## 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

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-

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

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-

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

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-

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

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

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

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-

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

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

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.