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"A Few Sheets of Paper Covered with Arbitrary Symbols": Formalism, Modernism, Mathematics

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Abstract

Modernism happened in mathematics, too. Between about 1890 and 1930, the field turned toward dramatic, previously unimaginable abstraction, as radical innovations dispensed with inherited conventions. The recognition of non-Euclidean geometry raised the strange notion that multiple, contradictory realms of mathematics could coexist and remain equally correct, even if some seemed non-representative of the world or nonsensical in human experience. Mathematicians grew frenzied in the effort to resolve newfound paradoxes. The turn of the century saw a series of attempts to rethink math’s most basic foundations, and the 1920s witnessed explosive controversies about the role of language in mathematics. These developments do not merely constitute change in one field; they demonstrate that modernism was a broader cultural phenomenon than literary scholars have realized.

This dissertation begins from the shared intellectual history of literary and mathematical modernism: a common attempt to rethink foundational axioms, a common ambivalence toward growing abstraction, and a common interest in and anxiety about form. This convergence, I argue, exposes an interpretive dilemma intrinsic to form in mathematics and writing alike. Modernist shifts in cultural assumptions gave rise to newly abstract forms that revealed form to have been always already abstract—so abstract as to estrange its own cultural function. Formalism, in literature, skates uncomfortably between profundity and superficiality, because attention to literary language in and of itself risks discounting language’s fundamental function: to express, refer, and communicate. But decades before formalism named a school of thought in literary interpretation, formalism—by that name—identified an understanding, and an interpretation, of mathematics.

Mathematical formalism has roots reaching back to the 17th century, but after a long history it reached its heyday in the 1920s, when the field’s escalating abstraction had led many to question math’s grounding in the real world, i.e., to doubt its basic capacity for meaning. The influential mathematician David Hilbert argued that such a grounding could be secured via attention to form—in effect, that formalism could rescue mathematical meaning by deferring that meaning, first treating mathematics as though it were only meaningless marks on paper. This dissertation uses Hilbert’s mathematical formalism to explain the apparent contradictions of literary formalism. I demonstrate that writers and mathematicians discovered the same solutions to an urgent common problem, developing technical apparatuses (variables, equivalence relations, axiom systems) through which meaning can develop directly from patterns and rules rather than terms and symbols. Mathematics has a remarkable relationship with reality, because it is both a maddeningly ethereal realm of thought and an undeniably descriptive tool that reliably predicts physical phenomena. I argue that the bond between mathematics and science is not only a metaphor for the relationship between literature and reality, but an imitable model that multiple modernist authors knowingly seized upon and manipulated. It is in this mathematical manner that the most abstract, formal, and seemingly unworldly moments in modernism acquired relevance to history, world, and culture. They did so by describing the world without making reference to the world, expressing the shapes of reality via pattern and form.

My first chapter, “Opposition, Polysemy, Pattern: Virginia Woolf’s Mathematical Generality,” begins from the more commonly understood relation between literature and mathematics: one of radical difference. In her early novel Night and Day Woolf resists the rigidity of mathematics, consistently imagining mathematical communication as a negative image of her own art, an inaccessible language that resists translation into English. But this opposition, I argue, also makes visible a likeness between the two fields, glimpsed in the “sacred pages of symbols and figures” that Woolf consistently uses to frame their common signs and signifying processes. The formalism of mathematical writing thus promises to explain many of the escalating formal complexities of Woolf’s later novels. I demonstrate that, even where she makes no direct reference to mathematics, Woolf uses the greater generality of mathematical modes of meaning to reinvent the force of ambiguity in Jacob’s Room and contradiction in To the Lighthouse, before settling into the enormous patterns that establish a formula for “the life of anybody” in The Waves.

In mathematics, such generality is most meaningful in conjunction with structured systems—the patterns that make unrestricted meaning possible. I take up the problem of defining and theorizing literary structure in the following two chapters. In “The Metaphor as an Equation: The Formal Abstraction of Ezra Pound’s Precisions,” I trace Pound’s repeated descriptions of image and metaphor as instruments of equation, and I demonstrate that technical mathematical definitions of equation characterize some of Imagism’s starkest innovations. Between 1890 and 1910, mathematical understandings of equation had shifted, reaching beyond number to become newly applicable to shapes, ideas, and words. Pound, I argue, harnessed recent mathematical redefinitions of equality to place a new weight on syntax: in his early poetry, semicolons unseat verbs, and ellipsis makes possible new forms of comparison. Pound’s early metaphors invented a newly symmetric and iterable relationship between tenor and vehicle, one capable of underlying more procreant forms. And form, here, acquires a newly interdisciplinary definition: the patterns and arrangements of relations between things that can exist independent of those things.

Pattern has to start from somewhere: a problem that consumed Eliot across his long career. In pure, modern mathematics, absolute knowledge and incontrovertible proof are built from patterns that always stand upon unproven assumptions—axioms. Eliot had examined those axioms in detail. In 1914, as a student at Harvard, Eliot took Bertrand Russell’s graduate course in advanced logic, intensively studying the modernist axiom systems of Gottlob Frege, Alfred North Whitehead, and Russell himself. In “From Axiom to Leap of Faith: T.S. Eliot’s Formal Systems,” I trace Eliot’s ambivalent preoccupation with logical assumption in “Prufrock” and The Waste Land and argue that, in Ash Wednesday and Four Quartets, Eliot ultimately used Russell’s formal systems to reimagine poems as infinitely intricate patterns built from finite starting points: an idea that already existed, in more troubled ways, in his earliest poetry. Eliot conceived of the most complex poetic forms as axiom systems, wherein whole aesthetic universes develop from mere handfuls of beliefs and linguistic links, just as, in mathematical axiom systems, complex meaning and sophisticated beauty develop from strikingly few assumptions, definitions, and rules.

Axiom systems refer, intrinsically and everywhere, to themselves, and in this way the patterns loop back upon themselves, at once referring to themselves and anticipating their own analysis. I conclude by turning from poetry and fiction to the literary scholarship and theory engendered by modernism, generalizing the ways in which the philosophy of mathematics reframes the critical stumbling blocks of form and reference. Mathematics was a regular trope, and often a conceptual source, in the first defining works of twentieth-century formalist criticism, evincing a continuing association between the logical analysis of form and the commingling of mathematics and language. The Russian formalists, British practical critics, and American New Critics all saw mathematics as a closed linguistic system, making it uniquely useful as a descriptive analogue for literary form: mathematics offered the prime example of a system that models isolated signifying processes while lying (by general consensus) utterly outside of literature.

I argue that the mathematics of form made these literary formalisms a natural and necessary response to literary modernism, determining their near-simultaneous development across different cultural contexts. When modernist form unsettled modernist content, a meta-content emerged: texts by Woolf, Pound, and Eliot all self-consciously refer to the interpretive difficulties they create. During exactly the same era, Hilbert pushed the intuitive sense of mathematics into a separate plane only to find that a new discipline derived from that distinction: metamathematics, which uses mathematics to study mathematics itself. Far from pushing the world out, formalist analysis indirectly drags the world back in, for writers and mathematicians alike. “A Few Sheets of Paper Covered with Arbitrary Symbols”: Formalism, Modernism, Mathematics thus culminates in the argument that the apparent paradoxes of formal literary interpretation—wherein formalist reading sees only the façade yet penetrates to the heart of the thing itself—are not coincidental to how we have practiced literary formalism. In fact, they are logically attendant on any application of formalist analysis, in any field. Formalism depends on the deferral of meaning but then, elaborating its own rules, bypasses that assumption to uncover a meaningful pattern, a meaning in pattern. Modernists from Woolf to Hilbert discovered and elaborated this interpretive loop. Our literary practice of formalism today remains dependent on that logical process of form.

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This item is under embargo until October 12, 2023.