No More Learning

As the vision of poets in Kant, which helps us to find the sublime, this vision, which helps us to find quantum mechanics, may be seen as material in the same sense. It may, however, no longer be seen as formal, unless in terms of (if one may use such an expression) a radically de-formalized form--that is, if we can, phenomenally, and especially geometrically, "see" anything in this way at all. It may not be humanly possible to do so, even though, in contrast to the ultimate constituents of matter or the ultimate effi- cacity of the data in question, the elements constituting these data are available to phenomenological apprehension. We do, however, now
Algebra and Allegory 67
68 Arkady Plotnitsky
treat these marks, even their collectivities, purely "formally" (without "form") rather than in any way configuratively. In particular, we divest them of their classical and hence configurable appearance (in either sense), even though they do form configurations, or what can be so seen in certain circumstances--say, wavelike patterns, or a trace of a particle in a cloud chamber. Even within a single phenomenon, where these marks appear, they must be divested of the possibility of being explained in classical terms and hence of their manifest classical con- figurativity. For example, they should not be seen either as points re- sulting from classically conceived collisions between "particles" and the screen or as forming a classically conceived wave pattern. Neither "picture" corresponds to what in fact occurs. At this stage, even the radical (Derridean) trace-like character of these marks is suspended, although this character will have to be given to these data in order to explain them in quantum-theoretical terms.
This suspension is necessary, and the vision that results is possible, for the following reasons. As we have seen, the mathematical formal- ism of quantum mechanics does not formalize or otherwise describe the configurations, individual or collective, of these traces as such, or for that matter any material physical process in the way classical physics would. Instead, it enables statistical (and statistically very precise) predictions concerning certain collective, even if not individu- al, configurations of material marks in certain circumstances. 19 Inso- far as one can interpret the situation and specifically the quantum- mechanical mathematical formalism as applicable to quantum objects themselves or their interactions with measuring instruments in terms of conventional physical (or indeed any) concepts, such an interpretation can only be allegorical in de Man's sense. First, this "allegory" does not at this level describe any physical configuration or process. Second, it does not (in contrast to a symbol, for example) relate the partial (complementary) descriptions involved to, or allow one to presuppose even in principle, any classical wholeness that would reduce their com- plementary character. Thus, in order for a theoretical formalization and interpretation of quantum physics to take place, these marks, while "visible," have to be divested of any form of mathematical and specifically geometrical representation. Classical physics is largely defined by the possibility of such representation or, more broadly, phenomenalization. 20 In quantum mechanics, neither is possible any longer. This impossibility is reflected in the trace-like or written (in Derrida's sense) character of the marks in question and the allegorical
(in de Man's sense) nature of the interpretation of these traces or of the mathematical formalism of quantum mechanics. All such classical rep- resentations may be seen (with caution) as forms of (aesthetic) ideolo- gy, even when they are divested of everything (tropology, figuration, reflexivity, semantic depth, and so forth) except for mathematical for- malization, including of the kind invoked by de Man in the context of the material sublime. At this point mathematization itself (such as that of the Galilean or Newtonian, or indeed Euclidean, program) may be seen as ideological, even if we see it as purely "formal" or purely "aes- thetic" and as devoid of technical, or standard ideological, dimensions of the practice of mathematics and science. The efficacious processes themselves will, again, be, in Bohr's terms, far beyond the reach of any pictorial visualization, phenomenalization, representation, conception, and so forth. Now, however, the manifest effects, the visible marks, in- volved, too, must be seen (in either sense) as divested of any geometri- cal structure consistent with classical physics. Either individually or collectively, these traces must be seen as allowing for no classical physi- cal description, as would be the case of the "radical formalism," at least in the mathematization and geometrization of science, if not of the Kantian sublime itself.
I cannot consider the subject here, but it may be argued that Heisen- berg's great paper introducing quantum mechanics appears to reflect the process just described. 21 First, it suspends the application of classi- cal physics to quantum data and the very possibility of configuring these data accordingly. Instead it treats them formally (but, again, without giving them form) as material and phenomenal effects, di- vorced from all classical configurativity. His introductory elaborations in the paper itself would suggest nearly as much. Heisenberg does not explore the epistemological consequences of the situation, of which he was only vaguely aware at the time. His main concern was to offer a mathematical formalism that would enable theoretical predictions in the situations where all previous attempts had failed. These conse- quences emerged in the subsequent developments, specifically in yet another great paper by Heisenberg himself, introducing the uncertain- ty relations, and in Bohr's work. 22 Heisenberg's invention of quantum mechanics, however, appears to have been partly enabled by the de- configurative phenomenology just discussed.
From this viewpoint, the ultimate "aesthetics" or "poetic vision" of physics is not that of coherence, harmonious wholeness, and other icons of classical aesthetic ideology, although these may apply at other
Algebra and Allegory 69
70 Arkady Plotnitsky
levels of quantum theory. Instead it is the aesthetics of the radical de- coherence, of the formal without form--the quantum aesthetics, if not the quantum sublime, although "the formal without form" would fit the sublime as well. And yet, for all that, the quantum-mechanical for- malism, or radical formalism or formalization elsewhere, may remain only another form of aesthetic ideology projected into the radical and nonformally formal "configuration" (a term henceforth inadequate) of material marks, if any nonaesthetic or nonideological perception of them is possible. I shall return to this topic later. My main point here is that, once divested of all mathematization and, especially, geometriza- tion, such as that of optics, the formalism (mathematical or conceptu- al) of quantum mechanics, for example, of quantum optics, can only conform to radical formalization. It does not conform to a radical for- malism analogous to the radical formalism of the material sublime, which is parallel to the formalization of classical physics, that is, again, short of "linguistic terms" of understanding it. Linguistically under- stood, it might be seen as subsumed by radical formalization.
Then, reflecting back, the material vision of the sublime would be seen as only appearing (in either sense) to be a classical (and classically architectonic) vision of the mathematization or geometrization of pure optics. In fact, however, as the vision of the sublime qua the sublime, it will have involved radically deconfigurative aspects that are analogous to those found in "contemplating" the data of quantum physics, as just considered. Would this be the vision necessary in order to find the sub- lime, as poets do, if not the vision of the sublime itself? The sublime appears to correspond to a vision of that which always escapes archi- tectonics, geometrization, and so forth, while appearing to be available to them. (We recall that, in contrast to the beautiful, this vision cannot be seen as having an object, but rather as making such an object im- possible. ) By contrast, in the quantum-mechanical vision the material marks constituting quantum data should be "seen," should be made to be seen, as altogether devoid of any conceivable architectonics, with- out phenomenally suggesting this evacuation of the architectonic. Inso- far, however, as it may not in fact be possible to see anything in that last way, this vision comes closer to the sublime, although it may still be different from what either Kant or de Man had in mind. For, on the other hand, if we apply this more radical deconfiguration or disfigu- ration to the sublime itself, we may enact, as it were, the material desublimation of the sublime. The latter would undermine the already "deconstructive" visions and "ideologies" based on the sublime, which
now appear as merely more sophisticated forms of aesthetic ideology. Perhaps once made "more intelligible," "understanding [the materiali- ty of the sublime) in linguistic terms" reveals the un-architectonic un- sublime thus suggested, although, and because, Kleist's and Shelley's disfigurative visions appear to offer its best model.
Although contextualized somewhat differently in relation to the vi- sion of poets ("the way the poets look at the oceans severed from their geographical place on earth"), something of this type is suggested by de Man's remark, via Montaigne. We must, de Man says, consider "our limbs," formally, "in themselves, severed from the organic unity of the body. " "We must, in other words, disarticulate, mutilate the body in a way that is much closer to Kleist than to Winckelman," or one might say deprive them of their geometry (AI 88). Within the ma- terial vision of the sublime thus understood, any, for example, linguis- tic, formalization of such "parts" is a form of "algebra. " Some of its aspects can serve to construct partial and ultimately inadequate alle- gories (some of them geometrical) of the materiality of the "body" in question, both that of the manifest effects (analogously to those of quantum-mechanical marks) or their efficacities. Indeed, the original "parts" or "limbs" are already such allegories, derived from the classi- cal view, and hence as supplementary as the body itself. Accordingly, a more radical disarticulation, mutilation, disfiguration of the (un)body is at stake, even at the level of manifest effects. Following the overall epistemology here discussed, the efficacity of these effects is, again, in- accessible in any way, no more by means of disarticulation, however radical, than by means of articulation. With respect to this efficacity, the dismemberment and disarticulation in question (at the level of the effects) itself reflects only this inaccessibility. This disarticulating dis- memberment of the body will be linked to the linguistic understanding of materiality and specifically to the disarticulation of tropes, as indeed the term (figure? trope? ) disarticulation suggests. 23
De Man's reference to Kleist here and in closing "Phenomenality and Materiality in Kleist" is crucial and confirms my overall point. The question, then, would be whether it is any more possible to experience such a vision or even assign it any geometry than to have any vision the (un)configurations of marks constituting the data of quantum me- chanics. What would such an experience, say, in a vision of a sea, be? Is it possible? Can it be shown to be rigorously impossible? For, to see something as the poets do, however, may correspond to the situation in question, if we can see "anything" here at all, even beyond the radically
Algebra and Allegory 71
72 Arkady Plotnitsky
inaccessible efficacity of all visible effects involved. The allegorical al- gebra enabled by this vision or un-vision would be quite enough in itself. We may call such a (nonvisualizable) sublime or unsublime the algebraic or the algebraic-allegorical sublime, as opposed to the geo- metrical or the geometrical-symbolic sublime. Indeed, the symbol may be argued to always remain geometrical, as is suggested by Coleridge's description of it in terms of translucent geometrical optics--a key starting point of de Man's investigation opened by "The Rhetoric of Temporality" (BI 192). This algebraic-allegorical sublime may corre- spond to the sense Kleist made of Kant's work, from The Critique of Pure Reason on. This sense also reflects subtle gradations of proximities and differences, between perhaps ultimately "algebraic" (and "quantum- mechanical") reason and the ultimately "geometrical" judgment of the sublime in Kant, even if in a Kleist-like reading.
Alternatively, the quantum-mechanical "vision" just outlined and the quantum-mechanical formalization overall may indicate the space of differences, infinitesimal and radical (and sometimes simultaneously both), between Kleist's aesthetic, literary formalization and Kant's philosophical formalization of the vision of the sublime, which self- deconstructs into Kleist's. Either way, here, too, in order to find this more radical formalization, we might want to do what poets do, now also as philosophers. The literary, now defined in accordance with, or indeed as, radical formalization, is part and parcel of this understanding, or reasoning, also in Kant's sense of "reason. " The scheme just out- lined must, again, be seen as fundamentally correlative to the irre- ducibility of techne ? --the technology of measuring instruments, the technology of writing, or of reading, or of painting, such as Ce? zanne's. I would argue that the interaction of materiality and phenomenality, individuality and collectivity, singularity and regularity, in late Ce? zanne, considered by T. J. Clark in this volume conforms and perhaps for the first time introduces this "quantum-mechanical" vision or un-vision in painting. This may indeed be "the [impossible] truth of painting" for Ce? zanne, the truth of painting as ultimately algebra, not geometry. 24 This--that is, always working with and through techne ? and the ir- reducibility of techne ? --is what poets and quantum theorists with a po- etic bent do, and how they ultimately find everything, for example, the sublime or quantum mechanics.
I would now like to discuss further features of radical formaliza- tion, first, the question of randomness and chance. It is famously in- voked at the end of "Shelley Disfigured," where de Man writes: "The
Triumph of Life warns us that nothing, whether deed, word, thought or text, ever happens in relation, positive or negative, to anything that preceded, follows, or exists elsewhere, but only as a random event whose power, like the power of death, is due to the randomness of its occurrence" (The Rhetoric of Romanticism 122; emphasis added). This formulation reflects the nonclassical concept of chance as correla- tive to the irreducible lawlessness--singularity--of individual entities, such as random and discontinuous ("quantum") events here invoked. This concept of chance links the thought of earlier figures, such as Kleist and Shelley, to the twentieth-century thinking, mathematical- scientific (specifically in quantum physics) or philosophical, to which de Man's and related nonclassical concepts of chance belong. It is worth, however, recalling the classical understanding of chance before defining the nonclassical understanding of it.
Classically, chance or, more accurately, the appearance of chance is seen as arising from our insufficient (and perhaps, in practice, unavail- able) knowledge of a total configuration of forces and, hence, of understanding a lawful necessity always postulated behind a lawless chance event. If this configuration becomes available, or if it could be made available in principle (it may, again, not ever be available in prac- tice), the chance character of the event would disappear. Chance would reveal itself to be a product of the play of forces that is, in principle, calculable by man, or at least God. In other words, in practice, we have only partially available, incomplete information about chance events, which are nonetheless determined, in principle, by a complete architec- ture of necessity behind them. This architecture itself may or may not be seen as ever accessible in full, or even partial, measure. The presup- position of its existence is, however, essential for and defines the classi- cal view as both causal and realist, for this assumption of the ultimate causal architecture underlying randomness and chance brings classical causality and classical reality together. For example, if we cannot exactly--rather than only in terms of probabilities--predict how the dice will fall, or fully explain why a particular outcome has occurred, it is because the sum total of all the factors responsible is in practice un- available to us (from a particular movement of a human, or perhaps divine, hand to minute irregularities in the material makeup of the dice themselves). In principle, however, a throw of dice obeys the laws of classical, Newtonian physics (or else chaos theory, which would not, however, change the essence of the point, since chaos theory is classical at bottom). If we knew all such factors, we could predict and explain
Algebra and Allegory 73
74 Arkady Plotnitsky
the outcome exactly by using these laws, which would describe both individual and collective behavior, and (lawfully) correlate them, in ac- cordance with the definition of classical law.
The nonclassical understanding of chance, correlative to radical for- malization, is fundamentally different. Nonclassical chance is irre- ducible and irreducibly lawless not only in practice (which may be the case classically as well), but also, and most fundamentally, in principle. There is no knowledge, in practice or in principle, that is or will ever be, or could in principle be, available to us and that would allow us to assume chance to be the product of the imperceptible workings of ne- cessity behind it. Nor, however, can one postulate such a causal/lawful structural necessity as unknowable (to any being, individual or collec- tive, human or even divine), but existing in and by itself outside our en- gagement with it. This qualification is crucial, since, as just explained, certain forms of the classical understanding of chance allow for or are even defined by the latter assumption. The nonclassical chance is not only unexplainable in practice and in principle, but is also irreducible in practice and in principle. It is irreducibly lawless.
We recall that nonclassical formalization or law does not account for individual events (again, understood as phenomenal effects) in the way classical formalization does, thereby also correlating individual and collective configurations they consider. As we have seen, classical formalization is defined by this concept of law. By contrast, it is a law of nonclassical formalization that individual events are, generally, not comprehended by its laws or by law in general, certainly not in the way they would be in classical physics. Nonclassical formalization allows for the concept of individuality or discontinuity at the level of the phe- nomenal effects. Indeed, taken to the limit of the irreducible singularity of the individual, this concept defines nonclassical formalization. At the same time, however, it offers no law that would enable us, in prin- ciple (rather than only in practice), to predict with certainty the out- come of individual events or when some of them may occur, or their ultimate nature and emergence (at least only certain, partial aspects of such events). Just about any outcome is possible, anything can happen in any given case, and each case is ultimately unique, singular. In this, the nonclassical world (even that of quantum mechanics) is very much like life, to which--or, one might say with Shelley, to the triumph of which--we must ultimately submit.
This world, however, also contains more order and richer orders than the classical one. Statistical or other (collective) patterns do emerge
within it--at the level of phenomenological effects (which is perhaps as much as one can hope for, even classically). Indeed, this emergence of patterns, or that which allows for patterns, is what is truly strange about life, or, again, quantum physics--in view of the irreducible uniqueness of all (rather than only certain) individual events. It also follows that in these circumstances the individual effects involved are in turn reciprocally, if equally enigmatically, affected by this "con- spiracy" and, hence, by the collective, even though--this is what is most mysterious and enigmatic--each remains singular and lawless even in these circumstances. Or rather, they would so remain, if one could, in these circumstances, trace them as individual effects. In other words, such lawful collectivities and lawless individualities appear al- ways to be mutually exclusive or, in Bohr's terms, complementary. In the nonclassical world, the irreducibly unknowable coexists with a greater and more multiple knowledge, orders, and so forth. All our ac- counts, or indeed conceptions, of what gives rise to these orders or these dis-orders or non-orders (they cannot be considered as disorders either) can only be ultimately inadequate allegories, correlative to the formal deformalization of material effects. But then, as de Man argues, this ultimately "algebraic" quality has defined the practice of allegory all along, in part as the practice of reading, reading books or nature, or both, and as each other. In this sense, all allegories may well be the al- legories of (epistemologically nonclassical) reading.
And yet, not only possible but very real events--such as, say, those described in Kleist's Penthesilea or Shelley's The Triumph of Life-- leave us no choice, any more than quantum physics leaves physicists a choice here. These events occur and their underlying, or indeed un- underlying, unreality--the lack of any conceivable reality underlying them--does not make them any less real, indeed makes them more real, than any (classical) reality we can conceive of. In de Man's terms, any such event manifests (the singularity of) radical material occurrence-- "an occurrence, which has the materiality of something that actually happens, that actually occurs" ("Kant and Schiller," AI 132). I would argue that at stake here is the radical materiality and/as singularity of events, corresponding to nonclassical epistemology. The latter, accord- ingly, would also disallow one to strictly locate when and where such events actually occur. What actually occurs does occur, but the point or moment itself of this occurrence is indeterminable. It always takes place, in Lucretius's remarkable and remarkably precise phrase, "incer- to tempore--incertisque loci" (at an uncertain time and at an uncertain
Algebra and Allegory 75
76 Arkady Plotnitsky
place), and, we may add, at an indeterminate juncture of the efficacious processes of occurrence. 25
It is, as I have stressed from the outset, crucial that, nonclassically, irreducible lawlessness defines and makes singular all constitutive indi- vidual events, including those composing (what is perceived as) ordi- nary events, rather than only certain absolutely extraordinary events-- such as, say, Penthesilea's final encounter with Achilles, the encounter that dismembers his body and Penthesilea's mind and language; Rousseau's encounter with "the shape all light"; or Shelley's dreamlike encounter with Rousseau, in The Triumph of Life. Ultimately, every event, specific configuration, or historical trajectory will prove to be unique--irreducibly singular and lawless. Or else each can always be nonclassically reconfigured as composed of certain singular, lawless in- dividual elements, on the one hand, and of certain lawful collectivities on the other; that is, in some situations, lawful individual elements of the classical type may, once refigured nonclassically, always be decom- posed into lawless individual constituents. Viewed from this perspec- tive, such elements will no longer be subject to classical law as such but instead will belong to a nonclassical lawful collectivity composed by lawless individual elements. It is worth noting that this decomposition need not be unique, even in quantum physics. From the perspectives centered on lawful collectivities, this lawlessness of the individual may not matter or be perceived at all, since certain patterns, statistical or other, allow us to disregard or make us miss this singularity and law- lessness in the classical, or classical-like fashion. To avoid this, we must, "quantum-mechanically," deconstitute this pattern as classical and complementarily engage both the perspectives--that of lawful col- lectivities and that of lawless, singular individuality. The classical view, as Blake understood so well, erases particulars as particulars (i. e. , ulti- mate individual constituents of such configurations), either by way of general concepts or by means of ethical, political, and aesthetic prac- tices. Indeed, the present argument as a whole may also be seen as a disfigurative reading or at least an extension of Blake's idea of "minute particulars. "
We can now give a more radical and more rigorous meaning to de Man's conclusion in "Shelley Disfigured. " "The Triumph of Life warns us that, [ultimately], nothing [and not only certain things], whether deed, word, thought or text, ever [and not only sometimes] happens in relation, positive or negative, to anything that preceded, follows, or exists elsewhere, but only as a random event whose power, like the
power of death, is due to the randomness of its occurrence," and hence to the radical, irreducible singularity and discontinuity of this event, and ultimately any individual event or particular historical trajectory. As it makes the allegory irreducible in any representation, phenome- nalization, knowledge, and so forth, death, or life-death, becomes a model or, better, allegory, perhaps the allegory, for the structure of every event of life. We may indeed define Romanticism in terms of this disassembling magnification or, more accurately (it is, again, not a question of magnifying the small), radicalization of any configuration, classically individual or classically collective, into the irreducibly sin- gular, unique constituents--minute particulars--and the nonclassical reconstitution or reassemblage of such minute particulars (from a nec- essarily different perspective or set of perspectives) into richly ordered multiplicities. We may, accordingly, also speak of radical organization along with radical lawlessness and singularity. We must, however, keep in mind the very different epistemological status of the nonclassical patterns and laws, the "algebra" and "allegory" of their functioning, as opposed to, one might say, the symbolic "geometries" of classical thought.
This is, then, what such literary texts as Kleist's and Shelley's, or such philosophical texts as Kant's and Hegel's, or Ce? zanne's paintings, do. They offer us new--efficaciously nonclassical--patterns, orders, or laws, and un-patterning, unordering, and unlawfulness, and new ways in which these relate to each other. Of course, we need to read and understand these texts in great detail in order to study how all this takes place in them. Such texts and such readings also question the philosophical, aesthetic, historical, and other roles and limits of the nonclassical. For, as I said, the latter may ultimately prove to be yet an- other case of aesthetic (or counteraesthetic) ideology. These complexi- ties are, I think, the main reason why de Man does not close "Shelley Disfigured" with the randomness of death as the final warning of Shelley's poem. Instead he adds the following:
[The poem] also warns us why and how these events [that is, all events as singularities] then have to be reintegrated in a historical and aesthetic system of recuperation that repeats itself regardless of the exposure of its fallacy. This process differs entirely from the recuperative and nihilis- tic allegories of historicism [or aestheticism]. If it is true and unavoid- able that any reading is a monumentalization of sorts, the way in which Rousseau is read and disfigured in The Triumph of Life puts Shelley
Algebra and Allegory 77
78 Arkady Plotnitsky
among the few readers who "guessed whose statue those fragments had composed. " Reading as disfiguration, to the very extent that it resists historicism [or aestheticism] turns out to be historically more reliable than the products of historical archeology [or aesthetic ideology]. To monumentalize this observation into a method of reading would be to regress from the rigor exhibited by Shelley which is exemplary because it refuses to be generalized into a system. (The Rhetoric of Romanticism 122-23; emphasis added)
The last clause must, I think, be read as indicating that Shelley's rigor refuses to be generalized into a system that would not allow for the nongeneralizable. Shelley's poem possesses a great power of gener- alization and offers us very general aesthetic, historical, and political laws, a whole constitution even. So does Kleist's aesthetic formaliza- tion in U? ber das Marionettentheater. In question in both cases, how- ever, are nonclassical organizations of "fragments" and (when pos- sible) the "algebra" and "allegory" of their nonclassical formalization. The latter relates to no underlying pattern ("geometry") of wholeness, and yet (similarly to quantum mechanics) it offers us a better guess as concerns the history or aesthetic (or otherwise cognitive) structure of the configuration in question. In question, again, is only the impossibili- ty of the ultimate knowledge, the knowledge of the ultimate efficacity of the events in question and at bottom of all events. By putting this impossibly into play, however, both a greater richness and a greater re- liability of a "guess" become possible as well. But then (which may be the main point of de Man's last sentence) each nonclassical reading may itself remain unique, singular. The lessons of such texts or of their reading or of their grouping together (which apply to de Man's own texts, such as those assembled, in either sense, in Aesthetic Ideology) are complicated accordingly. Thus, de Man's essay (via Shelley's poem) and his work in general teach us a lesson of great caution, or indeed issue a stern warning (the word that occurs twice in this passage). The success (or a failure) of any strategy, general (such as methodological) or singular, classical or nonclassical, is never guaranteed, except per- haps that, as the saying goes, "in the long run we are all dead. " In other words, ultimately nothing survives, even though in the shorter run (which may be long, even indefinitely long, but is always finite) certain strategies, such as that of Shelley's disfiguration, may be more effective, but even this cannot be certain. These are inevitable conse- quences, "effects," of the nonclassical efficacity here considered.
In short, the texts in question offer us allegories of nonclassical knowledge, which may also be seen as "reading," and hence the texts in question as "allegories of reading" in this sense as well. By a quali- fied analogy (considered earlier) with quantum mechanics, such texts may be seen as material signifying "surfaces" in which certain peculiar material effects manifest themselves and make possible certain mani- fest phenomenological effects. As manifest, these effects may be pro- cessed (and the corresponding linguistic clusters read) classically. Some of these effects, however, and, especially, their overall configuration are meaningless, and some of them (certain, to borrow Gasche? 's phrase, "linguistic atoms") are meaningless otherwise, and remain, or some- times are made, meaningless nonclassically. In a nonclassical reading, all of these effects will be convertible into a nonclassical configuration of singular marks or, again, "linguistic atoms," although in practice this program is difficult to follow through. This is why such texts defy classical reading and resist any reading. This resistance even to read- ing is ineliminable and defines nonclassical reading or knowledge. 26 Such texts also enact both nonclassical epistemological configurations and their, inevitably allegorical, analytical explorations. They non- classically and multiallegorically read themselves. Kleist's essay, by its very structure, also enacts the nonclassical grouping of particular texts and is read by de Man as such. It introduces textual particulars/ singularities at all levels, from the "linguistic atoms" of the signifiers to large textual and narrative units, which allegorize each other. In other words, the texts in question allegorize their own reading, which can it- self only be allegorical. In the process they offer us allegories of non- classical reading and, hence, teach us the latter and/as nonclassical knowledge.
De Man, in a nonclassical ensemble of his own individual texts, reads these texts as such "allegories of reading," partly classical and partly (and most fundamentally) nonclassical, partly general and partly unique, and so forth. For de Man, nonclassical configurations can only emerge by way of reading, each such reading being, again, unique, rather than in terms of a (independent) conceptual architecture. These readings do contain and enable the latter as well and make it nonclassical. De Man opens his reading of Kleist with a quotation from Schiller:
I know of no better image of a beautiful society than a well executed English dance, composed of many complicated figures and turns. A
Algebra and Allegory 79
80 Arkady Plotnitsky
spectator located on the balcony observes an infinite variety of criss- crossing motions which keep decisively but arbitrarily changing direc- tions without ever colliding with each other. Everything has been arranged in such a manner that each dancer has already vacated his po- sition by the time the other arrives. Everything fits so skillfully, yet so spontaneously, that everyone seems to be following his own lead, with- out ever getting in anyone's way. Such a dance is the perfect symbol of one's own individually asserted freedom as well as of one's respect for freedom of the other. (Friedrich Schiller, Aesthetic Education 300; The Rhetoric of Romanticism 263; emphasis added)27
This is, in present terms, a classical and classically "geometrical" description, or at least a description that allows for a classical reading. As such it can be, and is by de Man, contrasted to Kleist's nonclassical "algebraic" allegories (which may also be juxtaposed to Schiller's "symbol" here), which disallow classical readings.
Schiller's passage and his related elaborations are considered by de Man both in terms of the aesthetic formalization they offer and as Schiller's points apply to the formal structure of Schiller's text itself. The same strategy will be applied to Kleist's U? ber das Marionettentheater, with an exposure of the nonclassical character of the text and of its self-reading as an out- come. After a complex analysis, de Man arrives at a dance that is very different from the "strictly ballroom" dance of Schiller:
We have traveled some way from the original Schiller quotation to the mechanical dance, which is also a dance of death and mutilation. The violence which existed as a latent background in the story of the ephebe and of the bear now moves into full sight. One must already have felt some resistance to the unproblematic reintegration of the puppet's limbs and articulations, suspended in dead passivity, into the continuity of the dance: "all its members (are) what they should be, dead, mere pendula, and they follow the law of pure gravity. " (The Rhetoric of Romanticism 288)
The invocation of Newton's law of gravity, the paradigmatic classi- cal physical law, is of much interest and significance in the context of the present essay and in general. Both the question of the classical laws of physics and, hence, the formalization of nature, and how classical such formalization can in fact be are at stake. I cannot pursue these subjects here. I shall, however, return to the question of "falling," physically the defining phenomenon of gravity. In Einstein's general
relativity, his theory of gravitation, the fall is merely an aspect of the geometry of a space curved by gravitation. The analysis of this space, however, involves a very complex "algebra" (of the so-called tensor calculus) and the technology of rulers and clocks, which would open yet another chapter in the history of the book of nature, the role of al- legory in physics, and the reading of de Man. 28 De Man continues:
The passage is all the harder to assimilate since it has been preceded by the briskly told story of an English technician able to build such perfect mechanical legs that a mutilated man will be able to dance with them in Schiller-like perfection. "The circle of his motion may be restricted, but as for those available to them, he accomplishes them with an ease, ele- gance and gracefulness which fills any thinking mind with amazement. " One is reminded of the protests of the eyeless philosopher Saunderson in Diderot's Lettre sur les aveugles when, to the deistic optimism of the Reverend Holmes, disciple of Newton, Leibniz and Clark, he opposes the sheer monstrosity of his own being, made all the more intolerable by the mathematical perfection of his highly formalized intellect: "Look at me well, Mr. Holmes, I have no eyes. . . . The order (of the universe) is not so perfect that it does not allow, from time to time, for the produc- tion of monsters. " The dancing invalid of Kleist's story is one more vic- tim in a long series of mutilated bodies that attend on the progress of enlightened self-knowledge, a series that includes Wordsworth's mute country-dwellers and blind city-beggars. The point is not that the dance fails and that Schiller's idyllic description of a graceful but confined free- dom is aberrant. Aesthetic education by no means fails; it succeeds all too well, to the point of hiding the violence that makes it possible. (The Rhetoric of Romanticism 288-89)
At stake, then, is the possibility of formalization, aesthetic or other, under the condition of the radical, lawless, singularity and deformity-- monstrosity--that is quite manifest, materially and phenomenally. Both singularity and law--formalization--and their relationships and conflicts take a very radical form, parallel to the radical disfigurations of Shelley's The Triumph of Life. There is also a revealing textual par- allel. Here de Man invokes "a long series of mutilated bodies that at- tend on the progress of enlightened self-knowledge. " The essay on Shelley asks about our (according to the present analysis un-Romantic) aesthetic, historical, and other formalization of Romanticism: "For what we have done with the dead Shelley, and with all other dead bodies that appear in Romantic literature--one thinks, among many others,
Algebra and Allegory 81
82 Arkady Plotnitsky
of the 'dead man' that ' 'mid that beauteous scene / Of trees, and hills and waters, bold upright / Rose with his ghastly face . . . ' in Words- worth's Prelude (V. 470-72)--is simply to bury them, to bury them in their own texts made into epitaphs and monumental graves" (The Rhetoric of Romanticism 121). Thus, the Empedoclean algebra-allegory of the dismemberment of the body, (re)thought or (re)allegorized in "quantum-mechanical" terms, now applies within very broad limits. It can be further correlated with the quantum-mechanical allegories in- volved in the optics of Shelley's poem: the wave and particle imagery there; the manifest quantum-like discontinuity of events and textual atoms; the divestiture of marks and traces from all architectonics so as also to reveal their inaccessible (material efficacity); the radical materi- al aesthetics; the collapse of realism and causality; the "algebraic" and "allegorical" nature of whatever patterns or forms of order are left to us; and so forth. In other words, the nonclassical features of quan- tum physics and Shelley's poetic epistemology can be assembled and brought together in reciprocal allegories. 29 De Man's reading does not do this but is in part made possible by these reciprocities, as considered earlier in relation to quantum physics and its nonclassical formalization.
De Man explores the "algebra" of "the mutilated body" at some length in his late essays. The deepest and most significant instance may well be his analysis, considered earlier, of the Kantian architectonics in the third Critique. Accordingly, I shall only offer a few supplementary points. In a parallel gesture to his Kleist essay (cited by de Man), de Man invokes Diderot's Lettre sur les sourds and les muets in consider- ing the allegorization of the faculties of reason and imagination in terms of both the anthropomorphized dramatic conflict and the sacrifi- cially mutilated body. The invocation has Dionysian overtones and an invocation of the figures of Antigone and Iphigenia (AI 86-87). Then, he proceeds, via Kleist and Kant's first Critique, to a reading of Kant's architectonics and its self-de-architectonization in the Empedoclean terms of a mutilated body. The conclusion offers extraordinary elabo- rations on the allegorical algebra of Kant's text. De Man writes: "to the dismemberment of the body corresponds a dismemberment of lan- guage, as meaning-producing tropes are replaced by the fragmentation of sentences and propositions into discrete words, or the fragmenta- tion of words into syllables or finally letters" (AI 89; emphasis added).
One thus encounters the workings of radical materiality, or/as sin- gularity, both in the world and in the text. It would, however, be a mis- take to see both as merely (if at all) mirroring or mapping each other,
as de Man's usage of corresponds here might suggest, but should not. (It is difficult to be certain given the complexities of the concept and the very signifier of "correspondence" in de Man. )30 Instead, insofar as one wants to or can approach the world by way of a text (or a text by way of reading), the dismemberment or "decoherence" of language-- the divergence, ultimately irreducible and uncontrollable, of the mean- ing of figures, tropes, signifiers, and so forth, indeed of whatever carries meaning in a given text--manifests the irreducible inaccessibili- ty of the world or life through peculiar configurations of material and phenomenological effects. Accordingly, analogously to quantum- mechanical epistemology, the dismembered, decohered language or representation (i. e. , the configuration of the corresponding phenome- nal effects) does not map or otherwise represent them any more than "coherent" language and representations do, or reading represents a text. However, decoherent representations or allegories appear to be better suited to relate, via the algebra of allegory, to the world and life, or to read the kind of texts in question here. One might say that the radical (material) singularity of individual events of life and the radical inaccessibility of their efficacity find their proper expression or allegory in this circumstance of the dismemberment or decoherence of language and tropes.
Aesthetic formalization as radical formalization and the overall epistemological machinery in question also become, in an antithetical parallel with Schiller's classical text, enacted in Kleist's essay, at the level of figures or tropes. On the one hand, there is a certain "collec- tive" semantic field within which these figures and tropes function and which--that is, a more or less shared meaning or more or less coherent set of meanings--they obey. On the other hand, once rigorously con- sidered individually, or, again, in a certain ultimate decomposition, these figures and tropes can no longer be fully subsumed by such a meaning or a coherent configuration of meanings. Or, in the terms in- troduced earlier, they begin to decohere. Accordingly, one speaks of (an enactment of) a decoherence of figures and tropes, or of all lan- guage, in a nonclassical text, such as Kleist's, or Shelley's, or Kant's, if in the latter case, to some degree, against other forces, conceptual and tropological, of Kant's text. 31 This decoherence or dissemination (in Derrida's correlative sense) defines the functioning of virtually all fig- ures and tropes in these texts. They give the materiality of the signi- fiers, "linguistic atoms," a formal aesthetic structure or un-structure we encounter in the case of quantum mechanical marks, as considered
Algebra and Allegory 83
84 Arkady Plotnitsky
earlier. Or rather, the materiality of the signifier in de Man's sense is this un-structure, which then requires a very different "algebra" of for- malization. De Man writes:
[W]hen, by the end of the tale, the word Fall has been overdetermined in a manner that stretches it from the theological to the dead pendulum of the puppet's limbs to the grammatical declension of nouns and pro- nouns (what we call, in English, the grammatical case), then any com- posite word that includes Fall (Beifall, Su? ndenfall, Ru? ckfall (#46) or Einfall) acquires a disjunctive plurality of meaning.
C's story of the puppets, for instance, is said to be more than a ran- dom improvisation: "die A? usserung schien mir durch die Art, wie er sie vorbrachte, mehr als ein blosser Einfall. " As we know from another narrative text of Kleist ["U? ber die allma? hliche Verfertigung der Gedanken beim Reden"], the memorable tropes that have most success (Beifall) occur as mere random improvisation (Einfall) at the moment when the author has completely relinquished any control over his meaning and has relapsed (Zuru? ckfall) into the extreme formalization [emphasis added], the mechanical predictability of grammatical declension (Fa? lle).
But Fa? lle, of course, also means in German "trap," the trap which is the ultimate textual model of this and of all texts, the trap of an aesthetic education which inevitably confuses dismemberment of language by the power of the letter with the gracefulness of dance. This dance, re- gardless of whether it occurs as mirror, as imitation, as history, as the fencing match of interpretation, or in the anamorphic transformations of tropes, is the ultimate trap, as unavoidable as it is deadly. (The Rhetoric of Romanticism 289-90)
In introducing "the dismemberment of the body" toward the end of "Phenomenality and Materiality in Kant," de Man speaks of the word Glieder in Kant as "meaning members in all the senses of the word, as well as, in the compound Gliedermann, the puppet of Kleist's Marionettentheater" (AI 88). In the same paragraph de Man adds a playful reference to Montaigne's "cheerful" invocation of "Monsieur ma partie," further extending the multilingual decoherence--or again, coherence-decoherence--of tropes by dismembering all members in- volved in their constitution. "Fall" is a decisive figure and concept in Kleist, including in defining any stability, formal--linguistic or mathe- matical--or physical, for example, monumental. It is equally decisive for Shelley or Keats (whom de Man discusses in this context in "The Resistance to Theory"), or de Man, who brings all three together, al-
though, interestingly, he does not consider "fall" (or, again, dance), as he could, in Shelley, in the way he does in Kleist or Keats. It would not be possible to consider here the relevant physics, for example, the way gravity bends even light itself (which would bring all three figures and texts together in yet another way). These connections must be relevant to de Man's reading, even if only because from Newton to Einstein and beyond they changed our sense of fall or (they are perhaps ultimately the same) the world. Kleist once said of the arch, another great figural model or allegory: "the arch stands because all the stones want to plunge at the same time," and, I would add, with the preceding analy- sis in mind, each following its own trajectory. We know, of course, that a random, lawless event, such as an earthquake in Chile, can bring the arch down in any event. What Kleist tells us here, however, is that even the standing arch is a kind of dance in a gravitational field. We all know or assume, naively, that, in dancing, a fall is the least graceful event, or the least graceful--and the least formalizable--form of dance. It is more difficult to realize, as Kleist did looking at dancing mari- onettes, that dance is perhaps only a graceful form of falling (always commanded by many a gravitational field of our life, or death) and that grace itself is, in each case, a very singular, and very difficult, com- bination of fall and dance, just as is the grace of Kleist's or Shelley's writing--their dancing pens, without ever falling, except as a form of dance, albeit on thin ice. As Nietzsche tells us, however: "Thin ice is paradise for those who skate with expertise. "
NOTES
1. The use of the term concept requires caution here, especially in applying it to Paul de Man's work. In particular, it may not be possible to "abstract" these concepts from the thought and text of figures involved or indeed to make them "abstract"--free of particularities or even singularities, or, in de Man, (the practice of) reading. Although Jacques Derrida's "assemblages" ("neither terms nor con- cepts"), such as, most famously, diffe? rance, or "concepts" as defined by Gilles Deleuze and Fe? lix Guattari in What Is Philosophy? , trans. Hugh Tomlinson and Graham Burchell (New York: Columbia University Press, 1993), offer better mod- els, de Man's practice remains unique in this respect. Andrzej Warminski, "'As the Poets Do It': On the Material Sublime," and J. Hillis Miller, "De Man as Allergen" (both in this volume), and Rodolphe Gasche? 's The Wild Card of Reading: On Paul de Man (Cambridge: Harvard University Press, 1998), offer further guidance.
2. I will not be able to discuss in sufficient detail the secondary literature on de Man and other key authors to be considered here, for example (to give a very incomplete list), by such scholars as Jacques Derrida, Werner Hamacher, Rodolphe Gasche? , Carol Jacobs, Peggy Kamuf, J. Hillis Miller, and Andrzej Warminski. By
Algebra and Allegory 85
86 Arkady Plotnitsky
the time one comes to other figures the list of pertinent commentaries becomes practically interminable, although Jacques Derrida's, Jean-Franc? ois Lyotard's, and Jean-Luc Nancy's work on Kant is especially significant here, and especially diffi- cult to put aside. I also bypass two related topics--Derrida's analysis of "law," "event," and "singularity," including in his writings on de Man (and commentaries on these topics in Derrida by, among others, Richard Beardsworth, Rodolphe Gasche? , and Samuel Weber), and Gilles Deleuze's approach to these subjects (quite different from both that of de Man and that of Derrida). I am also grateful to Jacques Derrida, Rodolphe Gasche? , Samuel Weber, and the editors of this volume for helpful discussions.
3. Paul de Man, Aesthetic Ideology, ed. Andrzej Warminski (Minneapolis: University of Minnesota Press, 1996), 119-20; hereafter AI.
4. The epistemology becomes classical once such exclusion takes place. This point is crucial to Derrida's reading of Kant in "Economimesis" (Diacritics 11:3 ([1981]: 3-25).
5. An analogous argument would apply to other pairs of that type, such as the general and the particular, which similarly figure in de Man's work.
6. For Gasche? 's view of de Man's epistemology, see, especially, The Wild Card Of Reading (108-13, 181-83), and of formalism in de Man, the chapter "Apathetic Formalism" (91-113).
7. This point indicates that the rhetoric of allegory in de Man is indeed the rhetoric of temporality. I cannot consider the question of temporality here, al- though it is crucial in de Man and significant in quantum theory.
8. I am not sure to what degree one can speak of "materiality without matter" in de Man, as Derrida suggests in his "The Typewriter Ribbon: Limited Ink (2)" (in this volume). Some aspects of de Manian "materialism" may be conducive to such a view. However, the material visions of Aesthetic Ideology, including that of "the material sublime," as considered by Warminski's in "As the Poets Do It," appear to suggest that a certain economy (inscription) of matter, analogous to the general economy (also in Bataille's sense) of Derrida's diffe? rance (Positions, trans. Alan Bass [Chicago: University of Chicago Press, 1980], 64), is at stake in de Man's work, insofar as this economy relates to the ultimately inaccessible here in ques- tion, which makes "matter" yet another ultimately inadequate term and concept, perhaps having less strategic force than "materiality. "
9. Bohr appears to apply the term phenomena to the material configurations in question themselves rather than to their representation or phenomenalization. His thinking on the subject is, however, quite subtle and is closer to the present under- standing, certainly in terms of the epistemology at stake. I have considered this point and Bohr's quantum epistemology overall in a number of previous articles and books, to which I refer here and throughout this discussion, most pertinently, "Techno-Atoms: The Ultimate Constituents of Matter and the Technological Con- stitution of Phenomena in Quantum Physics," Tekhnema: Journal of Philosophy and Technology 5 (1999), and Complementarity: Anti-epistemology after Bohr and Derrida (Durham, N. C. : Duke University Press, 1994). For Bohr's own presenta- tion of these ideas, see his essays in The Philosophical Writings of Niels Bohr, 3 vols. (Woodbridge, Conn. : Ox Bow Press, 1987), hereafter referred to as PWNB.
10. On this point I refer again to Complementarity and "Complementarity,
Idealization, and the Limits of Classical Conceptions of Reality," in Mathematics, Science, and Postclassical Theory, ed. Barbara H. Smith and Arkady Plotnitsky (Durham, N. C. : Duke University Press, 1997). The connections with relativity emerge in view of the following key aspect of Einstein's theory (whose connections to quantum epistemology were especially significant for Bohr). Rather than being given independently of our instruments of observation, such as rulers and clocks, and, then, represented by means of these instruments, as Newtonian physics assumes, space and time become "products" or effects of instruments. In other words they are products of the technology of observation (and, in more complex ways, of our theories) and indeed represent or embody experimental and theoreti- cal practices.
11. I have considered this subject in "All Shapes of Light: The Quantum Mechanical Shelley," in Shelley: Poet and Legislator of the World, ed. Stuart Curran and Betty Bennett (Baltimore: Johns Hopkins University Press, 1995).
12. Paul de Man, "Shelley Disfigured," in The Rhetoric of Romanticism (New York: Columbia University Press, 1984), 94.
13. Most immediately, the latter express strict quantitative limits (defined by Planck's constant, h) upon any exact simultaneous measurement of both such com- plementary variables. In Bohr's interpretation, however, the uncertainty relations manifest the impossibility not only of simultaneous measurement but the simulta- neous determination or unambiguous definition of both such variables at any point. Once again, not even a single such variable can ever be unambiguously as- cribed to quantum objects themselves.
14. This statement cannot be seen as strictly defining allegory, which, as de Man says on the same occasion, is difficult to do (AI 51). If, however, there could be one (or any) such definition, the formulation just cited appears to come as close to it as possible. The feature itself indeed appears to characterize the practice of allegory, at least from Dante on. Galileo's project of the mathematical sciences of nature can be seen from this allegorical viewpoint, and connected to Dante, along these lines. (I refer the reader to an article by David Reed and the present author, "Discourse, Mathematics, Demonstration and Science in Galileo's Discourses concerning Two New Sciences," forthcoming in Configurations. )
15. The details of quantum-mechanical formalism and of the specific form of algebra (that of the so-called operators in infinite-dimensional Hilbert spaces) are not essential here.
16. Cf. de Man's use of "linguistic terms" in "The Rhetoric of Temporality," in Blindness and Insight: Essays in the Rhetoric of Contemporary Criticism (Minne- apolis: University of Minnesota Press, 1983), 203; hereafter BI.
17. The question of the particular architectonic involved in each case is com- plex, even though a certain geometrical architectonic is suggested by a kind of (pure) geometrical figure (rather than equation) defining Galileo's or Newton's sci- ence. Newton felt obliged to recast his mechanics in (Euclidean) terms of geometry rather than those of calculus in preparing Principia.
18. Immanuel Kant, Critique of Judgement, trans. J. H. Bernard (New York: Hafner, 1951), 24.
19. As will be seen, the nature of quantum probability is in turn nonclassical, and is not defined, as in classical physics, by, in practice, insufficient information
Algebra and Allegory 87
88 Arkady Plotnitsky
concerning the systems that, in principle, behave classically. As I have indicated, while quantum theory (at least in Bohr's interpretation) fully conforms to non- classical epistemology, it has its specificity. Accordingly, further qualifications con- cerning it may be necessary, which, however, would remain consistent with my overall argument here. In particular, in certain idealized cases, some among experi- mentally measurable quantities and, hence, some aspects of individual observable "events" involved can be predicted exactly, that is, with the probability equal to unity, by using the dynamical laws of quantum mechanics, such as Schro? dinger's equation. Hence, the prediction of such quantities may be seen as comprehended by these laws. Such predictions, however, would not allow us to define the outcome as an "event" (say, in the way we could in classical physics) even in idealized cases and hence to make overall individual events themselves subject to law. In this sense the conditions of radical formalization would still rigorously apply even in these cases. (I also leave aside for the moment that such predictions can only concern effects of the interaction between quantum objects and measuring instruments, and can never apply to "events" of the quantum world itself. ) In general, however, in quantum physics there are always "events" that cannot be comprehended by law even with respect to their partial aspects--in principle, rather than only in practice, which is possible in classical physics as well. Nor, in contrast to classical physics, can we ever be certain concerning the conditions under which an idealization of the type just described would apply, even though we, again, can estimate probabilities when it applies. This is part of the irreducibly statistical character of quantum theory, rather than (as classical statistical physics) its being statistical by virtue of the struc- tural complexity of the systems involved and, hence, our lack of sufficient informa- tion concerning them. Quantum theory predicts only correlations between events (and does so exceptionally well), but tells us at best only half a story concerning the correlata themselves. This is of course epistemologically extraordinary, but should not be surprising by this point. As I have indicated, in the field of quantum physics, anything can always happen and nothing is ever fully guaranteed, which, as will be seen, is also the principle of de Man's epistemology.
20. Whether this representation in fact corresponds to any "physical reality" is yet another question, which I shall suspend, since the negative answer would only make the present argument stronger.
21. Werner Heisenberg, "U? ber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen," Z. Phys. 33 (1925): 879-93.
22. Werner Heisenberg, "The Physical Content of Quantum Kinematics and Mechanics," Quantum Theory and Measurement, ed. John A. Wheeler and Wojciech H. Zurek (Princeton, N.