Patterned Ambiguities:Virginia Woolf, Mathematical Variables, and Form
This article argues that Virginia Woolf's most central, memorable symbols share the semantic properties of mathematical variables: markers that are designed to flexibly denote multifarious, undetermined meanings. Woolf uses the generality that characterizes pure mathematics to reinvent the scope and shape of ambiguity in Jacob's Room, and in turn variables allow for an understanding of form in terms of the patterns that characterize "the life of anybody" in The Waves. Mathematics offers its own definitions of form, tied to mathematical formalism, a revolutionary 1920s movement. Ultimately, mathematical attention to Woolf's patterns offers an understanding of what it is that we call literary form, an understanding built from pattern rather than particularity.
In his preface to the second edition of Seven Types of Ambiguity, William Empson writes, "I claimed at the start that I would use the term 'ambiguity' to mean anything I liked."1 That is, ambiguity is itself ambiguous. We know, in literary studies, that ambiguity is important and that it is everywhere. But I am not sure if we know what it is or how it works. When ambiguity's range is large, directed toward not two or three potential confusions but innumerable and unlimited possibilities, then how, exactly, does the symbol still symbolize?
Virginia Woolf tells us that her "ambiguity is intentional . . . as much elasticity as possible is desirable,"2 and her most central terms [End Page 73] are characteristically elastic, with a markedly open range of meaningful possibilities. Consider the lighthouse in To the Lighthouse (1927): frequently sought after but largely unknown, an imposing figure of significance that shifts meanings with its observers. In Three Guineas (1938), with weighted and yet undirected symbolism, Woolf shuffles guineas around as markers of prominently undetermined value. In The Waves (1931), waves signify so much that to name their significance, even mere examples of their significances, starts to seem reductive, like the undersized attempt to describe one drop of water in a whole ocean. These enormous central symbols mark the possibilities of symbolism even more than they actually symbolize. They telegraph the multiplicity of their meanings more than they attach to those meanings themselves. It is no coincidence that these symbols, at once expansively meaningful and surprisingly unknown, are also the organizing objects in Woolf's titles. Such symbolic entities naturally become titles because they already function like titles; they are labels for that which seems too slippery or unwieldy to be labeled, just as titles are shorthand for the piles of pages that cannot be reread every time they are discussed. Symbols such as these label undetermined expansiveness, providing us with terms for speaking about—but not determining—the indeterminate. Scholars have long since noted the fluidity and multiplicity of these symbols,3 yet no one has ever really broken down how they work.
This paper will characterize this breadth of meaning, characteristic of Woolf's work but common elsewhere as well, using models borrowed from mathematics—particularly, with the type of multiplicity of meaning that mathematical variables offer. In formal mathematics, variables do not label items yet to be determined, but instead mark that which we refuse to determine. Variables exist to mark very many meaningful possibilities, choosing no favorites from among their referents. As such, in mathematical parlance, variables offer generality, which implies no vagueness or imprecision, but multiplicity: the capacity for one word or phrase to describe many different things.4 Megan Quigley has argued that Woolf's writing is characteristically [End Page 74] vague, meaning that it depends on indefinite definition, on meaning that has blurry boundaries.5 By contrast, generality, as I use it here, indicates not blurry definition, but multiple definitions: each meaning, individually, might be utterly precise, but stands as only one among a glut of other potential meanings.
Historians of mathematics have established that, during Woolf's era, mathematics went through a modernist revolution of its own.6 That mathematical revolution comprised dramatic abstraction, a fraught relationship with empirical science and the day-to-day world, a reinvention of disciplinary assumptions, and profound controversy about the role of language in the field. A number of those in Woolf's orbit were working on, and writing about, modernist mathematics, including Bertrand Russell, Frank Ramsey, G. H. Hardy, and Alfred North Whitehead.7 However, influence cannot adequately [End Page 75] explain this convergence, both born of and constituting modernism itself.8 December 1910—the moment Woolf tells us that "human character changed"9—saw not only Roger Fry's post-impressionist exhibition but also the initial publication of Whitehead and Russell's monumental Principia Mathematica. 1922 witnessed not only Jacob's Room, Ulysses, and The Waste Land, but also David Hilbert's "The New Grounding of Mathematics." This paper will close with a contemplation of Woolf's direct reactions to mathematics,10 but my deeper argument is that she independently developed semantic structures that mathematicians of the same moment were examining. Here, Woolf's writing does not just happen to dovetail with mathematics; the mutual attention to patterns of ambiguity reveals intersections already intrinsic to the fields themselves, commonalities already embedded in formal languages (e.g., arithmetic, C++) and natural languages (e.g., English). Ann Banfield has argued that "[t]he 'solid, mathematical universe' . . . supplies an abstract, intangible support" to Woolf's writing. In other words, she claims that a "logical skeleton" lies at the core of Woolf's shimmery and fluid art.11 Implicit in that argument is the assumption that art cannot stand up on its own strength—hence the need for a logical skeleton. In contradistinction to Banfield, I do not argue that mathematics offered philosophical [End Page 76] support to Woolf, but that it offers a readerly apparatus to us. I argue that variables offer a uniquely descriptive model by which we can understand the multiplicities, ambiguities, and contradictions of Woolf's symbolism. Here, mathematics emerges not as an epistemic structure supporting Woolf's thought, but as an abstract language explaining that thought, one that characterizes Woolf's work and explains how her symbols function.
Woolf developed the same forms of generalizing symbolism that mathematicians were using and theorizing, in the same era, because she was tackling the same semantic problems that these mathematicians were: the problem of how to say much in limited space applies as much to mathematics as to prose, and the complexities of form and the promises of adaptable, far-reaching symbolism apply there as well. In the early twentieth century, newfound and sometimes paradoxical abstraction in mathematics forced mathematicians to rethink how their field pertained to the world, to explain how mathematical symbols symbolize. In this paper I read modernist mathematical texts not only as historical context, but as work in the theory of symbolism, language, and form. Woolf's generality ultimately forces a newly interdisciplinary conception of form, built from pattern rather than particularity. Generality allows for recognition of commonality, of echoes, consonances, and repetitions, and those networks of repetition—together with refusals of repetition—define form.
Naming the Undetermined
I meant nothing by The Lighthouse. One has to have a central line down the middle of the book to hold the design together. I saw that all sorts of feelings would accrue to this, but I refused to think them out, and trusted that people would make it the deposit for their own emotions—which they have done, one thinking it means one thing another another. I can't manage Symbolism except in this vague, generalised way. Whether it's right or wrong I don't know, but directly I'm told what a thing means, it becomes hateful to me.—Woolf, in a letter to Roger Fry, 192712
Among Woolf's characters, Jacob Flanders stands out as unknown, almost a placeholder into which "feelings . . . accrue." Always already dead, Jacob surfaces only in remembered glimpses, anecdotes, and bits of information, which never cohere into one stable person. Indeed, readers have long understood that Jacob himself is inaccessible in any finally decided form. In 1922, the very first published [End Page 77] review of Jacob's Room declared, "We do not know Jacob," and a host of scholars have argued that Jacob is indeterminate.13 Jacob's room is the most concrete emblem of his selfhood, but it, too, is undetermined: above the doorway there is "a rose, or a ram's skull"—we don't know which.14 Like Jacob, it is unfixed, shifting with him from Scarborough to Cambridge to London. In fact, this room is most accessible as a name: Jacob's Room. That exact title interjects itself oddly, surfacing repeatedly and identically inside the text: "in the middle of Jacob's room"—"What do we seek through millions of pages? Still hopefully turning the pages—oh, here is Jacob's room."15
Names and titles possess an odd prominence throughout Jacob's Room, with its overweening accumulation of names for unfamiliar characters. By my count, this little novel presents over two hundred named characters, proliferating more than one per page. To keep track of so many names in such a short text is impossible, so that in Jacob's Room names repeatedly highlight our ignorance. Each new name presents another unknown possibility for personhood. As the terms themselves multiply, the fabric of the text becomes increasingly opaque. Here readers perceive words more easily than meanings, and the increased possibility for meaning that comes with undetermined terms is accompanied by a step into formalism—a deeper attention to the forms of description than to what may be described—as readers give up on understanding the characters as subjectivities. Where reference is so multiple and so indefinite, we become self-consciously aware that the "eye goes down the print."16 So many unknowns drive us toward a self-conscious re-evaluation of symbolism itself.
As witnesses continually fail to capture Jacob's character, his name becomes the only consistent grip on him. The definitiveness of this [End Page 78] name sidesteps the undetermined potentiality that is Jacob himself, allowing us to, at least, discuss that which is unknown. And the novel repeatedly introduces Jacob by his name before he himself ever appears: the typographically emphasized and already doubled "Ja—cob! Ja—cob!" interjects three times on a single page at the beginning of the novel.17 From the first page, his name also preempts the story, announcing Jacob's death more directly than the narrative ever does: Flanders Field will swallow Mr. Flanders with all the inevitability of post-facto history. Woolf named Jacob not at birth but by death, and as an unknown soldier, his name is endowed with a symbolism too multiple to rest as a marker for just one person. Jacob Flanders might have been one hypothetical man, but "Jacob Flanders" marks the very possibility of signifying personhood. In this way, Jacob's constant, prominent name marks the innumerably unknown.
In an experimental novel where character is utterly unstable, the stability of name offers solid ground. This constancy in naming is far from being a foregone conclusion—James Joyce proved that with the relentlessly shifting names in Finnegans Wake. In fact, Woolf's secure (albeit numerous) use of name and description is among the only techniques binding Jacob's Room to more traditional form, allowing the sense in which any single page of Jacob's Room could have sprung from a Victorian novel, yet the accumulation of those pages is utterly strange and new. We do not know who Jacob is, but we know absolutely that he is called Jacob. This unknown significance marked by a stable signifier mirrors undecided values marked by variables: the symbols central to pure mathematics18 that allow single terms to serve as reliable markers for utterly undetermined referents. Amid such generality, reliable terms keep us from floating away in confusion. [End Page 79]
Jacob's name is not without confusions, but the degree of these confusions underscores my point. One character, Mrs. Papworth, repeatedly mistakes Jacob's surname: "Mr. Sanders was there again; Flanders she meant." However, here and elsewhere where Mrs. Papworth fails to correct herself, readers are able to identify the mistake; we could easily inform Mrs. Papworth of Jacob's true surname. Yet we look on in uncertainty during the many moments where characters are confused about Jacob's personality and identity rather than his name. When the narrator asks us, "But how far was he a mere bumpkin? How far was Jacob Flanders at the age of twenty-six a stupid fellow?" we are unable to answer.19
So many pages describe Jacob in very particular ways, but those particularities refuse to stabilize. Jacob is not one determinate person, yet neither is he totally indeterminate. Instead, Jacob is everywhere potentially determinate, but only potentially determinate. And, as Shankar Raman has noted, the introduction of mathematical variables constitutes "a shift from representing things—be they commodities, people, or algebraic unknowns—as determinate-but-unknown to representing them in their merely potential determinateness."20 This shift is crucial, because it marks the difference between the limited ability to represent any one example and the expanded capacity to demarcate and characterize entire fields of possibility. Jacob's personality and actions are to a remarkable extent only possibilities, and yet those possibilities are notably specific amid their multiplicity. He may run off with Sandra Wentworth Williams or turn back and marry Clara Durrant. He might yet acknowledge Bonamy's love. He may be one of those young men who "will soon become fathers of families and directors of banks," or he might actualize his bohemian impulses. He "might become stout in time"—or might not.21 The text offers markedly particular imaginings of Jacob and his future even as it refuses to ultimately endorse any of them. And these possibilities do not only pertain to the unrealized future; they crucially indicate the great degree to which we cannot precisely describe Jacob's present inclinations, personality, and selfhood. Individual readers are bound to have intuitions about which of these possibilities ring truest with Jacob as they understand him, but certainly Woolf has constructed the text so that they are all legitimate possibilities. [End Page 80]
Or at least, they would be if we could suspend Jacob's inevitable and overdetermined death. Jacob is one memorial to Woolf's brother Thoby Stephen, who died at twenty-six, and even such a conflicted and various representation as Jacob stands in competition with several other versions. Woolf's various reincarnations of Thoby—Jacob Flanders, Andrew Ramsay from To the Lighthouse, Percival in The Waves—are so multifarious, so mutually contradictory, and so generally opaque because, like variables, the unrealized futures of the young dead are always only potentially determinate. And here it is also worth remembering that this particular memory was already mathematical. In "A Sketch of the Past" (1939), Woolf recalls Thoby often studying mathematics with their father at the dinner table,22 and Quentin Bell describes the subject as Thoby's singular talent among the Stephen children.23 Like Andrew Ramsay, perhaps he "should have been a great mathematician."24
In some sense, the unknown, potentially determinate is exactly what those who die young leave behind them. In that way, like a variable, Jacob can potentially take on many mutually exclusive instantiations without generating actual inconsistency, because Jacob is not one indeterminate man so much as he is one name, one constant marker, for the multiple determinate men he might have been or become. Here, again, he exists as does his room, which is ornamented by "a rose, or a ram's skull"25: Woolf offers two determinate yet divergent possibilities. This is the multiplicity characteristic of mathematical variables, and this is the multiplicity characteristic of Jacob.
Variables allow mathematicians to defer determinations of meaning by providing a constant name that they can treat as if determined, allowing them to calculate whatever can be known independent of the particular identity of x. However, whereas mathematics is a realm of binary truth or falsehood, Woolf uses the same semantic mechanisms in a different context, and to work toward a different end. Variables such as Jacob allow Woolf to attend to particularity while refusing to commit to any single, monolithic reality.
Woolf did compare literary and mathematical symbols. Night and Day (1919) describes mathematics as "sacred pages of symbols and figures" with a kind of scarcely deferred reverence for poetry and novels—which, certainly, are also pages of symbols and figures.26 In To [End Page 81] the Lighthouse, Lily Briscoe aligns the engagements fostered by poetic phrases and mathematical symbols: "It was love, she thought, pretending to move her canvas, distilled and filtered; love that never attempted to clutch its object; but, like the love which mathematicians bear their symbols, or poets their phrases, was meant to be spread over the world."27 Woolf demands that we set symbols free, and her demand pertains as much to the relation symbols bear to their meanings as to the love we might bear to those symbols. Woolf's symbols, like mathematical variables, do not "clutch" at their referents, but "spread over the world."
In "Craftsmanship" (1937), Woolf explains that "[w]e have been so often fooled in this way by words, they have so often proved that they hate being useful, that it is their nature not to express one simple statement but a thousand possibilities." Then she defines the "useful statement" that words so detest as "a statement that can mean only one thing," hammering home the claim that words "hate anything that stamps them with one meaning or confines them to one attitude."28 This refusal of delimited meaning has its apotheosis in the variable—a sign that exists to generalize potentially infinite possibilities of meaning. Jacob's Room, I argue, is an entire novel constructed around not only the unknown or the indeterminate, but the everywhere potentially determinate; it is an entire novel constructed around a variable.
On Variables and Meanings
[S]he cast her mind alternately towards forest paths and starry blossoms, and towards pages of neatly written mathematical signs.—Woolf, Night and Day29
The variable is a semantic model not used in literary theory,30 but philosophers of language and linguistics have long since compared [End Page 82] the generality of variables to the ambiguities of words. Raman has examined algebra's influence on Renaissance culture, analyzing literary multiplicities alongside algebraic variables to great effect. Yet, during the Renaissance, algebraic variables were limited in the sense that they could only take on numbers as their values. By Woolf's era, the possibilities had exploded, after late nineteenth-century innovations by George Boole, Georg Cantor, and Gottlob Frege, among many others, expanded the range of semantic applications for variables, allowing this mathematical symbolism to describe non-mathematical things. Far from merely indicating numbers, by the modernist era variables could stand in for sets, propositions, objects, events, phrases, and, very generally, anything that might exist or be thought of at all. Russell was revolutionizing philosophy in large part through the use of variables and mathematical quantifiers to analyze and refer to descriptions in language.
Variables can be thought of as simply one name for many possible referents. Russell defines variables as terms that do not "have a meaning that is definite though undefined" but instead are truly "undetermined"31: "I take the notion of the variable as fundamental . . . x, the variable, is essentially and wholly undetermined."32 Joseph Shoenfield writes, "Unlike a name, which has only one meaning, a variable has many meanings. In an analysis text, a variable may mean any real number; or, as we shall say, a variable varies through the real numbers."33 Variables exist to mark very many—often infinitely many—possible meanings. Yet, from a mathematical vantage point, what exactly does meaning mean?
In 1892, the mathematician, logician, and philosopher of language Gottlob Frege divided meaning into Sinn and Bedeutung, often translated as sense and reference, or sense and denotation. Frege writes, "It is natural, now, to think of there being connected with a sign (name, combination of words, written mark), besides that which the sign designates, which may be called the Bedeutung of the sign, also what I should like to call the sense of the sign, wherein the mode of presentation is attained."34 Edward Zalta explains the distinction thus: "The expressions '4' and '8/2' have the same denotation but express different senses, different ways of conceiving the same number. The descriptions 'the morning star' and 'the evening star' denote the [End Page 83] same planet, namely Venus, but express different ways of conceiving of Venus and so have different senses."35 Frege writes that sense has more to do with the subtler connotations and evocations of literature: "In hearing an epic poem, for instance, apart from the euphony of the language we are interested only in the sense of the sentences and the images and feelings thereby aroused"—not the denotation.36 The interest of artistic description lies in its expansive implications more than in the literal objects, actions, or values referenced, and in literature the sense becomes staggeringly larger and more complicated than in the simple distinction between 4 and 8/2. In fact, I would say that the senses of artistic symbols can frequently be so complex and multifarious as to blur, eclipse, or swallow up their denotations entirely. In much of literature, it seems, to draw a line between sense and reference may be to risk generating a false binary.
The division has its difficulties elsewhere as well, underscored by continuing disagreement over any English translation of Frege's terms. Russell influentially translated Sinn as "meaning" and Bedeutung as "denotation," thus swapping out the most literal translation of Bedeutung to use it to indicate, instead, Sinn. In Russell's hands, the denotation developed extreme technical precision: in 1905 he argued that the phrase "All men are mortal" actually denotes the claim "'If x is human, x is mortal' is always true," while "'the father of Charles II. was executed' becomes:—'It is not always false of x that x begat Charles II. and that x was executed and that "if y begat Charles II., y is identical with x" is always true of y'." 37 In this manner, Russell claimed to be able to translate any denoting phrase into a logical statement. That theory of descriptions depends on variables, on the great generality that can attach to x. However, Russell does not only import the semantic breadth of variables: he brings mathematical and logical rules into language together with those variables. In the process he actually neglects multiplicities of meaning: for example, above, the fact that "men" is not necessarily gender neutral.
Woolf, on the other hand, insisted that meaning is multiple, fluid, and amorphous. In "Craftsmanship," she distinguishes between "surface" and "sunken" meanings, generating a boundary that roughly corresponds to Frege's line between Bedeutung and Sinn, insofar as the surface meaning seems to indicate more literal denotation, whereas sunken meanings are implications and associations. [End Page 84] But Woolf also makes it clear that, in natural language, such demarcations can never be stable: "At the first reading the useful meaning, the surface meaning, is conveyed; but soon, as we sit looking at the words, they shuffle, they change."38 Russell's logical system for denotation cannot apply here, because the words refuse to cling to any single logical structure. Describing a statement as simple as that of a signboard in the London Tube, she explains:
[I]t is the nature of words to mean many things. Take the simple sentence "Passing Russell Square." That proved useless because besides the surface meaning it contained so many sunken meanings. The word "passing" suggested the transiency of things, the passing of time and the changes of human life. Then the word "Russell" suggested the rustling of leaves and the skirt on a polished floor; also the ducal house of Bedford and half the history of England. Finally the word "Square" brings in the sign, the shape of an actual square combined with some visual suggestion of the stark angularity of stucco. Thus one sentence of the simplest kind rouses the imagination, the memory, the eye and the ear—all combine in reading it.
But they combine—they combine unconsciously together. The moment we single out and emphasize the suggestions as we have done here they become unreal; and we, too, become unreal—specialists, word mongers, phrase finders, not readers. In reading we have to allow the sunken meanings to remain sunken, suggested, not stated; lapsing and flowing into each other like reeds on the bed of a river.39
To "single out and emphasize" one meaning, setting it apart from the others, is to make it unreal. The distinction between surface and sunken meanings, between denotation and sense, may be apparent and useful, but it is also somewhat artificial, and logical indications cannot be categorically separated from fluid implications. Woolf is weighing in on the same boundaries in meaning posed by Russell and by Frege before him; she makes this clear by bringing in her friend Russell by name—in sense and sunken meaning if not direct reference. The ducal house of Bedford, for which Russell Square is named, was Bertrand Russell's family. The 6th Duke of Bedford was his great-grandfather, the 1st Earl Russell, his grandfather. Bertrand Russell grew up in that grandfather's home. As she passes Russell Square on the Tube, contemplating a relentless intermingling of meanings, Woolf tells us, beautifully and indirectly, that she is passing Bertrand Russell by. It is an appropriately elliptical allusion, since she is arguing that isolated direct reference is ultimately ineffective. [End Page 85] Here Russell's presence, like Woolf's preferred meanings,40 remains sunken, fluid, and intermingled.
Yet Woolf does distinguish between surface and sunken meanings even as she tears down any wall between the two. There is a difference even if there is no reliable distinction. On an abstract level, these two types of meaning can clarify how variables mean, and they can also help us characterize the particularly fluid semantics of Woolf's most central figures. Literary symbols usually involve a range of connotations and evocations. It is usually their sense, rather than their reference, that is most complicated, most multifarious, and most interesting. But, I argue, symbols such as Jacob, or the lighthouse, or Woolf's waves, do the opposite instead: like mathematical variables, they have relatively singular senses, but remarkably multiple referents. The really odd thing about Jacob is not that we associate him with many different kinds of men, but that we cannot seem to choose which man he is. Woolf's waves not only connote many things, but also denote many, many things. That is exactly the semantic mode embodied by variables in pure mathematics, where a single variable can point toward many referents or denotations, but it will have only one sense, only one name.
Literary symbols do not usually behave in this way. Allegory is one clear counterexample, offering symbols that are too far married to individual referents to allow for this level of variability. However, the greater gulf separating most literary symbols from Woolf's variables involves not the number of meanings but the paths by which symbols lead to those meanings. Variables choose no favorites from among their possible referents. Literary symbols typically claim some natural, or at least culturally predetermined, connection to their meaning, because authors create symbols that are relevant to their content, that naturally open it out for us, playing on cultural expectations. Usually, the symbol itself is indispensable to the meanings it comes to reverberate with, and the range of meanings usually ramifies via pre-existing cultural associations, the metonymic senses of the symbol. In "The Love Song of J. Alfred Prufrock," when Prufrock announces that he "shall wear the bottoms of [his] trousers rolled,"41 T. S. Eliot's symbolism relies on the assumption that we [End Page 86] will have known old men who wear their trousers rolled. There is certainly slippage and multiplicity to the meaning of this phrase, but it proceeds along lines of metonymy as well as metaphor, of sense more than reference. Its preexisting cultural associations mean that it reaches to complicate and ramify existing tropes more than to strive for generality. Mathematical variables exist at an opposite symbolic extreme: when I use x to designate some range of values, I do not assert that there is any natural or preexisting connection at all between the written mark and its referents. The object chosen to carry these meanings is arbitrary, and the multiplicity of x has nothing to do with the written letter. Any cultural associations we do have with that x become pale and irrelevant up against its references and denotations.42
Meanwhile, Woolf frequently seems to delight in choosing symbols that are to some extent arbitrary. Jacob often seems inadequate to the book that contains him, as though he might as well be any young man who died in the Great War. The first draft of The Waves was subtitled The Life of Anybody, and the final novel is interested in personhood more than in its characters specifically. To the Lighthouse tells us quite directly that the particular significance of that lighthouse is an accident of whim: in "Time Passes," we read of "some random light . . . from some uncovered star, or wandering ship, or the Lighthouse even," as though characters' attentions might just as well be directed elsewhere. Even in the midst of his own passionate quest to reach the lighthouse, Mr. Ramsay thinks, "The Lighthouse! The Lighthouse! What's that got to do with it?"43 The lighthouse is only one among many possible loci of meaning; great significance does settle on that particular object, but To the Lighthouse reveals that it could easily have been otherwise. Woolf does play on readers' preexisting associations when she presents Jacob, or the lighthouse, or her crashing waves, but those associations come to seem minor, even somewhat arbitrary, up against the denotations that Woolf presents anew.
As the symbolic object becomes arbitrary, we find ourselves cut off from possibilities for metonymy. Natural associations become questionable when the object itself has only questionable relevance. That [End Page 87] troubled relevance of metonymic contiguities leaves the possible referents floating freely, without any network or hierarchy among them. Instead they exist interchangeably, though each different from the others. Across her career, Woolf repeatedly created enormous, central symbols that rely on newly drawn comparisons more than preexisting associations, metaphor more than metonymy, reference and denotation more than sense or connotation. Variables allow for an infinity of potential referents, opening up unlimited ranges of potential meaning, but what most fully distinguishes them from standard artistic symbols is a kind of equality among their referents. Selection from among the field of possible meanings is free and unattached. Symbols such as these allow for more complete slippage among their referents.44
Variables retain the capacity for precision while also embodying this height of multiplicity in meaning. It may be tempting to recall variables in their most simplistic form, e.g., solve for x in 2x = 6. However, to do so would be dangerously misleading, because there is only one value for x that makes that equation true. Regarding the role of variables in pure, modern mathematics, it is necessary to imagine cases where many meanings are conceivable: for example, ax = b, where x may take an infinite number of meanings, but at the same time it is delimited by the values of a and b. In pure, formal mathematics, variables exist to generalize possibilities, not as mere placeholders for single solutions.
Mathematical variables are well defined in particular ways. First, the context provides certain bounds (in the example above, we can determine that x may be any number, rational or irrational, real or imaginary, but it cannot be a table or chair, even though there do exist other instances in mathematics and formal logic where x may well be such an object). Second, although a variable might possess the power to stand in for an infinite range of possible values, once decided it must remain constant and thus iterable within a given expression (in ax2 + bx + c = 0, x can be any number, but we always know that x will be the same number both times that it appears). Regarding their bounded iterability, linguists have often compared variables to pronouns. That is, "she" is a flexible term that can stand in for an infinite number of subject positions, but it remains bounded (it must refer to a female) as well as iterable (within any single well-understood phrase, "she" should always designate the [End Page 88] same female). And, like variables, "she" tends to shift denotations more than senses; when "she" refers to two different women in two different contexts, its reference shifts without its connotation changing. Variables model a more extreme form of deixis.
The importance of variables in mathematics can hardly be overstated. Near the beginning of An Introduction to Mathematics (1911), which Woolf and her husband Leonard held in their library,45 White-head credits the generality belonging to variables with the origin and basis for mathematics itself: "Mathematics as a science commenced when first someone . . . proved propositions about any things or about some things, without specification of definite particular things."46 Similarly, in the very first line of The Principles of Mathematics (1903), which the Woolfs also owned,47 Russell tells us that "Pure Mathematics is the class of all propositions of the form 'p implies q.'" Here p and q are variables denoting propositions that themselves must contain only variables and relations,48 because, as Russell explained elsewhere, "we do not, in [mathematics or logic] deal with particular things or particular properties: we deal formally with what can be said about any thing or any property."49 In this sense, variables can be regarded as the defining feature of mathematics. Variables are the means by which the forms of truths may be uncovered independent of their particulars.
Notice that, as Russell uses them here, variables need not point to numbers, and their range is emphatically general. In turn, the linguistic comparison of variables and pronouns is limited insofar as pronouns are typically used in sentences where they only have one appropriate referent. Regarding pure mathematics, this analogy [End Page 89] will be more informative if we contemplate scenarios where a pronoun indicates not one person, but (potentially) any number of people. Consider the sentence "reading Jacob's Room, one loses track of names." This sentence refers neither to one particular person nor to all people. Instead, it refers to as many people as read Jacob's Room, and the exact number of people involved has no effect on its truth or meaningfulness. Under Russellian analysis, this sentence has an if-then logical structure: "if x reads Jacob's Room, then x loses track of names." Here the important point is that the sentence may just as well refer to a million people as to one person, without that undetermined breadth compromising any of its accuracy or sensibility. That capacity for specificity, always operating within full generality, is at the heart of what makes variables so powerful.
A variable is one semantic model for how language can mean both one thing and another. Given that language must adapt to describe a potentially infinite variety of circumstances with only a finite number of words at its disposal, this generality is at the heart of language's very ability to refer. By extension, variables model how language works. And variables offer literary theory a semantic structure for the radical simultaneity of generality and specificity, indicating all possible entities in a manner wherein a specific entity can still be substituted at any time. In this sense, variables very literally exist to allow us to rigorously discuss everything or anything at once.
I opened this paper with Empson's avowal of the ambiguity of the word ambiguity. That word derives from the Latin ambi-(around, both) and agere (lead, drive, act): etymologically, it describes a situation that drives both ways, that leads one around in multiple directions. Today, ambiguity has two primary definitions: (1) the state of possessing two or more possible interpretations; or (2) an uncertainty or confusion arising from indistinctness. Mathematical generality can characterize and illuminate literary ambiguity when ambiguity has the former definition. Yet, for literary purposes, that distinction between the former and the latter may never be complete, because where there are multiple possible interpretations, confusion can so naturally result, and because when literature is confusing, it so often leads to multiple interpretations. Variables can and do confuse algebra students every day, but mathematicians can disregard that confusion as a factor of the human world, external to mathematics. Literature, on the other hand, exists within that complicated human world. I argue that Woolf's variables exhibit the first ambiguity more than the second, proffering generality more than indistinctness and multiplicity more than confusion. Nevertheless, anyone who has read Jacob's Room can attest to the confusions it causes. My [End Page 90] argument can never be, and should never be, so absolute as is the mathematics it draws from.
Form and Pattern
Like oars rowing now this side, now that, were the sentences that came now here, now there, from either side of the table.—Woolf, Jacob's Room50
Russell writes that "the absence of all mention of particular things or properties in logic or pure mathematics is a necessary result of the fact that this study is, as we say, 'purely formal.'" He goes on to explain what he means by form: "The 'form' of a proposition is that, in it, that remains unchanged when every constituent of the proposition is replaced by another. Thus 'Socrates is earlier than Aristotle' has the same form as 'Napoleon is greater than Wellington,' though every constituent of the two propositions is different."51 According to Russell, form is, by definition, never specific. Instead, it exists in the relationships between terms irrespective of the terms themselves. Recognition of form in and of itself thus depends on symbols such as variables, which are capable of lending full generality, and the analysis of form then becomes the analysis of the patterns of those variables' interrelation. Form becomes a recognizable arrangement of things that exists independent of those things.
Meanwhile, when in Jacob's Room Woolf describes a conversation as "sentences that came now here, now there, from either side," "like oars rowing now this side, now that,"52 we never learn what those sentences say or mean; she presents only the pattern of their presence. This is language existing as a formal balance of undecided terms, and Woolf thus portrays discourse as form in Russell's sense of the term, as a pattern with the ability to exist independent of its specific content. That kind of generalized form is common elsewhere in Woolf's work, where a more abstract and interdisciplinary conception of form is made visible by symbolic generality. Woolf's lyrical writing reveals the same conception of form that Russell defines for mathematics.
The generality belonging to variables naturally directs attention to patterns because, apart from the repetition of terms or the lack thereof, it leaves us with very little else to recognize or rely on. In turn, like Jacob's constant name marking a seemingly undetermined [End Page 91] and multiple personhood, Woolf's particular words for varying possibilities remain markedly constant and ordered. Above the doorway to Jacob's room we see "a rose, or a ram's skull," and then on the final page Woolf describes it yet again as "a rose or a ram's skull."53 While the object is variable, its description is static. That constancy is crucial to mathematical methods of expressing generality: while the variable x might stand for an infinite range of possible referents, once we have selected x as a signifier, our characterizations of that range must stay faithful to our chosen mark. Where reference is at least partially determined, descriptions can shift, but if all that we know about x is that it is marked by "x," if we shift (without notice) to calling it "y" we will have given up all knowledge entirely. We keep calling the undetermined "x" precisely because we have determined so little about it, in the same way that our uncertainties about Jacob are much of what renders his name so central. In this way, with repeating terms portraying a diverse reality, "multiplicity becomes unity, which is somehow the secret of life."54
Relying on stable signs for undetermined referents, Jacob's Room repeatedly reuses the same descriptions to designate remarkably varying possibilities. As the narrator uses the same words, in the same order, to describe the indistinguishable doorway to Jacob's room, Mrs. Durrant similarly repeats an observational contradiction with the same words in the same order: to her Jacob is "extraordinarily awkward. . . . Yet so distinguished looking" and then "Extremely awkward . . . but so distinguished-looking." Later, Sandra Wentworth Williams's description is nearly identical: she notes Jacob's "extreme shabbiness" as he remains "very distinguished looking." By the end of the novel, the words have been taken up by the narrator and by an unindividuated "they": "'That young man, Jacob Flanders,' they would say, 'so distinguished looking—and yet so awkward.'"55 By now, although the order shifts, the pattern has become so well established that the inverted phrase remains utterly familiar. While it is possible to imagine young men who are both awkward and distinguished looking, this remains a generally contradictory set of characteristics. Woolf highlights human inconsistency with consistent terminology. She emphasizes undecided multiplicity with singular descriptions. Far from being paradoxical, this can be cause and effect: descriptive variety is difficult when so little knowledge is available to describe, and rather than split one unknown into several, the narrator settles for the formal knowledge that repetition achieves. [End Page 92]
Where reference is truly and radically undetermined, repetition, or the absence of repetition, is actually all that remains to be analyzed. Even if we know nothing at all about what x may be, "x and again x" still tells us more than does "x" alone. Jacob's Room tells us twice over, one hundred pages apart, that "it is no use trying to sum people up."56 As an explanation of Jacob himself (which it is), this comment declares an absence of understanding—here the narrator is admitting that she does not know Jacob in any finally decided sense. But in repeating that uncertainty, verbatim, Woolf insists instead on the understanding of arrangements and commonalities that variables allow for: the capacity to investigate patterns independent of particulars. That understanding is both more flexible and more widely applicable than any summation of Jacob would be. In that sense, it may even be more meaningful. In The Waves, Louis contemplates how "[l]ife passes. The clouds change perpetually over our houses. I do this, do that, and again do this and then that. Meeting and parting, we assemble different forms, make different patterns."57 Repetition, pattern, and form offer something that can be described in generality. For Louis, they allow him to articulate what he and his friends share in common. For the narrator of Jacob's Room, pattern and form allow her to tell Jacob's story while acknowledging all that she does not understand about Jacob himself.
Here form slips into formula—a mathematical entity that can articulate patterns, allowing specific facts to be inserted at any time and yet existing just as fully independent of those specifics. Formulas, containing variables, can describe and embody the rules of a situation abstracted from the situation itself. That generality does come with some risk of apparent flattening. In common usage, formulas evoke excessive fixity and fettering, conventional rules because pattern ungrounded by particularity can feel reductive, like empty laws without basis in fact or experience. In the same way, Louis's truthful description of human life ("I do this, do that, and again do this and then that") can sound empty, as Jacob himself often feels empty: if he exists as many different people, he could just as easily be no one at all.58 However, as the abstraction of formula triggers an awareness [End Page 93] of the relative concreteness of the signs and symbols that litter the page, Woolf's repetitions direct our analysis toward formal properties of the text itself, prompting a re-evaluation of form. That form is its own kind of irreducible and complicated knowledge, one that accords as much with multiplicities of experience as with any singular, objective facts.
According to the Oxford English Dictionary, historical uses of the term "form" indicate at once "the visible aspect" and "the essential determinant principle of a thing"; that is, form is at once an outward appearance and an essential determining structure.59 The literary history of formalism similarly ranges from the vital to the superficial, encompassing both an attention to the richness of pure language and a danger of reductive reading that evacuates words of meaning. This doubled usage sometimes seems to render formalist analysis the most superficial kind of study, while at the same time reserving for it examination of the deepest, most intrinsically literary elements in writing. Scholars from Raymond Williams to Samuel Otter have written powerfully on that dichotomy.60
Unnoticed in literary histories of formalism is the fact that formalism's presence as a term for a theory and methodology of mathematics actually predates its existence as a term of artistic study.61 [End Page 94] In fact, the term has a strikingly parallel presence in mathematics. There it sometimes describes a basic method of pure mathematics, the careful, rule-bound approach that is fundamental to mathematical pedagogy and mathematical proof. Yet at other times it indicates the radical and potentially evacuative claim that mathematics is only a meaningless manipulation of marks on paper.62 Before the emergence of Russian formalism in literary studies, the inextricable persistence of both the vital and the superficial versions of formalism already existed in mathematical usages. As the mathematician L. E. J. Brouwer wrote in 1912,
For the formalist . . . mathematical exactness consists merely in the method of developing the series of relations, and is independent of the significance one might want to give to the relations or the entities which they relate. And for the consistent formalist these meaningless series of relations to which mathematics are reduced have mathematical existence only when they have been represented in spoken or written language.63
Brouwer sought to discredit formalism, painting it as an empty and reductive understanding of mathematics. Yet his own description opens up the possibility that formalism might be much more than that: the methodology whereby, when viewed as a system of rigorously related symbols, mathematics becomes a kind of language. For [End Page 95] scholars of language, the theories of syntax and semantics therein implied should not be ignored. This would be a strange language indeed, because it defers or even obviates its own meaning, but in the process it achieves a study of interrelations among terms that reaches beyond any particular example.
In 1922, the same year Woolf published Jacob's Room, the mathematician Paul Bernays wrote that abstract mathematics is not the sphere of any logical content, but is rather "the domain of pure formalism." He went on to say, "Mathematics turns out to be the general theory of formalisms."64 In opposition to Brouwer, very influential mathematicians were embracing formalism in the 1920s. Bernays, with his teacher and colleague David Hilbert, argued that attention to mathematical form could ultimately enable a more effective and multiple engagement with mathematical meaning. The claim was not that mathematics was meaningless, but that pursuing it as though it were meaningless would provide a better understanding of math's basis and structure—in effect, that by deferring mathematical meaning, mathematical meaning could be rescued. And as Hilbert wrote in 1922, "The solid philosophical attitude that I think is required for the grounding of pure mathematics—as well as for all scientific thought, understanding, and communication—is this: In the beginning was the sign."65
This account depends on some intrinsic abstraction, because form has little to do with mathematics as we intuitively understand it. Apposite that, think of Woolf's description of Lily Briscoe's painting in To the Lighthouse. That ekphrasis has often been read as a model of a modernist formalism that was interdisciplinary across the written and visual arts,66 yet the even broader formalism (and generality) intrinsic to Woolf's abstraction there has been overlooked. When she describes Lily's painting as a structure of two undetermined forms existing in balance with each other—"if there, in that corner, it [End Page 96] was bright, here, in this, she felt the need of darkness"—the terms "bright" and "darkness" could almost be "x" and "not x," insofar as their relation and opposition to each other seems to be of greater importance than the meanings of those particular terms.67 Woolf's analysis of pattern extends to a contemplation of difference and repetition in any form. Lily struggles to "achieve that razor edge of balance between two opposite forces."68 That problem of artistic balance is everywhere in Woolf's own forms. To the Lighthouse itself has the same structure, with two universes balancing each other, differing and yet each reflections of the other, together straddling the abstract detachment of "Time Passes."
As Henri Poincaré wrote in 1902, "Mathematicians study not objects, but relations between objects; the replacement of these objects by others is therefore indifferent to them, provided the relations do not change. The matter is for them unimportant, the form alone interests them."69 If Jacob is meaningfully undetermined, we can still meaningfully analyze his relationships. Here form concerns not the many meanings of Jacob or the lighthouse, but how those undetermined symbols relate to the work's other terms. Form, from this mathematical perspective, comes to have very much in common with syntax, with the patterns that link unspecified terms. Such patterns can be more easily perceived when the details are out of focus. However, by blurring those details, we might reach either the multiplicity of meaning or the absence of meaning. Polysemy can frequently seem reductive, and it often flirts with meaninglessness. Gazing at Lily's painting, William Bankes notes that "[m]other and child—objects of universal veneration . . . might be reduced, he pondered, to a purple shadow without irreverence."70 The purple shadow might be further reduced to x without irreverence, and potentially without loss of meaning; each step feels like a reduction, yet with the increasing generality, something is gained. Patterns are more easily perceived with light and shadow, with x and not x, less easily perceived with mother and child. Pattern emerges directly from the undecided terms that Brouwer sought to call meaningless. In any field, the vital, structural version of form can follow directly from the superficial version. The bifurcated meaning of form is no historical accident; it results from a necessary interdependence.
In Woolf's most experimental text, The Waves, pattern becomes [End Page 97] apparent because of generality, and generality becomes meaningful because of pattern. There Woolf uses repetitions, parallels, and consonances to balance experiences and subjectivities without pinning them down, articulating personhood within a dense network of commonalities. As she explained later, in "A Sketch of the Past," this is "what I might call a philosophy; at any rate it is a constant idea of mine; that behind the cotton wool is hidden a pattern; that we—I mean all human beings—are connected with this."71 The characters of The Waves often seem inadequate as individuals, like voices rather than people: it is in their chorus, and in their discord, that Bernard, Susan, Rhoda, Neville, Jinny, and Louis become most significant. They exist particularly as a collection. From this perspective, waves make obvious sense as the title that eventually overtook The Life of Anybody: to describe anyone, Woolf must have believed that everyone shared something in common—that something, somewhere, repeats reliably in all human lives. Hence the soliloquies that break, over and over, like waves, and the characters and the lives that themselves mimic waves, crashing over and over again. Mathematically speaking, periodic repetition is what makes any wave a wave. Whether an ocean wave or a sound wave, it reliably repeats, and that repetition (or, sometimes, the lack thereof) is what most interests The Waves in waves. The entirety of The Waves proffers discourse, thought, and personhood as a repeating pattern. Repetitions operate from different perspectives and on innumerable levels in The Waves. Louis describes how "People go on passing. . . . They pass the window of this eating-shop incessantly. Motor-cars, vans, motor-omnibuses; and again motor-omnibuses, vans, motor-cars—they pass the window." And then again: "They go on passing, they go on passing . . . I repeat."72
By taking up the periodic motion of waves, Woolf also describes the capacities of language. Language communicates an unlimited wealth of differing experiences with only a finite number of words, such that repetition is necessary to its function—the same words must be able to convey different things in different contexts. As Neville describes, "Now begins to rise in me the familiar rhythm; words that have lain dormant now lift, now toss their crests, and fall and rise, and fall and rise again."73 Neville's experience of writing involves a "familiar rhythm" in large part because he has, of course, used all these words before. If they come now in different combinations, [End Page 98] each individual use remains a kind of repetition as "words that have lain dormant now lift . . . again." The words themselves are waves, but Neville does not relate them to waves for archetypal or associative reasons but for their most scientifically defining quality; they "fall and rise, and fall and rise again." Neville enacts this repetition not only by describing it, but also by mimicking it, repeating the exact words that themselves describe repetition. As he conceives words as waves, his words continue to repeat themselves in a newly literal wave: "Words and words and words."74 Repetition is always crucial to language, but it is particularly central to the formal ingenuity of The Waves, wherein diversity is legible within commonality, and commonality emerges from repetition.
These reiterations are only a couple among many that populate the entirety of The Waves, and they operate on higher structural levels well beyond individual words. Consider the opening sentences of each and every one of the interludes:
The sun had not yet risen.The sun rose higher.The sun rose.The sun, risen, no longer couched on a green mattress darting a fitful glance through watery jewels, bared its face and looked straight over the waves.The sun had risen to its full height.The sun no longer stood in the middle of the sky.The sun had now sunk lower in the sky.The sun was sinking.Now the sun had sunk.75
This pattern is exhaustive and relentless, and no interlude is allowed to escape the necessity wherein every single one must begin by positioning the sun. Containing this predictability is also the predictability of the interludes themselves, with their regularly repeating italicized descriptions that recur like drumbeats throughout the book. Meanwhile, the sun's course across the sky is itself a natural event that must repeat again and again, just as the waves crash repeatedly and the lives of differing voices rise and fall and rise and fall. These repetitions may occur on different time scales, but Woolf seems to tell us that the time scales do not matter. The pattern matters. Repetition is what both allows for and itself constitutes the underlying structure of The Waves, because pattern is what allows pure form to take on meaning, to prevent it from becoming a mere mishmash of [End Page 99] indistinguishable and unrelatable factors. As generality makes repetition visible, repetition makes form possible.
The absolute reliability of the patterns in The Waves can seem mathematical, but Woolf's symbolism nowhere claims mathematical certainty or objectivity. In fact, Woolf condemns mathematics that claims certainty applied to realms where no such assurance exists. People are far too complicated to be reducible to simple figures, and while Mr. Ramsay makes the grave error of thinking he could understand the complexities of life (including his own family) so objectively as "If Q . . . then R,"76 Jacob's Room repeatedly tells us, "It is no use trying to sum people up."77 In Mrs. Dalloway, "proportion" is Sir William Bradshaw's primary justification for the cruel regulation of society's outsiders, and Woolf highlights the term's mathematical meanings by emphasizing the divisions that Bradshaw produces: "slicing, dividing and subdividing, the clocks of Harley Street nibbled at the June day, counselled submission, upheld authority, and pointed out in chorus the supreme advantages of a sense of proportion."78 Here, mathematics becomes the figurehead of false objectivity, the dictatorial proclaimer of doubtful absolutes. Literary ambiguity cannot claim mathematical certainty without undermining its own subtlety, insight, and reach.
However, elsewhere Woolf lays out another conception of mathematics: a mode of meaning rather than a guarantor of truth. The remarkably flexible semantics of mathematical abstractions become valuable artistic models even as the certainty of mathematics remains suspect. In Night and Day, mathematics is at once a symbol for literature and the self-conscious opposite of literary creation. The protagonist, Katharine Hilbery, strives after mathematics with passion, seeing it as a thing of beauty and the only language in which she can express herself. Woolf repeatedly emphasizes the written form that mathematics and literature share: both are "sacred pages of symbols and figures." Yet Katharine also insists that "mathematics were directly opposed to literature."79 Woolf thus unites differing paradigms under common symbolism, but she never merges them. The semantic intervention of mathematical generality allows for common analysis without conflation—it need never entail forgetting the differences among these multiple meanings. [End Page 100]
Woolf treated mathematical conceptions of meaning as both a cautionary tale and an example to be emulated. Like form, formula, and formalism, this beast has two very different faces. The first version of mathematical semantics exists in the rigid and senseless numerals that befuddle and then imprison Rhoda toward the beginning of The Waves, wherein "meaning has gone" for good.80 That mathematics exists just as much in Bradshaw's proportion, with his arrogant insistence that his meaning is the only meaning. Yet another kind of mathematics emerges more freely in the "love that never attempted to clutch its object; but, like the love which mathematicians bear their symbols, or poets their phrases, was meant to be spread over the world."81 The first mathematics claims exclusive objectivity, too often enabling false declarations of absolute knowledge; the second mathematics exists in flexibly meaningful patterns (whether Katharine's mathematics or Lily's painting), which deny the validity of the first. The first mathematics collapses into a flat, reductive, and meaningless formalism; the second blossoms into a living, polysemous formalism.
These two mathematical formalisms exist outside of Woolf's representations. They embody two versions of knowledge, one claiming objectivity and the other flexibility, one already determined and the other forever adaptable. Woolf's work both draws from and creates the second fluid understanding as it shuns the rigidity of the first. Her writing disdains any facile formulation of the relationship between symbols and their meanings, whether that formulation presumes to be able to permanently dodge all meaning or whether it presumes to jump instantaneously to the one true meaning. Either mistake amounts to the same kind of arrogance. Instead, pattern in Woolf's work leads to the great multiplicity of meaning, even as the multiplicity of meaning itself points back to pattern. The tremendous interdisciplinarity of that cycle points toward how utterly real this abstract process is. Those modernists who elaborated abstract patterns were often accused of formalism in a negative, superficial sense, but the deep, structural side of formalism can follow naturally from patterns, even superficial patterns, and that process can apply in any field. This is the value of generality, which retains the capacity for knowledge without necessitating the hegemony of singular knowledge claims. It opens up both multiplicity and commonality. [End Page 101]
Jocelyn Rodal is a visiting fellow at Penn State's Center for Humanities and Information. She received her PhD in English from the University of California–Berkeley in 2016, and was previously a postdoctoral fellow at the Rutgers Center for Cultural Analysis. She is currently at work on a book manuscript entitled Modernism's Mathematics: From Form to Formalism. That project reads literary modernism alongside a contemporaneous modernist movement in mathematics. Looking at authors such as T. S. Eliot, James Joyce, Ezra Pound, and Virginia Woolf, she argues that modernists were using mathematics to create and elucidate form—form that, in turn, engendered formalism in literary studies. Modernism's Mathematics uses modernist mathematical theories of syntax and semantics to understand form, arguing that formalism, as a reading practice, has structural and historical roots in mathematics.
1. William Empson, Seven Types of Ambiguity (New York: New Directions, 1966), p. viii.
2. Virginia Woolf, "Women and Fiction," in Collected Essays (New York: Harcourt, Brace & World, 1967), vol. 2, pp. 141–148, at p. 141.
3. See, for example, James Hafley, The Glass Roof: Virginia Woolf as Novelist (New York: Russell and Russell, 1963), pp. 79–80; and Jack F. Stewart, "Existence and Symbol in The Waves," Modern Fiction Studies 18:3 (1972): 433–447.
4. Generality is one of the greatest goals of mathematical research. Alfred North White-head: "The certainty of mathematics depends upon its complete abstract generality"; Bertrand Russell: "in mathematics... the greatest possible generality is before all things to be sought"; G. H. Hardy: "The propositions of logic and mathematics share certain general characteristics, in particular complete generality"; Henri Poincaré: "if we open any book on mathematics... on every page the author will announce his intention of generalizing." Alfred North Whitehead, Science and the Modern World (New York: The Free Press, 1967), p. 22; Bertrand Russell, "The Study of Mathematics," in Mysticism and Logic (London: George Allen & Unwin, 1963), pp. 58–73, at p. 56; G. H. Hardy, "Mathematical Proof," Mind 38:149 (1929): 1–25, at p. 6; Henri Poincaré, The Foundations of Science, trans. George Bruce Halstead (New York: The Science Press, 1929), p. 32.
5. Megan Quigley, "Modern Novels and Vagueness," Modernism/Modernity 15:1 (2008): 101–129.
6. See Herbert Mehrtens, Moderne, Sprache, Mathematik: Eine Geschichte des Streits um die Grundlagen der Disziplin und des Subjekts formaler Systeme (Frankfurt: Suhrkamp, 1990); Jeremy Gray, Plato's Ghost: The Modernist Transformation of Mathematics (Princeton, NJ: Princeton University Press, 2008); and Moritz Epple, "An Unusual Career between Cultural and Mathematical Modernism," in Jews and Sciences in German Contexts, ed. Ulrich Charpa and Ute Deichmann (Tübingen: Mohr Siebeck, 2007), pp. 77–99. Also important here is Amir Alexander, Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics (Cambridge, MA: Harvard University Press, 2010). While Mehrtens, Gray, and Epple all compare modernist mathematics to modernist art and literature, Alexander attributes modernity in nineteenth and twentieth century mathematics to the earlier cause of romanticism.
7. On Woolf's relationship with Russell, see Ann Banfield, The Phantom Table: Woolf, Fry, Russell and the Epistemology of Modernism (New York: Cambridge University Press, 2000). Woolf did discuss mathematics with Russell: see The Diary of Virginia Woolf, ed. Anne Olivier Bell with Andrew McNeillie (New York: Harcourt Brace Jovanovich, 1978), vol. 2, pp. 146–148. Ramsey, Hardy, and Whitehead traveled in Woolf's larger social circle. Meeting Ramsey at a dinner party in 1923, she described him as "like a Darwin, broad, thick, powerful, & a great mathematician" (Diary, vol. 2, p. 231). She knew the Whitehead family (see The Letters of Virginia Woolf, ed. Nigel Nicolson and Joanne Trautmann [New York: Harcourt Brace Jovanovich, 1977], vol. 1, p. 135; and vol. 2, p. 78). Leonard Woolf and Hardy were friends; he appears several times in Leonard's autobiographies (Leonard Woolf, Sowing: An Autobiography of the Years 1880–1904 [New York: Harcourt, Brace & Company, 1960], pp. 123–126; and Beginning Again: An Autobiography of the Years 1911–1918 [London: The Hogarth Press, 1964], pp. 18, 20, and 52). More broadly, a growing body of scholarship has established Woolf's interest in, and knowledge of, the sciences. See Gillian Beer, Virginia Woolf: The Common Ground (Ann Arbor: University of Michigan Press, 1996); Michael H. Whitworth, Einstein's Wake: Relativity, Metaphor, and Modernist Literature (Oxford: Oxford University Press, 2001); Holly Henry, Virginia Woolf and the Discourse of Science: The Aesthetics of Astronomy (New York: Cambridge University Press, 2003); and Christina Alt, Virginia Woolf and the Study of Nature (New York: Cambridge University Press, 2010).
8. Wonderful work has been done on the complex and often attenuated nature of influence across literature and the sciences. See, for example: Whitworth, Einstein's Wake (above, n. 7), pp. 13–19; Beer, Virginia Woolf (above, n. 7), pp. 112–113; and Daniel Albright, Quantum Poetics: Yeats, Pound, Eliot, and the Science of Modernism (Cambridge: Cambridge University Press, 1997), pp. 1–2. In terms of the broader convergence of literary and mathematical modernisms, some third cause must have been involved: the vast and multifarious cultural shifts that produced modernism across fields. This is the argument made by both Mehrtens and Gray (Moderne, Sprache, Mathematik and Plato's Ghost [above, n. 6]).
9. Virginia Woolf, Mr. Bennett and Mrs. Brown (London: Hogarth Press, 1924), p. 4.
10. I have addressed Woolf's direct engagements with mathematics in "Virginia Woolf on Mathematics: Signifying Opposition," in Contradictory Woolf: Selected Papers from the Twenty-First Annual International Conference on Virginia Woolf, ed. Derek Ryan and Stella Bolaki (Clemson, SC: Clemson University Press, 2012), pp. 202–208. See also Makiko Minow-Pinkney, "Virginia Woolf and December 1910: The Question of the Fourth Dimension," in Ryan and Stella, Contradictory Woolf, pp. 194–201.
11. Banfield, Phantom Table (above, n. 7), p. 149 (quoting Leslie Stephen), and p. 259.
12. Woolf, May 27, 1927, in Letters (above, n. 7), vol. 3, p. 385.
13. Unsigned review of Jacob's Room, Times Literary Supplement, October 26, 1922, p. 683, in Virginia Woolf: The Critical Heritage, ed. Robin Majumdar and Allen McLaurin (New York: Routledge, 1997), pp. 95–97, at p. 97. See also Anna Snaith, Virginia Woolf: Public and Private Negotiations (New York: St. Martin's, 2000), pp. 79–80; William R. Handley, "War and the Politics of Narration in Jacob's Room," in Virginia Woolf and War: Fiction, Reality, Myth, ed. Mark Hussey (Syracuse: Syracuse University Press, 1991), pp. 110–133; Judy S. Reese, Recasting Social Values in the Work of Virginia Woolf (Selinsgrove: Susquehanna University Press, 1996), pp. 121–122; Mark Hussey, The Singing of the Real World: The Philosophy of Virginia Woolf's Fiction (Columbus: Ohio State University Press, 1986), pp. 21–45; and Christine Froula, Virginia Woolf and the Bloomsbury Avant-Garde: War, Civilization, Modernity (New York: Columbia University Press, 2005).
14. Virginia Woolf, Jacob's Room (New York: Harcourt, 1978), p. 70.
15. Ibid., pp. 176, 97.
16. Ibid., p. 40.
17. Ibid., p. 8.
18. That is, pure mathematics as opposed to applied mathematics: here "pure mathematics" describes abstract math, as distinct from math instrumentalized to solve problems in science or the material world. Pure mathematics is characterized by defined axiom systems, complete proof, and the greatest possible generality. Applied math is more comfortable with approximation, and it does not necessarily demand generality. The designation of "pure" mathematics directs attention to the differences between math and science, but it often leaves unaddressed the matter of what distinguishes math from logic—the branch of philosophy that studies rules of reasoning and inference. A number of modernist mathematicians sought to prove that mathematics was an extension of logic, and those thinkers often gave definitions of pure mathematics that encompassed logic: see Bertrand Russell, The Principles of Mathematics (New York: W. W. Norton and Company, 1996), p. 3; Mysticism and Logic (above, n. 4), pp. 74–76; Introduction to Mathematical Philosophy (London: George Allen & Unwin, 1919), p. 192; and Whitehead, Science and the Modern World, (above, n. 4), pp. 20–21.
19. Woolf, Jacob's Room (above, n. 14), pp. 102, 154.
20. Shankar Raman, "Specifying Unknown Things: The Algebra of The Merchant of Venice," in Making Publics in Early Modern Europe: People, Things, Forms of Knowledge, ed. Bronwen Wilson and Paul Yachnin (New York: Routledge, 2010), pp. 212–231, at p. 213. This article is indebted to Raman's work.
21. Woolf, Jacob's Room (above, n. 14), pp. 151, 153.
22. Virginia Woolf, Moments of Being (San Diego: Mariner, 1985), p. 111.
23. Quentin Bell, Virginia Woolf, A Biography (Orlando: Harcourt, 1972), vol. 1, p. 26.
24. Virginia Woolf, To the Lighthouse (San Diego: Harcourt, 1981), p. 289.
25. Woolf, Jacob's Room (above, n. 14), p. 70.
26. Virginia Woolf, Night and Day (New York: Oxford University Press, 2000), p. 477.
27. Woolf, Lighthouse (above, n. 24), p. 47.
28. Woolf, "Craftsmanship," in Collected Essays (above, n. 2), pp. 245–251, at pp. 246, 247, 250–251.
29. Woolf, Night and Day (above, n. 26), p. 224.
30. Apart from Raman, exceptions include Steven Cassedy, Flight from Eden: The Origins of Modern Literary Criticism and Theory (Berkeley: University of California Press, 1990), which points out that attention to literary forms as relational systems has sometimes prompted the comparison of literary meanings and algebraic variables; and Jennifer Ashton, From Modernism to Postmodernism: American Poetry and Theory in the Twentieth Century (New York: Cambridge University Press, 2005), which argues that Gertrude Stein turned to literary variables in her late work in the attempt to write what Stein called "a history of every one" (Gertrude Stein, The Making of Americans [Normal: Dalkey Archive, 1995], p. 191 and passim).
31. Russell, Mathematical Philosophy (above, n. 18), p. 10.
32. Bertrand Russell, "On Denoting," Mind 14:56 (1905): 479–493, at p. 480.
33. Joseph Shoenfield, Mathematical Logic (Boca Raton: CRC Press, 2010), p. 7. Here, by "analysis," Shoenfield indicates the formalized version of calculus.
34. Gottlob Frege, "On Sinn and Bedeutung," trans. Max Black, in The Frege Reader, ed. Michael Beaney (Malden: Blackwell, 1997), pp. 151–171, at p. 152.
35. Edward Zalta, "Gottlob Frege," in The Stanford Encyclopedia of Philosophy, last revised August 4, 2015, http://plato.stanford.edu/archives/spr2016/entries/frege/.
36. Frege, "Sinn and Bedeutung" (above, n. 34), p. 157.
37. Russell, "On Denoting" (above, n. 32), pp. 481, 482.
38. Woolf, Collected Essays (above, n. 2), p. 246.
39. Ibid., pp. 247–248.
40. In deference to Woolf's intermingling of meanings, throughout this paper I use "meaning" as an all-embracing term, encompassing Sinn and Bedeutung, sense and denotation, sunken meaning and surface meaning. Where I refer specifically to the former, I use sense, connotation, or evocation. To indicate the latter, I speak of denotation or referent.
41. T. S. Eliot, "The Love Song of J. Alfred Prufrock," in Collected Poems: 1909–1962 (New York: Harcourt Brace & Company, 1991), pp. 3–7, at p. 7.
42. Mathematical variables cannot be completely without associations. E = mc2 offers cultural meanings that x = yz2 does not, even though the two equations are mathematically identical. Choices in mathematical notation can definitely involve cultural connotations. However, while mathematical language (like all language) inevitably carries some nondenotative associations, it generally involves far fewer such implications than most language does.
43. Woolf, Lighthouse (above, n. 24), pp. 126, 151.
44. Elsewhere in literature, symbols such as these are less common, yet when they do appear they tend to take over, frequently acting as titles and lying at the centers of texts. Other dominating examples might include the great whale in Moby Dick or the scarlet letter in The Scarlet Letter.
45. The volume was signed by Leonard, which may indicate that he purchased it. The edition is undated; see The Library of Leonard and Virginia Woolf: A Short-Title Catalog, ed. Julia King and Laila Miletic-Vejzovic (Pullman: Washington University Press, 2003), p. 242.
46. Alfred North Whitehead, An Introduction to Mathematics (New York: Henry Holt and Company, 1911), p. 15.
47. 1903 edition, signed by Leonard. Library (above, n. 45), p. 193.
48. Russell, Principles of Mathematics (above, n. 18), p. 3. Russell's opening sentence in full: "Pure Mathematics is the class of all propositions of the form 'p implies q,' where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants." Here logical constants are terms whose only semantic value involves matters such as relation or implication: examples include "and," "or," "if," "then," "not," "every," "some" "such that," and sometimes "is." Apart from terms such as these, for Russell the propositions of pure mathematics can contain only variables.
49. Russell, Mathematical Philosophy (above, n. 18), p. 196.
50. Woolf, Jacob's Room (above, n. 14), p. 57.
51. Russell, Mathematical Philosophy (above, n. 18), pp. 198, 199.
52. Woolf, Jacob's Room (above, n. 14), p. 57.
53. Ibid., pp. 70, 176.
54. Ibid., p. 131.
55. Ibid., pp. 61, 70, 145, 155.
56. Ibid., pp. 31, 154.
57. Virginia Woolf, The Waves (San Diego: Harcourt, 1978), p. 170.
58. The first time Leonard read Jacob's Room he called the characters "ghosts" and "puppets" (Woolf, Diary [above, n. 7], vol. 2, p. 186), and a substantial strain of scholarship on Jacob's Room has concurred, arguing that its characters' superficial, emptied-out existences satirize human failures. See Alex Zwerdling, "Jacob's Room: Woolf's Satiric Elegy," ELH 48:4 (1981): 894–913; Ann Ronchetti, The Artist, Society, and Sexuality in Virginia Woolf's Novels (New York: Routledge, 2004); Susan C. Harris, "The Ethics of Indecency: Censorship, Sexuality, and the Voice of the Academy in the Narration of Jacob's Room," Twentieth Century Literature 43:4 (1997): 420–438; and Froula, Virginia Woolf and the Bloomsbury Avant-Garde (above, n. 13).
59. Oxford English Dictionary Online, s.v. "form, n." (Oxford University Press, June 2017).
60. Raymond Williams, Keywords (New York: Oxford University Press, 1983), pp. 137–140; Samuel Otter, "An Aesthetics in All Things," Representations 104:1 (2008): 116–125. Also important here is Paul De Man, "Semiology and Rhetoric," Diacritics 3:3 (1973): 27–33, at pp. 27–28.
61. "Formalist" was used as far back as the seventeenth century to indicate a stickler for superficial rules. Raymond Williams identifies a couple intermediate usages suggesting how the term migrated from that meaning toward designating a theory of literature. The earliest uses of the term as a school of literary theory seem to belong to the Russian formalists, beginning around 1916. On the other hand, mathematicians recognized a formalist school of thought before that, particularly in German (formal, der Formalismus). Eduard Heine and Johannes Thomae outlined formalist philosophies of mathematics in the second half of the nineteenth century, and Frege influentially contested their formalism in 1903. In English, as early as 1900, Russell used "formalist" to describe a philosophy too entangled with rule-bound symbols to be useful; he compared that philosophy to the more effective formal methods of mathematics. In 1913, an English translation of a paper by L. E. J. Brouwer described formalism as a school of thought wherein mathematics could be realized only in language. See Oxford English Dictionary Online, s.v. "formalist" and "formalism" (Oxford University Press, December 2013); Williams, Keywords (above, n. 60), p. 138; an excerpt from Frege's Grundgesetze, "Frege against the Formalists," trans. Max Black, in Translations from the Philosophical Writings of Gottlob Frege, ed. Peter Geach and Max Black (Oxford: Basil Blackwell, 1960), pp. 182–233; Russell, A Critical Exposition of the Philosophy of Leibniz (Cambridge: Cambridge University Press, 1900), p. 170; L. E. J. Brouwer, "Intuitionism and Formalism," trans. Arnold Dresden, Bulletin of the American Mathematical Society 20:2 (1913): 81–96.
62. As the claim that mathematics is fundamentally the study and manipulation of formal terms (usually, but not necessarily, conceived as marks on paper), formalism has antecedents reaching back to G. W. Leibniz's work on the infinitesimal and George Berkeley's explanations of arithmetic. However, formalism was most influentially developed by David Hilbert in the 1920s. Hilbert made no claims about math's fundamental ontology, instead arguing that math could productively be treated as if it were only marks on paper. The multiplicity of mathematical formalism is underlined by the fact that Hilbert's movement itself went through identifiably different stages: see Paolo Mancosu, The Adventure of Reason: Interplay between Philosophy of Mathematics and Mathematical Logic, 1900–1940 (New York: Oxford University Press, 2010), pp. 125–158. Mathematicians also use formalism to describe a formal, rule-bound methodology of mathematics. This latter usage makes no ontological claims, but simply describes how mathematics can sometimes be done without holding its semantic content in mind, focusing instead on the syntactical rules governing relations between terms. See, for example, Russell, Mysticism and Logic (above, n. 4), p. 95; and Terence Tao, "There's More to Mathematics than Rigour and Proofs," What's New, https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/.
63. Brouwer, "Intuitionism and Formalism," inaugural address at the University of Amsterdam, 1912 (above, n. 61), p. 83.
64. Paul Bernays, "Hilbert's Significance for the Philosophy of Mathematics," trans. Paolo Mancosu, in From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s, ed. Paolo Mancosu (New York: Oxford University Press, 1998), pp. 189–197, at p. 196.
65. David Hilbert, "The New Grounding of Mathematics, First Report," trans. William Ewald, in From Brouwer to Hilbert (above, n. 64), pp. 198–214, at p. 202.
66. See, for example, John Hawley Roberts, "'Vision and Design' in Virginia Woolf," PMLA 61:3 (1946): 835–847; Jonathan R. Quick, "Virginia Woolf, Roger Fry and Post-Impressionism," The Massachusetts Review 26:4 (1985): 547–570; Christopher Reed, "Through Formalism: Feminism and Virginia Woolf's Relation to Bloomsbury Aesthetics," Twentieth Century Literature 38:1 (1992): 20–43; and Beer, Virginia Woolf (above, n. 7), pp. 37–38.
67. Woolf, Lighthouse (above, n. 24), p. 52.
68. Ibid., p. 193.
69. Henri Poincaré, Foundations of Science (above, n. 4), p. 44.
70. Woolf, Lighthouse (above, n. 24), p. 52.
71. Woolf, Moments of Being (above, n. 22), p. 72.
72. Woolf, The Waves (above, n. 57), pp. 92, 93.
73. Ibid., p. 82.
74. Ibid., p. 83.
75. Ibid., pp. 7, 29, 73, 108, 148, 165, 182, 207, 236.
76. Woolf, Lighthouse (above, n. 24), p. 34.
77. Woolf, Jacob's Room (above, n. 14), pp. 31, 154.
78. Virginia Woolf, Mrs. Dalloway (San Diego: Harcourt, 1981), p. 102.
79. Woolf, Night and Day (above, n. 26), pp. 477, 42.
80. Woolf, The Waves (above, n. 57), p. 21.
81. Woolf, Lighthouse (above, n. 24), p. 47.