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

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

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

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

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

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-

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

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

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-

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

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

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.