocations had the (unintended, if perhaps not unforeseen) consequence of expanding the horizon of possible posits (at least for the juridically fit dialectician) to a properly eccentric (acentric, non-centric, decentered) and infinitely manifold cosmos. Thence Cusanus comprehended the (neo-Hermetic) rhetorics of the circle and the sphere as metonymic metaphors for an omniscient and omnipresent divinity, a usage he adapted (exapted, translated, transposed) to a mechanical model of a contrary cosmos at once stably omnicentric and acentrically mobile:

The ancients did not attain unto the points already made, for they lacked learned ignorance. It has already become evident to us that the earth is indeed moved, even though we do not perceive this to be the case. For we apprehend motion only through a certain comparison with something fixed. For example, if someone did not know that a body of water was flowing and did not see the shore while he was on a ship in the middle of the water, how would he recognize that the ship was being moved? And because of the fact that it would always seem to each person (whether he were on the earth, on the sun, or on another star) that he was at the “immovable” center, so to speak, and that all other things were moved: assuredly, it would always be the case that if he were on the sun, he would fix a set of poles in relation to himself; if on the earth, another set; on the moon, another; on Mars, another; and so on. Hence, the world-machine will have its center everywhere and its circumference nowhere, so to speak; for God, who is everywhere and nowhere, is its circumference and center.

—Nicolas Cusanus, 1440

absolute relativity ≠ relativist absolutism

This oft-quoted passage from De Docta Ignorantia (On Learned Ignorance) is obviously problematic in more ways than one. To risk short shrift by a short circuit, we might venture that this “Learned Ignorance” consists (not in some tired ‘know-nothing’ hokum, but rather) in the triple transverse Cusanus cuts through cosmology, onto-theology, and epistemology—to the effect that every set of subjectively apprehended coordinates attains equality through displacement—or to put it another way, that every perspective is situated to enjoy the unobstructed view from nowhere. Here we should note that ‘absolute relativity’ does not coincide with historicized ‘relativist absolutism’ but rather reveals its ahistorical obverse.

Were we to artificially trisect out the epistemological purview, we could compare Lacan (1960) on the convergence of Selbstbewusstsein (self-awareness) in Hegel: “The antinomy the Scholastic tradition posited as principial is here taken to be resolved by virtue of being imaginary. Truth is nothing but what knowledge can learn that it knows merely by putting its ignorance to work.” In kind, compare Wittgenstein (1929): “Contradiction, one might say, vanishes outside all propositions: tautology vanishes inside them. Contradiction is the outer limit of propositions: tautology is the unsubstantial point at their centre.” This latter recapitulation short-circuits us back to Cusanus (1464), with “The concept of truth that rejects both opposites disjunctively as well as conjunctively is the more absolute.” That is, we are to find that truth conforms neither to either/or (contradiction) nor to both/and (tautology) vis-à-vis neither phenomenological percept nor propositional purport.

you can’t get there from here; you have to go around

The cosmological consequences of (1) the epistemic shift were (and for all but the ascetic, are) difficult to assimilate cognitively, such that the attendant mechanics resisted formalization until (variously) Galileo, Descartes, Leibniz, and Newton. The phenomenological consequences of (2) the visuospatial shift, as manifest in the painting of architecture (and the architecture of painting) could scarcely have been avoided but for virtual philistinism or actual blindness. Vesely offers a viewpoint crucially distinct from Harries, particularly vis-à-vis our receding view on the forward movement of the medievals (and for a more apt analogy than the double-pendulum, compare the dolly-zoom of Hitchcock’s 1958 Vertigo).

The conventional association of perspective with illusion is of a later date. During the fifteenth and sixteenth centuries, perspective shares in the primary ontological task: to reconcile the diversity of the natural phenomena with the universal order, the human with the divine, and terrestrial with celestial reality. At this stage, to see means not to view the surfaces of things but to think the depths of reality—how things are related amongst themselves and how they are structured by the unifying power of Being.

—Dalibor Vesely, 2004

While accounting for both retrospective views, we shall attempt to make a long story short (brutish, nasty, poor, and solitary): Early medievals favor (2A), a theocentric perspective on (particular) human figures—and by extension, on (universal) Humanity—a planar pseudo-omniscience from which relative scale corresponds to relative ‘degree of divinity’ and towards which grace is attributed (indexed and codified) by the pictorial device of the halo (being not so much a circle as a ‘flattened sphere’) radiating about the phantasmatic nexus of the graced subject. Throughout Christendom, the exoteric perspectival shift occurs in staggered, overlapping stages, but the net effects are definitive: Late medievals favor (2B), a human perspective on the mundane—on emblematic props, buildings, and landscape elements—but more crucially, on divinity. Prior to the shift, the key element is the golden (back)ground, on which Harries reckons thusly:

I called the gold background a metaphorical device meant to carry us beyond the familiar sensible world. It thus functions somewhat like the words “absolute,” “perfect,” or “infinite” added to predicates taken from the sensible world in order to make them more adequate to God. Such strategies only make sense only as long as there is an assumption of some continuity between the mundane and the divine, or at least some commensurability.

First, the empirical, material (back)ground contracts from (A), where it invariantly infills the negative space bounding figural elements to (B), where it circumscribes individual halos:

Among other indices, this figurative gesture indexes a contraction of divinity (in the fashion of a secondary quality) from (A), the omnipresent ideal, formal ground (that is, the basis) of material reality to (B), its inequitably distributed instantiation. The golden vestige is then further characterized as a secondary quality (and with it, grace,) by representing the halo not as an abstracted sphere, but as a disk in perspective—as if grace empirically radiated about the graced personages on an off-axis Euclidean plane:

[Harries cont’d:] As a new subjectivism began to assert itself in the concern with perspective that Alberti systematized, the use of gold backgrounds had to appear an increasingly hollow convention. And something similar holds for the presupposed analogy of being. This new anthropocentric art had to raise once again the old Platonic question: Given the self-consciousness that finds expression in the adoption of perspective and the transformation of the visible world into subjective appearance, how could art still claim to serve divine reality? Is art not tied by its very essence to appearance?

Lastly, the gold disappears altogether, leaving a vestige of a vestige, the transparent ‘wireframe’ ring to which we are all now accustomed—as in the cartoon convention used to index ‘angelic’ semblance—just as we are accustomed to the sectarian headgear panoply, nonconformity to which has grown no less perilous with our compartmentalized modernization. (George Carlin: “Catholic men and Jewish women—no hats. Catholic women and Jewish men—hats. Somebody’s . . . backwards, don’t you think?” )

Finally, we can gauge the shift in sophistication at a glance by way of Renaissance conceit: Given the full flowering of painterly realism—with deliberate foregrounding of techniques such as forced perspective and anamorphic occlusion—the reintegration of such outmoded tropes as gold ground and pie-plate halos presents a jarring stylistic conflict that verges on what we would now describe as pastiche:

[cont’d:] And the fault would seem to lie not just with one-point perspective but with the visible as such. We stand on the threshold of a conception of art that no longer places the work of art in the service of truth, but reduces it to a kind of entertainment. Similarly, we stand [i.e., in Alberti’s time,] on the threshold of a conception of science that no longer demands of itself adequacy to the things themselves, but is content with a mastery of representations.

—Karsten Harries, 2001

While this threshold was obviously crossed in full by artistic practice, in theory it was crossed less fully by science than by philosophy—namely, we took up Kant’s distinction of irrepresentable noumena (i.e., “the things themselves”) from phenomenal representations, then followed it out by discarding the former and consigning ourselves to the latter. Again, this puts science and philosophy in a bothersome bind while securing privilege of piety for art and religion. In any case, as we historically assess either shift—the esoteric (1) or the exoteric (2)—we cannot forego our current knowledge in order to reckon the impact on the medieval mind. (The go-to author on this point is obviously Eco.) To wit, it would be difficult to imagine never having seen a one-point or two-point perspective rendering in which a 3D volume is fixed to a 2D plane (with or without an explicit vanishing-point or horizon)—if not in the case of a painting, then certainly in a photograph:

from Time-Warner Center to Timeworn Ur-Center

When it comes to the problem of rendering impredicative specifications, we should like to make plain our preference for visuospatial figuration over algebraic notation and written phonology (‘writing’—or if you will, ‘phonography’), be it of an apodeictic or assertoric character. More particularly, the very process of transposition (translation, transcription) from one modality to another brings with it a host of difficulties that must be overcome (or scaffolded to comprehension) by dialectics of investigation and interrogation, visualization and ratiocination, apperception and apprehension. Find here the characteristic range of faculties germane both to premodern mnemotechné (see Mary Carruthers, Frances Yates, Paolo Rossi) and to the medial sciences (optics, harmonics, astronomy, mechanics). Where the latter tactical manifold was absorbed into modern physics, the former went the way of the dodo as expanding colonial empires pressed the machinery of academia into the service of rote repetition clerk manufacturing. (Humph!)

The notion of a homogenous Euclidean space is a modern invention; it largely coincides with the development of perspective, leading to the formation of the Cartesian space and eventually to the discovery of non-Euclidean geometries. The inadequacy of Euclidean space was acknowledged already at the beginning of the nineteenth century by Karl Friedrich Gauss [...] Only much later did Bernhard Riemann convince mathematicians that a non-Euclidean geometry might be the geometry of physical space and that we could no longer be sure which geometry was true.

—Dalibor Vesely, 2004

The rectilinearity apparent in a still photograph is the illusory byproduct of an optical fixation that persists through changes in scale and orientation. That is, unlike in an actual full-scale environment, you can scan your line of sight around a photograph to take in a fixed scenario (be it miniaturized or magnified) without incurring hyperbolic distortion. Conversely, perspectival embedding in an actual full-scale environment presents an exoteric constraint—an invariant ocular warp that incurs continuous contextual correction by the visual cortex—a constraint which, as Borges notes of his coin of no realm, can nonetheless be strictly inverted (but not eliminated) by esoteric visualization:

There was a time when I could visualize the obverse, and then the reverse. Now I see them simultaneously. This is not as though the Zahir were crystal, because it is not a matter of one face being superimposed upon another; rather, it is as though my eyesight were spherical, with the Zahir in the center.

—Jorge Luis Borges

To foreground the Euclidian artifice at work in painterly realism, simply square yourself up to any interior or exterior wall to see how straight the ‘straight lines’ are in your peripheral vision (i.e., without moving your head or your eyeballs). Your first instinct may be reckon such a view as ‘spherical’ in the manner of a fisheye lens (it being a fixation of only slightly less suspicious artifice than a photograph), but it is rather the opposite: As Borges shows, it is only in the ‘mind’s eye’—or more technically (if contentiously), by way of partially modularized (yet tightly integrated) dorsal-ventral visual pathways (working ‘offline’ so to speak)—that the ‘spherical view’ can be apprehended (and only by practiced faculties). More precisely, in an (‘online’) environment, the human optic mechanism is embedded in a planar pseudosphere (or tractricoid, which is sort of like the hyperboloid’s doppelgänger).

Bind two straight sticks together, one across the other, so as to form a big plus sign. Then bind a third stick to the two of them at their intersection, perpendicular to both. The three sticks are perpendicular each to each. Clearly this can be done. But now can a fourth stick be bound to the three, at their intersection, and perpendicular to all three? Clearly not. Here we observe a remarkably simple and basic law of nature: the tridimensionality of space. It is not a law of pure geometry; geometry works nicely for any number of dimensions. It is a physical law. [...] The man in the street, baffled by talk of a fourth dimension, blames his want of visual imagination. He might better credit his insight into nature. He can see, in his mind’s eye, that three is all there are. Barring, of course, double-talk. The mathematician generalizes, as is his wont.

—W. V. Quine, 1986

“You are here.” Meaning what: a point? a line? a plane? a volume? a vector? a tensor? a spinor? a twistor? You are where? Should we push this subtractive specification towards representing (by a 2D planar representation of a 3D Euclidian mesh representative of) not-exactly self-evident 4D (rather, 3 + 1) Minkowski space (spacetime, space-time, time-space), we would find Einstein’s special relativity accounted for by way of a ‘light cone’ (or ‘null cone’). As far as visualizations go, the Minkowski-Einstein light cone is useful, if traumatic. This topographical oddity takes several topological short-cuts, such that two cones (bounding absolute past and absolute future) represent two expanding circles (projected and retrojected at light-speed from the “null” point) that in turn represent spheres.

Well, this is fine and dandy so long as you are willing to accept (with or without consulting Hume) that once the manifold temporality of your lived experience (including both linear ‘run-time’ and non-linear reminiscence) is collapsed to a point, you and the person three feet away from you persist along causally incommensurable timelines. And if unwilling to accept such absurd consequence? “You are now.” Meaning what? Or, you are when?

anamorphic anamneses

While you might not wish to consult Hume, should you find the time to compare William James’s “spurious present” to Gerald Edelman’s “remembered present” and Henri Bergson’s “most contracted level of the past” (complete with cones and such), you may find the similarities and differences illuminating. For the moment, we shall set aside temporal matters in order to examine in brief Wolfgang Wildgen’s schematic spatial treatment of the problems posed by our environmental situation—our embedded perspective, our viewpoint or POV (cf. Jakob von Uexküll’s “Umwelt” and Kurt Lewin’s “Life-World”). In an effort to recapture the mnemonic-artificial Urbild of cognized space within the immediate constraints faced by our earliest human ancestors (Cro-Magnon and Homo Erectus), Wildgen configures what should by now be a familiar topology:

The representation of space has to do with frontiers (their transition) and perspectives. A first perspective is centrifugal, i.e., starting form the self and its basic bodily motions an experienced three dimensional space is cognized: in front of–behind (go), above–below (climb, fall), left–right (grasp with the left hand or right hand). This space of bodily motion with feet and arms defines the immediate space, where objects may be approached, reached and manipulated. The intermediate space depends on man’s ecology; it can be the housing (the cave, abri) or the village; the distal space contains roughly all possible itineraries (of hunting/gathering).

—Wolfgang Wildgen, 2001

As with Cusanus at sea (on Mars, etc.), we cannot help but take our exoteric perspective as a given, to say nothing of our subjectivity: wherever you go, there you are; whenever you go, then you are. So what happens when you go nowhere nowhen? As Cusanus shows, the simple apprehension of one’s own individual place as a movable viewpoint, from which one perceives and processes the world presupposes a reflexive viewpoint—or rather, as Wildgen details, two reflexive viewpoints: without venturing to place ourselves back in Cusanus’s boat shoes, we can grasp the immediate consequence of self-apprehension in Wildgen’s surmise as to our shoeless progenitors:

[cont’d:] The second perspective is centripetal, i.e., the self is seen as the place of effects triggered by external causes. The sky, the horizon (typical points where the sun sets or rises), the favored direction of winds, the ridge of mountains may be the external locus of orientation for the self, who is at the center of a force field or gradient implicit in these delimitations. Many myths and religions refer to this extreme locus of orientation as they interpret the fate of humans as standing under the control of such distant (and often invisible) forces.

Barring, of course, double-talk.

So, to recap our redoubled reflexivity: Our first (esoteric) self-reflexive viewpoint is invoked by/as/in self-awareness tout court —that is, taken as (not ‘understood as’) the awareness of being somewhere prior to, or irrespective of, being somewhen—that is, being in the volume, situated on the plane, rather than traversing the analogously temporal line. (To this, you might compare Heidegger’s Dasein, as the Da ‘Here/There’ of Sein ‘Being’.) Our second (exoteric) self-reflexive viewpoint is that of being-seen from an implicitly indeterminate elsewhere. (To this, you might compare Freud’s locative characterization of the unconscious as “ein anderer Schauplatz”—the ‘other scene’ or ‘stage’.) As such, the first self-reflexive viewpoint is automatically reciprocated in the second—that is, awareness of being-seen follows necessarily—not simply from mere seeing but from meta-awareness that oneself sees. (Finally, should you frown upon Teutonic tongue-twisters, you might compare such Anglo po-mo cog-sci jargon as “higher-order thought” and “theory of mind”.)

Pointing to primitive religious beliefs does not, however, find the most interesting location for theoretical conflict and potential conceptual change [...] one more likely to be affected by emerging cognitive theory in particular [...] is our current self-conception: our shared portrait of ourselves as self-conscious creatures with beliefs, desires, emotions and the power of reason. This conceptual framework [...] is often called “folk psychology” by philosophers, not as a term of derision, but to acknowledge it as the basic descriptive and explanatory conceptual framework with which all of us currently comprehend the behavior and mental life of our fellow humans, and of ourselves.

—Paul Churchland, 1995

“1 + 1 = 3 or more”

—being, as per the late great Josef Albers, sufficient to shew how the Bauhaus Master whittles the sour apple to its seminal core. (“And my architecture” quips Mies van der Rohe, “is almost nothing.” Yet more Germans—what can we say?) The “crucial point not to be overlooked here” (to borrow one of Žižek’s charming tics) is in fact neither a point nor a crux, but is rather an interval. But between what and what do we ‘find’ this intervallic lacuna—this space, this interstice, this gap, this delay, this suspension, this moment, this void, this hole, this blank? Modern English ‘interval’ comes (by way of Old French entrevalle) from Latin intervallum—as from inter- ‘between’ and vallum ‘rampart’—that is, the defensive structure found atop a castle wall or parapet—an architectonic archetype sure to be reiterated in perpetuity by every schoolchild given a box of crayons.

As such, in fully concrete terms, (1)—an interval is the negative ground between two positive figures; in the case of the rampart (unlike Albers’ free-floating pair of paper strips), all three require one materially substantial base. In fully abstract terms, (2)—an interval is a set of all real numbers (integers, fractions, decimals, roots, etc.) defined by a minimum and maximum. In ambiguous terms—i.e., bridging our folk-psychological notions of concrete and abstract, (3)—an interval is found on a graduated scale, such as degrees celsius, which is set to a reference frame (viz., the freezing and boiling points of water) or again, more to the topic at hand (and much like Albers’ free-floating pair of paper strips), (4)—an interval is a period of duration at fixed spatial coordinates (“so-and-so occurs every fifteen seconds”) or a distance in relativistic space-time (“from here to Alpha Centauri”).

While it may seem we’ve reduced ourselves here to either foreclosing “the God of the gaps” or disclosing the Gap of the gods, that would be a false choice on par with fideistic fallibilism. To put it another way, that would be to contrast (A), the terminus ante quem—be it cosmogonic or crepuscular—‘before which’ we may freely presuppose our choice of actantial vincula, with (B), the terminus a quo—be it conceptual or conceptive—‘from which’ we may (somewhat less freely) purport our choice of incommensurable measures between our vestigial being and our thinking of vestiges in the manner of Descartes’ double-Cogito. And for our next trick, we put Eco’s platypus in Lacan’s hat in order to pull out a rabbit.

When time is thus viewed, an enduring solid is seen as spreading out in four dimensions: (1) up and down, (2) right and left, (3) forward and backward, (4) hence and ago. Change is not thereby repudiated in favor of an eternal static reality, as some have supposed. Change is still there, with all its fresh surprises. It is merely incorporated. To speak of a body as changing is to say that its later stages differ from its earlier stages, just as its upper parts differ from its lower parts. Its later shape need be no more readily inferred from its earlier shape than its upper shape from its lower.

—W. V. Quine, 1986

As re (re-)retractions, lest Quine get snagged in a Hegelian bramble, he snips his quips from inscribing a specification (uncontroversially enough) to pricking a deflation (of Perennial purport; Wittgenstein: “If we take eternity to mean not infinite temporal duration but timelessness, then eternal life belongs to those who live in the present”) to deflating his deflation—a sophistical switchback which, while defying his allegiance to evolutionary empiricism (or “naturalized epistemology”), at least serves to alleviate our anxiety. But his is double-talk. In being an(y) organism with a central nervous system (if not a name), one (we) can very well warrant inferences as to its (our) spatially distributed parts—even if not a fortiori of its (our) temporally delimited whole—can it (we) not?

tempus locus hocus pocus

Should we submit to Kant our apophatic “Albers Interval” (if you will) with its constitutive ramparts as a set of coordinates for the ineradicable radix of reflexivity, he would likely deflect it into either (A) the antinomy of TIME+SPACE vis-à-vis unfathomable origin and illimitable extension (i.e., the cosmo(theo)logical problem now circumscribed by contemporary astrophysics), or into (B) TIME as the “purest” transcendental schema of the purest categorial concept, with SPACE as the perpetual runner-up, as the ‘impure’ categorial medium in which temporal schemata are necessarily applied (i.e., the onto(theo)logical problem of circumscription as such ), leaving us to grasp after an immanent apprehension of a transcendental apperception. So suspended, we would have failed to budge Heracleitus from Pythagoras—we would be constituted by (in, of, or as) TIME passing (falling, moving, flowing, trajecting, ‘being-thrown’) through SPACE. (Whatever that means.) So is this (a fortiori, are we, am “I”) an immovable object? A moving target? Both? Neither?

I will take as example a top—a game known to us all, even in practical terms. A boy pitches out a top; and as he does so, he pulls it back with the string which is wound around it. The greater the strength of his arm, the faster the top is made to rotate—until it seems (while it is moving at the faster speed) to be motionless and at rest. Indeed, boys speak of it as then at rest. So let us describe a circle, (B, C), which is being rotated about a point (A) as would the upper circle of a top; and let there be another circle, (D, E), which is fixed. Is it not true that the faster the movable circle is rotated, the less it seems to be moved?

—Nicolas Cusanus, 1460

Perhaps you and yours invest some particular quantity with magical qualities—and mark, this designation includes two-valued logic, three-personed divinities, tetra-grammatons (or winds, or corners), penta-cles (or -grams), hexagrams (unicursal, redoubled, or binar-i-ching), eightfold paths (or fists), ennea-ds (or -grams), dec-alogues (or -imals, Dewey or otherwise), innumerable dozens (and baker’s)—in sum, the integers—but excludes your pi, your phi, your various logs and roots. Whether or not you so invest, so long as we can formalize a continuously variable interval in the manner of Albers, we can scale any schematized subset of rational numbers up or down by an irrational increment—that is, by a problematic (if not ‘fuzzy’) quantity not unlike that by which the trefoil knot differs from the triplicated rings. While this may strike you as an obscure overgeneralization, one common discursive mechanism of this type would be Hegelian sublation (or dialectical synthesis).

While that may strike you as an obscure overgeneralization, some common reflexive, recursive or “reentrant” (Edelman) mechanisms of infinitesimal, integral, convergent, differential or différancial scaling would include: additive reiteration, recapitulation, comprehension, and projective-identification—and contrariwise, subtractive discretion, exception, evacuation, dissociation and disidentification—all of which we find manifest in, as, or by our characteristic contraction—from what we think of (in terms of naïve realism) as the “outside” to what we think of (in terms of folk-psychology) as the “inside” of our met(aphor)onymically bounded nutshell—the n-dimensional manifold in which, allegorically speaking, we could each “count [our]self a king of infinite space, were it not that [we] have bad dreams.” Of what? Of perpetual oscillation across the ramparts of the king’s keep—an inconquerable institute bounded by moat and bulwarked by pitch, whereabout hic sunt dracones and other fire-breathing oddities goaded forth from Borges’ Beastiary.

To put it in terms less dramaturgical and more mechanical, we can apprehend this positive-negative reciprocation in (of, by, as) the continuously gradable delay between action and attribution—that is, as the nominally temporal (1-D, ‘now’) run-time interval recursively propped open by sensorimotor ex-afference between the nominally spatial manifolds of on-line (3-D, outside, here, there) environmentally situated phenomena and off-line (n-D, inside, wherever, elsewhere) cognition—in short, the imagination (or ‘mind’s eye’). Thus, we reduce the problematic to a deceptively simple mechanism—the same mechanism by which we humans are equipped for sophisticated action planning, and crucially (if counterintuitively), by which we are predisposed to imitate rather than emulate the goal-directed actions of our kinfolk, thereby accruing the arbitrary (unnecessary, surplus, excess) surface features constitutive of conventionalized semiosis (aka ‘culture’).

the vicious curvature of “the very closed circle”

And the crucial point not to be overlooked here is that precisely on account of the notion of “absolute knowledge”, Hegel remains entirety within this Kantian horizon of finitude as ontologically constitutive. [...] every epoch experiences itself as the “end of history.” And “absolute knowledge” is nothing other than the explication of this historically specified field that absolutely limits our horizon: [...] Here, we can risk a topological specification of the Kant-Hegel relationship. The structure of the Kantian transcendental field is that of a circle with a gap, since man as a finite being does not have access to the totality of beings: [above, fig. A] However, contrary to the common view, the passage from Kant to Hegel does not consist in closing the circle: [fig. B] If this were the case, Hegel would simply return to a pre-Kantian, pre-critical metaphysics. Hegel does indeed “close the circle”, but this very closure introduces a supplementary loop transforming it into the inner eight of the Möbius band: [fig. C] In other words, Hegel definitely maintains the gap around which the transcendental field is structured: the very retroactivity of the dialectical process (the “positing of presuppositions”) attests to it. The point is just that he displaces it: the external limit preventing the closure of the circle changes into a curvature which makes the very closed circle vicious.

—Slavoj Žižek, 1992

or, going nowhere fast

Viciousness notwithstanding, you may find it worthwhile to compare Žižek’s specification to David Gamez’s (2007) schematization of the usual suspects via juxtaposition of “stable [and] collapsing [and] unstable hermeneutic circles”—to which end both authors offer interesting (if inadequate) diagrams (Žižek, focused argument; Gamez, explicative diagrams). In any case, whether “the circle” (of the ‘transcendental structure of subjectivity’) is considered “open” or “closed” it remains a metaphorical circle, a recirculating uroboros which—given the rotation and duration necessary to configure, conceive, contemplate or comprehend it—circumscribes more of a self-disclosing helicoid (that is, a surface) than a self-foreclosing band (that is, a fragment):

Of course, you can simulate a representation of a fragment of the Möbius surface by way of the cut-and-paste routine (above), such that you can hold in your hand a “one-sided, one-edged” band of paper—but then you would have to imagine infinitely extending the (both) edge(s) such that it intersects itself at every angle. Thus, if we attempt to formally adapt the figure to the Hegelian-post-Kantian foreclosure (that is, to elide or revoke the discursive remove afforded by Žižek’s metaphorical usage), we confront the Möbius specification as a surface boundary with neither an inside to “get outside of” nor an outside to “get inside of”—and that bears more than a passing resemblance to “the set of all sets that do not contain themselves as a member”—so, you can take it up with Zermelo-Fraenkel.

Regarding notional topologies (as per naïve realism) that ascribe an ‘outside’ and ‘inside’ to the n-dimensional phenomenal manifold, we inevitably face such peculiar scenarios as Stanley Cavell (1979) on Wittgenstein: “If I take the space I am in to be outer, I have to imagine for the other an inner space which I could not possibly enter. Which nobody could enter; for he didn’t enter it.” Yet we can also imagine such alternative topographies as concave/convex, whereby, in taking the manifold as (if) a surface, the embedded-situational Umwelt has a ‘convex’ character and the imaginative-intellective Innenwelt has a ‘concave’ character (cf. hyperboloid/pseudosphere). From here, we may consider problematic usages of the mirror as a metaphorical device (again, Eco: “the normal mirror is a prosthesis which does not deceive”) and subject them to distortions of scale and orientation—e.g., as when you look at your reflection in one side of a spoon, and then the other: do these two (malleable, gradable, flexible) ranges of view resolve to fundamentally incommensurable points, or might they simply be stubbornly uncoöperative perspectives? ... ... ...

de ludo globi

Once upon a time, in some out of the way corner of that universe which is dispersed into numberless twinkling solar systems, there was a star upon which clever beasts invented knowing. That was the most arrogant and mendacious minute of “world history,” but nevertheless, it was only a minute. After nature had drawn a few breaths, the star cooled and congealed, and the clever beasts had to die.

—Frederick Nietzsche, 1887

Bruno might have added that this gives still too much significance to the event. The tale would have been better told had Nietzsche spoken not of the most arrogant and mendacious minute, in the pathetic superlative, but of a happening that repeats itself over and over again.

—Karsten Harries, 2001

In what, then, did Bruno and Nietzsche disagree? In return versus no-return; in finitude versus true infinity; in limited power versus true omnipotence. Nietzsche would eventually prefer Eternal Return and perpetual circularity to Bruno’s eternal transcendence of immanence. Thus, for Nietzsche the key symbol of his metaphysics is the ring; whereas for Bruno it has to be the spiral...

—Ramon Mendoza, 1995

...not otherwise than in animals which we recognize as such, [the universe’s] parts are always in continuous alteration and movement and have a certain ebb and flow, always absorbing something from the exterior and emanating something from the interior [viz., cf. nails, fur, wool, hair, skin, hide, etc.] Thus, it is more than plausible that, since everything participates in life, many and innumerable beings live not only within us but also in all composite things; and when we see something which is said to die, we must not believe that thing dies but rather that it changes and terminates its accidental composition and unity...

—Giordano Bruno, c. 1583

Such a view of the cosmos, which has one root in Plato’s Timaeus, makes it difficult to take the individual, and therefore to take death and the need for salvation, very seriously. In proto-Nietzschean fashion, Bruno overcomes the sense of contingency and the nihilism associated with it precisely by denying the existence of a personal God. The other side of this denial is the deification of the cosmos, which is described with adjectives once reserved for God: it is now understood to be necessary.

—Karsten Harries, 2001

Both were grossly misunderstood by their contemporaries and by many of their modern interpreters as well [...] in the end, they both failed, for their radical ethical reevaluation and their total cultural revolution never took place. Moreover, centuries after their disappearance, the bulk of humanity continues to think and behave as if neither one of them had ever existed and had anything of real importance to tell us.

—Ramon Mendoza, 1995

This brings us to the optical limit of our ontopolography, as the foregoing invidious indivisibles cannot be presented but for visualizations more fatiguing for the imagination than Peirce’s QP sucked into Minkowski’s light cone. Taking the “Möbius band”, for example, to attempt to account for the evidently interminable flux (or temporal vicissitude) constitutive of being-semi-conscious (notwithstanding systematic evacuation by noetic “contraction” as per Plotinus, Eckhart, Cusanus, Bruno, Bergson, Beckett, Artaud, Deleuze,,,) you(/it) would not only have to intersect (your/)itself at every angle, it/you would have to draw you/it out over time (that is, by substituting our not-quite-metaphorical “Albers Interval” for an even-less-metaphorical “Möbius Interval”) to circumscribe a sort of anti-helicoidal Möbiunstrosity. Phooey!  Here’s some whirligigs. You do the math.

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