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Volume 3, Issue 2 (2006).
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Minute Particulars and Quantum Atoms: The Invisible, the Indivisible, and the Visualizable in William Blake and in Niels Bohr [1]

By Arkady Plotnitsky

By Way of a Prologue

Let us consider William Blake's 1808 drawing "The Vision of the Last Judgment" (Fig 1). If there could be a single feature that defines this work, including as against Michelangelo's Last Judgment that inspired it, it is a certain continuous density, thickness, of Blake's pictorial conception of the event, in particular as occurring in the human mind. The overall design suggests the shape of a human brain, where – as Blake's "The Human Abstract" tells us – all the trees of human thought grow, although other images are undoubtedly intimated as well. The figures "fill" the field of the work much more densely than they do in Michelangelo's fresco, making Blake's conception more akin to those of the Baroque and especially reminiscent of The Last Judgment and other works by Tintoretto, an affinity reinforced by portrayals of groups of human bodies in his major prophetic works. (Plate 46 of Jerusalem, for example, is very Tintorettoesque.) [2] This plenum-like (intensive-continuum) density suggests that the field of figures and groups of figures actually drawn by Blake is designed to only function as an initial configuration defined by certain graphic features, such as delineations of figures or groups, which make each into a certain concrete indivisible entity or, in Blake's language, a "minute particular." Blake appears to want our experience, our vision, of his work and of the event of the Last Judgment itself to infinitely expand the image into ever larger assemblages and to fill the resulting landscape with an immense continuous and dynamically expanding array of minute particulars. In other words, he appears to want our imagination to continue his art by mentally (phenomenally) drawing or delineating new minute particulars in our vision until it reaches a certain, Blakean, infinity. This infinity extends dynamically, rather than forms a finished static picture of the infinite.

It is crucial, however, that this vision must be or become a visionary work of art, rather than an (ultimately always limiting) abstraction, against which Blake warns us throughout his works, as in "The Human Abstract" (a uniquely fitting title) and its tree that "grows … the Human Brain" (24), where the Last Judgment takes place. Abstraction, against "the richness of being," has tempted us at least since Newton and Descartes, or Euclid, or Plato to whom Blake refers in his description, noting that Plato's imagination surrenders itself to abstraction, which makes him to deny knowledge and understanding to Poets and Prophets ("The Vision of the Last Judgment," 544).[3] Blake thus also reminds us that there are (at least) two Platos. The first (more familiar) is the mathematical one, who wants to make all thought mathematical, geometrical, as mathematical abstraction may serve as the ultimate form or paradigm of abstraction. The second is the Plato of Philebus, who, even if against himself, tells us that thinking is the work of an artist inhabiting our mind (38e-39b). Our thinking about thinking still negotiates between these two conceptions, for Blake by dispensing with the first, the mathematical one, nearly altogether.

This infinite expansion or, as Blake calls it, infinite vision, or, conversely outrincapacity to pursue this expansions (and we do tend to reduce, impoverish even the initial field of Blake’s work), is what Blake calls "The Last Judgment," the event that, he argues, occurs at every instance of our thought. "Its Vision is seen by the [Imaginative Eye] of Every one according to the situation he holds" ("The Vision of the Last Judgment") (E. 544). [4] As enacted in this work, our experience of Blake's vision is this experience. It is a test, the Last Judgment of our imagination, "each time unique" (as Jacques Derrida once said about the end of the world), that we undergo every time we experience it, and thus an instant of our vision or our lack of vision every single moment of our life.

Introduction

This essay offers a reexamination of one of Blake's great poetic and philosophical inventions, his concept of minute particulars, from the perspective of modern physics, and specifically the idea of atomicity, from Newton or even Democritus (the founder the concept of atomicity) and classical physics to Niels Bohr and quantum physics. My concerns is the architecture of the concepts involved, and the shared historical genealogy of this architecture. This hopefully will allow me to avoid anachronism as much as possible, apart from a certain irreducible reconstructive anachronism that no historical argument can avoid.

Blake's concept of minute particulars replaces the mechanical, Democritean concept of atomicity (in the Greek sense of not divisible any further), understood as a limited divisibility of matter – that is, of atoms discontinuously separated from each other. This view underlies the Newtonian vision of nature at the ultimate level of its constitution and of physics as a (mathematized) description of this constitution. The concept of minute particulars, I argue, plays a special role in Blake's case against Newton and in shaping Blake's alternative vision of the world, defined by the principle of the Poetic Genius, which is partially based on this concept. While Newton's vision concerns primarily nature, Blake's vision is that of the human world, apart from which there is no nature, according to Blake. One of Blake's key points is that Newton's mathematized vision of nature, or the very idea of nature, at the least of material or mathematical-material and specifically atomistic nature – the "matter" of modern physics – is also defined and indeed created by a particular (and from Blake's viewpoint, perverted) human vision. This vision, moreover, defines a certain paradigm of human thought, which by Blake's time had come to dominate modern philosophical thought, as defined by Descartes and (along more directly Newtonian lines) John Locke, and the very culture of modernity. This vision arises through the process of mathematical abstraction and results in further abstractions, such as and in particular the key elements of Euclidean geometrical vision – points, lines, planes, or three-dimensional space. Blake is hardly less critical of mathematical abstractions supplied by algebra or, especially, calculus, with its infinitesimal vanishing points.

Blake's vision is, by contrast, one of the world defined on the model of artists' work and, perhaps especially, drawings such as his own. I single out the art of drawing in view of the well-known significance of idea of the line, of an articulated delineation, in Blake's art, and its connection to engraving and hence the technologies of writing or inscription. In this sense of technological production of such entities (even when they are phenomenal), this vision may be seen as material. But this is very different from the Newtonian idea of physical matter; it is however akin to a certain technological materiality and accompanying conception of atomicity, due to Bohr, found in quantum physics, except for the mathematical (or mathematical-experimental) character of quantum theory, which the theory shares with classical physics.

Blake's vision and his concept of the ultimate human vision are defined on the model of the "thick" material artistic delineation, rather than disembodied abstractions of mathematicians and delineations they produce in mathematics and physics, governed by "demonstration" rather than "vision." No matter how thin an actually drawn line may be, it is still infinitely thicker and infinitely richer or more nuanced than the mathematical one-dimensional line. In quantum physics this technological materiality appears alongside another form of materiality, which is inaccessible to any technology, material or mental, and thus cannot be seen as in any sense mathematical. Hence, it is antithetical to Newton's vision of nature, and in this sense it is Blakean – but only in this sense. The irreducible inaccessibility of this materiality to any form of conception or vision (rather than only to a mathematical one) makes this materiality and the concomitant view of the capacities of human thought just as alien to Blake's vision as it is to that of Newton. Blake's view of human nature (our bodies and minds) is defined as much by the potential reach of our vision as by its richness. Blake sees both as equally infinite and thus as containing no inaccessible remainder – in particular the irreducibly, infinitely inaccessible remainder of all human vision found in quantum physics, at least in Bohr's view of it.

My argument in this essay proceeds, nevertheless, through the intersection between Blake's work and the way of thinking about the ultimate constitution of nature found in Bohr's interpretation of quantum mechanics, known as complementarity, especially his technological and techno-epistemological concept of "atomicity." This concept, too, replaces the Democritean conception of atomicity as a limited divisibility of matter and, by so doing, becomes arguably the closest available concept to that of Blake's minute particulars, except perhaps for Leibniz's monads. (I shall discuss Leibniz's concept of monad later, but it should be noted at the outset that, while both Blake's and Bohr's conceptions of atomicity are monadological, neither should be identified with Leibniz' monad and the degree of analogy between them is limited.[5]) In a further proximity to Blake and against Newtonian physics, to "Blake against Newton," Bohr's interpretation of quantum mechanics, "Bohr against Newton," implies that nature at the ultimate (quantum) level of its constitution is not subject to an abstract(ed) mathematical description, although, as noted above, it is not subject to any other description or even conception either.[6] Mathematics still plays a crucial role in this formulation, and it would be difficult to have physics in its modern disciplinary sense otherwise. As against that of classical physics, however, the mathematical formalism of quantum mechanics only serves to predict the outcome of relevant experiments (and it does so only in terms of probabilities) rather than describe nature even in an idealized way. Thus, at the quantum level nature only appears to allow the use of mathematical abstractions (more arcane than those of classical physics) to predict – again, statistically – but not to describe, even by way of mathematical idealizations of the kind one finds in classical physics. Perhaps this abstraction is one of the reasons why classical physics cannot describe the ultimate constitution of nature, and as such limits our interaction with and our vision of nature.

But then it appears that nothing else, however abstract or however concrete, however mathematical or however artistic, can do better on this score than does quantum theory. As against Blake's view or vision of human vision as capable of an infinite reach, Bohr's argument implies that this constitution or any law that may govern it (assuming the idea of law could still apply) is not subject to any description or even conception available to us, in short to any vision, however artistic or however scientific. This constitution cannot be visualized or even conceived of and hence cannot be artistically delineated. Our interaction with nature is irreducibly limited, even though and because (as against the classical post-Newtonian view of physics) the concept of this interaction, human and technological, is essential to this view. By the same token, as against Blake and closer to Newton, Bohr also upholds the status of nature or at least matter as independent of human imagination. His concept itself of matter is very different from that of Newton by virtue of the irreducible inaccessibility of its ultimate constitution to human thought. Bohr was well aware, just as Blake and perhaps Newton were, that any concept of nature is still a product of human imagination. Both quantum mechanics and Bohr's epistemology required a fertile and fantastically powerful imagination, just as Newton's physics did (as Blake realized). In this case, however, this imagination also takes us to a limit beyond which it cannot reach.

It is remarkable (and given the preceding history of physics, unexpected) that our understanding and a rigorous physical theory of the ultimate constitution of nature required a conceptual architecture or, again, a theoretical imagination that was in its key aspects closer to Blake than to Newton, qualified as this proximity may be. First, physically, there is no picture of material atoms or particles (idealized as massive points) or their motions, which can serve as an idealized model of quantum nature. Second, concomitantly, there is no mathematical description (of whatever kind) of this nature. In approaching nature at the ultimate level of its constitution, quantum physics appears to follow Newton as concerns its mathematical character, but only up to a point, because there is no mathematical description of nature any longer. It appears, however, to follow Blake as concerns its epistemological architecture, albeit, again, only up to a point because human vision of whatever kind cannot reach beyond certain limits. First of all, this architecture is defined by the interaction between nature and our (material) instrumental technology, and hence also by our interactions with nature, which interactions may be seen as corresponding to the inscriptive technologies of Blake's illuminated manuscripts. These technologies include those of our bodies and of our minds, which may be seen in technological terms as well, in the broad sense designated by the Greek word tekhne. This interaction does of course exist in classical physics but has no fundamental, constitutive (only an auxiliary) significance in the Newtonian view of physics. It can be neglected (at least, as concerns our instruments) in the disciplinary practice of classical physics itself, from classical mechanics to Einstein's relativity. (The latter is an epistemologically Newtonian theory, although very different physically and mathematically, and, as I said, as such it has its own proximities to Blake.)

As Bohr argues, however, this interaction is irreducible in quantum physics and, again, defines it physically and epistemologically, and by so doing brings quantum mechanics closer to Blake, at least, again, along these lines. On the other hand, Blake "shares" with Newton, as well as Einstein, the belief that it is possible for us to envision, at least in principle, the ultimate constitution of the world (nature for Newton and Einstein), although the vision Blake has in mind is, again, very different from Newton's mathematical vision of particles or atoms in the void. As will be seen, there are alternative views of quantum physics, in particular those associated with the Bohmian theories (after David Bohm), which give our vision more chance. They are, accordingly, closer to Blake's way of thinking, except for the mathematical nature of these views.

It would be difficult to expect a strict alignment between Blake's vision and any particular modern (mathematical) physical theory, from Newton's mechanics to relativity and quantum theory and beyond, and one must often navigate a rather serpentine path in following these relationships. On the other hand, there is much to gain by following the curves of this path and by tracing the inevitable and inevitably multifaceted complexity of the relationships of Blake's vision, physics (or mathematics), and philosophy. This complexity also reflects that of the history of the concepts involved, from Leibniz's monads to Blake's minute particular to Bohr's atomicity, which are part of a much greater historical-conceptual rhizome. The concepts that bring Blake and Bohr together – minute particulars in Blake and atomicity in Bohr (Bohr's "atoms" may be seen as "particulars" rather than Blake's "minute particulars") – are most significant for this article, but the differences are also important.

According to Bohr, in introducing the second volume of his philosophical works, Essay 1937-1958 on Atomic Theory and Human Knowledge:

The importance of physics science for the development of general philosophical thinking rests not only on its contribution to our steadily increasing knowledge of that nature of which we ourselves are part, but also on the opportunities which time and again it has offered for examination and refinement of our conceptual tools. In our century, the study of the atomic constitution of matter has revealed an unsuspected limitation of the scope of classical physical ideas and has thrown new light on the demands on scientific explanation incorporated in traditional philosophy. … The main point of the lesson given us by the development of atomic physics is, as is well known, the recognition of wholeness in atomic processes, disclosed by the discovery of the quantum of action [Planck's constant, h]. The following articles present the essential aspects of the situation in quantum physics and, at the same time, stress the points of similarity it exhibits to our position in other fields of knowledge beyond the scope of the mechanical conception of nature. We are not dealing here with more or less vague analogies, but with an investigation of the conditions for the proper use of our conceptual means of expression" (2: 1-2; emphasis added).

It is significant that Bohr speaks of similarities here, which also allows for and requires an exploration of differences in order to understand more rigorously conditions for the proper use of our conceptual means of expression, which conditions are often different in science, philosophy, and art. We develop a better understanding of such concepts and conditions of their proper use by "measuring" them against each other and by exploring their affinities and differences along different lines, and the different views of nature and mind to which they lead. The most crucial point, however, which defines the program of this essay, is Bohr's final point here, to the effect that "we are not dealing here with more or less vague analogies, but with an investigation of the conditions for the proper use of our conceptual means of expression," or of our thinking in general. Our thought is, as a product of our brains, physically born of atoms (the way we see it now, whereby atoms, such those of hydrogen or oxygen, are comprised by more elementary indivisible entities, such as quarks and electrons), but it also discovers various concepts of atomicity itself. With Blake and Bohr, we move from physical atoms, or quantum objects, such as elementary particles, to atoms as technology and thought. The differences between the two conceptions notwithstanding, this transition itself is one of the greatest achievements of human thought.

Bohr: From Atoms to Matter Beyond Visualization

It took quantum theory and Bohr a while – quantum theory about 40 years and Bohr about fifteen years (roughly between 1925 and 1940) – to replace the Democritean, mechanical conception of atomicity as a limited divisibility of matter with a new conception. The Democritean conception was initially used in quantum physics in the wake of Max Planck's discovery of the quantum nature of radiation, such as light, which inaugurated quantum physics in 1900. This discovery revealed that radiation, previously believed to behave as a continuous, wave-like phenomenon in all circumstances, could, under certain circumstances, exhibit discontinuous features. The limit at which this discontinuity appears is defined by the frequency of the radiation and a universal constant of a very small magnitude, h (Planck's constant) which turned out to be one of the most fundamental constants of all physics. The indivisible (energy) quantum of radiation in each case is the product of h and the frequency: ν, E = hν. Within a few years of Planck's discovery energy quanta were associated by Einstein in one of the great papers of his miraculous year, 1905, with particles of light, photons.

Quantum physics was troubled by major difficulties from the outset and quickly revealed new ones, the "trend" that has defined it throughout its history and continues into its second century, although by now mostly in higher-level quantum theories, dealing with physical processes at very high energies. First of all, in contrast to either Newton's earlier (strictly) corpuscular optics or to the (strictly) wave optics that replaced it by the nineteenth century, neither aspect of light could be disregarded, since light appears to exhibit wave-like features in some circumstances and particle-like ones in other. This posed a great difficulty, given the mutual exclusivity of these features, beginning with the respectively discrete nature of one and the continuous nature of the other, and the corresponding fact that we cannot conceive of entities that can be simultaneously both. Indeed, all other elementary constituents of matter, such as electrons (originally seen as particles), were found to exhibit the same wave-particle duality. These difficulties persisted even after a theory that was able to properly predict the outcome of the relevant experiments and now known as quantum mechanics was introduced by Werner Heisenberg and Erwin Schrödinger around 1925-1926.

Bohr eventually came to see both the Democritean (atomistic) and wave-like views of nature as inadequate for an understanding of quantum phenomena and for the interpretation of quantum mechanics itself. He introduced a set of new concepts, including, sometime in late 1930s, a new concept of phenomenon as applicable in the quantum mechanical situation, which helped him to refine his interpretation of quantum phenomena and quantum mechanics, as complementarity. It is this concept that he eventually developed into a new concept of atomicity. [7]

Complementarity is, chronologically, the first concept introduced by Bohr (around 1927) in building the architecture of his interpretation of quantum phenomena and quantum mechanics. The term 'complementarity' was defined initially as the mutual exclusivity of certain quantum phenomena (in the conventional sense of the term phenomena), such as those that are particle-like and those that are wave-like, and yet the necessity of considering both types in order to form a comprehensive description of the overall quantum-mechanical situation. Bohr initially proposed this definition without offering any particular epistemology underlying quantum phenomena. Eventually, however, he developed a special concept of phenomena, which then led him to his (essentially equivalent) concept of atomicity. These concepts grounded his ultimate interpretation of quantum phenomena and quantum mechanics as complementarity. The term 'complementarity' itself came to designate this interpretation, rather than only a mutual exclusivity of certain phenomena (although the latter concept retains its crucial significance in Bohr's scheme). Bohr's concept of phenomena has a complex architecture and was necessitated by the famously strange features of quantum physics, as represented, for example, in the so-called double-slit experiment, often seen as an archetypal quantum experiment. It may, accordingly, be useful to describe the experiment here.

The arrangement of the double-slit experiment consists of a source; at a sufficient distance from it a diaphragm with two slits (B and C), widely separated; and finally, at a sufficient distance from the diaphragm, a screen (a silver bromide photographic plate). A sufficient number (say, a million) of quantum objects, such as electrons or photons, emitted from a source are allowed to pass through the slits in the diaphragm and leave their traces on the screen. Technically (this circumstance became crucial for Bohr), we can never observe such objects themselves – nobody has ever seen a moving photon or electron – but only traces left on the screen, from which we deduce the existence of such objects. Indeed, all we have in any given event is a final trace of a "collision" on the screen, which is a very complex process involving millions of atoms and interactions between them. If properly amplified, each such trace would constitute a very complex picture, without however ever allowing us to detect the quantum object itself involved, which is always lost, destroyed in the cascade generating this picture. Everything else just mentioned – the emission of the object, its passing or not passing through slits, and so forth, is ascertained on the basis of other observations and measurements that we can perform in similar circumstances. Two set-ups are considered. In the first we cannot know through which slit each particle passes, in the second we can, at least in principle, which, as will be seen, is a qualification of considerable importance.

If both slits are open and no arrangements (such as particle counters) are made that would allow us to establish or even make it in principle possible to know through which slit each particle passes, a "wave-like" interference pattern will emerge on the screen (Fig. 2). This pattern will emerge regardless of the distance between slits or the time interval between emissions of the particles. The traces of the collisions between the particles and the screen will "arrange" themselves in a pattern even when the next emission occurs after the preceding particle is destroyed after colliding with the screen. The emergence of this pattern is enigmatic or mysterious since each individual event is completely unpredictable and random – this is indeed one of the great and defining mysteries of quantum phenomena: that individually random events can, under certain circumstances, form ordered or correlated collectivities. This pattern is the actual – visible – manifestation and, according to Bohr's and most standard interpretations, the only possible manifestation of quantum-mechanical "waves." In this type of interpretation, one can speak of "wave-propagation" or of any attributes of the classical-like phenomenon of wave-propagation (either associated with individual quantum object or with their behavior as a multiplicity) prior to the appearance of these registered marks only by convention, for the sake of convenience.

According to Bohr's interpretation, however, the same is also true as concerns the attributes of classical particle motion, in particular trajectories, and quantum objects are seen by Bohr as indescribable in any terms and inconceivable by any means available to us. We see on the screen only traces of the collisions between quantum objects and the screen. The latter themselves are destroyed in the process of this, as Bohr called it, "irreversible amplification" of all our encounters with quantum objects to the classical level at which we can observe things. Bohr's drawing of the apparatus is the second drawing shown in Fig. 2 (Bohr's "Fig. 4,") is designed to illustrate the fact that we only observe the traces of quantum events, shown on the screen on the left, and never quantum objects or their motions. Nothing is shown between the heavy parts of the apparatus. The first drawing or diagram shown in Fig. 2 (Bohr's "Fig. 3") presents a naïvely classical representation of the situation in terms of a wave picture, which is never observable. In Bohr's view, it is physically nonexistent and has only a symbolic or metaphorical significance.

If, however, there are counters or other devices that would allow us to check through which slit particles pass (merely setting up the apparatus in a way that such knowledge would in principle be possible would suffice), the interference pattern inevitably disappears. In other words, our knowledge concerning throughout which slit a given particle of such a multiplicity passes inevitably destroy the interference pattern. The emergence of order depends on how ignorant we or our technologies are (ultimately altogether) of how this emergence is possible.[8]

The situation just described, sometimes also known as the quantum measurement paradox, is remarkable, and it is certainly incompatible with our basic understanding of physical objects, such as Democritean atoms, and their motion, the concepts upon which classical, Newtonian physics is based. Common characterizations of this "incompatibility" include "strange," "puzzling," "mysterious" (and sometimes "mystical"), and "incomprehensible." The reason for this reaction is that, if one speaks in terms of particles themselves, individually or (which is hardly less troubling) collectively, they appear somehow to "know" whether both slits are or are not open, or whether there are or are not counting devices installed, and behave in accordance with such notion. Furthermore this lack of knowledge concerning through which slit each particle passes allows collective orders to emerge out of complete randomness of each individual event contributing to this order. It may be shown that any attempt to conceive of this behavior in terms of physical attributes of quantum objects themselves or their motion (on which, again, classical physics is based) would lead to a logical contradiction or to require difficult assumptions, such as attributing volition or personification to nature in allowing particles individual or collective "choices" or "communications." At the very least, such an attempt would lead to the so-called nonlocal behavior, an instantaneous action at a distance, in conflict with the finite limit (defined by the speed of light in the vacuum) upon all physical influences, which is a well established experimentally and provides the basis for Einstein's relativity theory.[9]

Bohr found a way out of the paradox. He noted first that, although the two types of phenomena in question (the one defined the interference pattern and the other by a random distribution of traces) belong to the totality of the observable phenomena and must, therefore, be both considered by quantum theory, they are always mutually exclusive and hence are never in any actual conflict with each other. He, accordingly, defined such phenomena (mutually exclusive yet both necessary for a comprehensive account of the overall situation) as complementary. Eventually, he interpreted the situation as signaling the impossibility of ascribing physical attributes, such as positions and motions, to quantum objects themselves or their behavior. This fact, however, does not affect the phenomena we actually observe and indeed allows one to define them without contradiction. Bohr wrote: "To my mind, there is no other alternative than to admit that, in this field of experience, we are [rather than with properties of quantum objects] dealing with individual phenomena [observed in our measuring instruments] and that our possibilities of handling the measuring instruments allow us only to make a choice between the different complementary phenomena we want to study" (PWNB 2: 51). That is, these possibilities allow us only to make a choice between two different types of effects of the interaction between quantum objects and measuring instruments upon those instruments, thus making the measuring technology an irreducible factor in all quantum phenomena. This leads him to a new definition of the term "phenomena" as applicable in quantum physics:

I advocated the application of the word phenomenon exclusively to refer to the observations obtained under specified circumstances, including an account of the whole experimental arrangement. In such terminology, the observational problem is free of any special intricacy since, in actual experiments, all observations are expressed by unambiguous statements referring, for instance, to the registration of the point at which an electron arrives at a photographic plate. Moreover, speaking in such a way is just suited to emphasize that the appropriate physical interpretation of the symbolic quantum-mechanical formalism amounts only to predictions, of determinate or statistical character, pertaining to individual phenomena appearing under conditions [of measuring instruments] defined by classical physical concepts. (PWNB 2: 64)

In this definition the term "phenomenon" refers only to registered observations or measurements, in other words, only to what has already happened and not to what may happen, even if the latter possibility corresponds to a rigorous prediction enabled by quantum mechanics. (Such predictions, however, could only be statistical and hence never fully guarantee a given outcome.) Thus, that what could be, even in principle, visible could only be found as manifest in our experimental technology, while quantum objects themselves are beyond any visualization or intuition. The German word Anschaulichkeit, often used by Bohr, who also speaks (in English) of "pictorial visualization," carries both meanings.

To this extent it is indeed true that "the observational problem is free of any special intricacy," as Bohr says. Given, however, that quantum objects and their behavior are completely left out of this picture, the epistemological consequences of this claim are radical. The whole realist (descriptive or/as visualizable) and causal view of nature defining and defined by classical physics, and the whole realist philosophy of nature are inevitable casualties. Quantum mechanics only predicts (and in general only predicts statistically) the outcomes of certain experiments on the basis of other already performed experiments, but it does and cannot (nothing can in this view) describe the behavior of quantum objects responsible for these outcomes. Accordingly, the collective order arising from random individual events in certain circumstances (which is, again, an experimentally well-confirmed fact) is left unexplained as well, and is rigorously put beyond explanation in order to give it a logical consistency. No underlying hidden order (which would eliminate the randomness of the individual events involved) is or can be assumed. Indeed, it may be shown (this is the content of the so-called Bell's theorem) that once such an assumption is made or even if one makes an assumption that quantum objects could be assigned physical properties on the model of classical physics, significant difficulties arise, specifically a violation of relativity.

According to Bohr's view, all individual quantum phenomena, that is, the outcomes of all individual experiments, are physically individual and discrete, and thus are particle-like, although not particles, in the following sense. They are defined by such physical entities as traces or inscriptions of collisions between individual quantum objects (or what we infer as such on the basis of the experimental evidence or phenomena) and silver-bromide photographic plates or equivalent physical macro-objects, such as cloud chambers, where such traces appear. Since no original event or ultimate process efficacious of such events can even be reached, the terms trace and inscription, "writing," can be taken here in Derrida's sense, defined by the same type of epistemology or techno- or texto-epistemology. (Derrida relates writing, the textual technologies of writing, to technology in the general sense of the term conveyed by the Ancient Greek word tekhne.)

Just as are Blake's minute particulars, defined by technologies of engraving-inscription, Bohr's phenomena or "particulars" are techno-phenomena, and Bohr's concept of atomicity is always that of a techno-atomicity. Each is defined by a feature analogous to a kind of trace a classical-like particle would leave in similar circumstances. The difference from the classical situation arises when the totality of the observable phenomena in question in quantum mechanics is considered. The character of this totality, however, leads to a suspension of assignment of any and all classical physical attributes to quantum objects themselves or their motion, particle-like (atomic) or wave-like. As such, quantum phenomena – but not quantum objects, which cannot be subject to any conception we can form – may be seen as a form of discrete particulars, especially because every event that gives rise to quantum phenomena is individual, unique and, in general, not repeatable, making any possible theory of quantum phenomena unavoidably statistical. Two qualifications to this description of Bohr's concept of phenomenon and its analogy to Blake's concept of minute particulars are in order. First, according to Bohr, each such phenomenon contains such a particle-like trace but is defined by the overall (rigorously) specified experimental set-up, where this trace occurs, e.g. the source emitting the quantum objects, the slit or slits through one of which it may pass, and the screen where the trace in question occurs. Second, strictly speaking, by "phenomenon" one must refer to our (conscious) perception, enabled by the "technologies" of our bodies, of the material object in question – the measuring apparatus or a certain part of it and a physical, material trace of its (past) interaction with a quantum object found in it. The above discussion, however, is self-evidently consistent with these qualifications.

This conceptual architecture could be seen to be monadological, insofar as it parallels the ways in which Leibniz's monads interact with the world. However, unlike that of Leibniz, Bohr's concept is quantum-phenomenal and quantum-mechanical, and does not aim at a broader metaphysics. Moreover, and more significantly, in terms of our perception there is no difference between classical and quantum phenomena, which is one of Bohr's aims in introducing his concepts of phenomenon and then atomicity. (The ultimate conception of quantum objects as inconceivable is what is different, and this is not classical.) All our perception and thinking are both classical and monadological. Bohr's concept implies and indeed reflects strict and strictly specifiable (via Planck's constant, h) universal limits, which close quantum phenomena, and do not allow us, whatever technology we can develop, to reach beyond these limits. This is not Leibniz's conception (although there are certain limits as to how far monads see or conceive of the world), nor is Blake's (for whom there are no such limits).

All wave-like phenomena found in quantum physics are comprised by a large accumulation of such individual phenomena and discrete rather than continuous, and hence are wave-like rather than are waves, which are continuous phenomena. These phenomena are wave-like because, in a certain circumstances, as explained above, this accumulation has a wave-like or wave-interference-like and, hence, ordered collective pattern. Accordingly, these phenomena may be seen as a form of organization or order, enigmatically formed, since each such event in itself is random. In other conditions such a pattern is a random accumulations of such traces, "dots." Note that these "dots" are such only in a low resolution and result from a complex process involving literally millions of atoms, which we can in fact trace – but only up to a certain limit – without ever reaching the level of quantum objects and processes themselves. The ultimate, quantum constitution of nature, leading to these "dots" or their more complex constitution as traces is separated from our knowledge and thought by an unbridgeable and hence irreducible, discontinuous abyss. Our perception of the world, what we see as the world, our thought gives as a continuous picture, against the background of which certain discrete configurations or sets of continuous configurations can appear. Something in nature allows and compels us to extract from this continuum a certain set of such separated configurations or phenomena, Bohr's particulars, discontinuous with respect to each other – that is, to infer from the specific character of the totality of these phenomena the existence of quantum objects and processes as something that is beyond any picture or conception. This interpretation allows one to avoid ascribing to quantum objects such mutually incompatible and hence contradictory properties as those of (discrete) particles and (continuous) waves, since these now appear only separately and never simultaneously in measuring instruments, and thus to avoid the contradictions that plagued quantum physics at the earlier stages.

In sum, Bohr defines phenomena, as applicable to quantum phenomena, technologically in terms of manifest (observable) effects of the interaction between quantum objects and measuring instruments upon those instruments. This is all that can possibly be observed or represented in quantum mechanics. Thus the concept is defined, first of all, by a certain "indivisibility" or "wholeness," insofar as the behavior of quantum objects cannot be considered apart from their interaction with the measuring instruments involved. As regards this behavior each such phenomena are "closed," and Bohr came to speak of "closed phenomena" in this sense, is a (individual) minute particular (PWNB 2, p. 73). Any attempt to break the "envelope" of a phenomenon in this sense would only lead to another phenomenon or set of phenomena of the same type, without ever bringing us closer to quantum objects themselves. (As against reaching the Newtonian/Democritean mechanical vision of atoms and their motions in the void.) By the same token, Bohr's concept of phenomenon also entails an irreducible epistemological "discontinuity" between quantum phenomena and the ultimate objects responsible for their emergence, which objects are placed beyond the reach of the theory itself or any possible conception. The nature of this "discontinuity" may be more radical and more subtle, insofar as it involves a further discontinuity between this epistemologically discontinuous scheme, which may be seen as a form of idealization of nature, and the possible actual constitution of nature. For the moment, the epistemological rupture just described becomes for Bohr the meaning of quantum discontinuity, and as such part of his interpretation of Planck's discovery of the quantum.

It is this thinking concerning quantum phenomena that leads Bohr to his concept of "atomicity," which retains the original Greek sense of the indivisibility of certain entities, atoms, but radically departs from the Democritean idea of atomicity as a limited divisibility of matter itself at the ultimate level of its constitution. Bohr's concept of "atomicity" is defined instead in terms of individual effects of quantum objects upon the classical world. In other words, Bohr's concepts of atomicity and phenomenon are essentially equivalent. This definition entails yet another form of discontinuity or, more accurately, discreteness, insofar as each such phenomenon or effect is singular and indeed unique, and as such is separated or isolated from any other such phenomenon or effect – that is, from other "atoms." Thus, all standard features of atomicity – indivisibility, discontinuity, and discreteness – are, however, retained, but also redefined, by Bohr's concepts. Bohr's atomicity is, thus, a new philosophical and, qualitatively, physical concept, which, as a concept, does not involve any mathematics. It can, however, be related, quantitatively, to the numerical experimental data and mathematical formalism of quantum mechanics, insofar as the latter can predict these data configured in terms of this concept. Bohr's concept obviously requires an adjustment, as against our usual understanding of atoms, insofar as the application of such terms becomes conceptual rather than physical. "Atomicity" now refers to materially complex and, hence, materially sub-divisible entities, rather than to indivisible physical entities, such as quantum objects ("particles") or even to traces left by the interactions between quantum objects and measuring instrument. For, as explained above, to quantum objects themselves one cannot ascribe such properties any more than other properties or attributes, while, such each such trace – say, a collision between a quantum object and a silver bromide screen – appears as discrete (a "dot") only at a low resolution. Physically, each such trace or the process that led to it is immensely complex. No finer resolution, no "zooming," however powerful, would ever erase the discontinuity or rupture in question, the abyss between our vision, however amplified by technology, and the ultimate (quantum) constitution of nature, left in the abyss of the invisible and unvisualizable. In other words, quantum objects are responsible for the epistemological discontinuity inhabiting Bohr's atoms, that between what we can possibly know or think and the ultimate constitution of nature.

The insurmountable nature of this discontinuity leads to the main difference between Bohr's particulars and Blake's minute particulars, and, through this difference, to the difference between their visions of the world. The first is that of nature and our interaction with nature, including by means of our thought and imagination, in Bohr and the second is that of human thought and imagination in Blake. Bohr's vision is akin to Blake's in that it leaves no place for Newton's in our understating of the ultimate constitution of nature and our interaction with nature – Newton's dream or sleep, including as Blake understood it, "Single vision & Newtons sleep" (“Letter to Thomas Butts” E. 693). Blake, however, would have been just as, perhaps more, disturbed by the uncircumventable limits to our imagination that Bohr's vision implies or creates; that black-hole-like "Ulro," as he calls it, might have appeared to him worse than that created by Newton's vision.[10] The idea that the ultimate constitution of the world may be altogether beyond the capacity of human thought and, especially, imagination to perceive or indeed create deeper reality and order, is not something that would appeal to Blake and is indeed antithetical to his vision of what our vision of the world could and should be.

Blake: From Minute Particulars to the Infinite Vision

According to Blake, through the workings of the Poetic Genius human imagination has the capacity to break the envelope (or, successively, envelopes) of individual minute particulars and reach the ultimate (infinite) vision of order of the world as a continuous assembly of minute particulars. It is as if we could actually see how a given phenomenal field, even a line or a point, is continuously, one by one, constituted by its ultimate constituents ("minute particulars"), to see the intensive infinity of the continuum, rather than only being able to see certain partial configurations continuous or discontinuous, or mixed, of such constituents points.

As noted earlier, Blake likes lines and the continuous articulation – delineation (in either sense) – they provide, apart, however, from their one-dimensionality (a mathematical abstraction), as against artistic concreteness and particularity. When they "take up" and over "the minute particulars" and "the articulation of a mans soul" mathematical abstractions harden minute particulars "into grains of sand," as against the expansion of a grain of sand into the infinite and artistic articulation made possible by the Poetic Genius (Jerusalem, Plate 45, 8-10, 20). No two actually, physically drawn lines are ever strictly the same, and even any two printed lines from the same engraved plate are always different, a circumstance that Blake of course knew very well and used in his artistic practice. Neither a dimensionless point nor a one-dimensional line is something that we can actually see or visualize, although it may be said that out visualization may create something peculiarly in-between actual ("thick") points lines and mathematical points and lines without thickness. Blake must have considered such entities as Euclidean or Newtonian and, hence, perverted abstractions, as against the contours of actually drawn or visualizable lines.

One must accordingly be cautious in using analogies between his lines and the intensive continuum of mathematics. This is in part why, although the vision and the very conception of a (merely) material nature is rejected, materiality itself remains crucial in its technological forms, whether material or mental. Blake's lines, or points, are drawn (inscribed or engraved) and have a thickness to them even when we imagine them, and, hence, again, are not mathematical. The "geometries" of delineation in a Blakean infinite vision are those of the artists' material designs not of mathematicians' de-materialized abstractions. For the moment, to the (limited) extent that the one can use any mathematical analogy here, there is in Blake a vision of the ultimate constitution of something like the continuum, of what the constituents are and how the come together, but these constituents, while minute and particular, are not points.[11] Blake's minute particulars are elementary or atomic in a sense similar to that of Bohr's atomicity. By the same token, however, they are not (mathematically) punctual, although they may involve artistic, "thick," points, just as Bohr's phenomena may involve the point-like traces of the collision between (unobservable) quantum objects and the classical world that we observe on silver-bromide photographic plates or in cloud chambers. These traces, too, are physical ("drawn" by nature and technology, as it were) rather than mathematical. Blake affirms the possibility of vision that transcends and transforms ordinary perception and its continuity, but through a continuously exuberant (Blake's term, which we might take literally here) expansion of the continuous artistic vision of the type suggested by his "The Vision of the Last Judgment" (Fig. 1), rather than an abstraction, refined as it may be, of a mathematical vision. This expansion gives more, even infinitely more, than that (continuum) which we see and it also gives a different form of refinement of our intuition – refinement by enrichment, rather than by a mathematical or philosophical reduction.

Blake's vision is that of the virtual, a form of virtual reality – as mathematics is – but it is not mathematical, any more than his visionary physics is physical or, again, mathematically physical. This assessment can, I think, be sustained, even though mathematics and physics do have their Blakean dimensions, sometimes against their own disciplinary grain. Blake's suspension of mathematics from his visionary mathematics and his suspension of (mathematical) physics from his visionary physics are parallel and correlative. Nature (at the very least, as something that is separate from the mind) is the concept that Blake rejects, to begin with, while such a concept is equally retained in and is crucial to Newtonian and non-Newtonian physics. In Bohr's view quantum objects exist independently of us, even though they are not only unknowable but also inconceivable.

Of course, very few even among the most optimistic of classically oriented physicists would believe it possible to (phenomenally) envision the ultimate constitution of nature. Einstein, for example, was cautious on that score, however much he believed in the capacity of our thought to come close to the ultimate workings of material nature. The dream of modern physics, from Galileo and, especially, Newton to Einstein and beyond, is to invent the mathematics that captures the ways nature works through mathematical models that may be envisioned in the place of nature. Quantum mechanics puts into question even this part of physics' dream and, along with other developments of twentieth-century mathematics and physics, makes us reexamine how envisionable almost any mathematics actually is – as opposed to being an abstracted fiction, as Blake suspected it was.

But then, Blake was not a physicist, only a visionary physicist; he could afford to deny the idea of nature, which is difficult, albeit not altogether impossible, in the case of the disciplinary practice of modern physics, which defined itself, from Galileo on, as a mathematical science of nature. Nor was he a mathematician, and he could, accordingly, bypass disciplinary requirements of mathematics, for example, those pertaining to space. His vision is closer to mathematics insofar as it need be seen a vision of nature as it is defined by modern physics (classical, relativistic, or quantum). Mathematical objects might be and sometimes are seen as forming a mental or ideal ("Platonist") reality extending far beyond the portion of it used by physics, although it difficult to predict what physics may borrow from mathematics. Mathematics, too, however, appears to place limits upon how far such concepts may go, insofar as what is ultimately (or ultimately ultimately) at stake in mathematics and what is responsible for these abstractions cannot be represented, mapped by them or even conceived of in their terms. Thus, we do not know how the mathematical continuum is ultimately constituted by its points – if indeed it is – a fundamental problem of set theory known as Cantor's continuum problem.[12] But, in any event, our mathematical envisioning of the ultimate constitution of the line as the intensive continuum or of other continuous objects appears to stop at a certain abyssal limit, not unlike the limit envisioned by Bohr in the case of quantum mechanics. In other words, the ultimate constitution of most mathematical objects, at least most infinite or continuous mathematical objects, appears to be beyond our conception.[13]

Of course, insofar as mathematics or mathematized view of the world, such as that found in modern physics, provides sources, positive or negative (and Newton was both for Blake) for the invention of new forms – "new visionary forms dramatic" – they can also serve the Poetic Genius in man (Jerusalem, Plate 98:35). But in doing so they inevitably move beyond themselves, become nonmathematical or nonphysical. One can locate such possibilities in post-Riemannian geometry and, especially, topology, the basis of Einstein's general relativity (his non-Newtonian theory of gravity) and, following it, modern cosmology, or related mathematical "abstractions," such as fiber bundles or chaos-theoretical objects, which appears to lead to more Blakean forms of spatiality. Such extensions to a poetic or artistic vision, defined by (material) delineations, of non-Euclidean spaces are found throughout Blake's illuminated manuscripts and are often suggested or entailed by his texts, as, again, "finite" figurations of the Blakean "infinite." The designs of Europe, of The Book of Urizen, Milton, or again, "The Last Judgment," are arguably the most spectacular examples of this approach.[14] Certain forms of extension of the mathematical or the physical beyond their limits are also found in the process of mathematical and physical invention, in the workings of the Poetic Genius in the practice of physics and mathematics. Conversely, such extensions cannot claim to be absolutely nonmathematical or nonphysical either and often contain within themselves, at least potentially, certain mathematical or physical concepts, as, for example, do Blake's non-Euclidean and non-Newtonian spaces, in that they are are close to those of Riemann or Einstein. Still, Blake, I think, would resist such a mathematical reduction of his spaces, lines, or points; and, as quantum mechanics appear to tell us, it may be that he was right even as concerns physics, insofar as nature resists our mathematical abstractions.

In contrast to Bohr's epistemology of quantum mechanics, the Poetic Genius in us would enable us to envision the insides of envelopes of particular phenomena without limits. This is possible because this poetic vision offers us a picture of the world (as a continuous plenum) that is not a Newtonian/Democritean mechanical vision of atoms in the void or, again, the mathematical, Euclidean vision of point, lines, planes, and so forth. Thus, Democritus and Newton allow us a greater capacity for envisioning nature in its ultimate constitution than does Bohr or, it appears, nature itself – but infinitely less than Blake thought possible through his idea or vision of the Poetic Genius. Bohr's interpretation of quantum phenomena, too, allows a structure of subdivision of each phenomenon and, hence, an expansion of our vision analogous to those found in Blake. In this case, however, we can only go so far, even in principle, since we are limited by Planck's constant, h. This process could only produce phenomena or atoms that may be and generally are different each time, but have the same indivisible structure with quantum objects and processes beyond our reach, since we can never bridge the quantum rupture and reach the ultimate constitution of nature. Nor could we ever overcome the discreteness of those phenomena that relate to the quantum constitution of nature, although our phenomenal perception of the visible macro-world, or that which creates this perceive world through both our senses and our minds, is, again, continuous. By contrast, Blake's vision is a continuous vision of the ultimate constitution of the world, a vision of a complex and complexly ordered continuum: plenum, but a plenum of minute particulars that are continuous and continuously delineated entities rather than only (thickly) punctual ones, which are rare in Blake.

From a certain point on, an expanding vision becomes a (continuous) sequence of opening continually leading one to a vision of "infinity in a grain of sand" and "eternity in an hour" ("Auguries of Innocence."). Once again, however, as in the case of "The Vision of the Last Judgment," such a continuum is comprised of minute particulars as irreducibly structured, thick and delineated, entities – body-like, book-like, city-like, etc. – rather than is point-wise of whatever dimension. It is a topological-like (topology is the ultimate mathematics of continuity) vision of how minute particulars join in a plenum; except, again, that the object of this vision would not be mathematically formalizable for Blake, although certain mathematical objects or rather certain visualized models of such objects may come close. But then such models are not rigorously mathematical, although they are of course crucial for the practice of mathematics. Blake would probably see mathematical topology as Newtonian or Leibnizean, and thus a perversion of poetic vision, especially insofar mathematics makes it impossible to ever see the ultimate constitution of its objects. In particular, it appears to bring our mental objects to the quantum-mechanical like limits of the irreducibly unthinkable, the uncircumventably unenvisionable.

While, thus, the type of vision Blake has in mind is in conflict with Bohr's vision of quantum phenomena, it does correspond, to a degree and with the qualifications given above (since these theories remain tied to mathematization and hence mathematical abstraction) to visions accompanying and permitted by other modern physical theories. (It is a separate question how far these theories actually permit such visions.) Among such theories are Einstein's general relativity (his non-Newtonian theory of gravity), based on Riemannian geometry; certain models of chaos and complexity theories; and the so-called Bohmian version or (there are several) versions of quantum mechanics, which are of particular interest here, given their contrast to the standard quantum mechanics in Bohr's interpretation.[15]

In Bohm's alternative theory of quantum phenomena, the ultimate ontology of nature combines wave and particle features, in part following Louis de Broglie's earlier ideas. The picture of the quantum worlds emerging in this ontology is not unlike the image of the peacock's tail invoked in Jerusalem: "And the dim Chaos brightened beneath, above, around! Eyed as the Peacock" (Chapter 4, Plate 97; E. 257), the order arising in the poetic vision to replace chaos, chaos or anarchy in part created by Newton's perverted vision of order. Such a vision is never possible in the standard quantum mechanics, at least in Bohr's and related interpretations. It is the presences of waves and hence, again, continuity that makes these theories different and more Blakean. There are particles in Bohm's theory but each is, as it were, guided or piloted by a wave. (Bohm's theory is sometimes called the pilot-wave theory). As I noted above, from the viewpoint of physics, Bohmian theories are problematic in view of their nonlocality, an infinite propagation of physical influences, incompatible with relativity. As models of thought, which (metaphorically) move with infinite speed and (metaphorically) in both a particle and a pilot-wave manner, they may be interesting, and once suitably de-mathematized, are close to Blake.[16] Once again, however, Blake would resist any mathematical vision, such as Bohmian or chaos-theoretical, with which his works have certain affinities and to which they are sometimes compared.

But then, again, mathematics is not ultimately (or fully) formalizable either, and in this sense the physical theories just mentioned can only take our vision so far. Perhaps, as I indicated earlier, Blake was right and the limitations of our (mathematizable) vision of nature or of the virtual world of mathematics itself arise from the mathematical or proto-mathematical (that is, to be potentially refined so as to become mathematical) character of this vision. It arises from our need to measure things to predict the behavior of things in measurable, numerical terms, something of which Blake was suspicious, although – as is clear especially in Jerusalem – he realized that it may be unavoidable and sometimes productive at certain stages in the life of human thought. However far mathematics may have moved beyond its connections to physics or nature into the ideal realm, that part of its origin(s) which establishes one or another connection of mathematical objects to numbers, still defines it. This is true even in topology, which suspends measurement, but which became mathematics only when it acquired a relation to numbers.

Plate 14 of The Marriage of Heaven and Hell offers a good example of an early work that launches Blake's program sketched here and defines the gradient of his future work. This juncture of the Marriage introduced Blake's minute particulars as defined by a conjunction or superposition of the book, the body, and the city, a conjunction that was to define Blake's vision from this point on.[17] Plate 14 links two "memorable fancies." The first (Plate 13) describes Blake's "dining" with the Prophets Isaiah and Ezekiel and (re)introduces Blake's concept of "Poetic Genius." It is the primary principle of human perception, found in every human being, but often dormant or inactive, or rather (since no perception would be possible without it) not properly put to work as an active principle. It is especially disabled in contemporary man, who is enchained by organized religion and analogous and complicit institutions, such as contemporary legal and political institutions, or post-Newtonian mathematics and science, which could be seen, via Deleuze and Guattari, as instances of "state" mathematics and science insofar as the state rigidly defines and controls their development.[18]

Plate 15, "A Printing House in Hell," offers an allegory and, more crucially, an enactment of the liberated or awakened, activated (in the literal sense of made active), workings of the Poetic Genius. It anticipates vaster allegories and enactments of this process found in Milton and Jerusalem, especially as defined in, as "awaken[ed]" by, Blake's "song of Jerusalem," in the closing lines of Jerusalem (Plate 98; E. 258). As books, they are allegories of human perception, creation, and transmission of thought and knowledge – as the production, printing and dissemination of books. Making (also in the original sense of poesis and tekhné) or reading such a book is, interactively, both a form of liberated perception, thought, and knowledge, and a model for the workings of the Poetic Genius, specifically as a (re)creation but never a repetition of the book itself.

Plate 14, which bridges the two plates and prepares the Printing House plate, ends with a call for "cleansing the doors of perception." The process would make "every thing … appear to man as it is, infinite" (E. 39, emphasis added). This reformed technology of perception works through opening up the envelope of each particular thing into the infinity that this envelope contains – that it envelopes, in other words – a minute particular potentially culminating in the infinite vision of the world. This expanded vision is contrasted to the un-reformed, closed or finite, vision "of all things thro' narrow chinks of his cavern." The book is thus conceived of and materially, technologically developed as a conglomerate of minute particulars, by Blake as a vehicle of expanded human perception leading to the infinite vision. Technologically, it is akin to what Derrida calls writing (écriture), but without an accompanying "end of the book." Instead it makes the Blakean book writing in Derrida's sense, which is, however, consistent with Derrida's argument, especially if one thinks of the book in terms of final closure or limiting containment, as against expansion and transformation to other minute particulars, other books.[19] As a conglomerate of minute particulars, a book (such as an illuminated manuscript) can only contain a potential of an infinite vision to be differently realized by a reader; ultimately all such visions could cohere into the collective or communal infinity of the human world, similarly, but not identically, to Leibniz's monadology. Each "minute particular" of writing/engraving, say, a "letter," or in Blake's own words in Jerusalem, "every Word & Every Character," is itself structured as a book and specifically as an illuminated manuscript. By the same token, each is a "Visionary form Dramatic," and is also "Human," and a human form, body and soul, "according to the Expansion and Contraction," and also the city, a kind of "Jerusalem" of its own (Plate 98: 35, 27; E. 257). Blake's persistent use of the shape(s) of the human body from ever more minute to ever increasing scale reflects the same type of machinery or technology of vision. The corresponding expansion of minute particulars culminates in the infinite body of Albion in Jerusalem, the city of Blake's vision and the poem (Jerusalem, Plates 98-99; E. 257-8), or the body of Christ, defining the ultimate cosmology Blake's Universe. The presence of the city architecture (in either sense, that of actual architecture and that of the concept of the city) is more complex, but the convergence of all three – the book, the body, and the city – in the visions offered by prophetic works, Jerusalem most especially, is unquestionable.

Blake's works, or his world, or our perception, to begin with, may be contracted into classical arrangements of primitive elements or impoverished or empty (rather than rich, minute-particular) singularities, such as the point particles of Newtonian physics. This would be effected by unregenerated perception or knowledge, which (re)assembles the world on the basis of this reductive (in either sense) vision. This process is allegorized in the "minute-particular" plates of Jerusalem (such as Plate 45 of Chapter 2 and Plate 55 of Chapter 3). Each "Minute Particular" of Albion is "hardened" by a Newtonian vision into a "grain of sand," from a superposition-fusion of the book, the body, and the city (each of these is clearly intimated in the plate) of the infinite vision (Jerusalem, Plate 45, 20).

The infinite vision would, conversely, expand a grain of sand into this type of infinity, beginning with thickening into a material, delineated and, thus, minute-particular-line point. In order to reach this vision it is necessary, first, to divest the text (verbal or visual) of or, again, liberate it from a reductive reading and, then, to reassemble, reorganize, it into a different text, which can open-up the infinite vision. Blake's infinite vision and his vision of the infinite are defined by this process of reorganization of minute particulars, divested of Newtonian vision (in the broad sense) that contracts them into dead, mathematical, dimensionless point-like elements, subject to strict mathematical law: "Single vision & Newtons sleep." Each minute particular is infinitely expandable or, as it were, re-expandable so that the same process may be infinitely reenacted, in each case possibly giving rise to a different (sub)universe of its own where an analogous organizational dynamics would apply. I say "analogous" because new minute particulars themselves may be quite different, thus making this dynamics fundamentally nonfractal – the pattern may change with the change of scale – although more fractal-like sequences are found as well in this process or, accordingly, in Blake's poetry and design. The process itself, however, is interminable or infinite, just as the iteration of fractals is. Blake might have found fractals interesting, but, I think, ultimately boring, however intricate a given set may be (such as that of the famous Mandelbrot set, to which his poetic method has been compared in recent years).[20] In Blake's vision, the constitutive parts of any pattern are uniquely singular, as they organize into the order, or possibly multiple orders, of the whole.

Accordingly, Blake's famous description of the ultimate poetic vision in "Auguries of Innocence" entails an ultimately nonfractal expansion (with, again, some fractal-like strains) rather than fractal iteration:

To see a World in a Grain of Sand And a heaven in a Wild Flower Hold Infinity in the palm of your hand And Eternity in an hour ("Auguries of Innocence," 1-4; emphasis added)

Most crucially, the "Newtonian" order, which contracts and "harden[s]" minute particulars, "the jewels of Albion," into the identical grains of sand and in which minute particulars and their assemblages are governed by the same law, subject to (mathematical) "Demonstration," would never apply (Jerusalem, Chapter 2, Plate 45:17, 20, 44).

The human vision of the Universe according to Blake, however, and the Universe itself is conceived so as to have more order than any Newtonian universe can possibly have. The ultimately infinite (unlimited and unending) interplay of minute particulars would entail immense (dynamic) order and organization, open to the expanded human vision. It is just that this order is, by definition, assembled out of minute particulars that cannot obey any Newtonian-like law; hence the overall order is not "Newtonian" either, in any reasonable sense of the term. This order is, again, Bohrian, quantum-mechanical, except that it expands into infinity and its continuous plenum, rather than being cut off by a finite limit defined by Planck's constant, h, is never an order of nature itself (although this is quantum-mechanical too, insofar as in quantum mechanics we always deal with our interactions with nature). At each stage, once all or some minute particulars are expanded into richer structures, the same organizational dynamics would apply to the whole new assembly of new minute particulars arising from each previous minute particular. The Blakean universe is the infinite limit of this process, which "limit" is unlimited, insofar as it remains interminably expandable. This, I would argue, also makes Blake's vision of the infinite, not reducible to any mathematical concept of infinity, intensive or extensive, and is a deliberate opposite of differential calculus, which would be for Blake, the calculus (in either sense) of the finite limit of the infinite, which is both its power and its (for Blake, Satanic) limitation.

From this perspective, chaos, or indeed Nature itself (materiality), would be merely an aspect of unregenerated vision. The Blakean organization gives light, illuminates, but thus also eliminates chaos and replaces it instead with the organized infinity. Jerusalem renders this process, with an end point represented by the exuberant picture mentioned earlier (figure 3): "And the dim Chaos brightened beneath, above, around! Eyed as the Peacock" (Chapter 4, Plate 97; E. 257). The order, organization, of minute particulars ultimately suspending or indeed organizing, ordering chaos into an immense order, "in fury of Poetic Inspiration/ … build[ing] the Universe stupendous: Mental forms Creating" (Milton, Plate 30:19-20; E. 129). Or perhaps one should speak of removing the veil, the illusion of chaos and even of nature, since the latter – that is, the Newtonian vision of the world as nature or as divided into nature and spirit, the soul and the body, and so forth – is complicit with Chaos. It is, as Shelley would have it, "a mask of anarchy."

Conclusion

With Bohr, atoms – Bohr's particulars – become technology and thought, through which and only through which we can rigorously approach the ultimate constitution of nature, as this is no longer possible in terms of classical physics and its (physical) atoms. We can, however do so only at the cost making nature inapproachable, inconceivable in its ultimate constitution, and limiting our interaction with nature to set of discrete techno-atomic phenomena in Bohr's sense, defined by the outcomes of the experiments we perform and the equivalent phenomena in nature. But then, Bohr's main concern is not to explore the limit of human imagination, but to offer the primary engine of scientific thought, an interpretation of such phenomena and of quantum mechanics, as a mathematical theory that predicts them.

With Blake, "atoms," as minute particulars, become thought and technology (inscription and writing) that allow us the ultimate vision of reality. There, nature, as conceived by physics, from Newton to Bohr and beyond, is no longer relevant or possible. Blake's works are both multiple allegories of this situation and, like Bohr's techno-phenomena or techno-atoms, "containers" of the ultimate constitution of the world. In Blake, however, these "containers" could be opened and, in their openness, are continuously linked and made to interact by imagination, thus leading us to an infinite, un-containable vision. Blake's main concerns are, concomitantly, a transformation of human perception/imagination and enabling the vision of the (human) world in order to transform not only our thought – quantum mechanics and Bohr's complementarity do this too – but also of our life.

I would argue, however, that the proximity between Bohr's and Blake's visions may ultimately be more significant than the differences between them, fundamental and irreducible as these difference are. It is difficult to overstate the radical nature of the idea that there are no material atoms of matter, but only singular phenomena: Bohr's particulars or Blake's minute particulars, through which and only through which we can, technologically, approach the ultimate constitution of the world – the world of matter for Bohr, the world of the mind for Blake. That this constitution is unavailable to any human vision, according to Bohr, and is reachable by a human vision defined by the Poetic Genius, according to Blake, does not, I would argue, outweigh the significance of this great idea (the singular may be appropriate here). This idea, and thus the link between Blake and Bohr is especially significant because, as I noted at the outset, this argument rigorously applies in physics insofar as it is, in Galileo's words, a mathematical science of nature. There, "Blake against Newton" and "Bohr against Newton" are joined.

It is, as I said, remarkable that "nature," in and by virtue of the quantum character of its constitutions, or at least our interaction with nature, "follows" Blake rather than Newton and his idea of particles and their motion in the void, Blake's primary target. How this fact is related to who we are as human or human/animal beings, to the nature of our bodies and of our minds, is an open and tremendously difficult question. There is, however, no question that, just as is the case in classical physics, the epistemology of quantum physics is related to what we are, to the nature of our bodies and of our minds. The world may exist apart from us, although Blake would not accept this. But we and only we give it a shape, or deprive it of any possible shape, as concerns how we can we see – envision and un-envision – it, including seeing it as the world. The last jury or, in Blake's language, the last judgment (which is to say, the next judgment) as to what nature may or may not permit our thought and imagination in quantum theory, is still out. Blake would reject such limits for imagination, the only true reality for him. The last judgment on this point is still out, too.

Notes

[1] I would like to thank Donald Ault, Roger Whitson, and Terry Harpold for their help in my work on this article.

[2] Cf., Gilles Deleuze in his The Fold: Leibniz and the Baroque, specifically on Tintoretto, and on the Baroque concept of the line, pp. 14-17.

[3] I borrow the expression "abstraction against the richness of being" from the title of Paul Feyerabend's posthumously published, Conquest of Abundance: A Tale of Abstraction vs. the Richness of Being, which offers a Blakean attack on, jointly, both Platonist (mathematical) formalism and Democritean (physical) atomism, and their extension to Einstein and beyond. Feyerabend sees both as complicit in the ideology of abstraction, their essential differences (i.e., mind vs. nature) notwithstanding. Niels Bohr, interestingly, enough is exempt by Feyerabend, for the reasons related to Bohr's ideas to be discussed here, especially his departure from Democritean atomism.

[4] All quotations from Blake are from the David Erdman edition of William Blake’s Complete Poetry and Prose. I ought to clarify that I am not claiming that Blake's text just cited refers to this particular version among his several (four) versions of the pictorial image of the Last Judgment, and indeed the text may itself form a separate (verbal) version of the work, rather than function as a supplement to any among the pictorial versions.   (I am grateful to Donald Ault and Roger Whitson for directing my attention to this last possibility.)  This particular quotation is applicable to all of Blake's versions of the work.  On the other hand, the particular version discussed here appears to me to manifest the Baroque aspects of Blake's art especially graphically (in the present sense of the Baroque and as here considered).  These aspects are found in other versions as well (his text just cited included), and, conversely, the difference between them could be productively explored from this perspective, for example, as representing different interrelationships between the Renaissance (Cartesian or Newtonian) and Baroque spatialities.  Unfortunately, the subject is beyond my scope here.

[5] It is also worth noting, however, that, as so much else in Leibniz, his monadology might be seen as "against Newton." It may also be related to his non-Newtonian view of space, as both relative and relational (vs. absolute) and as defined, vs. Newton's empty space, by the presence of mater in, which anticipates Einstein's relativistic spatiality.

[6] We often see Einstein's relativity, especially his general relativity theory, his non-Newtonian theory of gravity, as his main juxtaposition to Newton, a view, in part, reinforced by Einstein's own comments to that effect. This view is well justified and the relationships between Blake's and Einstein's ideas, especially those concerning spatiality and its relations to matter have been productively explored in literature on Blake, most especially in Donald Ault's Visionary Physics: Blake's Response to Newton – arguably still the best available treatment of Blake's confrontation with Newton – soon to be republished as Blake Newton, and Incommensurable Textuality. Barrytown: Station Hill Press, forthcoming, 2007. On the other hand, the relationships between Blake and quantum theory, and specifically Bohr's ideas, have been barely addressed. Leibniz's thought connects both problematics, Einsteinian and quantum-mechanical, and both, differently (although, again, not without interconnections) to Blake.

[7] While the sketch offered here is sufficient for my argument in this article and in order to follow this argument, it cannot be claimed to be comprehensive in general. I have considered the subject in detail in previously published works, in particular The Knowable and the Unknowable: Modern Science, Nonclassical Epistemology, and the "Two Cultures” and "Mysteries without mysticism and correlations without correlata: on quantum knowledge and knowledge in general." I permit myself to refer to these studies for further details and references.

[8] The fact that even the possibility in principle of knowing through which slit the particles pass inevitably leads to a disappearance of the interference pattern may be shown to be equivalent to Heisenberg's uncertainty relations, ΔqΔp = h, the most famous formula of quantum mechanics. (Here q is the coordinate and p is the momentum of a quantum object; Δ is the degree of deviation from the exact value of a given quantity, and h is Planck's constant.) The formula says that no measurement can ever allow one to ascertain both the position and the momentum of a given object at a given point. The possibility to always do so in classical mechanics allows one to give it both realist and causal character. As concerns the behavior of individual systems it considers, classical mechanics has a strictly deterministic character, allowing us to predict the behavior of the system at any point or fully trace the past of the system, once we know its position and momentum at a given point. The situation is more complex in the case of the systems considered in classical statistical physics or chaos theory, both of which are, however, realist and causal, even though they are not deterministic in this sense.

[9] This nonlocality is a problem of Bohmian quantum mechanics, which is a different theory of quantum phenomena, rather than a different interpretation of the standard quantum mechanics under discussion here.

[10] A concept of dead stars, analogous to that of black holes (insofar as nothing could escape their gravity), was current at the time and might have been known to Blake. The idea was also specifically linked to Newton's corpuscular optics (particles of light in the void), and was abandoned when Newton's optics was abandoned in favor of wave optics.

[11] This conception has a mathematical analogue in topology, which defines its continuous objects (such as lines or surfaces) in terms of covering it by other continuous objects, "neighborhoods "(of points), as they are called.

[12] For the discussion of the continuum problem in its proper context, see Joseph W. Dobben, Georg Cantor: His Mathematics and Philosophy of the Infinite, and for the popular exposition see John L. Casti's discussion of it in his Mathematical Mountaintops: The Five Most Famous Problems of All Time.

[13] I have considered this aspect of the epistemology of mathematics in The Knowable and the Unknowable (126-32).

[14] These and related connections between Blake and modern mathematics have been explored especially in Ault's work, including his more recent and unpublished works.

[15] For an exposition of Bohmian mechanics and accompanying philosophical outlook, see David Bohm's Wholeness and the Implicate Order.

[16] The continuity of waves is also something that Schrödinger initially attempted to bring to the standard quantum mechanics but failed to do, although his mathematical equation itself stands and can, and it appears must, be physically and epistemologically understood along the lines of Bohr's thinking.

[17] I have considered this subject in more detail in earlier approach to Blake's minute particulars in "Chaosmic Orders: Nonclassical Physics, Allegory, and the Epistemology of Blake's Minute Particulars." The present argument, however, departs from that offered in that article, most especially, as concerns the artistic, as against mathematical, nature of Blake's vision.

[18] See Chapter 12, "1227: Treatise on Nomadology: – The War Machine," of Gilles Deleuze and Félix Guattari's A Thousand Plateaus, which also links "state mathematics" to the Newtonian paradigm.

[19] See Jacques Derrida, Of Grammatology.

[20] For a more detailed discussion of fractality and nonfractality see "Chaosmic Orders."

References

Ault, Donald. Visionary Physics: Blake’s Response to Newton. Chicago: U of Chicago Press, 1974.

Blake, William. Complete Poetry and Prose. Ed. David V. Erdman. Berkeley, CA: U of California P. 1981.

Bohm, David. Wholeness and the Implicate Order. London: Routledge, 1995.

Bohr, Niels. The Philosophical Writings of Niels Bohr. 3 vols. Woodbridge, Conn.: Ox Bow, 1987.

Casti, John. Mathematical Mountaintops: The Five Most Famous Math Problems of All Time. Oxford: Oxford UP, 2001.

Deleuze, Gilles. The Fold: Leibniz and the Baroque. Tran. Tom Conley. Minneapolis: U of Minnesota P, 1993.

--- and Felix Guattari. A Thousand Plateaus. Tran. Brian Massumi. Minneapolis: U of Minnesota P, 1987.

Derrida, Jacques. Of Grammatology. Tran. Gayatri C. Spivak. Baltimore: Johns Hopkins UP, 1975.

Dobben, Joseph W. Georg Cantor: His Mathematics and Philosophy of the Infinite: Princeton: Princeton UP, 1990.

Feyerabend, Paul. Conquest of Abundance: A Tale of Abstraction vs. the Richness of Being. Chicago: University of Chicago Press, 2001.

Plotnitsky, Arkady. Chaosmic Orders: Nonclassical Physics, Allegory, and the Epistemology of Blake's Minute Particulars," Romantic Circles Practice Series. March 2001. 22 January 2007. <http://www.rc.umd.edu/praxis/complexity/plotnitsky/plotnitsky.html>.

---. Mysteries without mysticism and correlations without correlata: on quantum knowledge and knowledge in general," Foundations of Physics 33.11. (2003): 1649-89.

---. The Knowable and the Unknowable: Modern Science, Nonclassical Epistemology, and the "Two Cultures." Ann Arbor: U of Michigan Press, 2002.

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