PART III— RELATIONALISM

# PART III— RELATIONALISM

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## Chapter Seven— Numbers, Systems, Functions—and Essences

Sometime in the nineteenth century, mathematics underwent a profound change that was to have lasting effects in a variety of fields.[1] Ernst Cassirer described the change in the last volume of The Philosophy of Symbolic Forms .[2] There were two major trends in modern mathematical thinking, Cassirer says. One is associated with Leibniz, the other with Kant. The trend associated with Leibniz is formalistic, whereas the trend associated with Kant is intuitionistic. A formalistic mathematics is one that emphasizes functions and ordering operations over the empirical reality of the elements that are ordered. What the elements are is not important—they can be numbers, symbols, apples, almost anything. An intuitionistic mathematics recognizes ordering operations and may even regard them as existing a priori, that is, as existing in the mind prior to experience, but it emphasizes the agreement between these ordering operations and the elements that they order. It regards these elements either as being empirical facts or as referring to empirical facts. "Thoughts without content are empty, intuitions without concepts are blind," said Kant in what has become probably the most famous single sentence of the Critique of Pure Reason .[3] For Kant, all experience is ultimately grounded in sense intuition (the process by which we receive, or "intuit," sense data), and this means that the concepts we use to order the elements of our experience, even if those concepts logically precede experience, must always be inseparably tied to experience.

Modern mathematics, Cassirer says, has followed Leibniz rather than Kant. The elements that mathematical propositions deal with are now

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seen as purely ideal, that is, as having no necessary grounding in any sensuous, empirical reality; and no matter what these elements are, the emphasis is not on them but on the structure of relations between them. Developments in nineteenth-century mathematics illustrate this fact. Non-Euclidean geometry, for example, helped turn mathematics "more and more into a hypothetical-deductive system, whose truth value is grounded purely in its inner logical coherence and consistency, and not in any material, intuitive statements" (3:364). Modern symbolic logic, developed by the English mathematician George Boole in the mid-nineteenth century, shows how "the validity of analytical processes is dependent not on the interpretation of the symbols that occur in them but on the laws of their combination" (3:364). In other words, it is not important to know what the symbols are or what real things they stand for; it is important only to know how they are related to each other. By the end of the nineteenth century, Cassirer says, the whole notion of truth and validity in mathematics has been reformed. It is no longer necessary to seek the truth value of a mathematical proposition in the contents of its elements since these elements are seen as purely ideal. Now, instead, we have to look for that truth value in a "complex of relations." The meaning of the purely ideal elements of mathematical propositions, Cassirer says, "can never be disclosed in particular representations directed toward a concrete, intuitively tangible object, but only in a complex network of judgments" (3:400).

This is a startling development. Before the period Cassirer talks about, mathematics was thought of as always ultimately making claims about a sensuous reality. The symbols of mathematics were just that—symbols of something. This something might not be specific and might not be named, but the idea was that something concrete could always be filled in for the symbols. Now, though, it is not the symbols that are important; it is the relations between them. Hence the most important concept of the entire era, namely the concept that we can call generically the system. Cassirer uses the expressions systematic context, complex of relations, and complex network of judgments . By the early twentieth century everyone is using this terminology: there is talk of systems, networks, sets, structures, fields, anything that sounds like a collection of elements in which the way the elements are related to each other is more important than what the elements are.

And there will be talk of groups . The mathematical group is the system concept par excellence, the great testimony of the nineteenth century to the new worship of pure abstraction. Group theory even has a ro-

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mantic history to it, a history that, thanks to the efforts of a spoil-sport historian, has been shown to be false but that survived for a century and a half unmolested.[4] The theory was discovered by a Frenchman, Evariste Galois, and legend had it that he made the entire thing up in 1829, when he was twenty, on the night before a duel in which he knew he might perish (and in which, of course, he did). The real story is actually almost as exciting as the legend. Galois did discover group theory by the time he was twenty. He did die in a duel, and he did spend his last night feverishly working on the theory. It's just that he had discovered it and written about it earlier and was now, the night before his death, merely revising a manuscript for publication.

The precise definition of a group is technical, and its finer details are not worth going into here. Simply put, it goes something like this: A group is a collection of elements that are subject to a combinatory operation. It doesn't matter what the combinatory operation is; it can be an arithmetic operation like addition or multiplication, or it can be an entirely different sort of operation, like a rule for arranging cake crumbs in the squares of a checked tablecloth. There are two requirements that must be met for a collection of elements with its combinatory operation to be a group. The result of combining any two elements (the sum if the operation is addition, the product if it's multiplication, and so on) must always be a member of the original collection of elements, and certain algebraic properties must hold. These requirements give the group what is called closure, which means that the group is always selfcontained because it can never generate anything but its own elements.

An example of a group is the set of all integers with multiplication assigned as the combinatory operation. The set is infinite, but that doesn't matter because the product of any two integers is always another integer, and the algebraic properties all hold. Even though it contains an infinite number of elements, this group still has closure because only integers are admitted, and only integers can be generated by the combinatory operation. Another example is the toy known as Rubik's cube, which was popular in the early 1980s. The cube consists of a number of smaller cubes of various colors that can be twisted and rearranged relative to one another. The stated goal is to arrange the small cubes so that the colors on each face of the large cube will be the same. How is Rubik's cube a group? The elements here are the small cubes, or to be more precise, the positions of the small cubes. The combinatory operation is the twistings—determined by the physical properties of the toy—that rearrange the small cubes. The same physical properties that define

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the combinatory operation give the group its closure: no twisting of the toy (that is, no twisting that doesn't break it) will render anything but a new set of positions (these are the elements of the group), and the various algebraic properties are all observed.

Group theory is a perfect illustration of Cassirer's remarks about nineteenth-century mathematics. The very definition of a group places the emphasis not on the elements of the group, that is, not on what these elements are, but instead on the relations between them—their configuration, their structure, the principles by which they are ordered. Cassirer shows in other writings how extensively the concept was ramified and how influential it had become by the end of the nineteenth century. One ramification was responsible for introducing group theory into the logic of a great many fields in the natural, social, and human sciences. Developments in geometry led mathematicians to ponder the behavior of what they called transformations . A transformation is any operation that takes the members of one group of elements and generates from them a new group of elements. The field known as projective geometry is especially concerned with transformations. Projective geometry studies, for example, what happens to geometric figures when they are "projected," as if by a light source behind them, from their original space onto a new space. The transformation thus takes the arrangement of the points of the original figure (the elements of the first group) and "transforms" it into the arrangement of the points of the new figure (the elements of the second, "transformed" group).

What especially interested mathematicians about transformations was those instances where certain elements were left unchanged by the transformation. These elements were called invariants . Let's take as an example a standard Cartesian coordinate system (that is, a graph with an x-axis and a y-axis). The transformation will be defined as any rotation of this graph around the point where the axes intersect. Any such rotation will clearly produce a change in the graph relative to a fixed observer. But certain things will remain unchanged. For example, no matter how much we spin the graph on its central point, the distance between any two points on the graph will remain the same. The distance function, that is, the mathematical expression of the distance between any two points on the graph, is thus said to be invariant.

Group theory was taken to a higher level of abstraction as transformations of the sort I have described were themselves treated as elements of groups. Like the elements of a group, transformations can be anything at all, and that is why the theory had such wide applications. For in-

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stance, by the early twentieth century the concepts of groups, transformations, and invariants were pervasive in perceptual psychology. Cassirer studied this subject in a number of articles that he wrote in the last years of his life.[5] The central problem was how human subjects order the flood of sense impressions they receive so as to retain a sense of constancy in the objects before them. When we look at an object and then change our position, the sense data we receive from it in our second position are quite different from those we received in our initial position. If we imagine the retina as a film that simply records light impressions two-dimensionally without interpreting them three-dimensionally, then we realize that what we actually see when we look at, say, a cube is a quadrilateral—formed by light reflected off the cube—whose shape differs depending on the position of the cube relative to the retina. If we look at the cube from two different positions, the film impression we get is of two different quadrilaterals. If we look at a sphere of uniform red color, the film impression will show not uniform redness but an infinite range of shades of red. If we change our position, then the distribution of shades changes. And yet in both cases we remain convinced that we are dealing with an object whose actual shape or color is unchanged. In both cases we act against the sense data to remain convinced that what we see from several angles in the first instance is a cube, not a variety of quadrilaterals, and what we see in the second is a uniformly red sphere, not a collection of circles each with various shadings of red.

Cassirer shows how the theory of groups and transformations allowed perceptual psychologists to construct a conceptual model for this phenomenon. The idea is that as human subjects we automatically perform mental transformations of the type I described in projective geometry, and these transformations allow us to orient ourselves in space. (I am simplifying and shortening Cassirer's argument.) The transformations can be thought of as imaginary geometric reorderings of the visual field that are accomplished instantaneously and involuntarily; that is, they are part of the very process of object perception. In the theories of some late nineteenth-and early twentieth-century scientists these transformations exist as groups and follow the mathematical logic of groups. The principle of invariance is what permits us to establish a sense of constancy in the otherwise chaotic mass of sense impressions we receive. Thus the act of perceiving a red sphere involves both a complicated group of transformations, which are made to counteract the direct film impression of the sphere, and a principle of invariance, which

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leaves the red color fixed in the course of the transformations. Henri Poincaré, the French mathematician and philosopher, went so far as to assert that the group concept is a priori, that is, that it actually preexists in our minds and allows us to understand geometry, which he defines as the study of groups of transformations.[6]

Perceptual psychology was not the only field outside of mathematics where groups, transformations, and invariants put in an appearance. If we can believe Roman Jakobson, the concept of the invariant turned out to be the vital principle not only in mathematics but also in linguistics. He wrote on this subject a number of times. For example, in an article for a special issue of Scientific American in 1972, he talks about the important era of discovery for modern mathematics and linguistics and locates that era in the 1870s. He mentions the German mathematician Felix Klein, who in the 1870s proposed a new geometry based on the notion of invariant properties in transformations. Jakobson then demonstrates how similar principles arose in linguistics around the same time, and he cites the work of the most prominent linguists of the age, including Jan Baudouin de Courtenay and Ferdinand de Saussure. The link that Jakobson sees between mathematics and linguistics in that fruitful era is the shared concern for "the conjugate notions of invariance and variation" and for how these notions "brought forth the corollary task of eliciting relational invariants from a flux of variables."[7]

How does the shared concern in linguistics and mathematics show up in linguistics? The idea in the late nineteenth century and early twentieth century was to overthrow the prevailing approach to linguistics, that of the so-called Neogrammarians. This group of linguists viewed language primarily in a temporal sequence—in other words, historically, in its gradual and ceaseless unfolding from one stage to the next. The new, "avant-garde" linguists, as Jakobson calls them, were more interested in language as a relational system now, in the present. Relationalism and relativity were key concepts for this crowd, and Jakobson documents the parallels between the early school career of Albert Einstein and the developing ideas of the Swiss-German linguist Jost Winteler, with whom Einstein roomed as an adolescent when he was at school near Zurich. Winteler had always believed in something he called configurational relativity, and Jakobson intimates that this theory might well have played a role in the development of Einstein's own theory of relativity.

That suggestion might sound a bit extravagant, especially to those who accept the standard mythical view of Einstein as the half-mad sci-

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entist who knew no peers and whose ideas sprang to life spontaneously and unbeckoned in his disheveled head. There is no doubt that Jakobson is attempting to establish the respectable standing of twentieth-century linguistics by placing it on a level with the science that gave us the nuclear age. But he is also saying something serious about linguistics, namely that relationalism (of a sort that possibly bears analogy with relativity in physics) became dominant in it starting late in the nineteenth century. I mentioned in chapter 2 that even Potebnia's theory of language is relational.

What does relationalism look like in linguistics? Consider two prominent ideas in modern linguistics, ideas that, by chance, are among those that Jakobson himself championed during a good part of his career. The first is what Jakobson calls the markedness-unmarkedness distinction. In any linguistic system (in the relational era one speaks of "linguistic systems," not languages) features are distinguished by being marked (present and noticeable) or unmarked (absent and not noticeable, neutral). For example, in some linguistic systems masculine gender in a noun referring to a living creature does not necessarily indicate masculinity in that creature, whereas feminine gender does indicate femininity. In this instance, grammatical masculine gender is "unmarked," that is, it is neutral and carries no positive information about biological gender. Grammatical feminine gender, by contrast, is "marked," that is, it is nonneutral and carries positive information about biological gender.

The other prominent idea is "the continual, all-embracing, purposeful interplay of invariants and variations."[8] Jakobson had contemplated the tremendous variety of sounds in the languages of the world and had attempted to come up with properties that were common to all sound systems. He found the solution to his problem in the theory that there are certain invariant structures in the production of vowels and consonants, like the oppositions between "compact" and "diffuse" (qualities of the resonance chamber in the mouth), "grave" and "acute" (terms describing the kind and degree of occlusion of air as sounds are produced), and "voiced" and "unvoiced." These oppositions are what Jakobson calls distinctive features.[9] Let's say we were comparing the sound systems of French and English. Under the old way of thinking we would have tried to liken similar sounds, and we would have discovered that, for instance, an English b is different from a French b and that an English p is different from a French p . In both instances the French consonant would be far more tense and explosive than its English counterpart. The English and French equivalents are far from identical, and this sort of

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comparison would soon show that the two sound systems had relatively little in common. If, however, we examine the relation between b and p in each language, then we discover something that really is identical (invariant) in both, because both languages contain an opposition between a voiced consonant (b ) and an unvoiced consonant (p ) that are formed in roughly the same way (namely by the lips). What is invariant is not the individual sound p, treated in isolation, but the opposition in French and in English between p and its voiced counterpart, b . Thus no matter how widely the actual sounds of different languages may vary, there will always be a meaningful opposition between factors like the ones Jakobson mentions.

What distinguishes this approach from former approaches is precisely that it is relational. In both the marked-unmarked distinction and the interplay of invariants and variations truth is defined relatively. The isolated sign has no meaning by itself. The marked feature signifies what it signifies only because there is an unmarked feature that stands next to it, and the unmarked feature signifies what it signifies only because there is a marked feature that stands next to it . There is no universal trait in the languages of the world that signifies gender. Similarly, there is no way of reducing all the sound systems of the world's languages to a limited set of sharply defined, individual features. Consonants that we classify as guttural in one language are likely to sound completely different from those that we classify as guttural in another. A classification that is based on isolated qualities like guttural, palatal, and labial is doomed to be only approximate and therefore unsatisfactory. But we can safely talk about the universality of oppositions because when we do this, we are saying nothing about the absolute qualities of the individual sounds that are being opposed to one another. Each sound signifies what it signifies only because its opposite stands next to it, and this opposition has meaning only in the context of the entire language viewed as a system. Meaning is relational.

Jakobson wrote about this creed in many places. In Part II of this book I mentioned the "Retrospect" in which Jakobson describes his own intellectual genealogy by referring to various turn-of-the-century artists. In that essay he twice mentions the relational view of meaning. Here, too, he sees parallels between diverse fields. "Those of us who were concerned with language," he says, referring to the early years of his career, "learned to apply the principle of relativity in linguistic operations; we were consistently drawn in this direction by the spectacular development of modern physics and by the pictorial theory and practice

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of cubism, where everything 'is based on relationship' and interaction between parts and wholes, between color and shape, between the representation and the represented" (SW, 1:632). He approvingly quotes Georges Braque's phrase, which is similar to many others of the age: "I do not believe in things, I believe only in their relationship" (SW, 1: 632). Later on, he describes what he calls the topological approach to linguistics, one in which definitions are "purely relative and oppositive." This time he invokes the Scottish-American mathematician E. T. Bell, who said essentially the same thing as Braque: "It is not things that matter, but the relations between them" (SW, 1:637). In a lengthy article that Jakobson wrote in 1967, "Linguistics in Relation to Other Sciences," he talks about the interaction between linguistics and a number of sciences, including mathematics. Again Jakobson focuses on the branches of mathematics most concerned with relationalism and the system concept—set theory, Boolean algebra, topology, statistics, calculus of probability, theory of games, and information theory, which, he says, "find a fruitful application to a reinterpretative inquiry into the structure of human languages in their variables as well as their universal invariants. All these mathematical facts offer an appropriate multiform metalanguage into which linguistic data may be efficiently translated" (SW, 2:661).

Some of Jakobson's most extensive comments on the historical connection between mathematics and the rise of modern linguistics can be found in his article "The Kazan' School of Polish Linguistics." The article is largely about the Polish linguists Jan Baudouin de Courtenay and Mikolaj[*] Kruszewski, whom Jakobson always considered to be trailblazers in modern linguistics. Jakobson says here, as in the other essays, that the crucial era was the 1870s, since that is when the concept of invariance "became the dominant principle in mathematics" and when "the first glimmerings of the theory of linguistic invariants also showed up" (SW, 2:412). It is the time when Baudouin made his "first attempts at uncovering the phonemic invariants lying beneath the fluctuating surface of speech, which is filled with countless combinatory and optional phonetic variations" (SW, 2:413). Jakobson places Baudouin in the illustrious company of two of the greatest mathematicians of the nineteenth century, Nikolai Lobachevsky and Carl Gauss—Lobachevsky because, like Baudouin, he had published bold new ideas in the bulletins of the backwater Kazan University, and Gauss because all three men had been subjected to intimidation from contemporary critics. As in other passages, Jakobson appears to be craftily suggesting connections that

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probably aren't there. Let's be honest: in the history of Western thought, no matter how great his contribution to modern liguistics may have been, Baudouin de Courtenay is simply not in the same class as Lobachevsky and Gauss. Who besides linguists and some literary critics has ever heard of him? Nor does Jakobson demonstrate any kind of connection between Lobachevsky and Baudouin. Lobachevsky proposed a non-Euclidean geometry, a theory that has enormous importance for the trend in mathematics I've been talking about, so it would be nice to show a connection between the two men. But Baudouin doesn't appear to have taken his idea of invariance from Lobachevsky.

Still, the important thing is that the idea of relationalism is present in Baudouin and that it coincides historically with similar ideas in mathematics. There's no doubt that Jakobson is right. In the same article he mentions Jost Winteler and shows how the notion of invariants appears in his work in the 1870s. Moreover, as in the later "Retrospect," he draws the parallel between linguistics and physics, using E. T. Bell as his authority to say that the concept of invariance was not fully understood until it was supported by the theory of general relativity (SW, 2:427).

Jakobsonian linguistics played an important role in twentieth-century criticism, particularly in the rise of structuralism, as I explained in chapter 6. Many others besides Jakobson have recounted the rise of structuralism in general and structural linguistics in particular, and a number of writers have pointed to the role of mathematics and relationalism in this story. Once again Ernst Cassirer comes to mind. In 1945, a few months before his death, he delivered a lecture titled "Structuralism in Modern Linguistics," which was subsequently published in the journal Word .[10] Cassirer returns here to a theme that had become familiar in his writings: the change in the logic of sciences that took place in the late nineteenth century. The change could be described as the rejection of the old mechanistic models of physical phenomena in favor of systems models. The best example is the quest for a model that would explain electromagnetic waves. Traditional mechanical models, which attempted to provide a purely material explanation of electromagnetic phenomena, did not succeed because they became too intricate, says Cassirer. The solution was the concept of the electromagnetic field, in which individual parts have no independent meaning but exist only functionally, in relation to the entire system. Thus in this model an electron "is embedded in the field and exists only under the general structural conditions of the field" (p. 101). Cassirer speaks of similar developments in other fields and shows how linguistics developed along similar lines. The em-

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phasis is always on the system concept, which Cassirer illustrates with examples from the Danish linguist Viggo Bröndal and Jakobson. The passage excerpted from Bröndal reads like a definition of the system concept: it's all about "system" and "structure," "coherent sets" and the "interdependence of elements" (p. 104).

The most unequivocal statement I know of the mathematical origins of structuralism was made by Jean Piaget, the Swiss psychologist. In his book on structuralism he devoted an entire chapter to the concept of structure in mathematics. He asserts in that chapter that although we cannot tie someone like Lévi-Strauss directly to mathematics, structuralist thinking is nonetheless directly adapted from algebra. Piaget launches into an extensive (at least for the general reader) mathematical discussion of groups and transformations as a basis for an understanding of modern structural thinking.[11]

No one can beat Michel Serres, however, for the elegance of his description of the modern structural theory of meaning and its origins in mathematics. It is the subject of a number of essays in Hermès ou la communication, which Serres wrote during the 1960s and published in 1968, at the height of the structuralist fever in France.[12] Mind you, Serres does not claim to be writing about the structuralist movement as such. He's not interested in the specific expressions of the type of thinking he analyzes, so he doesn't spend his time talking about Lévi-Strauss, Roland Barthes, A. J. Greimas, Tzvetan Todorov, and others associated with that movement. Rather, he is interested in communication, as the title of his book suggests, and communication has to do with meaning. It's not always clear exactly what Serres has in mind historically when he refers to things that are modern. Sometimes he appears to be talking about his own age; at other times he appears to be talking about the twentieth century and its intellectual origins in the middle and late nineteenth century.

In any case, his argument, simplified, runs something like this: The traditional view of meaning, until modern times, has been a linear one. That is, meaning was thought of as tied, in a direct, "linear" fashion, to things. Serres uses the term symbolism in a rather uncommon sense to designate this notion of meaning. In symbolism objects of meaning "stand for" things, and things directly give meaning. A symbol is always linked in a linear way with its sense. The modern age changed the old way of thinking, Serres says. Instead of a linear view, in the modern age we have a "tabular" view of meaning. That is, we have replaced the "symbolist" view with a view that places the emphasis not on things

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themselves but on the relations between them. Meaning in the modern age is an affair that can be visualized as a chart or table (hence "tabularity"), a chessboard, on which only relative positions are important. We have accomplished this change precisely by importing the idea of structure from mathematics, and thus we have benefited from the same revolution in mathematics that Cassirer describes, namely the revolution that changed mathematics (to use Serres's terminology now) from a "symbolic" science (in other words, Cassirer's intuitionistic mathematics, where symbols always stand ultimately for sensuous objects) to a formal science. Tabularity has resulted in the notion of structure, which Serres defines thus (putting the whole passage in italics): "A structure is an operational group with indefinite meaning . . . that groups elements, which exist in any number and whose content is unspecified, and relations, which exist in finite numbers and whose nature is unspecified" (p. 32).

The idea of structure has thus given us a whole new method for classifying things. Previously things were grouped, according to principles of similarity, around a single archetype. In the modern age, instead of families of things grouped around an archetype we have families of models, and instead of an archetype we have what Serres calls a "structural analogon of form," which he identifies as the "operational invariant that organizes [the families of models], all content having been abstracted away." The true triumph of the modern age consists in our ability "to construct a cultural entity by taking a form and filling it with meaning." Structural analysis "dominates, constructs, gives" meaning, where in the past we were always at the mercy of preestablished meanings (p. 33). This is the glory of the modern age of abstraction.

It's not important to pause and define all the terms that Serres uses. He writes in a dense style that makes it difficult to excerpt from him without losing a meaning that comes with the context (in this sense he is an illustration of his own subject). But he really has said all that can be said about relationalism and its logical origins in mathematics. Serres is fond of mathematics and sees mathematical ideas at the origin of lots of different things. In his discussion of modern structural thinking there is no doubt that he is right, provided we don't assign too strict a historical sense to what he says. Structural thinkers didn't go to mathematicians, learn about group theory, and then come up with interpretations of novels. But the logical origin is certainly in mathematics, and it is not purely a matter of coincidence that relationalism showed up in so many different fields at around the same time. This is an age when Braque's

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phrase, "It is not things that matter, but the relations between them," is echoed by many different writers. Everybody is suddenly interested in sets and groups, systems of relations, invariants, models, complexes, fields, gestalts, combinatory operations, and similar concepts expressed with various other phrases. In literary theory it is the age of impersonality, where the essence of a literary work of art is not the author, not even some kind of eternal truth; now the essence is a complex of relations, exactly as it will be when the structuralist critics come along to give their stamp of approval to ideas that had been around for a century. Many of the same writers who, in a backward turn to a theological era, spoke of essences and treated literary texts as religious objects seem at the same time to have come up with a paradoxically secular answer to the question of what those essences are. No one is comfortable any more with the idea of being overtly religious and theological. The religious and the theological survive in the form of myths of essence, and now, as if to disavow any sort of metaphysical status to that essence, modern thinkers make it an abstraction, a relational structure ostensibly emptied of all symbolic (in Serres's sense) meaning.

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## Chapter Eight— Descartes in Relational Garb

"Why do we eat?" begins the popular French cookbook Je sais cuisiner (I know how to cook), by Ginette Mathiot. "Everyone has a pretty good idea. 'We need to eat in order to live.'" The question is how to do it. "In the present era, when the scientific spirit is increasingly asserting itself, nutrition often continues to be the victim of custom and prejudice and is governed by chance or caprice. But knowing how to nourish oneself is a science that one cannot scorn without detriment to one's health and to the family budget." Mme. Mathiot then goes on to explain that food responds to two "essential needs of our organism": the need for matter and the need for energy. Foods can thus be divided into two categories, depending on whether they provide energy or matter.[1] A few dozen pages later the reader has all the necessary information, together with the fundamental principles, to prepare a nutritious, economical, and tasty meal for any occasion that may arise.

A little less than three hundred years before the publication of Je sais cuisiner, René Descartes published his renowned Discourse on Method . In that work Descartes proposed a new "method" of thinking or problem solving. His method consisted of four rules, which Descartes writes in the informal, first-person style of the entire Discourse : (1) "to accept nothing as true that I did not know to be evidently so," (2) "to divide each of the difficulties I was examining into as many particles as I could," (3) "to conduct my thoughts in order, beginning with the simplest objects and those that are easiest to understand, and progressing, as if by degrees, to the understanding of the most compound," and (4)

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"always to carry out enumerations so complete and reviews so general, that I would be certain of having omitted nothing."[2]

Let's see how Descartes's method applies to a given problem—eating. Why do I eat? That's easy: I need to eat in order to live ("to accept nothing as true that I did not know to be evidently so"). How do I determine what to eat? Subdivide: there are two different categories of food, and each of these may be divided into various actual species of edible things ("to divide each of the difficulties I was examining into as many particles as I could"). What to do with all these edible things to make them nutritious, economical, and tasty? Start with the basic principles of nutrition, move to the basic principles of cooking, and continue building from there ("to conduct my thoughts in order, beginning with the simplest objects and those that are easiest to understand, and progressing, as if by degrees, to the understanding of the most compound"). What do I do, now that I'm ready to head for the kitchen and prepare something to eat? Stock my kitchen with all the utensils necessary for successful cooking (Mathiot lists them on pages 51–52) and consult a list of thousands of recipes ("to carry out enumerations so complete and reviews so general, that I would be certain of having omitted nothing").

I don't know anything about Ginette Mathiot's upbringing and can't say whether or not she was a scholar of Descartes. But the method is unquestionably in her cookbook. How many American authors of cookbooks start out with the basic question and the most clearly evident truth about their subject and then reason in so systematic a fashion to the real substance of their books, namely the recipes?[3] The Cartesian method is pervasive in French thought. One sees again and again in writing of all sorts this urge always to start back at the beginning of a thing and then carry it through its steps to the general and abstract conclusions that follow from it. All American students of French know this style of thinking, just as French schoolchildren do, because when they learn to write those infuriating and tedious exercises in literary analysis called explications de texte, they are taught always to start with the facts, to reason from the concrete to the abstract, from the simple to the complex, from the specific to the general.

In his late work The Crisis of European Sciences and Transcendental Phenomenology Husserl made the claim that Descartes is the "original founding genius of all modern philosophy."[4] Descartes's contribution to the modern spirit was to expand Galileo's "mathematization of nature" into a global notion of philosophy as a "universal mathematics" (Universalmathematik ).[5] Descartes's entire method was based on mathe-

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matical principles and designed to bring thought itself into accord with these principles. In the posthumously published "Rules for the Direction of the Mind" Descartes had written, several years before the Discourse, that to eliminate any obstacles in our efforts to discern the true from the false, we must follow an orderly method, like the one we see in the mathematical sciences. "Those who seek the straight path of truth must not concern themselves with any object about which they do not have certainty equal to [that afforded by] Arithmetic and Geometry."[6] In the Discourse and elsewhere Descartes talks repeatedly of the "certainty" and the "evidence" (that is, the state of being evident) of mathematical reasoning. I can't say for sure whether or not Descartes was the "original founding genius of all modern philosophy," but there is no mistaking the traces of his method in French thought.

### Mallarmé and the Light of Reciprocal Reflections

In the first part of this book I suggested that Mallarmé was satirizing the Cartesian method in his introduction to English Words . If that is true, then it is no surprise to see the young Mallarmé in 1869 dreaming up a project for a book in which all the factors I've been talking about come together. It seems that in the late 1860s Mallarmé was considering taking a degree in linguistics, and in a fragmentary note somehow related to this plan he writes:

A strange little book, very mysterious, a bit in the manner of the Fathers, very distilled and concise—this in places that could give rise to enthusiasm (study Montesquieu).

In others, the great and long period of Descartes.

Then, in general, some La Bruyère and some Fénelon with a hint of Baudelaire.

Finally, some me [du moi ]—and some mathematical language.
(OC, , p. 851)

A paragraph or two farther on, he mentions the Discourse (misquoting the title) and then says, "We have not understood Descartes, foreigners have taken possession of him: but he did arouse French mathematicians" (OC, p. 851). Mallarmé never explains the intriguing little phrase about mathematical language. If by chance he meant something like the rigorous, methodical mode of exposition that Descartes's principles suggested, then we know that the closest he ever came to adopting the Cartesian method in his own writing was to parody it. And perhaps the

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effort to parody that method was made only as the result of Mallarmé's having first internalized it.

The mathematical imagination had clearly seized hold of Mallarmé. But it is the "modern" kind of mathematics, the relational kind, that shows up time and again in his writings. Earlier I said that the theory Mallarmé appears to be proposing in "Crisis in Verse" led to a relational notion of language. In fact, if we look at that essay again, we can see that Mallarmé, without knowing it, is essentially giving us an illustration of Serres's argument about linear and tabular meanings. The "uniquestamp" view of meaning, where words turn out to be "materially truth itself," is really the linear view, or the symbolist view (in Serres's sense of symbolism). In this view one can draw a straight line from the signifying object to the thing it signifies. But in Mallarmé's enlightened view of language, meaning comes from the "reciprocal reflections" of words, and this sounds like Serres's notion of tabular meaning.

The idea of a group of relations appears several times in Mallarmé in one form or another. Later in "Crisis in Verse" he refers to music as "the set [ensemble ] of relations existing in everything" (OC, p. 368). Virtually the same phrase occurs in "The Book, Spiritual Instrument." Just after Mallarmé says, "Everything, in the world, exists to end up in a book," he lists the qualities that will be required in his book: "hymn, harmony and joy, like the pure set, grouped in some fulgurating circumstance, of relations between everything" (OC, p. 378). These phrases were composed later in Mallarmé's life. But even as early as 1866 he had adopted a relational view of poetic language. In a letter to François Coppée he says, "What we need to aim for above all in the poem is for words . . . to reflect on one another to the point where they appear no longer to have their own color but to be only the modulations of a scale ."[7] In a curious passage in a letter to another friend that same year, Mallarmé uses an image that suggests the same model. The young poet writes that he has just cast the plan for his entire oeuvre, after having discovered his own "center," "the center of myself, where I sit like a sacred spider, on the principal threads that have already come out of my mind and with whose help I will weave at the points of contact marvelous lacework."[8]

The place where we see Mallarmé's mathematical imagination at its best is his writings about the Book. In chapter 4 I described the manuscript notes that have been published under the title Le "Livre" de Mallarmé . It is full of numbers, calculations, and geometric designs. Some of the calculations appear to have to do with the number of spectators

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at a performance of this mysterious work. Others have to do with the arrangement of pages in the work. Still others are about as apparently mundane a question as the amount of money the author will be able to collect from ticket sales at the performance. Sometimes all three appear together (see illustration).[9]

But even though it's hard to say what these manuscript notes are, we can see that the numbers are an essential part of the work that Mallarmé was contemplating. This is not just scratch paper on which the author figured his monthly budget. The recurrence of certain significant numbers, usually multiples of four, shows that all these details were integral to the work. Whatever this "work" was supposed to be, we can say with some confidence that its numerical properties were not going to be left to chance. In fact, one has the impression that the "content" could never be more precisely spelled out than it was because the essential nature of the work was not the content but exactly these numerical properties. Consider again the drawing on page 95. What is it? Not just doodling, because there are other designs in the manuscript that look similar and are accompanied by various terms that have occurred in lots of other places in the manuscript notes. It appears thus to have something important to do with what the work is. It's as if Mallarmé had set out to determine a set of geometric coordinates for his work and once that was done had decided nothing more was necessary.

How can a work have geometric coordinates? For one thing, it can be a performance in which the disposition of seats, spectators, and performers (the "operator") is an integral part of the work. In that case the work has coordinates in the literal sense, coordinates that belong to actual points in the space in which the performance takes place. Or it can be the sort of thing where there are mystical numerical correspondences between numbers of spectators and the amount of cash they pay for admission. Then the coordinates cannot be assigned to actual physical locations but exist instead as abstractions, and these abstractions are diagrammed in the manuscript notes. Or it can be a book whose pages may be shuffled and reshuffled in any number of combinations to generate as many different "works." In this "mobile-pages" conception the pages of the book are distributed in various determined locations of the space where the book is performed. In any of these cases the essential point about the work is that it is a relational structure.

One of the strangest things Mallarmé wrote is the text called Un coup de dés (A throw of the dice), published in 1897. There's no exact word for what this work is. In fact it's not even clear that Un coup de dés is

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From Jacques Scherer,  Le "Livre" de Matllarmé: Premières recherches sur des documents inédits
(Paris: Gallimard, 1957), pp. 37(A)–38(A).

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the title. The text, twenty-one pages long, contains the sentence "Un coup de dés jamais n'abolira le hasard" ("A throw of the dice will never abolish chance") spread out in four pieces over seventeen pages ("Un coup de dés / jamais / n'abolira / le hasard"). This central sentence is printed in oversize type. In between its fragments are numerous phrases and sentences written in a variety of smaller types. The "poem," if that's what it is, is of course tricky to read. The main sentence gives it a kind of syntactic completeness and serves as a unifying device. The problem is what to do with all the other words. It is tempting to read them as a highly complex network of parenthetical and dependent clauses, where the clauses printed in smaller type depend on the ones printed in larger type. But even if you try to read it in this way, you get hopelessly bogged down in twenty-one pages of interlocking grammatical dependence and soon give up. Paul Valéry, who claims to be the first human being (other than the author) ever to see this work, says that Mallarmé first read it to him "in a low, even voice, without the slightest 'effect,' almost to himself."[10] The text actually may be read in many different ways: you may read one type size at a time, you may read through the text in the order in which the words are printed, or you may simply read in any order you choose. Once again we have a shuffling game, a relational scheme whose most prominent feature is precisely its refusal of linearity. This refusal is both syntactic and semantic—syntactic because a linear reading of the text is almost impossible, and semantic because this is one place where even the most old-fashioned thinker will see that meaning does not arise word by word in a linear fashion.

There is also no linear connection between the text and the person who wrote it. How do you establish an authorial voice for a thing that can't really be read in any of the traditional senses? no, there can be no sign of the author's presence, and strikingly enough, the disappearance of the author (a much-touted idea in Mallarmé's age) seems to be explained by the relational quality of the text. Think back to the passage in "Crisis in Verse" in which the notion of reciprocal reflections is introduced: "The pure work implies the elocutionary disappearance of the poet, who instead yields the initiative to words, mobilized by the clash of their inequality; they light up from their reciprocal reflections, like a trail of fire on gems, taking the place of that palpable breath in the lyric inspiration of yore or the enthusiastic personal direction of speech" (OC, 366). Once words start reflecting off each other, they can no longer reflect back to the author, or so the author appears to be suggesting. Right before the passage where Mallarmé talks about the "pure set . . .

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of relations between everything" he says that the volume should "require no signatory" (OC, p. 378). In his autobiographical letter he speaks of a "text speaking of itself and without an author's voice" (OC, p. 663). And in Scherer's manuscript notes we read on the next-to-last page about a volume "for whose sense I am not responsible—not signed as such," this in the midst of a flurry of calculations determining the order and placement of pages (201 [A]).

"Things exist, we don't have to create them; all we have to do is grasp the relations between them; it is the threads of these relations that form verses and orchestras" (OC, p. 871). A sentence like this looks very structuralist avant la lettre . In fact, this and other passages convinced James Boon to write From Symbolism to Structuralism, in which he points out the affinities between the thought and poetics of French symbolist poets (Baudelaire, Mallarmé, Rimbaud, Verlaine, and, oddly, Rousseau and Proust) and the ethnology of structural anthropologist Claude Lévi-Strauss. For Boon, the passage I just quoted is emblematic of the whole worldview he sees in Mallarmé. Everything in Mallarmé's universe is relation, analogy, connection, and structure, just as it is in Levi-Strauss. The emphasis in both writers is always decisively shifted away from content. The one thing Boon doesn't discuss is the mathematical foundations of this type of thinking, but we know it's there for Mallarmé as it is for the entire structuralist movement.[11]

### Valéry and the Discourse On His Method

Paul Valéry lived his life as though he had been put on earth to provide the rest of us with a caricature of the French mind. "Ce qui n'est pas clair n'est pas francais," goes the timeworn eighteenth-century expression that our French teachers taught us in high school. But clair in French means both "clear" in the sense of comprehensible and "light" in the sense of bright, and Valéry's poetry is filled with radiant Mediterranean sunshine. The mind strives for the clear and comprehensible because what is comprehensible is radiant. Valéry worshipped the human mind, especially his own. The "hero" of his prose work Monsieur Teste (Mister Head) says something that perfectly describes the author and his characteristic pose:"Je suis étant, et me voyant; me voyant me voir, et ainsi de suite . . ." ("I am [in the process of] being, and seeing myself; seeing myself see myself, and so on . . .").[12]

Valéry loved admiring the workings of his own mind. Wrapped up

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with this fascination for the mind was a fascination for mathematics that went far beyond the numerical fantasies of Mallarmé. Unlike Mallarmé, Valéry actually fancied himself a mathematician, and he devoted an astounding amount of time and energy to studying contemporary mathematical theory and then trying to find applications for it in the most extraordinary fields. His friend Pierre Féline introduced him to a number of areas of mathematical study, among them group theory and transformation group theory. Valéry was apparently fond of using his knowledge of mathematics to dazzle his other friends, people like André Gide, who knew very little about such things.[13] The twenty-nine-volume set of Valéry's notebooks is a startling record of a mind obsessed with numbers, with itself, with its obsession with itself, with a mathematical expression of its own workings, and so on—a kind of monument to neo-Cartesian narcissism.[14]

Valéry's notebooks show a curious attitude toward "the original founding genius of all modern philosophy." To begin with, even though he devoted a substantial number of pages to ruminations on philosophers, Valéry was fond of playing the part of someone who is foreign to hard-core philosophy and doesn't quite understand it all. So when it comes to Descartes, he likes to talk about things that fussy, scholarly people would consider trivial. In one fairly long notebook entry he mentions the qualities that have struck him personally in various philosophers—this after saying that philosophers are boring to read and that their language is antipathetic to him. Of Descartes, Valéry can say only that it is the "individual" that "appears" to him, meaning, I assume, that he is attracted by the personal style of Descartes's writing.[15] He returns over and over again to the subject of Cartesian doubt and the most famous phrase from the Discourse on Method, "I think, therefore I am." And he speaks of the "insignificance" of the principles in the Discourse, saying that the charm of the work is "above its substance" (C, 16:728 [1:673]).

Like Descartes, Valéry sought clarity, certainty, evidence (again in the sense of "state of being evident"), and he sought it in mathematics. "Descartes. No occult qualities—made the greatest effort for Clarity, " Valéry muses at one moment in his notebooks (C, 10:103 [1:595]). "On Descartes: Clear and distinct ideas, " he writes a number of years later (C, 20:508 [1:700]). But the model he used was always the structural one, the one that placed the emphasis on the relation between elements rather than on the elements in and for themselves. Valéry saw the mind as essentially a relational system whose operation he attempted to de-

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scribe in the language of group mathematics. "Every act of understanding is based on a group," he says (C, 1:331). "My specialty—reducing everything to the study of a system closed on itself and finite" (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one "group" undergoes a "transformation" and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: "The S[elf] is invariant, origin, locus or field, it's a functional property of consciousness" (C, 15:170 [2: 315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind's momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. "Mathematical science . . . reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind" (O, 1:36). "Psychology is a theory of transformations, we just need to isolate the invariants and the groups" (C, 1:915). "Man is a system that transforms itself" (C, 2:896).

The Notebooks are not the only place where Valéry indulges in this kind of speculation. His mathematical theories turn up in his published writings, too. In an article in Mercure de France in 1899 he wrote about what he called the reversibility of states of consciousness. "The transformations that any given system undergoes are reversible when the system is able to return from a certain state to an earlier state, passing through the same states during the return as it had on its way here, only in reverse order. This definition, though its origin is in physics, is sufficiently general that one can attempt to apply it to the mind and view the mind as a system of transformations" (O, 2:1459). In a piece called "A Few Words about Myself," published in 1944, Valéry describes how he discovered at the age of twenty "that man is a closed system with respect to his cognition and his acts" (O, 2:1518). Later in the same essay he describes the entire credo of his youth:

There was a time when I saw.

I saw or wanted to see the figures of relations between things and not the things.

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Things made me smile from pity. Those who paused to consider them were to me sheer idolaters. I knew that the essential thing was figure .
(O, 2:1532)

The term system has at least two important senses for Valéry, and several of his notebook entries show that the elaboration of the idea of systems was associated with Descartes. I mentioned earlier that Mallarmé had a metaphysical crisis in his twenties, which led him to some important philosophical discoveries that were to occupy his mind for the remainder of his life. In 1892 Valéry, who had just met Mallarmé the year before, had a "crisis" of his own. On the face of it, it was a rather trivial matter—a case of unrequited love. It did, however, lead him to a major turning point: he renounced the emotional life in favor of something that he would call the System (with a capital S ). The System was an ideal program of intellectual contemplation whose chief purpose was to submit all important phenomena of mental life to rigorous, dispassionate analysis, the sort of thing we see piecemeal in Valéry's notebooks, which in a significant sense are the result of this program.[16] But system (with a lowercase s ) also means relational system in the sense I've been using. When Valéry talks about the System he often feels the need to mention systems and to bring up Descartes:

Grosso modo the System has been the quest for a language or a notation that would make it possible to treat de omni re just as analyt[ic] geo [metry] allowed Des Cartes [sic ] to treat all figures.

The human body (viewed as a syst[em] of variables) must be able to reveal this secret.
(C, 9:82 [1:812])

In another long entry, written about fifteen years after this one, Valéry's subject is once again the System: "It used to be, would have been, is, was, and would be a kind of method à la Descartes—I mean Geometry—because it would have to do with a sort of systematic translation of the diversity of objects and the transformations of consciousness or the mind into elements and modes of functioning (observable or probable) of this mind." And a little later in the same entry he brings up once again the notion of system:"In short, it seemed ever more strongly to me that what appears almost always and necessarily, like things, world, ideas, cognition, was somewhere else, the product of a functioning—that is, a bounded, closed system, forced to return to itself" (C, 20:290–292 [1:846]).

Perhaps the most intriguing of the notebook statements on the System

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and its Cartesian analogy is the one in which Valéry writes in adjacent columns about Descartes and himself. On the left side of the page are four brief phrases having to do with Descartes: "Discours de ma méthode" (instead of "Discours de la méthode"), "Story 1892" (the year of his crisis), "The finite—reduction to my system," and, in a reference to the Latin version of Descartes's famous sentence, "Instead of Cogito and Sum, my formula." On the right side of the page he writes this: "The System—is not a 'philosophical system'——instead, it's the system of me —my potential —my coming and going—my way of seeing and returning" (C, 18:55 [1:841]). As this entry and many others show, Valéry regarded Descartes's method not precisely as being similar to his own but as being a kind of analogical model, much as Kant invoked the name of Copernicus to call attention not to the content of his own new philosophy but to its revolutionary character.

The last entry I quoted is especially valuable because it hints at the reason for the difference between Descartes and Valéry. Descartes, too, founded a "system." In fact, his philosophy is often referred to as systemlike because of the coherence of its parts and its "closed" nature. But for Valéry, it is still just a philosophical system, that is, a corpus of writings on "philosophical" subjects. Valéry's System is different because its very logic is different. In Valéry we see the logic of relational systems, the "system of me." And what could be more systematic than a system of thought whose primary characteristic is that it is systemlike?

Valéry was consistent, and we find the system concept everywhere. In a world where the mind is a system, it is natural that the things the mind makes should look like systems, too. "I have eternally sought to define or construct a system of variables (that is, a systé[em] of notations)—and of the relations of the conditions among them that would make it possible to represent tangible life," reads an entry in the Notebooks (C, 18:608). One important category of things the mind makes is of course artworks, and not surprisingly, these behave like the systems (the minds) that created them. There's a reason for this, but it has to do with the subject of Part IV of this book, so I will put off talking about it until then. For now, let's just observe how the system concept works in Valéry's vision of artworks.

To a certain extent Valéry's remarks on artworks are historical. That is, he sees the relational quality of art partly as a phenomenon peculiar to the modern age. In an early article called "On Literary Technique" Valéry writes about the modern conception of the poet: "He's no longer the disheveled, delirious man, someone who writes an entire poem in a

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night of fever; now he's a cold scientist [savant ], almost an algebraist, in the service of a refined dreamer" (O, 1:1809). Mallarmé is partly to be credited with the modern spirit in poetry, as Valéry says in another place: he was "the first writer who dared to envisage the literary problem in its full universality . . . He conceived as algebra what all the others have thought about only in the particularity of arithmetic."[17] But certain types of artwork for Valéry are intrinsically structural, regardless of the historical period during which they were created. Valéry was a firm believer in the poetry-prose distinction. In "Poetry and Abstract Thought" the emphasis is on the difference between poetic language and the language of prose. Valéry felt that the distinction extended to the entire work of an, which is to say that a poem as a whole is different from, say, a novel as a whole. The distinguishing factor is the relational quality of poems, as he explains in his "Homage to Marcel Proust": "And while the world of the poem is essentially closed and complete unto itself, being the pure system of the ornaments and possibilities [chances ] of language, the universe of the novel, even of the fantasy novel, is connected with the real world." (O, 1:770). One of Valéry's favorite art forms was dance, because it so clearly embodied the relational principles he wanted to see in everything. In an essay originally given as a lecture in 1936, "Philosophy of the Dance," he says: "No exteriority! The dancer has no outside . . . Nothing exists beyond the system that she forms through her acts" (O, 1:1398). And a paragraph later he describes dance as "a group of sensations that makes an abode for itself, . . . that emits from the depths of itself this beautiful series of transformations in space" (O, 1:1398). Valéry's personal dream was to create the perfect, mathematically determined work of art, something he mused about in his notebook the year before he died:

To arrive at the completion of a work by means of formal conditions accumulated like functional equations——

in such a way that the possible contents are more and more circumscribed

Subject, characters, situations result from a structure of abstract restrictions—
(C, 28:468 [1:314–15])

If ever there was a prestructuralist thinker, it was Valéry. I say pre structuralist not only because Valéry's thought looks like structuralism even before there was any such field as structural linguistics, anthropology, or criticism but also because it shows the logical foundations of struc-

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turalism in all their naked, unabashed glory. Few structuralists later in the twentieth century felt the need to expose the mathematical foundations of their thought or repeat the by then timeworn credo about how the importance is in the relations and not in the elements themselves. They had gone beyond this credo, had taken it all for granted as something that didn't need to be said in their elaborate discussions of structures and systems. But Valéry did feel the need to say it—over and over and over again. For him, the real truth—the real essence, to be more accurate—always had to be anchored firmly in the neo-Cartesian logic of relationalism. And the credo was always worth repeating, as if that would keep the relational essence from suddenly slipping away, leaving behind the most dreaded monster of all: things themselves .

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## Chapter Nine— How Numbers Ran Amok in Russia

For the French, mathematics is always tied up with the Cartesian ideal of intellectual clarity and rigor, even if that ideal is there for no other reason than to be ridiculed. But leave it to the Russians to turn mathematics into a forum for impassioned political and theological debate. I gave a simple definition of group theory in chapter 7, and most people would be hard put to figure out how that theory could have anything to do with politics or religion. Alexander Vucinich, who wrote a two-volume history of the role of science in Russian culture, tells the fascinating story of how group theory caught on in Russia at the end of the nineteenth century, after first having been introduced there in the 1860s, and how it came to represent one camp in a struggle among mathematicians for the soul of the motherland.[1] The passion in this struggle appears to have been concentrated lopsidedly in the opposing camp. Classical mathematics, meaning differential and integral calculus in particular and algebraic calculations in general, promoted a mechanistic view of the universe, one in which causality reigned supreme and in which there was no room for "noncontinuous," free phenomena. In other words, classical mathematics is essentially materialistic, or so the argument ran in this camp. Ever since the 1860s, when the philosophy of nihilism came to dominate the political left in Russia—the philosophy, that is, whose basic premises were the material character of all natural phenomena, the nonexistence of free will, and the continuity of the animal and the human kingdoms—a pitched battle had raged between secular materialists on the left and conservatives on the right.

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Since the right supported the autocracy, and since conservative nationalist sentiment in Russia was always bound up with Russian Orthodox Christianity, the opponents of secular materialism championed the cause of philosophical idealism, if not outright religious fundamentalism. And so by a leap of logic that will seem astonishing today, the antialgebraists embraced such fields of mathematics as (they felt) supported idealism and freedom of will, two things that the obstreperous materialists on the left had no patience for. "Arithmology," or the theory of discontinuous functions (that is, functions not susceptible of causal explanation, thus "free"), became the favored field, and to free Russia from the scourge of materialism and make the world safe for imperial autocracy, these brave mathematicians set about to arithmetize mathematics.

How, you might ask, could anyone take this program seriously? It's hard enough to understand what idealism and materialism have to do with arithmology and algebra. It's even harder to understand how anyone can translate the study of fields as removed from the physical universe as these into the realm of nationalist politics. The trouble is that the antialgebraists were led by one of the foremost academic mathematicians of the day in Russia, none other than Nikolai Vasil'evich Bugaev, dean of the natural-science faculty at Moscow University and well known in intellectual circles. He was also Andrei Bely's father. Bugaev, as it happened, did not even believe in God. Bely always portrayed him as a severe, contemptuous, and demanding skeptic. He was an odd figure to be supporting ultranationalist causes in a country where ultranationalism was so tied up with the church. But apparently for him, idealism even without religion was close enough to the essential spirit of Russian autocracy, and so he embraced his cause with extraordinary fervor.

Most historians agree that the truly important developments in nineteenth-century mathematics were in precisely the areas that Bugaev and his supporters opposed, those areas, incidentally, that Jakobson considered so fruitful for the development of modern linguistics and that Cassirer considered so fruitful for twentieth-century thought in general. The division between the opposing camps in Russia quickly became blurred in the first decades of the twentieth century, when followers of Bugaev contributed to discoveries in such areas as quantum mechanics and drew on set theory—an algebraic field if ever there was one—for their work.[2] In any case, it would have been hard for nonmathematicians to take sides in a struggle like the one between Bugaev and the algebraists since most people would have trouble understanding the philosophical and political implications of mathematical theory. But that did not stop a

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great many artists and literary figures from becoming consumed by an overpowering interest in mathematics, and Bely was among them. Few of them could be described as really familiar with current mathematical theory of the sort that Bugaev and his friends were arguing about, but the number mania that took hold in the literary and visual arts (which were often hard to distinguish from each other) in the first three decades of the twentieth century ended up producing the same cult of relationalism in the arts as we find in human sciences like linguistics. And relationalism invariably meant a form of essentialism.

### Bely's Baskets, Roofs, And Rhombuses

Nikolai Vasil'evich Bugaev was a respected mathematician, and he and others like him drew all kinds of extravagant inferences about the connection between mathematics and politics. In such a climate perhaps some of the ideas his son came up with aren't so odd. Bely was a student of the natural sciences for a brief period, and at one time he contemplated the possibility of constructing an exact science of aesthetics. In an essay called "The Principle of Form in Aesthetics" he proposed one approach to the problem.[3] The notion was that the various art forms (music, poetry, painting, sculpture, and architecture) are distinct expressions of something single and universal (which we may call art, for the sake of simplicity). Bely wondered if a set of a priori principles could be discovered that would show why art in a certain instance can manifest itself, say, only as music or only as painting. An exact science of aesthetics would demonstrate that the various art forms are actually subject to a kind of logical necessity, not only in the sense that each an form exists as an art form by logical necessity (before the world existed, one could have predicted that there would be music, poetry, and so on) but also in the sense that any work of art comes by logical necessity to be expressed in the form in which it is expressed.

There's nothing amazing about that idea, and Bely is certainly not the first person to have thought of it. But next he takes it into his head to find a model for his theory in thermodynamics, and he spends the rest of the essay working out the details. If there is a principle of conservation of energy in physics, Bely thinks, then there must be a principle of conservation of creative energy in art. Using terms like quantity, tension, and kinetic creative energy, Bely bombards his reader with equations and calculations that make what he writes look like a chemistry

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textbook. The difference between what Bely writes and a chemistry book, though, is in the words between the equations. For example, at one point Bely says:

While composing large-scale artworks, Ibsen, for instance, attempted at first to expend a certain quantity of energy

,

but then, through corrections in his manuscript, heightened the tension of the expended effort:

This is how Goethe wrote Faust .
(S, p. 189; SE, p. 217)

Of course I've quoted this passage out of context, and some of the terms have been used earlier in the essay. But this is not just a cheap trick to make Bely look like a lunatic. The leap from physics to Ibsen and then to that extraordinary final comment on Goethe, which concludes a whole section of the essay and is never developed further, is every bit as wild when it's read in context.

"The Principle of Form in Aesthetics" was a youthful attempt at something Bely did not pursue in later years. Three years after he wrote it, however, he hit on another angle of the scientific aesthetic, one that was to prove much more fruitful. In 1909 he wrote four studies on poetic meter and rhythm in which he set out a method of verse analysis that combined elementary arithmetic and geometry. The result was a highly relational conception of the poetic work of art. What distinguished his approach in these studies from the one he proposed in "The Principle of Form in Aesthetics" was the role of the researcher. In the earlier study the researcher's task was to find, by purely logical deduction, the principles that precede the existence of any actual works of art. In the verse studies, however, the researcher's task is empirical and descriptive, and the conclusions are based on data taken from real works of art. In the earlier essay the method was deductive; in the verse studies it is inductive.

The first of the studies on meter and rhythm is titled "Lyric Poetry and Experiment" ("Lirika i èksperiment") (S, pp. 231–85; SE, pp. 222–73). Almost half of the essay is given over to a discussion of the empirical, descriptive method and its importance for establishing an exact science of aesthetics. The remainder is devoted to the elaboration of the specific

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method of verse analysis Bely will use in all four essays. Russian, like English, has a syllabotonic versification system; a line of Russian verse consists of a fixed number of syllables with a regular distribution of accented syllables. Because the distribution of accented syllables is regular, Russian verse can be divided into metrical feet. Russian verse accommodates a greater variety of feet than English, and all the basic combinations can be easily found: iambic, trochaic, anapestic, dactyllic, even amphibrachic.

By far the most common of Russian meters is iambic tetrameter, and it is the one Bely concentrates on. Bely was struck by something that no one had paid much attention to before, although it was perfectly obvious. Russian words generally contain only one accent each, and many of them are quite long. In fact, the average proportion of accented to total syllables in Russian prose, it was later shown, is 1 to 2.8.[4] This means that it is impossible to write truly iambic verse in a sustained way without resorting exclusively to the use of short words. What actually happens in Russian verse, of course, is that a great many positions that should be occupied by an accented syllable are not; so when we describe a particular set of verses as being written in iambic pentameter, we are not speaking with strict accuracy. In a given line of iambic pentameter there are likely to be one or more pyrrhic feet (both syllables unaccented). This tendency leads to the capital distinction Bely makes between meter and rhythm. Meter is the regular pattern a poem is meant to conform to, and terms like iambic pentameter describe it. Rhythm, by contrast, is the actual pattern we find in a poem. Since poems don't conform to the ideal pattern of a meter, their rhythm is really a pattern of violations. Bely called these violations either half-accents, because we tend to give a slight accent to a position in a verse that should be accented even when no accented syllable occurs on it, or accelerations, because in these half-accented positions the lack of a full accent has the effect of speeding up our reading.

The interesting part of Bely's analysis comes next. He decides that a good way to characterize the rhythm (not meter) of a poem is to draw a graph showing all the lines of verse and the four feet in each line. He places a dot in every position where there is a violation (an acceleration) and then forms designs by connecting dots that occur either in the same line or in consecutive lines. Any poem has numerous lines that are metrically regular, and dots are not connected over these, so the figures formed by this connect-the-dots game are usually small. Bely, carried away with enthusiasm by his pictures, then names them. If one line of

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verse contains a single acceleration and the following line contains two, we get an upright triangle. If the first line contains two and the next line one, we get an upside-down triangle. There are crosses, roofs, rhombuses, baskets, M 's, Z 's, and many other patterns.

Of course, a descriptive system like this must rest on the assumption that the geometric figures correspond to something perceptible to a listener or reader. An upright triangle, for instance, must come across as a rhythmic pattern of gradual acceleration, since it consists of a line with only one acceleration followed by a line with two. And when we say that baskets and rhombuses are particularly frequent in the verse of a certain poet and so are characteristic of that poet's rhythm, we must be using the words baskets and rhombus only as a kind of shorthand for something we can hear when we listen to the verse of the poet in question.

But after a short while Bely seems not to care much about whether his system corresponds to anything the listener hears, and his account of rhythm and versification is given over to charts and pictures. One has to ask at this point what the object of Bely's study really is. Toward the end of "Lyric Poetry and Experiment" Bely uses his geometric figures to make a distinction between "rich" rhythm and "poor" rhythm. Rich rhythm is characterized by a relatively large number and variety of geometric figures; poor rhythm is characterized by a relatively small number and variety of them. At one point, having charted the rhythmic patterns of selections from several poets, he says: "Comparing the examples of rich rhythms with the examples of poor rhythms, we see that the rhythmic figures for the rich rhythms are distinguished by greater complexity. The lines here are broken rather than straight, and simple figures join together here to form a series of complex figures" (S, p. 271; SE, p. 260). Remember that broken lines and straight lines correspond to something that should ultimately be perceptible when we listen to or read the poetry in question. But Bely largely stops talking about that and focuses instead on the designs themselves, which have now become a measure of the worth of a poet's verse. The second of Bely's four essays, "Toward a Characterization of the Russian Iambic Tetrameter" ("Opyt xarakteristiki russkogo cetyrexstopnogo[*] jamba") (S, pp. 286—330), consists almost entirely of statistical charts and descriptions of geometric figures. For example, we read this passage: "The roof is one of the most typical rhythmic devices. Pushkin uses it less often in his Lyceum poems than subsequently. Thus in 596 lines of verse from the years 1814 and 1815 only two roof figures occur. In a corresponding number of lines

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of verse from the years 1828 and 1829 we encounter the device in question 8 times. Could this be accidental? When I take another 596 lines of iambic tetrameter from the poems of 1824–1827, I find the corresponding device 6 times. I conclude from this that the more frequent use of this figure by Pushkin corresponds to a strengthening of his rhythm" (S, p. 309). And Bely goes on to provide statistics for the occurrence of the "roof" in other Russian poets.

This is not at all to say that Bely's system is without value. In fact, he pulls off a real coup in "Lyric Poetry and Experiment" by showing that his scientific definition of rich rhythm is borne out by common notions of the worth of poets. He takes a selection of Russian poets ranging from great to mediocre and examines 596 lines of verse by each, adding up for each poet the total number of geometric figures. It turns out that those with the greatest number are those commonly considered to be the greatest poets, and those with the lowest number are those most Russians would agree are second-rate or worse (S, pp. 273–75; SE, pp. 263–65).

Earlier I said that Jakobson gave Bely credit for inspiring him to undertake the analytic study of verse. Jakobson makes his remarks in the "Retrospect" to the fifth volume of his Selected Writings .[5] It is testimony to Bely's true importance in our story that Jakobson, writing not too long ago, begins the "Retrospect" with a discussion of the very essays I've been talking about. Jakobson disputes Bely's rather exaggerated view of his own importance in the history of verse studies, but he goes on to say that "beyond any doubt, Belyj's inquiry was the first to throw light on the Russian iambic tetrameter, its manifold accentual variations, and significant modifications which this favorite Russian measure underwent from the eighteenth to the early twentieth century. He discerned diverse and formerly unnoticed particulars and posed many questions of wider scope" (SW, 5:569). After this, Jakobson tells of how he himself attempted to apply Bely's method when he was still a teenager. He tells of the critique that another Russian symbolist poet, Valerii Briusov, wrote of Bely's verse studies in 1910. Jakobson mentions the work of the Moscow Rhythmic Circle, which Bely founded and which made certain advances over Bely's pioneering work. And he describes his own role in this history, how the Moscow Linguistic Circle, of which he was a member, systematically revised the work of both Bely and Briusov in 1914 (SW, 5:570).

I can't help thinking, however, that the true legacy of Bely's work is to be found not in Jakobson's studies on versification, meter, and rhythm

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but instead in the study of Baudelaire's "Les Chats" that Jakobson coauthored with Lévi-Strauss. On the surface Jakobson and Lévi-Strauss's study appears to be completely different from Bely's verse studies. Jakobson and Lévi-Strauss are analyzing a poem, to be sure, but the focus is almost exclusively on the grammatical characteristics of Baudelaire's verse rather than on such purely formal aspects as the occurrence and position of accentual irregularities. But if we take a closer look, we see that the method and the results are quite similar. Jakobson and LéviStrauss, toward the end of their essay, after having found many different patterns by which the poem may be organized according to grammatical features, say this: "As we now reassemble the pieces of our analysis, let us try to show how the different levels on which we have situated ourselves blend together, complete each other, or combine, thus giving the poem the character of an absolute object."[6] Their idea was to take the poem apart and put it back together again, and the two authors have done so several times over. The result is a whole new object, something different from the poem we started with, something better, something to replace the poem. What we end up with is a wondrous relational web that ties together all the different related points, a kind of transcendent schema that leaves Baudelaire's cats, with the "mystical pupils of their eyes" and their "fecund loins . . . full of magical sparks," in complete obscurity.

This is exactly what Bely had done, too, only he did it in 1909. He was a structuralist long before Jakobson and his friends ever dreamed of the kind of analysis that we see in the Baudelaire study and even before Jakobson began using the concept of structure in his earlier writings. Bely has taken thousands of lines of poetry and replaced them with boxes, rhombuses, baskets, roofs, crosses, and zigzags. He's forgotten what all the poems were about. There's nothing about the poor clerk whose fiancée has drowned in a Petersburg flood, nothing about the cheap pathos of the death of a peasant, nothing about the Georgian beauty singing her sad songs. Just baskets.

When it comes right down to it, Bely is also doing the same thing he did in "The Emblematics of Meaning." He's insisting that all the things around us that signify something are just surfaces hiding an essence. Bely never wants to sound too much like a religious man (how could he, coming from the home he came from?), so he always calls the essence something nonreligious. In "The Emblematics of Meaning" it was "value." In the verse studies it is a geometric system, a relational abstraction, a structure.

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I'm not saying that Jakobson learned structuralism from Bely or that we can trace a line directly from "Lyric Poetry and Experiment" to the essay on "Les Chats." What I am saying is that the method was there for Jakobson to see at a time when his ideas were only beginning to take shape, that the method is really the same as the one that Jakobson was to follow later, and that Bely's essays have been largely unknown for decades, whereas Jakobson's writings are known the world over. Would there have been literary structuralism without Bely? Of course. The most ardent Bely enthusiast would never be so audacious or foolish as to claim otherwise. But structuralism might well not have been the same had it not been for him.

### A Story of Squares, Rays, and Exhausted Toads

Mathematicians like Bugaev, no matter how outlandish their ideas about the applications of mathematics, were at least firmly rooted in whatever branch of mathematics they had ideologically committed themselves to. Even Bely, whose geometric figures take on a life of their own, started out with the perfectly respectable goal of using exact methods to analyze certain properties of verse. With the Russian Futurists, however, the connection with any goal as tangible as that becomes increasingly remote, and numbers in all senses—as abstract quantities, as members of relational systems, as printed figures representing quantities—become the object of an almost mystical fascination.

The term Futurist is not very precise. It would probably be more accurate to refer to the group of artists I have in mind as members of the Russian avant-garde, where avant-garde is used in a broad and unofficial sense. Still, Futurism is used loosely to refer to a large group of writers and visual artists who flourished from about 1910 through most of the 1920s. Any standard work on Russian Futurism will explain that the movement (if we can call it that) was divided into several different groups with strange names like Hylaea, Cubo-Futurism, the Mezzanine of Poetry, and Centrifuge, that the groups were usually at odds with each other over issues that would strike anyone from the outside as exceedingly bizarre, and that the members of individual groups were often at odds with each other, with the result that membership in the different groups was highly fluid and unstable.[7]

One writer has observed that the central feature of modernity is how the new comes to be seen in it as an absolute value.[8] This is especially

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true for the various movements that make up what we call Russian Futurism, which saw the new as a source of human salvation in the twentieth century. The naughty boys of this period were fond of defiling the images of all the classic figures of Russian culture, saying things like, "Throw Pushkin, Dostoevsky, Tolstoy etc. etc. from the Steamship of modernity." They heaped abuse on even contemporary writers they considered old-fashioned (many of them associated with Bely) in manifestos with titles like "A Slap in the Face of Public Taste" and "Go to Hell."[9] Russian Futurism lasted through the revolution, and many figures in the movement embraced the new regime and gave it many of its most lasting images in poetry and in the visual arts. This participation in the new political order is yet another mark of the modernity of Futurism, in several senses. To begin with, art was placed in the service of revolutionary struggle, and what better example could there be of worshiping the new as an absolute value? In addition, a new idea among the Futurists was to tear down the boundaries dividing the different an forms from one another. Bely, in "The Principle of Form in Art," had continued to subscribe to the outmoded idea that some sort of a priori principle obliges us to express ourselves artistically in one of a limited number of mutually discrete media of artistic expression. Many Futurists rejected this notion, seeking forms of art that would combine the traditional media. One of the results is that people in the movement were commonly poets and visual artists at the same time, writing poems, painting pictures, and producing works of art that are located somewhere in between poetry and painting. Vladimir Mayakovsky, undoubtedly the most noticeable member of the whole movement, produced hundreds of propaganda drawings for the revolutionary regime and included on them slogans and bits of verse printed in such a way as to make the words part of the drawings. Anyone who has seen an exhibit of Russian avantgarde art will have noticed how often the paintings include letters and words as prominent parts of their visual fields.

Another new thing was numbers. Maybe the members of the Russian avant-garde considered numbers to be part of a modern trend toward abstraction; or maybe after 1917 they saw them as symbols of the technological and industrial revolution that was to fortify and carry on their recent political revolution. In either case, numbers in this era became the object of a special cult, which expressed itself in some rather peculiar ways. Velimir Khlebnikov, one of the pioneering members of this movement, had studied mathematics at the university and had then developed his fascination for numbers into his own mystical system, which he

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writes about in many of his short essays and manifestos. For instance, he believed that there were certain key numbers that determined momentous events in human history. The quantities 365 + 48 and 365–48 were particularly important in this respect, and Khlebnikov fills whole pages of his prose writings with calculations designed to show how units of time based on various multiples of 413 and 317 separate certain key happenings. Of course, anyone who reads this immediately begins to suspect that Khlebnikov came up with his mystical number first and then went looking for facts to support the accuracy of his theory, rather than the other way around. Numbers pervade Khlebnikov's work. If you look through his collected prose writings, you'll see whole sections given over to a veritable riot of numbers and figures that have become, like Bely's shapes, an end in themselves. In fact, the visual impact of Khlebnikov's math mania together with the outlandish ideas he proposes make it difficult to classify his prose writings as essays or theoretical writings in the usual sense. They begin to look like a cross between prose poems and graphic art, the way Mayakovsky's propaganda posters do and the way so much of Futurist visual art does.

But there are a couple of serious messages here, just as there were in Bely. One has to do with the "mode of being" of numbers. The other concerns the relation between numbers and things and has implications for the relational-essentialist view of the literary artwork. Khlebnikov was fond of the fantasy that some sort of universal determinism governed world events and that numbers were its measure. This determinism expressed itself through time (multiples of 413 or 317 years), and so it was accurate to say that "time is the measure of the world," as Khlebnikov titled one of his essays.[10] Khlebnikov loves numbers so much because there is a necessary and determinate relation between them and what they stand for. Hence a comparison suggests itself with language since the necessary and determinate relation that exists between numbers and what they stand for is notoriously lacking between words and what they stand for. That was the whole reason behind Khlebnikov's and Kruchenykh's efforts to design a "transrational" language in which this problem would be overcome. Khlebnikov dreamed of the possibility of having numbers replace words as a means for thinking and communicating. In "Time Is the Measure of the World" he says:

In verbal thinking no basic condition of measurement is present—no constancy in the units of measurement, and the Sophists Protagoras and Gorgias

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were the first steadfast helmsmen to point up the dangers of navigation upon the waves of the word. Every name is merely an approximate measurement, a mere comparison of several quantities, of certain equals signs. Leibniz, in his exclamation, "The time will come when people, instead of engaging in abusive disputes, will calculate" (will exclaim: calculemus ), Novalis, Pythagoras, and Amenophis IV all foresaw the victory of numbers over the word as a method for thinking."[11]

Again and again Khlebnikov came back to this comparison between numbers and words, often in a way reminiscent of Mallarmé. For instance, shortly after the passage! just quoted, Khlebnikov says, "Being an antiquated implement of thought, the word will nonetheless remain for the arts since it is useful for measuring man through the constants of the world. But the major portion of books have been written because people have wanted, by means of the 'word,' to think about things that may be thought about by means of numbers."[12] In another place he suggests that we assign to all the thoughts of the earth a number since there are, after all, so few thoughts around. Then "languages will remain for art and will be freed from an insulting burden."[13] In other places he uses a scheme similar to the one Mallarmé had used in English Words and shows that certain initial sounds of words naturally conjure up the idea of certain mathematical operations. But what is most reminiscent of Mallarmé is the suggestion that since words fail in a function in which numbers succeed, namely the function of ideally signifying what they signify, they ought rightly to be left to art. This sounds much like Mallarmé's remark in "Crisis in Verse" that without the imperfections of language "verse would not exist : it, philosophically compensates for the shortcoming of languages, superior complement."[14]

Whatever we might call Khlebnikov's prose writings, Khlebnikov himself certainly did not refer to them as poems. There were other poets, however, who did incorporate numerals into their poetry and even provided theoretical reasons for doing so. A relatively minor figure, Ivan Vasil'evich Ignat'ev, wrote experimental poetry in which he used mathematical symbols for their visual impact. David Burliuk (1882–1967), one of the most noticeable members of the movement, though not one of the most talented, used mathematical symbols in his poetry. Burliuk was a painter and a poet, like many others of his generation, and much of what he did was for effect. If he thought it would be visually shocking to use mathematical symbols in poetry, he also must have thought it would be intellectually shocking to call the collection of poems in which

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these symbols appeared "The Milker of Exhausted Toads."[15] Nikolai Burliuk (1890–1920), brother of David, provided the theoretical justification for the use of mathematical symbols. In an essay called "Poetic Principles" he talks about the "graphic life of letters":

How many signs, musical, mathematical, cartographic, and so forth, there are in the dust of libraries. I understand the cubists, when they introduce numbers into their pictures, but I don't understand poets, who remain foreign to the aesthetic life of all these

[16]

The person who can probably be credited with using mathematical symbols and images to their greatest visual effect was El Lissitzky (1890–1941). Lissitzky is normally thought of as a visual artist, not a poet, but his compositions show the same enthusiasm for typography as we find in many of his contemporaries, and some of his works actually contain narrative elements. Since in this era people in the arts have to be placed on a gamut that runs from "pure" verbal art at one end to "pure" visual art at the other, with the entire range of combinations in between, maybe it would be most accurate to say that Lissitzky belongs a little closer to the visual end than, say, Khlebnikov (Khlebnikov, as it happens, produced some fairly good visual art of his own). In 1920 Lissitzky created (how do you say "wrote and drew" in one word?) a work (a story-drawing) called "Of Two Squares." Actually, the title as it appears on the cover is not "Of Two Squares," since Lissitzky spells out only the word translated as "of." "Two" is the numeral 2, and "squares" is a picture of a red square (only one square because Russian uses a singular form of the noun with the numbers two, three, and four and their compounds). On the title page, however, Lissitzky gives the title in words as "suprematist tale [skaz ] of two squares in 6 constructions." The constructions are the individual compositions that make up the work, so "a tale in six constructions" appears to be like "a play in five acts" or "a novel in six parts." They are geometric drawings in which the exploits of the heroes of the story, a red and a black square, are depicted. Accompanying the drawings, in letters that are characteristically arranged so as to be part of the entire visual effect, is the narrative: "They fly to earth from far away," and so on. What kind of work is this? On the inside of the back cover we are told that it was "constructed" (not written or drawn) in 1920, but this doesn't tell us much.[17] Later, in 1928, Lissitzky made some sketches for a children's book called "Addition, Subtraction, Multiplication, Division," in which the char-

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acters performing the "action" of the four arithmetic operations are numbers and letters drawn to look like various Soviet types: workers, peasants, and Red Army soldiers.[18]

Earlier I mentioned Michel Serres and his analysis of the modern cult of abstraction. In the modern age, Serres says, the emphasis is on structure, models, and relations, not on content. Number mania is just one symptom of the same trend operating in Russian modernism and the Russian avant-garde. Bely's system of verse analysis provided him with the means for abstracting away all content from the literary works he was investigating, for "taking a form and filling it with meaning," as Serres puts it, instead of relying on preestablished meanings.[19] Khlebnikov's funny proposal to number all thoughts and use only the numbers in referring to them is another example of the tendency to abstract away content and leave only a relational structure waiting to be filled with meaning. The same may be said of Lissitzky's typographic experiments, which show a playful approach to the process by which abstraction overtakes traditional content.

Abstraction was the order of the day in Russian art, just as it was in West European art. Cubism developed a real following in Russian art. The Cubo-Futurists are evidence of it, as are the numerous theoretical writings devoted to cubism in that era. The general trend in Russian art from around 1910 through the 1920s is toward increasing abstraction. We can find this trend in pictures that some of the most prominent artists of the period painted between 1909 and 1914. Natal'ia Goncharova (1881–1962) is a good example. After having produced traditional paintings like her iconic "Madonna and Child" in the years around 1905, she adopts a primitivist mode around 1909, devoting her compositions to rustic subjects like Picking Apples (1909), Peasant Picking Apples (1911), and Fishing (1910). These pictures contain recognizable human figures but are composed in "primitive" fashion: the figures are stiff, there is little depth, and the treatment of perspective is noticeably and intentionally childish. Around 1911, however, things begin to change again. Goncharova's major composition of 1911 and 1912 is Cats . There are not really any cats in this picture, just the feeling of their scratchiness and the crackling static electricity of their fur, qualities rendered pictorially by patterns of sharply drawn lines, or "rays." Soon Goncharova will be painting pictures with titles like The Clock (1911), The Cyclist (1912–13), and Dynamo Machine (1913), in which elements of the object or objects suggested in the title are arranged on the canvas in new and unrecognizable patterns. Human figures are now

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separated into fragments of faces and bodies distributed here and there according to rules very different from those followed in classical portraits. The work of another famous Russian painter, Kazimir Malevich (1878–1935), shows the same progression, from Renoir-like treatments of young women in the first years of the twentieth century, to cubist borrowings from Picasso around 1912, to paintings, starting around 1913, consisting of nothing but geometric shapes painted in black or red on a white background.

I'm not pretending that everything happening in the visual arts in Russia at this time was unique. It wasn't. As usual, many Russian artists relied heavily on their contemporaries in Western Europe for inspiration. The first Futurist movement was Italian, not Russian, and even though the Russians hated to admit it and went to great lengths to distort the truth, they borrowed a great many of their themes and ideas from the Italians. One striking feature of the Russian movement, however, was the degree of interpenetration between visual art and literary art. The move toward abstraction in the visual arts is difficult to characterize any more precisely than I've just done if we are limited to an empirical description of pictures. But because so many painters were also poets; because so many poets were also painters; because so many artworks of the era combined elements from both artistic media; and because so many artists wrote theoretical works on their painterly, poetic, and painterly-poetic techniques, we actually can document a move toward the kind of relational abstraction I've been talking about, and we can do so without just describing pictures.

"It has been known for a long time that what is important is not the what, but the how, i.e., which principles, which objectives, guided the artist's creation of this or that work!" proclaims David Burliuk in 1912 in his essay "Cubism (Surface—Plane)." In the same essay the painterpoet breaks down painting into its "component elements"—line, surface, color, and texture—and claims to have provided, in his epigraph, the "mathematical conception" of surface.[20] Natal'ia Goncharova's cat picture was composed in the "rayonist" manner, and starting in 1913 she and fellow rayonist Mikhail Larionov (1881–1964) published declarations on the principles of their new style. The idea was to get away from concrete forms: "Long live the style of rayonist painting that we created—free from concrete forms, existing and developing according to painterly laws!" How exactly does one go about making a rayonist painting? "The style of rayonist painting that we advance signifies spatial forms arising from the intersection of the reflected rays of various

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objects, forms chosen by the artist's will." Everything is combination and relation—"the combination of color, its saturation, the relation of colored masses, depth, texture"—and the goal is "a self-sufficient painting."[21]

Another Futurist, Sergei Bobrov, a member of the Centrifuge group, was fond of mathematical terms and concepts. One of his ventures was to continue Bely's statistical work in verse analysis. His theoretical writings are filled with references to various theorems and formulas, references whose application is often difficult to guess. In an essay on poetry called "The Lyric Theme," published in 1913, Bobrov invokes Newton's binomial theorem, the concept of the arithmetic mean, and various principles from geometry. Bobrov's purpose in using these concepts remains obscure, but the main point seems to be the rejection of all the traditional frameworks in which poetry is written and read—especially such content-centered frameworks as metaphysics and religion—in favor of some sort of pure idea of poetry. "The lyric," Bobrov says, "has a direct tie with the idea of the poem. Not, however, with the thought of the poem." Bobrov is after the essential quality of the lyric, which he calls "lyricity" (liricnost'[liri&!;nost ) and which appears to be separate from content.[22]

The rejection of content, nature, and objectivity in favor of the pure essence of either painting or poetry became the trademark of much Russian aesthetic theory beginning in the years before the revolution. Kazimir Malevich, who was a prolific writer of manifestos in addition to being a prolific painter, championed the cause of the "nonobjective" in art. In a little book titled From Cubism and Futurism to Suprematism: The New Painterly Realism he asserts the importance of keeping painting separate from nature. "The artist can be a creator only when the forms in his picture have nothing in common with nature," he says. Nature must be seen only "as material, as masses from which forms must be made that have nothing in common with nature." The whole purpose was to attain "pure painterly essence" and "nonobjective creation." "Painters should abandon subject matter and objects if they wish to be pure painters," he proclaims. "Our world of art has become new, nonobjective, pure."[23] In the service of this creed Malevich painted many of the canvases that he termed—then or later—"Suprematist": paintings with titles like Black Square (consisting of just that, a black square on a white background), Black Square and Red Square, and just Suprematist Composition .

Benedikt Livshits (1886–1939) was a poet who was interested both in the relational conception of art and in the connections between paint-

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ing and poetry. In an essay called "In the Citadel of the Revolutionary Word" he discusses poetic language and says "the highest type of structure is for me the one where words are matched according to the laws of inner affinities, freely crystallizing on their own axes, and do not look for an agreement with the phenomena of the external world or of the lyric self."[24] Later on, in a book of memoirs called The One-and-a-Half-Eyed Archer, Livshits recalls a time when he was pondering the problem of combining different art forms, above all painting and poetry. He had come to realize a basic truth about this matter, which was that the features one could hope to transfer from one art form to an "adjacent" art form are "relationships and mutual functional dependence of elements."[25] Once again, in the case of poetry, the effort is to remove language from its signifying function as traditionally conceived and to make it part of an abstract relational system.

One of the most fascinating subplots in the drama of Russian modernism and its cult of abstraction involves the mathematics of the "fourth dimension." I won't tell this story in any detail; Linda Dalrymple Henderson, an art historian who specializes in this period, has already done so at great length.[26] The nineteenth century, as Henderson explains, had provided two significant new challenges to traditional, Euclidean geometry. The first, non-Euclidean geometry, would eventually concern itself with the characteristics of shapes and forms in curved, rather than planar, spaces. The second was geometry of n dimensions, a field that got its start early in the nineteenth century. The basic notion was mathematically to characterize "spaces" that contained four or more dimensions. By the end of the century mathematicians were talking about such things as "hyperspaces" and "hypersolids." What are hyperspaces and hypersolids? Well, if we can generate a three-dimensional solid, say a cube, by assembling two-dimensional components (that is, the planes that form the surfaces of the cube), then surely we can generate a four-dimensional hypersolid by assembling three-dimensional components like cubes. Or we can picture a hypersolid as a thing whose surfaces are formed from spheres instead of points. Naturally, a hypersolid will need a space to exist in, and so we arrive at the notion of a hyperspace, one that will accommodate hypersolids and similar objects. The only problem is how to visualize spaces and objects like these. Mathematicians were no help here, so the job had to be done by artists, who set about to solve this problem as the notion of four-dimensional geometry took hold of the popular imagination in the late nineteenth century. To be sure, no one figured out a way to draw a figure whose hyperplane surface was

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made up of spheres instead of points, but, as Henderson shows, the idea of the fourth dimension caught on in Russia as a form of mysticism that translated itself into literary and visual artwork in a variety of ways.

The history went something like this, in Henderson's account. A minor Russian philosopher, Peter Demianovich Ouspensky (1878–1947), later known in the West for his role in disseminating the beliefs of the famous mystic Gurdjieff, wrote of the fourth dimension in a book modestly titled Tertium Organum: A Key to the Enigmas of the World (1911). In this book Ouspensky proposed a Promethean view of man in the universe, basing his ideas in part on a mystical notion of the fourth dimension. Next, a painter named Mikhail Matiushin (1861–1934) adopted certain principles from the French cubists and modified them in accordance with his understanding of Ouspensky's fourth dimension. In 1913 Matiushin collaborated on a number of projects with Malevich, Kruchenykh, and Khlebnikov, to whom he introduced Ouspensky's ideas. Henderson has shown that the Ouspenskian notion of the fourth dimension was actually a decisive factor in Kruchenykh's elaboration of zaum ' theory. In the essay "New Ways of the Word" Kruchenykh rhapsodizes about transrational language and speaks of a new "fourth unit" of psychic life, which he calls "higher intuition," citing Ouspensky's Tertium Organum . Higher intuition was the form of superior mystical knowledge that Ouspensky associated with the fourth dimension. A couple of pages later, Kruchenykh praises the false perspective found in the work of contemporary painters, saying that it gives their work a "new, fourth dimension."[27] References to the fourth dimension then appear in Larionov's articles on rayonism. The fourth dimension is particularly important to Malevich, who refers to Ouspensky's ideas in his theoretical writings and applies them to his painting. And in the 1920s El Lissitzky used his understanding of the fourth dimension, which he eventually came to identify with time (as many others did following a tremendous rise in the popularity of Einstein's theories around 1919), to develop a coherent theory of painting.

It would be ridiculous to assert that the painters and writers of the Russian avant-garde were interested in the fourth dimension for the same reason as mathematicians or that most of them had a truly mathematical understanding of it. To the artists the fourth dimension meant an escape from the concrete world and from the obligation of always representing or signifying it. This is the thought that turns up repeatedly in discussions of the fourth dimension. When it comes right down to it, the fourth dimension is just another abstraction based in mathematics, like Bely's

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baskets, Khlebnikov's numbers, and Jakobson's grammatical structures. Goncharova's painting of cats intentionally overlooks all the physical features we expect in a painting of cats, just as Jakobson and Lévi-Strauss overlook the physical features of cats in Baudelaire's poem about cats. Goncharova wanted the abstract essence of cat in her painting; Jakobson and Lévi-Strauss wanted an abstract essence that had to do with grammar.

Something particularly curious is going on with the Futurists. Their fascination with numbers and mathematical concepts inevitably leads to fundamental questions about the artwork, questions like what and where . What is an artwork, once it has been reduced to an abstraction? Where does it exist, what sort of space does it or its represented world occupy once that space has been made, by conscious effort, to resemble the familiar space of our world as little as possible? What sort of thing are we dealing with here, anyway? Everything the Futurists did seems designed to raise this question by challenging all our secure notions about art. It's a poem, but then it's like a painting, too. It's a painting, but then it's like a poem, too. It contains recognizable signs, but arranged in unrecognizable ways. The big question is a question of being . What is the mode of being of this thing? art of this period continually forces us to ask ourselves. The question seems to have its origin in speculation that is to a significant extent mathematical. At the same time it reflects the essentialist impulse Russian thinkers never seem to escape. Strictly speaking, it is a question of ontology, and that is the subject of the final part of this book.

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PART III— RELATIONALISM