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PALGRAVE STUDIES IN LITERATURE,
SCIENCE AND MEDICINE
Mathematics in
Postmodern
American Fiction
Stuart J. Taylor
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Palgrave Studies in Literature, Science and Medicine
Series Editors
Sharon Ruston
Department of English and Creative Writing
Lancaster University
Lancaster, UK
Alice Jenkins
School of Critical Studies
University of Glasgow
Glasgow, UK
Jessica Howell
Department of English
Texas A&M University
College Station, TX, USA
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Palgrave Studies in Literature, Science and Medicine is an exciting, prize-­
winning series that focuses on one of the most vibrant and interdisciplinary areas in literary studies: the intersection of literature, science and
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and emerging topics as well as established ones.
Editorial board:
Andrew M. Beresford, Professor in the School of Modern Languages and
Cultures, Durham University, UK
Steven Connor, Professor of Living Well with Technology, King’s College
London, UK
Lisa Diedrich, Associate Professor in Women’s and Gender Studies, Stony
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Peter Middleton, Professor of English, University of Southampton, UK
Kirsten Shepherd-Barr, Professor of English and Theatre Studies,
University of Oxford, UK
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University of Oxford, UK
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State University, USA
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Cardiff University, UK
Karen A. Winstead, Professor of English, The Ohio State University, USA
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Stuart J. Taylor
Mathematics in
Postmodern American
Fiction
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Contents
1 Introduction 1
2 Topological Structures and Allusion in Ratner’s Star 53
3 Algebraic Structures and Metaphor in Gravity’s Rainbow119
4 Ordered Structures and Cognition in Infinite Jest185
5 Conclusion: Literary Legacy of Mathematical Structures231
Works Cited273
Index301
vii
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List of Figures
Fig. 2.1
Fig. 2.2
Fig. 2.3
Fig. 3.1
Fig. 3.2
Fig. 4.1
Fig. 4.2
Fig. 4.3
Möbius strip
59
Billy’s Ladder and ‘Adventures’|’Reflections’ chapter structure
82
Ratner’s Star and ‘Frame Tale’ as Möbius strips
85
Parabola and straight line (intersection points circled)
131
V-2 missile’s deviation from Assigned Course—Source of
Pynchon’s Second Equation. With illustration and explanations
of some angles and parameters used in the equation for yaw
control168
Characters of Infinite Jest diagram (detail)
188
Partially ordered set diagram
189
Partially ordered set diagram. A zigzagging path through whole
and rational numbers showing one-to-one correspondence
between both sets
204
ix
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CHAPTER 1
Introduction
A mathematician, like a painter or a poet, is a maker of patterns.—
G. H. Hardy.1
This study examines intersections of mathematics and postmodern fiction of the United States, specifically within encyclopedic narratives by
Don DeLillo, Thomas Pynchon, and David Foster Wallace. My interdisciplinary approach draws upon the case of Nicolas Bourbaki, whose ‘encyclopedic’ treatise, Éléments de mathématique, provides an important
cultural touchstone for contemporary visions of mathematics as a totalised
system.2 The pseudonym for a group of world-leading French mathematicians working in the middle of the twentieth century, Bourbaki attempted
to create a definitive mathematical textbook from three foundational
structures. Bourbaki’s article ‘The Architecture of Mathematics’, often
considered a manifesto for the group, details three ‘great’ or
‘mother-­
structures’—topological, algebraic, and ordered structures—
which together encompass the entirety of mathematical activity and
1
G. H. Hardy, A Mathematician’s Apology (Cambridge: Cambridge University Press,
1992), p. 84.
2
Michael Harris, ‘Do Androids Prove Theorems in Their Sleep?’, in Circles Disturbed: The
Interplay of Mathematics and Narrative, ed. by Apostolos Doxiadis and Barry Mazur
(Princeton, New Jersey: Princeton University Press, 2012), pp. 130–82 (p. 156).
1
S. J. Taylor, Mathematics in Postmodern American Fiction,
Palgrave Studies in Literature, Science and Medicine,
https://doi.org/10.1007/978-3-031-48671-5_1
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2
S. J. TAYLOR
theory. While playing an important part in Bourbaki’s project to unify,
encircle, and totalise mathematics, these structures also reveal how encyclopedic narratives utilise the figurative efficacy of mathematics to challenge such epistemological exhaustion. By regarding the topological,
algebraic, and ordered structures of mathematics as modelling DeLillo,
Pynchon, and Wallace’s figurative strategies—respectively, of allusion,
metaphor, and cognition—three critical problems of the interplay between
mathematics and encyclopedic narratives can be addressed. As a self-reflexive figure, the Möbius strip is emblematic of postmodern literary exhaustion; by regarding the figure as a topological structure, Ratner’s Star can
be read as a postmodern system of allusion whereby mathematical and
literary intertexts are not exhausted but continually replenished. The difficulties of reading narratively inscribed mathematical symbolism, such as
the equations in Gravity’s Rainbow, are mitigated by understanding such
inscriptions in terms of algebraic structures within a mathematical model
of Pynchon’s metaphorical strategy. Exemplified by the novel’s use of endnotes, ordered structures clarify the specifically mathematical architecture
of Infinite Jest and illustrate how such structures emphasise to the reader
the cognition of conscious choices in negotiating hierarchies of narrative
containers. These three representative U.S. novels are, in chapters two,
three, and four, further contextualised through comparative analyses of,
respectively: topological themes and structures in the works of Lewis
Carroll and John Barth; algebraic structures in the ‘hard’ science fiction of
Catherine Asaro; and ordered structures in the narratives of the OuLiPian
mathematician-turned-author Jacques Roubaud. In so doing, my methodology enables a tracing of larger mathematical currents in the field of
literature.
The height of Bourbaki’s project parallels the rise of a wider structuralist imperative across the social sciences and humanities. The resulting
post-structuralist backlash provides a vital context for postmodern literature. At this point, the United States of America becomes something of a
crucible: the influence of Bourbaki was manifested in U.S. educational
reforms of New Math contemporaneously with Yale-School literary criticism’s deconstruction of the notion of a static, completed text. This shift
was accompanied by cultural responses to the novel that favoured New
Criticism’s narrative comprehension over traditional holisms—a movement away from the definitive Great American Novel (Moby-Dick or The
Making of Americans) to the more critical and reflexive encyclopedic novel
(Gravity’s Rainbow, Infinite Jest, Underworld). In 2004, Norman Mailer
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1
INTRODUCTION
3
articulated his belief that ‘[t]he Great American Novel is no longer writable’: any single work ‘can’t cover all of America now. It’s too detailed …
Now all the details get in the way of an expanse of a novel’.3 Mailer’s views
align with a critical consensus that ‘the encompassing American book has
proved an elusive entity’.4 Today ‘encyclopedic narrative’ has ‘replaced the
Great American Novel as the book about everything in the American
genre system’.5 Part of this displacement is due to the way details are
organised such that they don’t get in the way of what Mailer calls the
‘expanse of a novel’ yet simultaneously culminate in a work ‘less totalising’
than any single Great American Novel.6
While, for Edward Mendelson—one of the earliest scholars of the encyclopedic novel—the epistemophilic narrative inclusion of mathematics in
terms of a ‘technology or science’ was an acknowledged quality of the
form, the influence of Bourbaki’s uniquely ‘encyclopedic’ mathematical
architecture on literary encyclopedism has been underappreciated.7 To
recall the title of David Foster Wallace’s history of mathematics, the encyclopedic novel ‘about everything’ involves interacting with everything and
more—including highly abstract aspects of technical subjects. Bourbaki’s
mathematical structures, with their claim at an all-encompassing
3
Margo Hammond, ‘Norman Mailer on the Media and the Message’, Book Babes, 2004
http://poynter.org/2004/norman-mailer-on-the-media-and-the-message/20881/
[accessed 10 September 2023].
4
Luc Herman and Petrus van Ewijk, ‘Gravity’s Encyclopedia Revisited: The Illusion of a
Totalizing System in Gravity’s Rainbow’, English Studies, 90.2 (2009), 167–79, (p. 167).
5
Herman and van Ewijk, p. 168.
6
Ibid.
7
Edward Mendelson, ‘Encyclopedic Narrative: From Dante to Pynchon’, MLN, 91.6
(1976), 1267–75 (p. 1270). Indeed, while Mendelson’s term at least evokes comparison
with Bourbaki’s encyclopedism, Franco Moretti’s view of such narratives as a development of
a classical rhetorical category into ‘modern epic’ elides interdisciplinary appreciation. Though
such texts, as Moretti notes, draw attention to the ‘discrepancy between the totalizing will of
the epic and the subdivided reality of the modern world’, by considering this subdivision as
purely geographical, he neglects important distinctions and similarities between literature
and science—Franco Moretti, The Modern Epic: The World-System from Goethe to García
Márquez, trans. by Quintin Hoare (London: Verso, 1996), p. 5. Similarly, in Leo Bersani’s
reading of ‘encyclopedic fictions’, a monolithic view of the world’s ‘Real Text’ is made
redemptive only through a similarly monolithic literary surrogate. Bersani’s reading of the
‘redemptively dismissive encyclopedism’ of literature that does not deeply engage with science thus cannot consider texts such as Gravity’s Rainbow as encyclopedic and so cannot
adequately account for their interdisciplinary interests—Leo Bersani, The Culture of
Redemption (Cambridge: Harvard University Press, 1990), p. 198.
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4
S. J. TAYLOR
mathematics, are highly relevant to the project of literary encyclopedism.
Works by Don DeLillo, Thomas Pynchon, and David Foster Wallace—
particularly mathematically literate authors of complex encyclopedic fiction—provide important test-cases to make sense of the cultural
connections between mathematical structures in encyclopedic literature.
Bringing these interdisciplinary connections to light will show how ‘encyclopedic’ mathematics is used in the literary context of the encyclopedic
novel to, contradictiously (or ironically), resist the comprehensive delineation of Bourbaki’s project. As such, insofar as they can be considered encyclopedic, these authors’ works highlight the potential of combining
mathematics and fiction as a critical engagement with comprehensive
knowledge, while offering their readers a means of freedom within and
resistance to apparently totalising structures.8
1.1 Encyclopedic Architectures: Narrative
and Mathematics
Mathematics is implicated, albeit implicitly, in the cataclysmic architectural
demolition that concludes one of the earliest postmodern examples of the
encyclopedic novel from the United States, William Gaddis’s The
Recognitions (1955). In the final pages of the novel, musician Stanley is led
to the pipe organ’s keyboard in the church at Fenestrula. The priest warns
Stanley against playing ‘strane combinazioni di note [strange combinations of notes]’ as ‘La chiesa è così vecchua che le vibrazioni, capisce,
potrebbero essere pericolose [The church is so old that the vibrations, you
understand, could be dangerous]’.9 Presumably, the priest would permit
the consonant combinations of an octave, a perfect fifth or a perfect
fourth. These intervals, as Pythagoras found, were mathematically propor
2 3
2
tionate to the holy tetractys: , , and , respectively.10 Unfortunately,
1 2
3
8
Following Gödel, Barbara Herrnstein Smith and Arkady Plotnitsky argue that the question as to whether mathematical knowledge is different to other epistemological systems
(philosophy, literature; or generally between science and art) is ‘undecidable in general …
although decidable (and evidently decided) in specific situations’—Mathematics, Science, and
Postclassical Theory, ed. by Barbara Herrnstein Smith and Arkady Plotnitsky (London: Duke
University Press, 1997), p. 12.
9
William Gaddis, The Recognitions (London: Penguin, 1985), pp. 955–6.
10
Catherine Nolan, ‘Music Theory and Mathematics’, in The Cambridge History of Western
Music Theory, ed. by Thomas Christensen (Cambridge: Cambridge University Press, 2002),
pp. 272–304 (pp. 272–274).
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1
INTRODUCTION
5
Stanley ‘did not understand’ the priest’s words and so could not heed their
warning. The chord he plays is two notes ‘with three black keys between
them’—that is, F natural and B natural—with an unsightly mathematical
25
proportion of
, producing a tritone of diabolical dissonance.11 After
18
‘wringing that chord of the devil’s interval’ the church ‘walls quivered…
Everything moved, and even falling, soared in atonement’—ultimately the
building ‘collapse[d]’, killing Stanley.12 The ‘unstable’ mathematical ratio
demolishes the church, yet leaves what Stephen J. Burn describes as the
‘intricate architecture’ of Gaddis’s encyclopedic novel unscathed.13 In the
following survey of literary encyclopedic criticism, I will suggest that there
is a gap in appreciating the role of mathematics in the ‘architectures’ of
recent encyclopedic novels. This gap, I argue, can be filled by an interdisciplinary appreciation of the modern ‘Architecture of Mathematics’, a
structural science foregrounded in three exemplary postmodern U.S. encyclopedic novels: Don DeLillo’s Ratner’s Star, Thomas Pynchon’s Gravity’s
Rainbow, and David Foster Wallace’s Infinite Jest.14
In his preface to The Tragic Muse, Henry James criticises William
Makepeace Thackeray’s The Newcomes (1854–5), Alexandre Dumas’s The
Three Musketeers (1844), and Leo Tolstoy’s War and Peace (1869), for
their inclusion of the ‘waste’ of life’s ‘sundry things’—of ‘queer elements
of the accidental and the arbitrary’—that renders them ‘large loose baggy
monsters’.15 In contrast to such monsters, James pledges allegiance to narratives resembling a ‘mighty’ and ‘complete pictorial fusion’, analogous to
Tintoretto’s exemplary Crucifixion (1565). Here, all elements are infused
with meaning and vitality and, as such, collectively model the life that
11
‘List of Intervals’, Huygens-Fokker Foundation http://huygens-fokker.org/docs/intervals.html [accessed 10 September 2023].
12
William Gaddis, p. 956.
13
John Franceschina, Music Theory Through Musical Theatre: Putting It Together (Oxford:
Oxford University Press, 2015), p. 87; Stephen J. Burn, ‘The Collapse of Everything:
William Gaddis and the Encyclopedic Novel’, in Paper Empire: William Gaddis and the
World System, ed. by Joseph Tabbi and Rone Shavers (Tuscaloosa: University of Alabama
Press, 2007), pp. 46–62 (p. 48).
14
Don DeLillo, Ratner’s Star (London: Vintage, 1991) (first publ. in 1976); Thomas
Pynchon, Gravity’s Rainbow (London: Vintage, 2000) (first publ. in 1973); David Foster
Wallace, Infinite Jest (London: Abacus, 1997) (first publ. in 1996)—subsequently cited in
parentheses as RS, GR, and IJ, respectively.
15
Henry James, ‘Preface to “The Tragic Muse”’, in The Art of the Novel: Critical Prefaces
(London: University of Chicago Press, 2011), pp. 79–97 (p. 84).
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S. J. TAYLOR
‘count[s]’.16 While James’s ‘counting’ refers to meaningful value (or valuable meaning), its connotations of a mathematical process are evocative
and, for this study, important. Following James’s gesture towards literary
encyclopedism—a spectrum encompassing the monstrously diffuse to the
comprehensively fused—Northrop Frye’s anatomical criticism considers
the importance of mathematics to encyclopedic narrative forms.
In Frye’s reading, the manifestation of encyclopedic forms may be
regarded as a more abstract ‘mighty pictorial fusion’ of literature and
mathematics—that of the geometric circle.17 In his conclusion to Anatomy
of Criticism, Frye contemplates this encompassing interdisciplinary figure:
The arts might be more clearly understood if they were thought of as forming a circle, stretching from music through literature, painting and sculpture
to architecture, with mathematics, the missing art, occupying the vacant
place between architecture and music. The feeling that mathematics belongs
to science rather than art is largely due to the fact that mathematics is an art
that we know how to use. The difference between mathematics and literature on this point will be greatly reduced when criticism achieves its proper
form of the theory of the use of words.18
Thus, while Frye considers ‘the circle of learning’, his focus is less on
the ‘educational… encyclopaedic compilations of myth, folklore, and legend like those of Ovid’ and more on the encyclopedic circle: universal,
circular narratives which invoke a ‘total cyclical mythos’; where the Bible’s
‘gigantic cycle from creation to apocalypse’ is reproduced in the more
mundane ‘cycle of human life’.19
At a pure, abstract level, these universal circles, Frye suggests, ‘may be
an interlocking set of mathematical formulas’.
What this means is surely that pure mathematics exists in a mathematical
universe which is no longer a commentary on an outside world, but contains
that world within itself. Mathematics is at first a form of understanding an
James, pp. 84–85.
Etymologically from ἐγκύκλιος παιδεία [encyclical education], ‘the circle of arts and sciences considered by the Greeks as essential to a liberal education’, ‘encyclopaedia | encyclopedia, n.’ OED Online (Oxford University Press, June 2018) http://oed.com/view/
Entry/61848 [accessed 10 September 2023].
18
Northrop Frye, Anatomy of Criticism: Four Essays (Princeton, New Jersey: Princeton
University Press, 1971), p. 364.
19
Frye, p. 54; pp. 316–7.
16
17
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1
INTRODUCTION
7
objective world regarded as its content, but in the end it conceives of the
content as being itself mathematical in form, and when a conception of a
mathematical universe is reached, form and content become the same thing.
Mathematics relates itself indirectly to the common field of experience,
then, not to avoid it, but with the ultimate design of swallowing it…
[F]inally the physical or quantitative universe appears to be contained by
mathematics.20
For Frye, ‘the curious similarity in form … between the units of literature and of mathematics’, suggests a correspondence between the mathematical universe and an encyclopedic ‘literary or verbal universe’.21
Furthermore, if ‘in every age of literature there tends to be some kind of
central encyclopaedic form, which is normally a scripture or sacred book
in the mythical mode, and some “analogy of revelation,” as we called it, in
the other modes’ then, in the information age that emerged in the late-­
twentieth century, the contemporary form might well be found in the
abstract ‘scriptures’ of mathematics.22
While Frye offers intriguing avenues to discussing interdisciplinary
crossovers in encyclopedic writing, historically the vast majority of accounts
of the turn towards encyclopedism in contemporary literature tend to
begin with Edward Mendelson’s 1976 essay ‘Encyclopedic Narrative:
From Dante to Pynchon’. Mendelson aimed to describe ‘encyclopedic
narrative’ in order to appreciate the then recent publication of Thomas
Pynchon’s Gravity’s Rainbow in 1973. In developing a recognition of
itself as a singular delimited body, ‘[e]ach major national culture in the
west,’ writes Mendelson, delivers a single ‘encyclopedic author, one whose
work attends to the whole social and linguistic range of his nation, who
makes use of all the literary styles and conventions known to his countrymen, whose dialect often becomes established as the national language’.23
Yet Mendelson’s argument becomes problematic when he identifies two
Americans—Herman Melville and Pynchon—as encyclopedic authors.
Mendelson attempts to reconcile his system’s inconsistency by differentiating the scope of both writers. The plurality of American encyclopedic
authors is permissible, he says, because it highlights a late-twentieth century shift in American fiction: from a conception of America ‘as a separate
Frye, p. 352.
Ibid.
22
Frye, p. 315.
23
Mendelson, ‘Encyclopedic Narrative’, p. 1268, original emphasis.
20
21
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8
S. J. TAYLOR
entity’ to its current, dominant position in a transnational data network.24
For Mendelson, ‘Pynchon’s international scope, his attention to cartels
and communications-networks that ignore national boundaries’ support
his designation as ‘the encyclopedist of that newly-forming international
culture’.25 This culture can be distinguished by ‘its introduction of an
order based on information, of data, instead of the old order built on
money and commercial goods’.26
Mendelson’s criteria are highly restrictive, especially compared with
those presented in Robert Swigger’s earlier article. In ‘Fictional
Encyclopedism and the Cognitive Value of Literature’ (1975), Swigger
lists Broch, Borges, and Queneau as three exemplary producers of writing
that problematises the human or ‘cognitive’ valuation of knowledge
through encyclopedism. The works of these and other ‘encyclopedists’,
including ‘Pynchon, Pirsig, Barth, Vonnegut, Nabokov, Günther Grass,
Butor, Italo Calvino, and others’, are categorised by a ‘Rabelaisian gusto
of learning and expert elaboration’. For Swigger, then, ‘several modern
and contemporary works’ embody the encyclopedic ‘tendency … to
embrace everything there is, and to offer satisfactions, or, failing that,
appropriate displacements of the impatience for Erkenntnis [Knowledge]’.27
Clearly, Swigger’s ‘encyclopedists’ out-number Mendelson’s. That recent
accounts of the turn towards encyclopedism in contemporary literature
tend to begin with Mendelson indicate a stronger reaction to his restrictions than the inspiration to provide a more systematic account of Swigger’s
fictional encyclopedism. Hilary Clark is one such critic who notes that
differentiating Melville and Pynchon by their respectively national and
international scope and concerns, as Mendelson does, risks overlooking
their distinctive encyclopedic stylistic narrative traits. In her 1990 study,
The Fictional Encyclopaedia, Clark pursues a more exact study of such
traits. She first identifies the encyclopedic ‘impulse betrayed in a text when
it gathers and hoards bits of information and pieces of wisdom following
the logic of their conventional (metonymic) associations in the writer’s
Ibid.
Mendelson, ‘Encyclopedic Narrative’, p. 1271.
26
Mendelson, ‘Encyclopedic Narrative’, p. 1272.
27
Ronald T. Swigger, ‘Fictional Encyclopedism and the Cognitive Value of Literature’,
Comparative Literature Studies, 12.4 (1975), 351–66 (p. 353). Anna Sigridur Arnar reads
this problematisation as an impulse to continue critical discussions outside the text as part of
an encyclopedic dialogic exchange—Anna Sigridur Arnar, Encyclopedism from Pliny to Borges
(Chicago: University of Chicago Library, 1990).
24
25
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1
INTRODUCTION
9
culture’.28 Regarding Melville’s stylistic traits as encouraging ‘not so much
ironic as nostalgic’ effects, Clark argues that the inclusion and hoarding of
cetological trivia in Moby-Dick—as a substitute for Ahab’s physical capture
of the white whale—betray a ‘nostalgia arising from the encyclopaedist’s
despair over the impossibility of attaining to perfect knowledge’.29 By contrast, in his own ‘encyclopedic farrago’, Pynchon’s ‘magpie-instinct to collect facts’, particularly those inscribed in mathematical notation, results in
a reflexive critical irony that challenges the epistemological authority of
mathematics.30
With her conception of the fictional encyclopedia, Clark begins to
broaden Mendelson’s scope (in which Gravity’s Rainbow is one of only a
handful of ‘encyclopedic narrative[s]’) by returning to the root of the
word and its formal lineage. In so doing, analogies between mathematical
and literary encyclopedism are put into greater relief. The O.E.D. describes
how the original Greek term ἐγκύκλιος παιδεία (‘encyclical education’)
became corrupted into the compound ἐγκυκλοπαιδεία. This goes some
way to explain how the original sense of a dynamic process of the ‘circle of
learning’ as ‘a general course of instruction’ is more commonly associated
with its secondary sense of a ‘literary work containing extensive information on all branches of knowledge, usually arranged in alphabetical
order’.31 Clark argues that the static, didactical encyclopaedia (exemplified
in the Encyclopædia Britannica) ‘encircles’ knowledge—demarcating and
containing it within a totalised system—while the dynamic, dialogic encyclopedia (such as Diderot’s Encyclopédie) functions as an interface between
authors and readers, and so permits knowledge to grow outward like the
roots of a tree. This organic outward growth, in resistance to artificial
geometric corralling, is an important facet of the encyclopedic novel.32
28
Hilary Clark, The Fictional Encyclopaedia: Joyce, Pound, Sollers (London: Garland
Publishing, 1990), p. vii.
29
Clark, Fictional Encyclopaedia, p. 37. Clark also describes how the ‘architecture’ of the
fictional encyclopaedia is a structure in tension between a holistic, over-all coherence and the
‘episodic order of the oral epic’ where ‘Digressions and fragments are given as much emphasis as parts of the text contributing to a linear narrative. In the fictional encyclopaedia, then,
a thematic (or symbolic) or episodic order conflicts with and breaks into a linear or teleological order; the latter nonetheless functions as a base in the reader’s expectations’—Clark,
Fictional Encyclopaedia, pp. 43–44.
30
Frye, p. 311.
31
Clark, Fictional Encyclopaedia, pp. 17–18.
32
Hilary Clark, ‘Encyclopedic Discourse’, SubStance, 21:1.67 (1992), 95–110 (pp. 98–99).
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S. J. TAYLOR
For Clark, the two potential forms of encyclopedic discourse stem from
an impulse found in the Foucauldian archive. ‘Between the language
(langue) that defines the system of constructing possible sentences, and
the corpus that passively collects words that are spoken’, Foucault defines
what he terms ‘the archive’ as a level of
a practice that causes a multiplicity of statements to emerge as so many regular events, as so many things to be dealt with and manipulated. It does not
have the weight of tradition; and it does not constitute the library of all
libraries, outside time and place … It is the general system of the formation
and transformation of statements.33
Clark regards this archive or system as ‘another term for the encyclopedia’—particularly a ‘practice that “encircles,” encompasses, delimits
knowledge’. As a literary act, she finds the encyclopedia resonating in both
the ‘experience of difference and multiplicity’ in the sense of ‘vertically
divisible’ writing space. This space, for example, in the novels of Philippe
Sollers, constantly finds its limits pressed.34 For Clark, the encyclopedic
text ‘seeks order in the chaos of things to be known and said; it categorizes
and divides while amassing, excludes while including’.35 Observing the
encyclopedic text as ‘a special type of discourse’, she notes that it is not
limited to select ‘from a range of material’ but may plunder ‘from the
entire domain of human knowledge’.36
In considering mathematics in literary works, it is this conception of
encyclopedic discourse—as one evoking geometric encircling and an
‘impulse’ to collect and order both in a manner resembling the encyclopedic enterprise and one critical of any static totalisation of knowledge—that
I find more useful than recent moves away from the ‘encyclopedic’ narrative towards other terms like, ‘mega-novels’ or literary ‘maximalism’.37
Thus, while Levey has recently argued ‘that the encyclopedic label is not
wholly suitable for describing many of these texts because their fixations
33
Michel Foucault, The Archaeology of Knowledge and The Discourse on Language, trans. by
A. M. Sheridan Smith (New York: Pantheon Books, 1972), p. 130, original emphasis.
34
Clark, ‘Encyclopedic Discourse’, p. 97.
35
Clark, ‘Encyclopedic Discourse’, p. 96.
36
Clark, ‘Encyclopedic Discourse’, p. 98, original emphasis.
37
cf. Frederick R. Karl, American Fictions 1980–2000: Whose America Is It Anyway?
(Bloomington, Indiana: Xlibris, 2001), p. 155; Nick Levey, Maximalism in Contemporary
American Literature (London: Routledge, 2017).
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1 INTRODUCTION
11
are often on information and knowledge that resist the conferred significance that any fact included in an encyclopedia—being precisely the collection of privileged facts of the world—necessarily undergoes’ and
includes more trivial details to empower marginalised voices, this is what
Clark’s conception of encyclopedic discourse nevertheless achieves.38 In
both Levey’s maximalism and Clark’s encyclopedism ‘it is not what is
being represented that counts … so much as the activity of representing’
and ‘the implied process of its construction [is] one of its strongest effects,
rather than the content of its descriptions’.39 In this study, I will remain
faithful to the term ‘encyclopedic’ as it suggestively figures totalised
knowledge geometrically as a circle—a figure that also symbolises
Bourbaki’s hermetically sealed mathematics—which in encyclopedic novels is dismantled by literary techniques as modelled by mathematical structures. Ratner’s Star, Gravity’s Rainbow, and Infinite Jest all ‘demonstrate
an encyclopedic breadth, but one filtered by the politics of postmodernism
and poststructuralism’, and I aim to use mathematical structures to tease
out their figurative idiosyncrasies.40 In my conclusion, I will show how this
encyclopedic impulse (which Levey calls ‘maximalism’) ‘is in no way exclusive to big books’ (as Levey argues, ‘If anything, it is more of a phenomenological and epistemological standpoint… than a claim to physical
dimension’) by continuing to explore the strategies of DeLillo, Pynchon,
and Wallace in their (occasionally) shorter texts.41
Descriptive of a literary act rather than prescriptive of a condition for
writing, Clark’s understanding of encyclopedic discourse lifted the rather
arbitrary restrictions of Mendelson. From ‘that impulse betrayed in a text
when it gathers and hoards bits of information and pieces of wisdom following the logic of their conventional (metonymic) associations in the
writer’s culture’, encyclopedism manifests as an encyclopedic mode of discourse incorporated into the generic characteristics of the fictional encyclopedia.42 Following Clark, Jed Rasula notes that ‘the sort of narratives
associated with encyclopedism are the very ones most insistently cited for
their burlesque heterogeneity; and, inclining to pastiche’ they are
Levey, p. 9.
Levey, p. 3, original emphasis.
40
Levey, p. 9.
41
Ibid.
42
Clark, Fictional Encyclopaedia, p. v; pp. 4–5.
38
39
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12
S. J. TAYLOR
‘paradigmatically postmodern’.43 Stephen J. Burn also identifies generic
characteristics of this ‘vital postmodern form’ in the three long novels of
William Gaddis: The Recognitions, J R, and A Frolic of His Own.44
Burn also differentiates the modernist ‘kind of encyclopedia’ exemplified by Ulysses from the postmodern encyclopedic novel.45 As Joyce
acknowledged, the encyclopedic urban detail of Ulysses provides ‘a picture
of Dublin so complete that if the city one day suddenly disappeared from
the earth it could be reconstructed out of my book’.46 In contrast to the
reconstructive encyclopedism of Ulysses, Burn argues, is the deconstructive
impulse of Gaddis’s novel, which concludes with the architectural ‘collapse of the church at Fenestrula’. For Burn, this
symbolizes the crushing weight of information (expanding since the eighteenth century) because it is caused by an increase in the number of cultural
artifacts (the new composition by Stanley cannot be borne, significantly, by
the architecture), but also because the collapse arises from the inability to
process enough data.47
Thus, while the architecture of Ulysses functions as ‘a cultural storehouse
against apocalypse, Gaddis’s encyclopedic narratives typically confront
engulfment by culture at their climax. Supporting structures (churches,
business empires, sanity) collapse at the end, and culture overwhelms’.48
Burn’s evaluation of how the ‘intricate architecture’ of The Recognitions
both supports and is threatened by its ‘vast and layered erudition’ allows
us to appreciate the particularly postmodern characteristics of recent encyclopedic novels.49 It also allows us to consider the influence of another
encyclopedic architecture, Nicolas Bourbaki’s ‘Architecture of
43
Jed Rasula, ‘Textual Indigence in the Archive’, Postmodern Culture: An Electronic
Journal of Interdisciplinary Criticism, 9.3 (1999), 1–39 https://doi.org/10.1353/
pmc.1999.0022.
44
Burn, ‘The Collapse of Everything’, p. 51.
45
James Joyce, Selected Letters of James Joyce, ed. by Richard Ellmann (London: Faber,
1975), p. 271.
46
Joyce in Frank Budgen, James Joyce and the Making of Ulysses and Other Writings
(London: Oxford University Press, 1972), p. 69; Burn, ‘The Collapse of Everything’, p. 52.
47
Burn, ‘The Collapse of Everything’, p. 57.
48
Burn, ‘The Collapse of Everything’, p. 57. This overwhelming condition is an aspect of
‘total war’ described by Paul K. Saint-Amour—Paul K. Saint-Amour, Tense Future:
Modernism, Total War, Encyclopedic Form (Oxford: Oxford University Press, 2016),
pp. 7–10; pp. 214–218.
49
Burn, ‘The Collapse of Everything’, p. 48.
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1 INTRODUCTION
13
Mathematics’, and its three foundational ‘mother-structures’. Comparing
the relation between the encyclopedic architectures of mathematics and
narrative offers a corrective to the problem that, to date, none of the
accounts of the encyclopedic novel has fully considered the role of mathematics in such an epistemological project, especially with regard to the
encyclopedic narrative systems of mathematically literate authors such as
Don DeLillo, Thomas Pynchon, and David Foster Wallace.50 The
­encyclopedic works of these writers offer sites of interdisciplinary reflexivity that utilises mathematics to interrogate supposedly totalising systems of
knowledge and writing. From an understanding of mathematics in terms
of Bourbaki’s structures, we can regard topological, algebraic, and ordered
structures as models of figurative strategies in encyclopedic architectures.
1.2 Methodological Foundations
In their ‘critique of the admittedly nebulous Zeitgeist … “postmodernism”’, Sokal and Bricmont attack the ‘repeated abuse’ by thinkers and
writers in the arts and humanities ‘of concepts and terminology coming
from mathematics and physics’.51 To avoid opening the present study to
such charges of ‘abuse’ as they define it, I am committed to carefully considering the meaning, and justifying my use, of the scientific and mathematical terminology contained within. In addition to not abusing such
50
In more recent criticism, Luc Herman and Petrus van Ewijk return to evaluating
Gravity’s Rainbow as an illusory encyclopedia which ‘might stimulate a reassessment of the
imposed structure’ of totalising knowledge, while David Letzler investigates the encyclopedic function of David Foster Wallace’s endnotes in Infinite Jest—Herman and van Ewijk,
‘Gravity’s Encyclopedia Revisited’, p. 178; David Letzler, ‘Encyclopedic Novels and the
Cruft of Fiction: Infinite Jest’s Endnotes’, Studies in the Novel, 44.3 (2012), 304–24. As will
be discussed in the methodology section, I will approach the problematics of mathematics in
the encyclopedic novel by drawing on Tom LeClair’s systems theory, N. Katherine Hayles’s
‘field’ concept and nonlinear dynamics, and Gregory Bateson’s ecological conceptions of
mind, illustrating what Troy Strecker has suggested to be the efficacy of a combinative
approach between these critically variant methodologies of (or approaches to) encyclopedic
narrative—Troy Strecker, ‘Narrative Ecology and Encyclopedic Narrative’, in Avant-Post:
The Avant-Garde under “Post-” Conditions, ed. by Louis Armand (Prague: Litteraria
Pragensia, 2006), pp. 281–98 (p. 283). Strecker’s opening of the closed-system conception
of ecology, such as that of Karl Kroeber, suggests an analogous means of approaching the
encyclopedic novel—Karl Kroeber, Ecological Literary Criticism: Romantic Imagining and
the Biology of Mind (New York: Columbia University Press, 1995).
51
Alan Sokal and Jean Bricmont, Intellectual Impostures: Postmodern Philosophers’ Abuse of
Science (London: Profile Books, 1998), p. 4.
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14
S. J. TAYLOR
exotic terminology for its ‘intoxicat[ing]’ effect, the sober consideration
of mathematics in critical literary analysis necessitates the selection of an
appropriate methodology from the approaches available to literature and
science studies.52 The ensuing discussion will thus survey some different
approaches to analysing the relationship between literature and
mathematics.
Alice Jenkins has recently evaluated the efficacy of a number of
approaches available to literature and science scholars focusing on mathematics, including: One-Culture/two-way-traffic models; literature- or
mathematics-dominant two-culture models; historicist studies that emphasise actor’s categories; theoretical and philosophical incorporations; and
linguistic studies which combine several of the above.53 Perhaps the most
influential critic of the first of these approaches, Gillian Beer argued that
systemically inseparable relations between literature and science are not
binary but are those of ‘interchange rather than origins and transformation rather than translation’.54 In her inaugural and seminal contribution
to the field of literature and science studies, Darwin’s Plots (1983), Beer
argued that nineteenth-century discourse operated as a ‘two-way’ ‘traffic’
between the scientific and the more conventionally literary.55 This
Sokal and Bricmont, Intellectual Impostures, p. 4.
Alice Jenkins, ‘Mathematics’, in The Routledge Research Companion to Nineteenth-­
Century British Literature and Science, ed. by John Holmes and Sharon Ruston (London:
Routledge, 2017), pp. 217–34. Jenkins also considers a sixth category of quantitative statistical methodologies, exemplified by Moretti’s work.
54
Gillian Beer, ‘Translation or Transformation? The Relations of Literature and Science’,
Notes and Record of the Royal Society of London, 44.1 (1990), 81–99 (p. 81). As part of this
transformation, ‘texts which might seem “cultural context” to other fields’ are brought ‘into
literary critical attention’—Alice Jenkins, ‘Beyond Two Cultures: Science, Literature, and
Disciplinary Boundaries’, in The Oxford Handbook of Victorian Literary Culture (Oxford:
Oxford University Press, 2016), pp. 401–15, (p. 403).
55
Gillian Beer, Darwin’s Plots: Evolutionary Narrative in Darwin, George Eliot and
Nineteenth-Century Fiction, 3rd edn (Cambridge: Cambridge University Press, 2009), p. 5.
Prior to Beer, Aldous Huxley observed ‘the traffic of learning and understanding’ between
literature and science ‘flow[ing] in both directions’, while George Levine described this as an
appreciation that ‘to get to the heart of the culture one can travel the road of science, the
road of literature or—better—both’—Aldous Huxley, Literature and Science (London:
Chatto & Windus, 1963), p. 62; George Levine, ‘One Culture: Science and Literature’, in
One Culture: Essays in Science and Literature, ed. by George Levine and Alan Rauch
(Madison: University of Wisconsin Press, 1987), pp. 3–32, (p. 25).
52
53
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1 INTRODUCTION
15
conception can prove problematic, however, if the science in question is
mathematics which, in its pure form, is uniquely hermetical. Indeed,
mathematics is often seen as a ‘hindrance’ to studies of literature and science; Alice Jenkins argues that this could instead be considered a constructive disruption forcing a reconsideration of how mathematics can be
understood and presented in linguistic systems such as narrative.56 A pertinent reconsideration revolves around our understanding of
(post)structuralism. As will be seen, the modern hermetic conception of
mathematics owes much to the influence of the Bourbaki group, for whom
mathematics was wholly, purely isolate. Yet, in their reconstruction of
mathematics, Bourbaki became centred in structuralist movements that
swept across a variety of non-mathematical disciplines.57 Moreover,
Bourbaki’s influence on literature, initially and primarily through the
OuLiPo group who consciously emulated these mathematicians, suggests
that while the ‘traffic’ of discourse was not exactly ‘two-way’ it did in a
sense resonate harmonically—perhaps more akin to what Beer later identified as ‘curious crossplays’.58
Founded in 1960 by mathematically trained writers Raymond Queneau
and François Le Lionnais, the OuLiPo’s explicit engagement with
Bourbakian mathematics lends itself to mathematically dominant critical
56
Jenkins, ‘Beyond Two Cultures’, p. 414. Jenkins also acknowledges that ‘mapping direct
or indirect influence of one text on another is not always possible, and indeed is not always a
satisfactory approach. Literature and science studies often has recourse to less clear-cut explanations for textual analogies. Referring the connection to the zeitgeist won’t do…, and with
a fairly widespread shift away from the “one culture”, we have an opportunity now to investigate other historicist means of distinguishing between analogies in our own criticism that
are useful and productive, and those which are temporary or contingent. It is not enough to
say that we need to learn to be better historians, to become more determined in seeking,
adept at using, and scrupulous in judging evidence about connections between texts. While
we do need to do these things, we also need to go back a step in the process and develop
better understandings of analogies and similarities in our primary texts and the cultural field
surrounding them’—Jenkins, ‘Beyond Two Cultures’, p. 413.
57
Peter Caws, Structuralism: A Philosophy for the Human Sciences (Amherst, New York:
Humanity Books, 2000), pp. 14–16.
58
Beer, ‘Translation or Transformation?’, p. 81. OuLiPo is the familiar abbreviation of the
loose collective ‘Ouvroir de littérature potentielle’ [‘Workshop of Potential Literature’] discussed in greater detail below.
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S. J. TAYLOR
analyses.59 This approach has most recently been undertaken by Nina
Engelhardt, who observes that ‘[i]n cultural studies of mathematics the
two-way approach is not common’.60 Although Engelhardt does not
examine the OuLiPo, her method of analysing narrative fictions which
‘include… mathematics in an encyclopaedic attempt to present the
assumptions underlying Western culture’ produces a persuasive argument
for the fecundity considering cultural ‘interrelations’ of mathematical
practice and theory from ‘a literary perspective’—an approach that, while
utilising the tools of literary criticism, results in mathematics-dominant
readings.61 What Engelhardt describes as ‘interrelations’, Carlos Bovell
(following Jane Korey) calls ‘linkages’. Bovell’s combination of
mathematics and theology forms ‘an interdisciplinary dialogue’ which
­
looks ‘less for the most important concepts than for productive points of
linkage’ between mathematics and the humanities.62 In all these approaches,
Gillian Beer’s imperative must be maintained ‘The questioning of meaning in (and across) science and literature needs to be sustained without
seeking always reconciliation’.63
For Barbara Fisher, the solution to the particular difficulties of sustaining critical irreconciliation in literature and mathematics studies is found
by acknowledging a cultural distinction through literary-dominant
59
François Le Lionnais trained as a chemical engineer and developed as a mathematical
practitioner, historian, and artist: his interdisciplinary history of mathematics and culture, Les
Grands Courants de la pensée mathématique, was published in 1948, the same year that the
poet Raymond Queneau joined the Société mathématique de France—cf. Corinne François,
Les Fleurs Bleues: Raymond Queneau, 2nd edn (Paris: Bréal, 2000), p. 27; Raymond
Queneau, Stories and Remarks, trans. by Marc Lowenthal (Lincoln: University of Nebraska
Press, 2000), p. 151; Olivier Salon, ‘François Le Lionnais, Un Érudit Universel’, Images Des
Recherche Mathématiques: La Recherche Mathématique En Mots et En Images, 2009 http://
images.math.cnrs.fr/Francois-Le-Lionnais-un-erudit.html?lang=fr [accessed 10 September
2023]; Maurice Mashaal, Bourbaki: A Secret Society of Mathematicians (Providence, Rhode
Island: American Mathematical Society, 2006), p. 73; Jane Alison Hale, The Lyric Encyclopedia
of Raymond Queneau (Ann Arbor: University of Michigan Press, 1989).
60
Nina Engelhardt, Modernism, Fiction and Mathematics (Edinburgh: University of
Edinburgh Press, 2018), p. 12.
61
Engelhardt, Modernism, Fiction and Mathematics, pp. 19–20.
62
Carlos R. Bovell, Ideas at the Intersection of Mathematics, Philosophy and Theology
(Eugene, Oregon: Wipf & Stock, 2012), p. 6. Jane Korey, ‘Dartmouth College Mathematics
Across the Curriculum Evaluation Summary: Mathematics and Humanities Courses’, 2000.
https://math.dartmouth.edu/~matc/Evaluation/humeval.pdf
[accessed
10
September 2023].
63
Beer, ‘Translation or Transformation?’, p. 97.
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1 INTRODUCTION
17
analysis. In her 1997 collection of studies, Noble Numbers, Subtle Words,
Fisher proposes ‘that mathematical objects can be used as shaping instruments for the purpose of vivid storytelling’. Defining these objects, or
‘elements’, as aspects of ‘number, geometry, or mathematical abstraction’
Fisher illustrates how, within a literary text, they invariably ‘contribute to
the linguistic force of the work’ in which they are present. These elements
can be foregrounded in the text’s form as well as its content, able to be
‘used to formally shape the structure of a literary work’, or to ‘inform a
text in singular ways as agents and counter agents, simple devices or transcendent abstractions’.64 As such they can influence the transmission of
meaning from the framing of the narrative to the development of character and much in between. Though apparently distinguishing two ‘modes’
of generation containing these influential linguistic elements (as either
‘intuitive or ‘deliberate’) Fisher’s conception is more nuanced than an
instance of Intentional Fallacy. Her second mode, ‘where number, numerical series, and geometric figuration’ innovatively ‘contain and preserve
powerful emotion’, highlights the distinction between a passive incorporation of the mathematical and an active engagement with its elements.65 In
this active case, Fisher maintains, ‘the mathematical structuring functions
somewhat like a lead crucible that confines a radioactive substance’.66 In
her studies of Shakespeare, Milton, James, Borges, and Morrison, Fisher
has marked success in convincing the reader of a conception of ‘number,
algebraic letter-coding, geometric figuration, and mathematical abstractions as practical literary instruments’.67 Maintaining the linguistic-dominant figurations of these writers, though, Fisher ultimately subsumes their
active engagement with mathematical elements as ‘a natural aspect of language’ utilised to its highest literary, emotive potential.68
While mathematically dominant critical readings can restrict the flexibility of literary analysis, Fisher’s approach results in attention paid only to
‘the unpremeditated use of mathematics as a narrative tool … in the hands
64
Barbara M. Fisher, Noble Numbers, Subtle Words: The Art of Mathematics in the Science of
Storytelling (Madison: Fairleigh Dickinson University Press, 1997), p. 11.
65
Though not many narratives are actively explicit with their mathematical incorporations,
arguably any text of even the slightest degree of verisimilitude must commit to at least a passive incorporation: even singular and plural noun forms, say, depend on a world that distinguishes between the values 1 and 2.
66
Fisher, Noble Numbers, p. 12.
67
Fisher, Noble Numbers, p. 21.
68
Fisher, Noble Numbers, p. 13, original emphasis.
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18
S. J. TAYLOR
of a supremely gifted author’, such as Henry James, or Toni Morrison,
while ignoring ‘contemporary experimentation with number in literature’
as epitomised by the OuLiPo.69 A balance between these approaches is
found in historicist studies that consider actor’s categories of influence and
production. Helen Pycior’s 1984 article on nineteenth-century British
algebra specifically links ‘the symbolical algebra of Charles Babbage and
George Peacock to the nominalism of George Berkeley and Dugald
Stewart’.70 By arguing that these writers and thinkers were engaged in ‘a
discussion whose mathematical and philosophical elements were intimately and perhaps inseparably connected’, Pycior ‘illustrates the fertility
of suspending presentist distinctions between internal and external in the
pursuit of an understanding of earlier mathematical subcultures’.71 Mary
Poovey’s A History of Modern Fact undertakes a similar ‘double reading’—
a process which finds a dynamics balance between two distinct yet not
entirely separate cultures.72 Her methodology is a departure, both ‘from
intellectual historians in insisting that ideas cannot be separated from
modes of representation’ and ‘from Foucauldians in resisting any historical
account that privileges ruptures or focuses only on discourses’.73
While actor-dominant historicist studies offer a useful balance between
literary and mathematical practices, the fact that the two disciplines share
underlying ideas has been a staple of ‘the very long-established and productive field of the philosophy of mathematics’.74 This approach ‘has provided support for postmodernist and theoretical approaches’ to literary
and mathematical interconnections. Notably ‘literary theory has a long-­
standing interest in the relationship between mathematics and language,
particularly whether poststructuralist ways of understanding language also
apply to mathematical expressions’ as in the works of Derrida and Badiou.75
As Engelhardt has argued, ‘over the course of the twentieth century
Fisher, Noble Numbers, p. 13.
Helena M. Pycior, ‘Internalism, Externalism, and Beyond: 19th-Century British
Algebra’, Historia Mathematica, 11 (1984), 424–41 (p. 424).
71
Pycior, p. 428, p. 425.
72
Mary Poovey, A History of the Modern Fact: Problems of Knowledge in the Sciences of
Wealth and Society (London: University of Chicago Press, 1998), p. 22.
73
Poovey, p. 17.
74
Jenkins, p. 221. Cf. Richard Pettigrew, ‘Platonism and Aristotelianism in Mathematics’,
Philosophia Mathematica, 16 (2008), 310–32; John P. Burgess, ‘Mathematics and Bleak
House’, Philosophia Mathematica, 12 (2004), 37–53.
75
Jenkins, p. 221.
69
70
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1 INTRODUCTION
19
epistemological questions regarding mathematics and fiction yields to
ontological interests’—a related interdisciplinary transition that lead
Badiou to declare ‘mathematics = ontology’.76 The fundamentally cordial
relationship between mathematical and literary philosophical theories is
further emphasised by Vladimir Tasić. In Mathematics and the Roots of
Postmodern Thought (2001), Tasić aims at ‘reconstructing’ poststructuralism and deconstruction ‘from a mathematical point of view’.77 Viewing
these recent movements of critical practice through the ‘peculiar hybrid’
lens of mathematics (as part art, part science) Tasić presents postmodern
thought as a ‘deeply divided edifice’. This division results from the twin
influences (rooted in the history of mathematical philosophy) of (a)
romanticism on logical reductionism and (b) the formalistic rejection of
romantic humanism.78 Jeremy Gray’s Plato’s Ghost (2008) is another
important mathematical-­literary study that, like Tasić, considers literature
and mathematics as part of a shared cultural history. In the vein of Joan
Richards’s 1988 work on how mathematics became interwoven through
the ‘fabric’ of Victorian culture, Gray investigates the popular conception
of mathematics in the early twentieth century.79 In particular, Gray argues
that ‘the period from 1890 to 1930 saw mathematics go through a modernist transformation’.80 In doing so, Gray moves away from the consideration of ‘modernist’ as an object solely of artistic Modernism, emphasising
broader social, cultural, and philosophical interrelations. One site of such
interrelations is the analogy of ‘anxiety, a well-established theme in writing
about modernism’; Gray finds ‘that in mathematics, too, anxiety was a
growing presence’, as was an interest in the history of its own subject.81
Gray’s caveat warns against the claim ‘that the modernization of mathematics was part of a broader cultural push, animated by concurrent changes
in the arts’. Despite the fact that changes in mathematics ‘and the better-­
known artistic ones happened independently’, he does argue that ‘the
76
Engelhardt, Modernism, Fiction and Mathematics, p. 18; Alain Badiou, Being and Event,
trans. by Oliver Feltham (London: Continuum, 2005), p. 6.
77
Vladimir Tasić, Mathematics and the Roots of Postmodern Thought (Oxford: Oxford
University Press, 2001), p. 4.
78
Tasić, p. 5.
79
Joan L. Richards, Mathematical Visions: The Pursuit of Geometry in Victorian England
(Boston: Academic Press, 1988), p. 11.
80
Jeremy Gray, Plato’s Ghost: The Modernist Transformation of Mathematics (Oxford:
Princeton University Press, 2008), p. 1.
81
Gray, Plato’s Ghost, p. 4.
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20
S. J. TAYLOR
changes were similar in kind and were helped along by a growing diversity
and specialization in all walks of cultural and intellectual life’.82
With an equally philosophical concern, the essays collected by Barbara
Herrnstein Smith and Arkady Plotnitsky in Mathematics, Science, and
Postclassical Theory (1997) are united by their engagement with ‘quite
general but problematic concepts, notably knowledge, language, objectivity, truth, proof, reality, and representation’.83 The collection proceeds
from a ‘dissatisfaction of practitioners’ of ecology and neuroscience, for
example, ‘who have found that familiar or classical accounts of knowledge,
proof, truth, reality, and so forth do not cohere with empirical descriptions or mathematical [(that is, quantitative or statistical)] analyses’. Nor
can these accounts ‘capture the complex dynamic processes’ of which we
are made increasingly aware as investigative technologies improve observational powers.84 This dissatisfaction renders ‘postclassical theory’—an
umbrella term collecting the ideas more commonly known as ‘postmodernism’ and ‘deconstruction’—an important source of inspiration in navigating any new approaches to engaging with complexity. Smith and
Plotnitsky emphasise mathematics here to challenge its common role ‘as
an exception, prohibitive limit, or clear counter instance to the more radical reaches’ of postclassical theory: the postmodern subject may exist in a
Baudrillardian simulated hyperreality, yet two-and-two still makes four.85
Instead of being based on limitations, they argue that the interconnections
between mathematics and postclassical theory ‘are on the whole quite cordial and that, even where those relations are complex, they do not involve
any wholesale refutations or underminings in either direction’.86
In his own work, Arkady Plotnitsky takes an ‘asymmetrical’ approach to
Beer’s two-way model, arguing that while the ‘interactions between nonclassical thinking and mathematics and science … proceed in both
Gray, Plato’s Ghost, p. 14.
Mathematics, Science, and Postclassical Theory, ed. by Barbara Herrnstein Smith and
Arkady Plotnitsky (London: Duke University Press, 1997), p. 1, original emphasis.
84
Smith and Plotnitsky, p. 2.
85
Ibid.
86
Smith and Plotnitsky, p. 3. Following Smith and Plotnitsky, the essays in Apostolos
Doxiadis’s and Barry Mazur’s Circles Disturbed continue to pursue a theoretical approach,
most successful in Uri Margolin’s narratological taxonomy, discussed in greater detail below.
82
83
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1 INTRODUCTION
21
directions’, analysis must acknowledge its disciplinary dominant.87 Thus,
while his theoretical framework is dependent on Niels Bohr’s quantum-­
mechanical concepts of complementarity and general economy—which
model reality as non-totalising systems of relationships—Plotnitsky explicates his approach as ‘suited to the requirements of the humanities and
social sciences, rather than to those of the natural and exact sciences’.88
Nevertheless, Plotnitsky’s framework illustrates how even the ‘more complex’ products of mathematics and science can be considered ‘interactions … very much within the field of [his] study, the aim of which is to
develop a historico-theoretical framework capable of accounting for-­
complementary-­interactions of that type’.89 This complexity is compellingly utilised by Matthew Handelman, whose The Mathematical
Imagination is animated by ‘the productive tensions between mathematics and critical theory—as often competing but not necessarily opposed
ways of approaching the cultural problems of the present’ through a reading of ‘negative mathematics’ in the twentieth century.90
In another approach to accounting for such relational complexity, Brian
Rotman’s Signifying Zero (1987) probes particularly resonant interconnections between mathematics and literature through a linguistic focus
that balances historicist research with philosophical implications. In his
Foucauldian ‘archaeology’ of zero, Rotman investigated the extent to
which the mathematical term carried ‘contemporary intellectual or cultural charge’. Rotman’s investigation identifies ‘patterns of similitude,
homology, structural identity, parallelism, and the like between … mathematics, painting, money, and, to a lesser extent, written texts’. Significantly,
though, Rotman admits that these ‘signifying systems and codes’ are not
sufficiently distinct spaces housing epistemes of zero, but nevertheless their
87
Arkady Plotnitsky, The Knowable and the Unknowable: Modern Science, Nonclassical
Thought, and the “Two Cultures” (Ann-Arbor, MI: University of Michigan Press,
2002), p. xv.
88
Arkady Plotnitsky, Complementarity: Anti-Epistemology After Bohr and Derrida,
Durham, NC: Duke University Press, 1994), p. 13.
89
Ibid.
90
Matthew Handelman, The Mathematical Imagination: On the Origins and Promise of
Critical Theory (New York: Fordham University Press, 2019), p. 19.
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22
S. J. TAYLOR
important interconnections resonate around that mathematical term.91
Rotman’s investigation provides one route into probing how the ‘charge’
of mathematics can be traced in the encyclopedic novel’s ‘structural
identit[ies]’ between mathematical and narrative elements. In his later
semiotic analysis of mathematical activity, Mathematics as Sign, Rotman
takes a linguistic rather than historical approach and, in so doing, unites
formalist, intuitionist, and Platonist aspects of mathematics. While
acknowledging that such a unification and understanding may be beyond
the interests of any audience of mathematicians, Rotman’s semiotic model
permits an approach to textually inscribed signs and symbols of and as
mathematical activity. Thus, Rotman understands numbers and other such
mathematical objects as resulting ‘from an amalgam of two activities,
thinking (imagining actions) and scribbling (making ideal marks), which
are inseparable: mathematicians think about marks they themselves have
imagined into potential existence. In no sense can numbers be understood
to precede the signifiers that bear them; nor can the signifiers occur in
advance of the signs (the numbers) whose signifiers they are. Neither has
meaning without the other: they are coterminous, co-creative, and
cosignificant’.92 Rotman’s conception of mathematical activity as thinking/scribbling is a particularly useful model when considering the function of mathematical inscriptions in fictional narratives. Plotnitsky’s
historico-theoretical approach, on the other hand, is important for considering broader cross-cultural interactions, especially between highly abstract
mathematics and the postmodern, (post-)structuralist intellectual climate
in which DeLillo, Pynchon, and Wallace developed and published.
Situating my own analysis between Plotnitsky’s epistemological complementarity and Rotman’s mathematical semiosis thus enables a broad
91
Brian Rotman, Signifying Nothing: The Semiotics of Zero (London: Macmillan Press,
1987), pp. ix–x. Rotman is using the word ‘homology’ in its sociological or psychological
sense of structural resonance, not in its mathematical sense, which denotes a specific associative procedure on mathematical objects. For comparable approaches which blend historicist
and theoretical interests while maintaining a focus on language, cf. Steven Connors’s
‘Afterword’ and Matthew Wickman’s Literature After Euclid—Steven Connor, ‘Afterword’,
in The Victorian Supernatural, ed. by Nicola Brown, Carolyn Burdett, and Pamela
Thurschwell (Cambridge: Cambridge University Press, 2004), pp. 258–77; Matthew
Wickman, Literature After Euclid, Before Scott (Philadelphia: University of Pennsylvania
Press, 2016).
92
Brian Rotman, Mathematics as Sign: Writing, Imagining, Counting (Palo Alto, CA:
Stanford University Press, 2000), p. 39.
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1 INTRODUCTION
23
understanding of the various levels and degrees of mathematico-literary
engagement in key works of postmodern American novelists, specifically
DeLillo, Pynchon, and Wallace.
Such culturally sensitive emphasis on overt deployments of interdisciplinary resonating patterns is also found in the work of N. Katherine
Hayles. Better known for her celebrated contributions to posthumanist
theory, Hayles began her critical career outlining the cultural connections
between science and literature. Her debut monograph, The Cosmic Web:
Scientific Field Models and Literary Strategies in the Twentieth Century,
suggested that twentieth-century developments in the sciences and in literature proceeded along a similar, culturally embodied trajectory. The
Cosmic Web of Hayles’s title is a metaphor for the interdisciplinary concept
of field, which emerged and gained influence throughout the early-to-­
mid-twentieth century. Hayles brings together scientific and literary applications (united as cultural embodiments) of the field concept, with
particular emphasis on where its interconnected and self-referencing aspects
can be seen in twentieth-century culture. In doing so, like Beer, she highlights the two-way traffic of the field’s conceptual development:
a comprehensive picture of the field concept is much more likely to emerge
from the literature and from science viewed together than from either one
alone. In this sense the literature is as much an influence on the scientific
models as the models are on the literature, for both affect our understanding
of what the field concept means in its totality.93
The Cosmic Web’s argument for the field as a totalising organisational
principle was followed by its ‘sequel and complement’ Chaos Bound:
Orderly Disorder in Contemporary Literature and Science.94 In Chaos
Bound, Hayles charts the cultural pivot away from structural, ‘totalising
theories’, and towards disorder as seen in the complex systems of modern
physics, the fractal geometries of mathematics, and concurrent literary
postmodern experiments.95 As will be seen in my third chapter, Hayles’s
93
N. Katherine Hayles, The Cosmic Web: Scientific Field Models and Literary Strategies
in the Twentieth Century (London: Cornell University Press, 1984), p. 10.
94
N. Katherine Hayles, Chaos Bound: Orderly Disorder in Contemporary Literature and
Science (London: Cornell University Press, 1990), p. xiii.
95
Hayles, Chaos Bound, p. xii. With a dozen interdisciplinary collaborators, Hayles
expanded this project with her edited collection Chaos and Order: Complex Dynamics in
Literature and Science (London: University of Chicago Press, 1991).
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24
S. J. TAYLOR
consideration of such nonlinear, dynamical systems presents an important
critical entry-point to thinking about mathematics in Gravity’s Rainbow.
Analysing comparable developments of mathematics and the arts—in
theory, philosophy, practice, and cultural dissemination—also underpins
Robert Tubbs’s approach in Mathematics in Twentieth-Century Literature
and Art: Form, Content, Meaning, a significant contribution to analysing
mathematics in literature and science criticism. Tubbs identifies four categories of mathematically influenced artistic or literary creation: one
involves creative mathematical analysis of artistic works (for example,
Raymond Queneau’s use of geometric axioms to analyse literature, or
Troels Andersen’s examination of artist Kazimir Malevich’s squares);
another in which chance and randomness situate meeting points of artistic
and mathematical interest (e.g. Tristan Tzara’s Dadaist compositions from
strips of newspaper text); that in which ‘mathematical imagery, shapes,
forms, or methods’ were employed to express ‘highly nonmathematical
aesthetic ideals’ (poet Charles Bernstein’s use of mathematical symbols in
‘Erosion Control Area 2’, for example); and instances where mathematical
ideas provided creations with ‘innovative structures’ (as seen in John
Barth’s Möbius-strip narratives).96 Symbols and structures will be of particular interest to the present study as they both utilise a cultural-historical
approach to mathematics that identifies its appeal to literary concerns of
style and philosophical engagement. The artists and writers who engaged
with mathematics in the way Tubbs describes did so ‘to provide alternatives to the artistic ideals that had been dominant for a millennium’:
‘infused with mathematical content’ these innovations offer new avenues
to explore and understand literature and ‘related mathematical ideas’.97
The developing twentieth century saw a shift in public perceptions of
mathematics: not only were ‘mathematical ideas … widely accepted as
being relevant to our understanding of both the physical universe and our
place in it’, but also ‘Mathematical thinking was no longer the private
domain of mathematicians’—their ideas, ‘though not necessarily their
technical details’, becoming injected ‘into the daily discourse of artists and
intellectuals’.98 Tubbs’s study features an important comparison between
96
Robert Tubbs, Mathematics in Twentieth-Century Literature and Art: Content, Form,
Meaning (Baltimore: Johns Hopkins University Press, 2014), pp. ix–x.
97
Tubbs, p. xi.
98
Tubbs, pp. x–xi.
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1 INTRODUCTION
25
literature and mathematics in terms of Bourbaki. Each of Bourbaki’s
‘mother-structures’ is involved in Tubbs’s ‘examination of mathematical
concepts in twentieth-century literature’.99 Tubbs’s approach is particularly effective in showing the two ways, according to Nina Engelhardt, in
which a work of literature can engage with mathematics: by incorporating
mathematical elements into the narrative as part of ‘innovations in literary
form’, and by explicitly discussing ‘the place of maths in historical and
cultural contexts’.100 As is apparent from Tasić’s mathematical history of
postmodern theory, however, since the peak of poststructuralism in the
latter half of the twentieth century, these crossovers and interconnections
engage with different concerns of modernisation. Brian McHale famously
formulated this as a ‘shift of dominance from epistemology to ontology’.101
Three key U.S. fiction writers regularly associated with McHale’s ontologically dominant postmodernism and the encyclopedic novel—Don
DeLillo, Thomas Pynchon, and David Foster Wallace—also engaged with
mathematics in both of the intersectional ways Engelhardt identifies, inviting a consideration of their works through a cultural-historical appreciation of mathematics. The cultural-historical approach, which places
Bourbaki’s encyclopedic architecture of mathematics at the foundation of
late-twentieth-century thought, justifies the use of mathematical terminology in the following study of encyclopedic novels by DeLillo, Pynchon,
and Wallace. Drawing on Tubbs’s important interdisciplinary study, and
building upon interdisciplinary frameworks by Plotnitsky, Rotman, and
others, I will extend the discussion to postmodern American novels which
betray the influence of—and, I argue, can be elucidated by—Bourbakian
mathematics and related American educational reforms, from the New
Math to the New Criticism. The relevance of mathematical terminology
to this study is further validated by the fact that DeLillo, Pynchon, and
Wallace explicitly used mathematics in their encyclopedic works and had
varying, though significantly above-average, levels of understanding and
experience with mathematics.102
By dramatising the history, philosophy, and practice of mathematics in
encyclopedic narratives which—through mathematics’ ability to evoke,
Tubbs, pp. 38–9.
Engelhardt, Modernism, Fiction and Mathematics, p. 18.
101
Brian McHale, Constructing Postmodernism (Routledge, London, 1992) 8. McHale’s
original formulation was in his 1989 Postmodernist Fiction.
102
Sokal and Bricmont, p. 4.
99
100
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S. J. TAYLOR
articulate, and manipulate abstractions while constructing knowledge—
facilitate explorations of both epistemological concerns and what Brian
McHale has called postmodernism’s ‘ontological doubt’, their works are
particularly rich sites in which to explore the ‘embedding’ of mathematics
in contemporary culture.103 The protagonist of Don DeLillo’s Ratner’s
Star is a mathematical prodigy who joins a team of Nobel laureates trying
to decode an extra-terrestrial code at a top-secret research facility. The
plot’s picaresque comedy is given critical charge by DeLillo’s unique formal strategy. The novel consists of two parts ‘Adventures’ and ‘Reflections’,
nominations that explicate the major literary intertext of Lewis Carroll’s
Alice books. Chapter by chapter, ‘Adventures’ is structured on the history
of mathematics, from ancient Mesopotamian and Egyptian concepts of
number in the first chapter to Cantor’s set theory in the twelfth. The
cumulative effect brings into dialogue and confrontation ideas of space,
subjectivity, and knowledge in a self-reflexive system powered by both literary and mathematical elements. This system provides an intersection of
issues that are fundamental to our understanding of postmodernism and
the Cold War culture in which the novel was published. Similar intersectional attributes can be found in Thomas Pynchon’s quintessential postmodern novel, Gravity’s Rainbow. In its evocation of military technology,
the novel presents as ‘expert in ballistics, chemistry and some very advanced
mathematics’.104 In particular, the three mathematical inscriptions in
Gravity’s Rainbow have long been considered shorthand for both the novel’s imposing complexity and its encyclopedic quality.105 While foregrounding the mathematics underpinning the technologically facilitated
violence of the Second World War in this novel, Pynchon also suggests
processes of resistance: throughout Gravity’s Rainbow, his metaphors
resemble transformative mathematical operations. Influenced by both
DeLillo and Pynchon, David Foster Wallace was almost immediately
103
Brian McHale, ‘Modernist Reading, Post-Modern Text: The Case of Gravity’s
Rainbow’, Poetics Today, 1.1/2 (1979), 85–110 (p. 90); Engelhardt, Modernism, Fiction and
Mathematics, p. 28. Cf. Brian McHale, Postmodernist Fiction (London: Routledge,
2004), p. 10.
104
Mendelson, p. 1270.
105
Mendelson, ‘Encyclopedic Narrative’, p. 1270; Javaid Qazi, ‘Source Materials for
Thomas Pynchon’s Fiction: An Annotated Bibliography’, Pynchon Notes, 2 (1980), 7–19,
(p. 7); Stephen P. Schuber, ‘Textual Orbits/Orbiting Criticism: Deconstructing Gravity’s
Rainbow’, Pynchon Notes, 14 (1984), 65–74 (p. 65).
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1 INTRODUCTION
27
considered the ‘heir apparent’ to postmodern writing.106 His undergraduate training in mathematical logic and literary theory is evident in his first
novel The Broom of the System. He has described his magnum opus Infinite
Jest as ‘structured like something called a Sierpinski gasket, which is a very
primitive kind of pyramidal fractal’ and compared the novel’s end to the
geometry of ‘[c]ertain kind of parallel lines [that] are supposed to start
converging in such a way that an “end” can be projected by the reader
somewhere beyond the right frame’, while the novel contains a plethora of
mathematical topics and inventions (the novel’s tennis academy is structured as a cycloid, espouses a Cantorian tennis philosophy ‘mathematically
uncontrolled but humanly contained’, and hosts a teenage game whose
score is calculated by complex algorithms).107 Following the success of
Infinite Jest, Wallace was commissioned to write a biographical primer on
Georg Cantor’s set theory, Everything & More: a Compact History of
Infinity. Like DeLillo and Pynchon, Wallace’s writing and his passion for
mathematics collide and combine in encyclopedic narratives that simultaneously utilise mathematics to enhance their literary aesthetic and provide
a cultural site for, and means of, critical evaluation of mathematics in society. Later works by these authors—Zero K (2016), Bleeding Edge (2013),
The Pale King (2011, posthumous), respectively—continue to contemplate the developing abstraction of the individual subject in mathematical
algorithms of business and state.
106
Douglas Kennedy, ‘Oblivion: Stories by David Foster Wallace’, The Times (London, 24
July 2004) https://www.thetimes.co.uk/article/oblivion-stories-by-david-foster-wallace7qsvxdc5hxq [accessed 10 September 2023].
107
Michael Silverblatt, David Foster Wallace: Infinite Jest, Bookworm, KCRW (11th April
1996)
http://kcrw.com/news-culture/shows/bookworm/david-foster-infinite-jest
[accessed 10 September 2023]; D. T. Max, Every Love Story Is a Ghost Story: A Life of David
Foster Wallace (London: Granta, 2012), p. 319n19. In Euclidean geometry, which is bound
by (among other axioms) the parallel postulate, parallel lines do not converge; Wallace is
referring to the perspectival illusion of, for example, parallel railway lines that appear to meet
at the horizon’s disappearing point. Euclid’s mathematical theory of this optical phenomenon was developed by Vitruvius—whose work influenced many important architects including Leon Battista Albert—who describes perspective as ‘the correspondence of all lines to the
vanishing point, which is the centre of a circle’—Euclid, ‘The Optics of Euclid’, trans. by
Harry Edwin Burton, Journal of the Optical Society of America, 35.5 (1945), 357–72;
Vitruvius, On Architecture, trans. by Frank Granger, Loeb Classical Library (London:
William Heinemann Ltd, 1955), vol. i. bk. I.c.II, p. 27; The Mathematical Works of Leon
Battista Alberti, ed. by Kim Williams, Lionel March, and Stephen R. Wassell (Basel:
Birkhäuser, 2010), pp. 1–7.
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S. J. TAYLOR
By closely examining three test cases—Ratner’s Star, Gravity’s Rainbow,
and Infinite Jest—I will show how mathematics, considered as great structures which model literary techniques, is explicitly thematised and implicitly supportive in encyclopedic novel-systems. Moreover, as the final
chapter considers the legacy of such structures in their authors’ most
recent narratives, I will expand the temporal boundaries of the important
interdisciplinary works of Tasić and Tubbs to consider the intersection of
mathematics and literature from around 1960 to the present day. Not only
does this timeframe cover the rise and transformation of postmodern literature, which Tasić argues is intimately tied with mathematical philosophy, it also covers the rise and development of a particularly modern
conception of mathematics, supremely influenced by the Bourbaki group’s
project to unify mathematics. This study will use the unification of mathematics in terms of ‘mother-structures’ as a guiding thread through
neglected aspects of contemporary encyclopedic literature. My approach
roughly corresponds to identifying ‘architectonic patterns’ in the third of
‘six areas of significant contact or meaningful comparison’ between mathematics and narrative as identified by Uri Margolin—where ‘the use of
numerical or geometrical formulas, procedures, or patterns to determine
the composition or architecture of a narrative, and their occasional predominance over mimetic or thematic factors’.108 While Margolin’s methodology seeks intentional patterning—such as Dante’s law of three, or the
constraints of the OuLiPo—I shall use mathematical structures as
interpretative devices to better analyse mathematical patterning in
­
DeLillo’s, Pynchon’s, and Wallace’s encyclopedic architectures.109
108
The other approaches being: a literary portrayal of a mathematician; the use of a mathematical element as a key dramatic element; the use of mathematical notions, such as infinity,
as key thematic elements or situations; analogous fundamental concepts or conceptual issues;
mathematical concepts, models, and methods in theories of narrative—Uri Margolin,
‘Mathematics and Narrative: A Narratological Perspective’, in Circles Disturbed: The Interplay
of Mathematics and Narrative, ed. by Apostolos Doxiadis and Barry Mazur (Princeton, New
Jersey: Princeton University Press, 2012), pp. 463–84, emphasis added. Though the ‘architectonic’ approach is most appropriate for examining multiple, broadly consistent uses of
mathematics in the subjects of this study, all of Margolin’s interdisciplinary uses can be seen
in these encyclopedic novels. In Infinite Jest for example, James Incandenza’s supposedly
fictional theory of infinite annulation is geometrically inscribed in the novel, while his meta-­
artistic directorial projects are informed by a crossover of technical expertise and avant-garde
film theory. This inclusivity can be regarded as another feature of Wallace’s encyclopedic
impulse.
109
Margolin, pp. 465–466.
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1 INTRODUCTION
29
1.3 Mathematics: Foundational Crisis,
Structural Rebuild
In order to explore the ways in which DeLillo, Pynchon, and Wallace
approach and engage with the discipline in a manner that unravels its historical determinacy, it is first necessary to provide an overview of the recent
history of mathematics—a development symbolised by architectural
images. José Ferreirós and Jeremy J. Gray have noted that ‘architecture
has often been the source of analogies applied to science in general and
mathematics in particular’. Ferreirós and Gray note that Francis Bacon
and René Descartes used the ‘architectural metaphor’ to emphasise the
exemplary ‘foundations of mathematics: the totality of this science is
thereby compared to an edifice, with its higher and lower stages, and of
course sound foundations to support its weight’.110 These foundations
ruptured at the end of the nineteenth century, threatening the mathematical edifice with demolition, and mathematics’ supposedly smooth historical continuity, an accretive development from Pythagoras to Euler, was
fundamentally compromised.111 This architectural imagery aids our understanding of how DeLillo, Pynchon, and Wallace incorporate mathematics
into their own encyclopedic narrative constructions: their novels, which
both ape totalising systems and critique their totalitarian impulse, invite
comparison with the paradoxical state of mathematics—an exact discipline
and language in which the ‘grand book’ of the universe is written, yet one
in foundational ‘crisis’.112 The following survey will examine this foundational crisis, before tracing how, from its rubble, Bourbaki sought to
rebuild the architecture of mathematics and, in the process, initiated a
transatlantic structuralist movement in the shadow of which manifested
the postmodern poetics of DeLillo, Pynchon, and Wallace.
Rather than a single, singular event of the 1920s, as Ferreirós argues,
the ‘“crisis” was a long and global process, indistinguishable from the rise
of modern mathematics and the philosophical and methodological issues
110
José Ferreirós and Jeremy J. Gray, The Architecture of Modern Mathematics: Essays in
History and Philosophy (Oxford: Oxford University Press, 2006), pp. 2–3.
111
For an authoritative history, cf. Carl B. Boyer and Uta C. Merzbach, A History of
Mathematics, 2nd edn (New York: Wiley, 1989), pp. 3–613.
112
Galilei Galileo, ‘The Assayer’, in Discoveries and Opinions of Galileo, trans. by Stillman
Drake (London: Doubleday, 1957), pp. 231–80 (pp. 237–8).
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30
S. J. TAYLOR
it created’.113 The late-nineteenth century saw scrutiny placed on developments such as non-Euclidean geometry and complex analysis—including
Carl Friedrich Gauss’s curved geometries, Richard Dedekind’s algebraic
number theory, Bernhard Riemann’s complex analysis, Georg Cantor’s
transfinite set theory, and others—and their appropriate place within the
limits of mathematical activity. In particular, conventional mathematical
practices, exemplified by Leopold Kronecker, were challenged by
P.G.L. Dirichlet’s insistence ‘to put thoughts in the place of calculations’.114 This conceptual turn, illustrated by the achievements of Dedekind
and Riemann, was found ‘particularly shocking’ due to their advocation
‘of the view that mathematical theories ought not to be based upon formulas and calculations—they should always be based on clearly formulated general concepts, with analytical expressions or calculating devices
relegated to the further development of the theory’.115 In the 1880s,
oppositional stances were clarified, with Cantor’s set-theoretical proofs
becoming ‘quintessential examples of the modern methodology of existential proof’ and Kronecker’s public explication of his reactionary conservative beliefs through criticisms of Dedekind and Cantor.116
The modern methods of Cantor and Dedekind enjoyed growing esteem
towards the close of the eighteen-hundreds. Yet, while ‘the modern viewpoint in general, and logicism in particular, enjoyed great expansion’, the
early foundational ‘victory’ of this faction of mathematical foundationalism was soon undermined by the discoveries of ‘so-called logical paradoxes’.117 Cantor’s development of set theory in the late 1890s gave a
wholly original and useful conception of the continuum, and of how to
treat infinities mathematically.118 However, this exposed problems within
ordinal numbers and their resultant paradoxes. One particularly abrasive
José Ferreirós, ‘The Crisis in the Foundations of Mathematics’ in The Princeton
Companion to Mathematics, ed. by Timothy Gowers, June Barrow-Green, and Imre Leader
(Princeton University Press, 2008) 142–56, p. 142. For a fuller account of this ‘crisis’ in
context, cf. José Ferreirós, Labyrinth of Thought: A History of Set Theory and Its Role in
Modern Mathematics, 2nd Edition (Basel: Birkhäuser, 2007).
114
‘Gedanken an die Stelle der Rechnung zu setzen’—P.G. Lejeune Dirichlet, G. Lejeune
Dirichlet’s Werke (New York: Chelsea, 1969) vol. 2, p. 245, translated in Ferreirós,
Labyrinth, p. 28.
115
Ferreirós, ‘Crisis’, pp. 143–4.
116
Ferreirós, ‘Crisis’, p. 144.
117
Ferreirós, ‘Crisis’, pp. 144–5.
118
Boyer and Merzbach, pp. 630–2.
113
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1 INTRODUCTION
31
and tenacious semantic paradox dates back to Ancient Greece. A Cretan
named Epimenides woke from something of a hibernation to declare ‘All
Cretans are liars’, in essence saying ‘I am lying’. Was he telling the truth?
Today this paradox is perhaps more popularly formulated as the Barber
Paradox. Consider a barber ‘who shaves all those, and those only, who do
not shave themselves’. Does the barber shave himself? The answer can
only be a contradiction: if he does he doesn’t, if he doesn’t he does.
Perhaps the most famous of the paradoxes resulting from Cantor’s 1895
theory of sets (now called, rather unkindly, naïve set theory), this problem
of the impossible barber was articulated by Bertrand Russell and still carries his name in popular accounts. Cantor defined a mathematical set as an
‘aggregation [Zusammenfassung]’ of ‘definite, distinct objects of our perception or of our thought’ (these objects becoming ‘elements’ of the
set).119 In the examples above, then, Epimenides and the barber were
intended to be analogous to sets. Russell asks us to consider the set of all
sets that are not members of themselves (R). He points out that, if R is
not a member of itself, it must then contain itself. There then arises the
set-theoretical paradox: R cannot contain itself without also being a member of itself, which we have established it is not.120
By the twentieth century such paradoxes exacerbated existing frictions
that ultimately pitted logical and axiomatic methods against those of individual human intuition, leading to the foundational ‘crisis’ proper.121 In
1908 Ernst Zermelo attempted to circumvent such paradoxes by developing an axiomatisation of Cantor’s set theory.122 Yet, despite later improvements by, amongst others, Fraenkel (1921–22) and von Neumann (1925),
Zermelo’s axiomatic set theory is not demonstrably consistent. The anxiety
over foundations characterised the first three decades of twentieth-­century
mathematics. The response to this foundational crisis, what Jeremy Gray
119
‘“Unter einer Menges” verstehen wir jede Zusammenfassung M von bestimmten
wohlunterschiedenen Objecten m unsrer Anschauung oder unseres Denkens (welche die
‘Elemente’ von M gennant werden) zu einem Ganzen’—Georg Cantor, ‘Beiträge Zur
Begründung Der Transfiniten Mengenlehre’, Mathematische Annalen, 46.4 (481AD),
461–512 (p. 481), translated by Christian Voigt.
120
All three formulations are found in Bertrand Russell, The Philosophy of Logical Atomism
(London: Routledge, 1972), pp. 101–2.
121
Ferreirós, ‘Crisis’, pp. 142–3.
122
Morris Kline, Mathematical Thought from Ancient to Modern Times, 3 vols (Oxford:
Oxford University Press, 1972), iii, p. 1186.
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32
S. J. TAYLOR
sees as the development of ‘mathematical modernism’, is generally conceived in the coalescence of three competing groups of mathematicians
and philosophers.123 Later known as adherents to logicism, intuitionism,
and formalism, agents of these three groups believed that the consistent
foundations of mathematics were to be found by considering mathematics
as an activity of logic, intuition, and form, respectively, as will now be
summarised.
After Gottlob Frege aimed to ‘build mathematics as an extension of
logic’ (1879–1903), Bertrand Russell and Alfred North Whitehead’s
Principia Mathematica (1910–13) employed the symbolism of philosophical logic, which they believed preceded the axioms of mathematics.124
Critical of the arbitrary decisions taken by Russell and Whitehead—and of
the resultant obscure and incomplete system they considered Principia to
be—the intuitionists developed a radically alternative approach. Kronecker
found in Cantor’s work on the transfinite numbers and sets ‘not mathematics but mysticism’, and so declared Russell and Whitehead’s project to
be doomed from the start.125 Following Kronecker, in 1913 Henri
Poincaré wrote that ‘true mathematics, that which serves some useful purpose, may continue to develop according to its own principles’ and not
those of logic.126 These non-logical principles pertain to the intuitionist
appreciation of the mathematical continuum—that is, the set of real numbers—as conceivable only through self-evident operations of cognition
rather than a pre-existing logical construct. Following Poincaré, Brouwer
considers the continuum in terms of the passage of time. For Brouwer, the
intuitionist ‘considers the falling apart of moments of life into qualitatively
different parts, to be reunited only while remaining separate in time, to be
the fundamental phenomenon of the human intellect’. Thus, Brouwer
finds, from the flowing ‘apriority of time’, the fundamental intuition of
mathematics. This foundational intuition resides in the abstraction of the
idea of what he calls ‘bare two-oneness’ which ‘creates not only the numbers one and two, but also all finite ordinal numbers … [this] process may
be repeated indefinitely’ yielding new mathematical subjects. Moreover,
Gray, Plato’s Ghost, p. 4.
Kline, Mathematical Thought, p. 1191–3, original emphasis.
125
Kline, Mathematical Thought, p. 1197.
126
Henri Poincaré, ‘Science and Method’, in The Foundations of Science: Science and
Hypothesis, The Value of Science, and Science and Method (New York: The Science Press,
1913), pp. 358–546 (p. 480).
123
124
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1 INTRODUCTION
33
this process ‘gives rise immediately to the intuition of the linear continuum’. This mathematical continuum—which Brouwer calls ‘the
“between”’—is distinct from the classical time continuum from which life
moments fall apart. Brouwer’s mathematical continuum ‘is not exhaustible by the interposition of new units and therefore can never be thought
of as a mere collection of units’.127 Crucially, such an unending process
resulting from the inexhaustible ‘between’ cannot always be meaningfully
constructed by mathematicians. In such instances the possibility of a conclusive decision is denied.128
Distinct from the competing logistic and intuitive schools—which
regarded logic as, respectively, prior to and independent of mathematics—
is the early-twentieth-century approach to the foundations of mathematics
known as formalism, which deals with logic and mathematics simultaneously. Led by David Hilbert in the early 1900s, mathematical formalism
aimed ‘to provide a basis for the number system without using the theory
of sets and to establish the consistence of arithmetic’.129 In 1926, Hilbert
states that mathematics is ‘a stock of two kinds of formulas: first, those to
which the meaningful communications of finitary statements correspond;
and secondly, other formulas which signify nothing and which are the
ideal structures of our theory’.130 Hilbert continues to laud the benefits of
‘the logical calculus’—‘a symbolic language which can transform mathematical statements into formulas and express logical deduction by means
of formal procedures’. Just as in the ideal structural mathematical objects,
once ‘the signs and operation symbols of the logical calculus’ are regarded
‘in abstraction from their meaning’, the result is not ‘material mathematical knowledge which is communicated in ordinary language’, but ‘a set of
formulas containing mathematical and logical symbols which are generated successively, according to determinate rules’.131 Thus, in formalism
‘[a]ll signs and symbols of operation are freed from their significance with
respect to content’ and ‘all meaning is eliminated from the mathematical
127
L. E. J. Brouwer, ‘Intuitionism and Formalism’, trans. by Arnold Dresden, American
Mathematical Society Bulletin, 20 (1913), 83–96 (pp. 85–6).
128
Kline, Mathematical Thought, p. 1202.
129
Kline, Mathematical Thought, p. 1204.
130
David Hilbert, ‘On the Infinite’, in Philosophy of Mathematics: Selected Readings, ed. by
Paul Benacerraf and Hilary Putnam, 2nd edn (Cambridge: Cambridge University Press,
1983), pp. 183–201 (p. 196, original emphasis).
131
Hilbert, ‘On the Infinite’, p. 197.
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34
S. J. TAYLOR
symbols’.132 This means that mathematics is viewed systemically: it is ‘not
a subject about something, but a collection of formal systems, in each of
which formal expressions are obtained from others by formal transformations’.133 Between 1920 and 1930 the reach of the formalist approach
extended, through contributions by Ackermann, Bernays, and von
Neumann, into a Beweistheorie (‘proof theory’) or ‘metamathematics’ that
sought to establish the consistency of any formal system.
However, Hilbert’s confident belief that his proof theory ‘form[ed] the
necessary keystone of the doctrinal arch of axiomatics’ (that is, in the
structure of mathematics), redeeming it once and for all from paradoxes
was ultimately undermined not by subsequent contradiction, but by latent
metamathematical inadequacy.134 As this ‘keystone’ image suggests,
Hilbert resorted to architectural analogies to alleviate lingering foundational anxieties. In 1905, during one of his lectures on ‘The Logical
Principles of Mathematical Thinking’, he argued that mathematics did not
need Descartes’s firm foundations, but rather needed only ‘to secure as
soon as possible comfortable spaces to wander around’. This dynamic
environment is not anchored by an immovable, all-pervading bedrock;
indeed, it need only establish firm foundational support locally when ‘the
loose foundations are not able to sustain the expansion of the rooms’.135
Yet even this flexible construction could not be maintained. If the foundations of mathematics were in a state of crisis, by 1931 they were in one of
collapse, with Kurt Gödel’s proof that metamathematics was a logical
framework insufficient to establish the consistency of number theory.
More damningly, Gödel’s achievement, now known as his ‘incompleteness
theorems’, ‘proved that any system embracing number theory must contain an undecidable proposition’. Consequently ‘no system of axioms is
adequate to encompass, not only all of mathematics, but even any one
significant branch of mathematics, because any such axiom system is
incomplete’. This is the ‘death blow’ he dealt to ‘comprehensive
axiomatisation’.136
Despite Gödel’s ‘death blow’, mathematics continued to flourish under
the formalist ideology. Practically, this involved regarding formalism as
Kline, Mathematical Thought, p. 1204.
Kline, Mathematical Thought, p. 1205.
134
Hilbert, ‘On the Infinite’, p. 200.
135
David Hilbert, Logische Principien des mathematischen Denkens, quoted in Ferreirós and
Jeremy J. Gray, p. 3.
136
Kline, Mathematical Thought, p. 1207.
132
133
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1 INTRODUCTION
35
‘less a real faith than a Sunday refuge for those who spent their weekdays
working on mathematical objects as something very real’.137 This mid-­
century flourishing—where mathematical activity was pursued under
Platonism, while foundational epistemological issues, as and when they
arose, were countered with a more contemporary formalism—was particularly evident in the Bourbaki group. On the 10th of December 1934, a
few years after Gödel’s incompleteness theorems were published, a group
of young mathematicians gathered at the Parisian-Latin-Quarter Café
A. Capulade to plan a new university textbook on analysis. This group,
consisting of Henri Cartan, Claude Chavalley, Jean Delsarte, Jean
Dieudonné, and André Weil (all from l’École Normale Supérieure),
observed a requirement for an updated textbook that would attempt to
rebuild mathematics in the wake of Gödelian collapse. Operating under
the pseudonym Nicolas Bourbaki, they realised this would entail the articulation of a ‘collection of mathematical tools “as robust and universal as
possible”’. Moreover, in order to simplify and educate, they would need
to ‘determine the real substance of these tools and to present the most
general, and therefore universal, versions of them’.138 Ultimately their
efforts would inaugurate ‘a new vision of mathematics, a profound reorganization and clarification of its components, lucid terminology and notation, and a distinctive style’.139 With this rigorous reappraisal of
mathematics, Bourbaki gave the world ‘a modern way of teaching it and
even of doing it’.140 Within the recent history of mathematics, the case of
Nicolas Bourbaki offers both a cultural-historical source vital to our understanding of contemporary mathematics and, with their project to write a
‘encyclopedic textbook’ of modern mathematics, a model of considering
mathematical, encyclopedic, and literary interrelations.141
To some extent, logicism, intuitionism, and formalism each represented
what Alistair MacIntyre distinguishes as an ‘encyclopedic’ version of moral
enquiry, motivated by an ‘architectonic ordering of the sciences’. If
unchallenged, this version would result in a grand ‘Encyclopaedia…
displac[ing] the Bible as the canonical book, or set of books, of the
Ferreirós, ‘Crisis’, p. 155.
Mashaal, p. 9.
139
Mashaal, p. 2.
140
Mashaal, p. 4.
141
Harris, ‘Do Androids Dream’, p. 156.
137
138
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S. J. TAYLOR
culture’.142 Within this coalescence, Herbert Mehrtens identifies subtle
cooperative and competitive political tensions between formalism and
intuitionism, groups which he labels, respectively, ‘modernist’ and
‘counter-­modernist’ mathematical self-conceptions.143 While the former is
associated with a highly abstract, professionally exclusive internationalism,
counter-modernism’s concrete realism celebrates the common-sense of
populist nationalism manifest, at the extreme end, in the work of Ludwig
Bieberbach, ‘the infamous racist ideologue of a “Germanic” mathematics’.144 Thus, as Handelman notes, Horkheimer and Adorno ‘explicitly
implicated mathematics in the “horror” of Nazism that had stripped Jews
in Germany of their rights and citizenship’.145 Mehrtens concludes that,
while ‘the ideology of sound, realist common sense had crossed the border
into Germany in the early thirties’ as part of National Socialist counter-­
modernism, ‘mathematical modernism took the opposite route with a
group of scholars that were to name themselves “Bourbaki” and to become
the last high priests of a mathematical modernism before the post-modern
era’, a time at which the U.S. began to take over geographical dominance
of scientific enterprise.146 In this way, the existential stakes—of both mathematics and humanity—are understood as incredibly high, with Bourbaki’s
innovations aligned with collaborative rather than dominating international figurations.
Bourbaki’s presentation of mathematics in Eléments d’ Histoire des
Mathématiques ‘is characterised by uncompromising adherence to the axiomatic approach and to a starkly abstract and general form that portrays
clearly the logical structure’. In doing so they aimed ostensibly to amend
the problems they found in curricula following the foundational crisis,
hoping that their ‘emphasis on structure [would] effect a considerable
142
David Corfield, ‘Narrative and the Rationality of Mathematical Practice’, in Circles
Disturbed: The Interplay of Mathematics and Narrative, ed. by Apostolos Doxiadis and Barry
Mazur (Princeton, New Jersey: Princeton University Press, 2012), pp. 241–72, (p. 246);
Alastair MacIntyre, Three Rival Versions of Moral Enquiry: Encyclopedia, Genealogy, and
Tradition (Notre Dame, Indiana: University of Notre Dame Press, 1990), p. 19.
143
Herbert Mehrtens, ‘Modernism vs. Counter-Modernism, Nationalism vs.
Internationalism: Style and Politics in Mathematics 1900–1950’, in Mathematical Europe:
History, Myth, Identity, ed. by Catherine Goldstein, Jeremy Gray, and Jim Ritter (Paris:
Éditions de la Maison des sciences de l’homme, 1996), pp. 517–29.
144
Mehrtens, p. 519.
145
Handelman, p. 45.
146
Mehrtens, p. 527.
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1 INTRODUCTION
37
economy of thought’.147 Against the accretive conception of mathematics
and mathematical development, Eléments sought, in its ‘survey of all
worthwhile mathematics’, to build a contemporary account of mathematics from the rubble of foundational crisis.148 It also entailed the cooperation of a group of French mathematicians coming into prominence in the
wake of the First World War, as ‘almost half of the mathematics students’
who had matriculated at l’École Normale Supérieure ‘between 1911 and
1914 died’ in the conflict.149 In both senses of rebuilding, then, ‘it was
Bourbaki that saved French mathematics from extinction’.150 ‘The modesty of the word Eléments was misleading’, notes David Aubin, as ‘the
parallel with Euclid’s Elements reveal[ed] the extent of Bourbaki’s
ambition’.151 Its first volume was published in 1939. To this day, at over
seven thousand pages, it is still incomplete—such is the scope of its ambition.152 More than a work in progress, the Eléments is generally considered
‘a worldwide success’ which, along with the group’s idiosyncratic collective writing and organisational procedures, made Bourbaki ‘famous’.153
Bourbaki’s project and its influence have perpetuated an understanding
of mathematics as the most comprehensively rigorous and abstract, universalising technical discourse. The subject’s imperative for coherent unification is a model replicated by science in general. Yet, beyond their
scientific context, the metaphors which comprise Bourbaki’s vision have
regularly been used to describe the literary attributes of the encyclopedia.
Boyer and Merzbach, p. 706.
Ibid. Boyer includes Eléments in his own history’s general bibliography, though he sets
it aside from historiographic orthodoxy: Eléments is ‘Not a connected history but accounts
of certain aspects’ of mathematical thought, especially of modern times’—Boyer and
Merzbach, p. 716.
149
Mashaal, p. 44. The operation of the group under a singular pseudonym emphasises
‘the theme of the self-construction of the mathematician, the debate about personal and collective identity, about the We and the I of the mathematician’ Mehrtens finds in twentieth-­
century mathematical writing: ‘in opposition to the individual who constructs himself as part
of a higher order … be it God, Nature or Evolution’ stands the ‘modern, autonomous,
independent and “free” creator’, exemplified in Bourbaki’s collective ‘I’—Mehrtens,
pp. 525–6.
150
Jean Dieudonné, quoted in Mashaal, p. 45.
151
David Aubin, ‘The Withering Immortality of Nicolas Bourbaki: A Cultural Connector
at the Confluence of Mathematics, Structuralism, and the Oulipo in France’, Science in
Context, 10.2 (1997), 297–342 (p. 303).
152
The latest volume—Théories spectrales: Chapitres 3 à 5—was published by Springer
in 2023.
153
Mashaal, p. 10.
147
148
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38
S. J. TAYLOR
In Semiotics and the Philosophy of Language, Umberto Eco examines definition, and the connection between language and knowledge, in the literary systems of the dictionary and the encyclopedia. He notes that ‘the
current opposition “dictionary/encyclopedia” is traced back to the classical models of the tree and the labyrinth’.154 Like Eco’s linguistic analysis of
the encyclopedic as a ‘semantic concept’, conceptualisations of mathematics frequently resorted to metaphors of tree and labyrinth.155 Throughout
the early-to-mid-twentieth century, the tree metaphor typically illustrated
mathematical practice and its accumulating knowledge (see, for example,
Glenn James’s 1957 The Tree of Mathematics).156
Bourbaki sought alternative metaphors in their contribution to François
Le Lionnais’s 1948 The Great Currents of Mathematical Thought, ‘The
Architecture of Mathematics’. Here, Bourbaki asked whether contemporary mathematics is singular or pluralistic: ‘do we have today a mathematic
or do we have several mathematics?’ If the latter; what accounts for the
‘splintering’ of the field’; if the former, what unifies it.157 Bourbaki’s position would already have been clear to anyone familiar with their grand
project Eléments d’ Histoire des Mathématiques—where the title’s
‘Mathematique’ is ‘unusually for the French, singular, for this was how
[they] had come to see mathematics as a whole’.158 As it was for Hilbert
before them, for Bourbaki ‘The unity of mathematics was taken for
granted; and if it was not unified, then the goal was to strive for unity’.159
Mixing natural and architectural metaphors somewhat (considering a
developing yet unified mathematics as both an ‘organism’ and ‘Tower of
154
Umberto Eco, Semiotics and the Philosophy of Language (Bloomington, Indiana: Indiana
University Press, 1986), p. 2.
155
Eco, p. 85.
156
Felix Klein likens mathematics to ‘ein Baum [a tree]’ in Elementarmathematik Vom
Höheren Standpunkte Aus (Berlin: Springer-Verlag, 1968), i (p. 16) (first publ. in 1908).
Hermann Weyl passionately follows Klein, calling mathematics a ‘stolzen Baum [proud tree]’
in ‘Uber Die Definitionen Der Mathematischen Grundbegriffe’, in Gesammelte
Abhandlungen, ed. by K. Chandresekharan (Berlin: Springer-Verlag, 1968), ii (p. 304) (first
publ. in 1910).
157
Nicolas Bourbaki, ‘The Architecture of Mathematics’, trans. by Arnold Dresden, The
American Mathematical Monthly, 57.4 (1950), 221–32 (p. 221) (first publ. as ‘L’architecture
Des Mathématiques’, in Les Grands Courants de La Pensée Mathématique, ed. by François Le
Lionnais (Marseilles: Cahiers du sud, 1948), pp. 35–47). As David Aubin notes, this chapter
is readily taken to be ‘Bourbaki’s articulation of his own program’—Aubin, p. 305.
158
Aubin, p. 303.
159
Aubin, p. 305.
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1 INTRODUCTION
39
Babel’) Bourbaki aimed to establish this unified yet dynamic view of mathematics on fundamental arrangements which they called structures.
Probably more than any other investigation into Bourbaki, Leo Corry’s
has criticised the efficacy of their theory of structure within mathematical
practice. Instead, Corry identifies the real value of the concept of structure
in Bourbaki’s historiographic appreciation of mathematics that enables
them to consider a structural image of mathematics, rather than any philosophy, epistemology, or methodology underpinned by notions of structure.160 This issue of image over body develops the view of Giorgio Israel,
who also considers structure to be an ideological concept of Bourbaki.161
Yet Bourbaki’s structural image was a vital tool in their rebuilding of
mathematics. Through structures, Bourbaki were able to better develop
Hilbert’s architectural metaphor. With a structurally imagined formalism,
Bourbaki managed to remain ‘[m]ore faithful’ than Hilbert ‘to the complex panorama of twentieth century mathematics’—including the ability
to actually do mathematics in spite of foundational anxieties—by conceiving of ‘mathematics as a polis’:162
a big city, whose outlying districts and suburbs encroach incessantly, and in
a somewhat chaotic manner, on the surrounding country, while the center
is rebuilt from time to time, each time in accordance with a more clearly
conceived plan and a more majestic order, tearing down the old sections
with their labyrinths of alleys, and projecting towards the periphery new
avenues, more direct, broader and more commodious.163
Just as the ‘encyclopedia is built up’, this architectural mathematics is
constructed as a network of structuralist ‘buildings’.164 This metaphor
emphasises the efficiency of the modern urban environment, envisioning
architectural innovation on prime real estate. As Peter Galison writes,
Bourbaki presents here ‘a simile of high modernity, Bourbaki as
Haussmann—that mid-nineteenth-century urban bulldozer who tore
160
Leo Corry, Modern Algebra and the Rise of Mathematical Structures (Basel: Birkhäuser,
2004), p. 302.
161
Giorgio Israel, ‘Un Aspetto Ideologico Della Matematica Contemporanea: Il
“Bourbakismo”’, in Matematica e Fisica: Struttura e Ideologia, ed. by E. Donini, A. Rossi,
and T. Tonietti (Bari: De Donato, 1977), pp. 35–70.
162
Ferreirós and Gray, p. 4.
163
Bourbaki, ‘Architecture’, p. 230.
164
Saint-Amour, pp. 175–176; Caws, p. 5. Caws understands this building as both a ‘structure’ and ‘a set of relations among entities that form the elements of a system’—Caws, p. 13.
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40
S. J. TAYLOR
through the ramified ancient neighborhoods of Paris, sending open, radial
avenues out from the core’. Yet, rather than a metaphor of open-system
expansion, Bourbaki’s city is subject to the abstract laws that render mathematics as a ‘human-made object’, a grand architecture in which ‘there is
no universally acknowledged time ordering’ such as that implied by the
growth of a city. The story of mathematics Bourbaki represented with the
city metaphor was a paradoxically ‘non-narrative narrative, the account
outside time, a structure, an architecture to be contemplated as it ordered
“mathematic” from set theory on out’.165 As Galison notes:
True, a building must stand the second floor on the first, the first on the
foundation; but the completed edifice itself stands as a whole, not as a temporally developed sequence… As a partially realized vision of mathematics,
[Bourbaki’s] is a picture of a narrative outside time, a structure of structures
voided not only of the physicality of objects but even of the specific, purely
mathematical referentiality of mathematical entities. Here was supposed to
be relations of relations to be contemplated out of time and out of space.166
Despite their images of city and architecture—which somewhat resemble contemporary digital sites of ‘postmodern, information-based technoculture’167—the world of Bourbaki’s Éléments comes to be appreciated, by
critics such as Galison, as ‘fiercely impersonal, voided of heuristics, stripped
of images’.168 The ‘Architecture of Mathematics’ suggests that, in the
words of Pierre Cartier, ‘The Bourbaki were Puritans’ while revealing the
totalitarian vision of ‘the particularly, peculiarly Bourbakian narrative of
modernity’.169 With it, Bourbaki seeks to construct a new mathematics
differing (explicitly) from the bewildering routes of the ‘labyrinth’ and
(implicitly) from the entangled roots of the ‘tree’, yet purified of the
humanistic and organic qualities such metaphors suggested. Obversely,
opponents to this vision could often be identified by their use of the
orthodox metaphor. In his cutting attack on the New Math, Morris Kline
165
Peter Galison, ‘Structure of Crystal, Bucket of Dust’, in Circles Disturbed: The Interplay
of Mathematics and Narrative, ed. by Apostolos Doxiadis and Barry Mazur (Princeton, NJ:
Princeton University Press, 2012), pp. 52–78 (p. 53).
166
Galison, p. 57.
167
Alan Clinton, ‘Conspiracy of Commodities: Postmodern Encyclopedic Narrative and
Crowdedness’, Rhizomes: Cultural Studies in Emerging Knowledge, 5 (2002), ¶¶ 1–28 (¶ 3).
168
Galison, p. 72.
169
Galison, p. 85; p. 66.
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1 INTRODUCTION
41
sought to preserve the ‘trunk’ of the ‘great tree’ of mathematics from the
destruction advocated in Bourbaki’s manifesto.170 As these metaphorical
connections suggest, mathematical discourse comes to form an important
part of our understanding of the structural strategies of encyclopedic literature. Bourbaki’s encyclopedism views mathematics simultaneously as
an enclosing city and as an eternal, crystallised architecture; following
Bourbaki’s interdisciplinary influence—from structuralism to poststructuralism—the incorporation of mathematics in the postmodern encyclopedic novel enables an artistic evocation of totalising hermeticism and a
challenge to totalitarian epistemologies. Thus, our understanding of these
novels as ‘sizeable demarcation[s] within a much larger whole [whose]
structure[s are] only a means to achieve the idea of a neat totality’ is
enhanced by investigating related structural concerns of mathematics.171
1.4 (Post)Structuralism and Postmodernism:
Bourbaki to OuLiPo
Following the end of the Second World War—in a globe reconfigured by
Yalta and fungoid clouds of Little Boy and Fat Man—Bourbaki’s ‘formalist style dripped down into undergraduate teaching and even reached kindergarten, with preschool texts on set theory’—a program of New Math
which illustrates the particular influence of Bourbaki’s mathematics on
late-twentieth-century American thought.172 Bourbaki’s ‘modern way’ of
what Kline calls ‘Mathematising’—its emphasis on set theory and fundamental structures—was the basis of the New Math, a convenient, yet
unstable shorthand for diverse reforms of ‘new mathematics curricula’.
New Math spread throughout the West during the middle of the twentieth century, when the Occident’s industrial, scientific, and cultural developments contributed to significant global reorientations of power. The
wave of reforms was further provoked by competition, in all these arenas,
with the Soviet Union, a developmental antagonism that rapidly accelerated after the 1957 launch of Sputnik. Worried about falling behind in
these developments, Western nations—particularly the United States,
where New Math would be most energetically pursued—emphasised the
170
Morris Kline, Why Johnny Can’t Add: The Failure of the New Math (New York: Vintage
Books, 1973), pp. 91–92.
171
Herman and van Ewijk, p. 178.
172
Reuben Hersh, What Is Mathematics Really? (London: Vintage, 1998), p. 164.
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42
S. J. TAYLOR
necessity of training new citizens from a rigorous mathematical foundation. These technically adept citizens would benefit from policies that promoted a ‘solid knowledge of mathematics’ as ‘necessary for all scientists
and engineers’—whether they were employed as such or served in a kind
of home-front reserve unit of technicians. Thus, ‘The West’ is more than
a convenient shorthand—the coincidence of the reforming contexts and
development of New Math occurred relatively concurrently in France and
in the United States. Bizarrely, however, this turn towards applicability
went against Bourbaki’s abstracting impulse. As James Gleick writes,
Bourbaki’s argument that ‘Mathematics should be pure, formal, and austere’ was not ‘strictly a French development’:
In the United States, too, mathematicians were pulling away from the
demands of the physical sciences as firmly as artists and writers were pulling
away from the demands of popular taste. A hermetic sensibility prevailed.
Mathematicians’ subjects became self-contained; their method became formally axiomatic. A mathematician could take pride in saying that his work
explained nothing in the world or in science. Much good came of this attitude, and mathematicians treasured it… With self-containment
came clarity.173
The explicit political stakes driving this mathematical reform had a very
different effect in the humanities. This was particularly conspicuous in the
United States, as Mark Walhout has argued: ‘American inability to comprehend ambiguity’, or geopolitical uncertainty, during the Cold War
exacerbated an anxiety of impotence despite the country enjoying physical—military, technological, and scientific—supremacy.174 In this climate,
the New Criticism became the primary academic response to literature.
Bearing striking terminological parallels with the New Math, the New
Criticism is today considered ‘the poetics of the Cold War’.175 Whereas the
developments in mathematical pedagogy were part of training and mobilising scientifically capable and active citizens, the New Critical approach is
regularly viewed as a withdrawal from the world outside the work of art.
For Terry Eagleton, this insulation (the close reading of a closed text)
James Gleick, Chaos: Making a New Science (New York: Viking, 1987), p. 89.
Mark Walhout, ‘The New Criticism and the Crisis of American Liberalism: The Poetics
of the Cold War’, College English, 49.8 (1987), 861–71 (p. 866).
175
Walhout, p. 871.
173
174
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1 INTRODUCTION
43
‘meant committing yourself to nothing’.176 However, others saw in the
New Criticism a potential antidote to contextually determined readings.
Lionel Trilling advocated that, instead of reading with terminology like
idea, with its cold, detached New Critical ‘abstractness’, we should appreciate literary ideas as ‘not those of mathematics or of symbolic logic, but
only such ideas as can arouse and traditionally have aroused the feelings—
the ideas, for example, of men’s relation to one another and the world’.177
Although comprehensive comparison is beyond the scope of this study, it
is useful to note that these two major pedagogical reform movements—
New Math and New Criticism—were distinct (though arguably not fundamentally opposed) responses to the geopolitical situation from the
1950s onwards.178 At the very least their comparable responses to structuralism support the case for situating the observation of cross-disciplinary
pollinations between mathematics and literature in U.S. culture.
The most influential U.S. reform group for mathematical pedagogy was
the School Mathematics Study Group (SMSG). Under the leadership of
mathematician Edward Begle, this group of professional mathematicians
and educators received the largest share of federal funding and ‘effectively
created the “official” version of new math’ in America.179 Integral to its
initial success was Begle’s belief that mathematical understanding is fundamental to ‘intelligent citizenship’.180 Thus, in ‘defining and shaping
national characters’—where the Cold War provoked a distinction between
the ‘American self’ and the ‘Soviet personality’—Begle directed the
demands that ‘elite mathematical practices’ become considered ‘a desirable component of American intellectual training’.181 This renewed fervour
176
Terry Eagleton, Literary Theory (Minneapolis: University of Minnesota Press,
1983), p. 50.
177
Lionel Trilling, The Liberal Imagination (Garden City, New York: Doubleday, 1957),
pp. 277–8.
178
In another case of nominal parallelism, the title of Morris Kline’s attack on what he
regarded as the impenetrably hopeless New Math Why Johnny Can’t Add is an allusion to
Rudolf Flesch’s Why Johnny Can’t Read which advocated a return to simple phonetics, ‘the
alphabetic method’, in developing literacy—Rudolf Flesch, Why Johnny Can’t Read – And
What You Can Do About It (New York: Harper & Bros., 1955), p. 8.
179
Christopher J. Phillips, The New Math: A Political History (Chicago: University of
Chicago Press, 2014), p. 2.
180
E. G. Begle, ‘The School Mathematics Study Group’, NASSP Bulletin, 43 (1959),
26–31 (p. 28).
181
Phillips, p. 13.
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44
S. J. TAYLOR
of mathematical-intellectual training occurred concurrently with the belief
that secondary school mathematics at the time ‘was worn out and poorly
adapted to the modern demands of economics, technology, science and
culture’.182 As Povey and Adams have recently summarised, such pedagogical initiatives are not ideologically neutral, but rather are a recurring
aspect of the instrumentalising understanding of mathematics as ‘a “motor
of progress”’, or ‘the exploitation of the earth to generate the never-­
ending accumulation of capital’, through ‘the making of certain types of
people, those who have technological skills and can contribute to economic growth’ that simultaneously ‘de-sensitises us to human and ethical
values and erases our connection with the “other”’.183 In the late twentieth century, SMSG pursued such a (re)‘making’ of citizenry through an
incorporation of traditional, yet outdated, mathematical ideas into a new
‘“total picture” of mathematics’ that emphasised its ‘basic unity’ while
foregrounding its characteristic structure.184 As Christopher Phillips writes
Structure was meant both in the sense that mathematics was the study of
well-defined objects and properties and in the sense that mathematics was a
highly systematic discipline, more concerned with methods of reasoning
than with any particular set of objects or facts. These structures and systems
distinguished SMSG’s math for “intelligent citizenship” from math as an
exercise in rote memorization.185
The proposed reformation would incorporate a new, modern view of
mathematics as a unified democratic subject, a ‘universal language’ which,
by emphasising concepts and relations, ‘required no cultural prerequisites’. School curricula informed by a desire to use ‘“modern” math for
the shaping of “modern” students’ emphasised the perceived need ‘to
prepare citizens for modern society, for a world of complex challenges,
seemingly rapid technological changes, and unforeseeable future conflicts’.186 By giving American (and Western) students a foundation in
Mashaal, p. 138.
Hilary Povey & Gill Adams, ‘Disordering mathematics, citizenship and socio-political
research in mathematics education amongst the “rubble of words”’, Research in Mathematics
Education (2021) 23:3, pp. 306–322.
184
Phillips, p. 14.
185
Ibid.
186
Phillips, p. 15.
182
183
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1 INTRODUCTION
45
reason and free-thinking, the theory was they would be given the intellectual advantage over the perceived closed, retrograde, and monolithic
mindset thought to be the ‘Soviet personality’.187 Less abstractly, from
such citizens the technological and military developments of the West
would accelerate and outstrip those of the East. It was believed (and legislated), then, that the Cold War would be won with a totalising understanding of a modern unified mathematics fostered from childhood: as a
result, a whole generation—specifically that which was tellingly named in
a manner evoking a mathematical expression for an unknown value,
Generation X—was ‘subjected to the New Math’.188
Bourbaki’s work was a foundation, not only for the structural educational changes of New Math; more broadly it initiated a cross-disciplinary
movement towards structuralist concerns throughout the ‘soft’ sciences,
most clearly evident in Claude Lévi-Strauss’s anthropology and Jean
Piaget’s conception of developmental psychology. From Bourbaki’s focus
on the three fundamental structures—topology, algebra, order—Jean
Piaget ‘decided that children’s mathematical ideas are built from the same
three elements’. This structuralist coincidence Hersh calls a ‘Bourbaki-­
Piaget’ philosophy.189 Piaget’s theory resembles Ludwig Wittgenstein’s
1953 discussion of a child learning a rule, wherein ‘the practice of ostension [e.g. learning “cat” by someone pointing to a picture who says “kh.
ah. tuh…. khaht.”] presupposes abilities on the part of both teacher and
learner. They must already be able to recognize the sorts of things being
ostended’.190 In the arts, structuralism was advanced in linguistics by
Ferdinand de Saussure and Roman Jakobson, and developed through the
works of Roland Barthes, Michel Foucault, Jacques Derrida.191 Adjacent
to these critical-theoretical developments, yet not entirely independently,
a particularly Bourbakian brand of literary structuralism led to the
Phillips, p. 12.
David Foster Wallace, Everything and More: A Compact History of Infinity (London:
Phoenix, 2005), p. 239n23 (first publ. by W.W. Norton (2003)). Subsequent citations will
appear, in parentheses, in main body as ‘EM’.
189
Hersh, p. 226.
190
Stewart Shapiro, Thinking About Mathematics: The Philosophy of Mathematics (Oxford:
Oxford University Press, 2011), p. 276.
191
Cf. François Dosse, History of Structuralism, trans. by Deborah Glassman, 2 vols
(Minneapolis: University of Minnesota Press, 1997), i, pp. 23–24, 75, 82–3, 149–150, 219,
243–244.
187
188
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S. J. TAYLOR
foundation of the literary group OuLiPo.192 Indeed, that the ‘fifties and
sixties were the years of structuralism’ is largely due to the prominence of
Bourbaki and their emphasis on mathematical structures.193
In the years following the Second World War, then, there arose a complex and mutually supporting dynamic between mathematics and conceptions of structures in the sciences and humanities. The belief around this
time that ‘mathematics is everywhere’, that it ‘is an essential part of everyone’s academic and cultural knowledge’, seems to be ‘based partially on
the fashion of structuralism’. After Bourbaki’s influential fashioning, then,
mathematics was regarded as the epitome of structural science par excellence—the yardstick of all structural investigations of the world and daily
life.194 In 1997, David Aubin probed further into ‘the intersection of three
arenas’ that resonate with Bourbaki’s name: ‘mathematics; the structuralist and postmodernist discourses; and so-called potential literature’. In
doing so, Aubin argues that Bourbaki ‘acted as a cultural connector’.195
Moreover, as will be explored in greater detail, ‘the notion of structure
always remained problematic for Bourbaki’s enterprise, yet at the same
time central to his discourse’.196 Aubin finds ‘the seed of a lasting cultural
connection’, or a ‘cross-breeding’, between anthropology, linguistics, and
mathematics in a ‘fortuitous encounter’: ‘the intersection of Lévi-Strauss,
Jakobson, and Weil, in New York in 1943 … helped make structuralism
possible’.197 That Bourbaki’s structuralism is fundamental to structural
developments in the humanities, from Levi-Strauss’s Elementary Structures
of Kinship through to the New Critics presents multiple lines of interdisciplinary inquiry. Where necessary I will invoke this critical lineage in isolated readings. Beyond the scope of this study, however, such invocations
may allow us to better understand Bourbaki’s passive (post)structuralist
influence on contemporary authors, notably Derrida’s influence on
Wallace, and how Piaget’s Bourbaki-flavoured developmental cognitive
theories may be comparable to Bourbaki’s structures in child-centred narratives from Carroll’s Alice’s Adventures in Wonderland, through DeLillo’s
Ratner’s Star to more recent works such as Helen DeWitt’s The Last
Mashaal, p. 73.
Mashaal, p. 84.
194
Mashaal, p. 139.
195
Aubin, p. 300.
196
Aubin, p. 306.
197
Aubin, p. 311.
192
193
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1 INTRODUCTION
47
Samurai. What is specifically pertinent to the present study, though, are
the consequences of how mathematics was taught following Bourbaki.
Corry considered that, while the Bourbaki manifesto, ‘Architecture of
Mathematics’, articulates ‘the picture of mathematics as a hierarchy of
structures’, this picture ‘is nothing but a convenient schematic sketch’—a
non-formal totalising image of mathematics.198 However, this picture
helps galvanise a vision of mathematics that experienced its ‘golden age’ in
the 1960s.199 It would become so influential as to ‘spread to the world of
mathematicians and then to that of higher education; from there, it spread
to high school teachers who proposed using it to revolutionize secondary
math education’, the movement seen above as New Math.200
Perhaps the most culturally important structuralist offspring of
Bourbaki’s architectural mathematical image is the OuLiPo, an experimental literary group that directly ‘models itself on Bourbaki’.201 Its leading figure, Raymond Queneau, became involved after having acrimoniously
left the Surrealists. From their inception OuLiPo were interested in pushing ‘the overlap between, or intersection of, mathematics and poetry’.
Influenced by Queneau, the group ‘undertook a vast programme of investigation into the formal devices used by writers over the centuries (“analytic OuLiPo”) and into the literary potential of patterns that could be
cannibalised from formal languages such as mathematics, logic, computer
science, and—why not?—chess (“synthetic OuLiPo”)’.202 Having mathematicians such as Jacques Roubaud and Claude Berge within its ranks,
OuLiPo also benefited from direct contact with Bourbaki, such as
Queneau’s attendance at the 1962 Bourbaki conference. Queneau explicated the identical outlook of both groups with regards to mathematical
or literary creation in his ‘The Foundations of Literature According to
David Hilbert’, where points, lines, and planes operate in literature as
words, sentences, and paragraphs. As Queneau’s title suggests, this continues a conversation with Hilbert on the subject of creativity in literature
and mathematics. The story goes that, in Göttingen, a mathematician quit
Corry, Modern Algebra, p. 334n101.
Mashaal, p. 130.
200
Mashaal, p. 145.
201
Alison James, Constraining Chance: Georges Perec and the Oulipo (Evanston, Illinois:
Northwestern University Press, 2009), p. 124.
202
David Bellos, Georges Perec: A Life in Words, Rev. ed. (London: Harvill Press, 1999),
pp. 348–9.
198
199
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48
S. J. TAYLOR
to become a novelist, which Hilbert explained as ‘He did not have enough
imagination for mathematics, but he had enough for novels’.203 This is
important to consider as it helps us avoid cultural stereotyping: both of
mathematicians as anthropomorphised calculating machines and of literary artists as capricious fops aloof from logical discipline. Between Bourbaki
and OuLiPo, as representatives of mathematics and literature, imaginative
creativity was certainly a common ground. The Bourbakian imprint can be
seen clearly on perhaps the most famous Oulipian novel—George Perec’s
Life a User’s Manual. In the original French the title reads La Vie mode
d’emploi—an overt nod to the fact that each book in Bourbaki’s Élements
included a ‘user’s manual’ supplement titled mode d’emploi du traité.204
Above all, the two groups were united in ‘humor, taste for secrecy, and use
of structures [i.e. constraints]’.205 For Alison James, Bourbaki gave OuLiPo
‘a partial justification for the arbitrariness of the literary constraint’.
Considering mathematics as a Bourbakian ‘serious game’, Oulipian ‘writing can operate as a game of constraints and yet remain rigorous in its
practice’. The tension between play and procedure prompts James to consider OuLiPo’s aesthetic as a ‘paradoxical poetics of necessity grounded in
arbitrariness’.206
As can be seen from the OuLiPo’s literary application of Bourbakian
mathematics, Bourbaki’s mathematical structuralism developed in the tradition of G. H. Hardy. In his A Mathematician’s Apology (1940), Hardy
famously wrote that:
A mathematician, like a painter or a poet, is a maker of patterns … The
mathematician’s patterns, like the painter’s or the poet’s must be beautiful;
the ideas like the colours or the words, must fit together in a harmonious
way. Beauty is the first test: there is no permanent place in the world for ugly
mathematics.207
Constance Reid, Hilbert-Courant (New York: Springer-Verlag, 1986), p. 175.
Anne-Sandrine Paumier and David Aubin, ‘Polycephalic Euclid? Collective Practices in
Bourbaki’s History of Mathematics’, in Historiography of Mathematics in the 19th and 20th
Centuries, ed. by Volker R. Remmert, Martina Schneider, and Henrik Kragh Sørensen
(Basel: Birkhäuser, 2016), pp. 186–218 (p. 192).
205
Mashaal, p. 73.
206
James, p. 125. As will be seen in Chap. 2, a key example of the Oulipian constraint, S+7,
uses a mathematical operation to probe the assembly of mechanical structures.
207
G. H. Hardy, A Mathematician’s Apology (…) paragraph 10.
203
204
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1 INTRODUCTION
49
With their application of the axiomatic method through three ‘mother-­
structures’, ‘Bourbaki paints a picture of a mathematical universe’ organised as a ‘hierarchy of structures progressing from simple to complex and
from general to specific’.208 Following the early-twentieth-century crisis
within mathematics, it is important to emphasise that ‘Bourbaki believed
that set theory was the foundation for all mathematics’.209 This, above all
else, ensures that Bourbaki’s structuralism is consonant with the contemporary practice of mathematics, where set theory ‘is taken to be the ultimate court of appeal for existence questions’. This is due to the theory’s
comprehension: the ‘set-theoretic hierarchy is so big that just about any
structure can be modelled or exemplified there’, thereby settling ontological questions over mathematical objects.210 This all-encompassing potential of Bourbaki’s set-theoretic architecture, composed of three fundamental
structures—topological, algebraic, and ordered—has implications for literature beyond the explicit use of Bourbakian mathematics by the OuLiPo.
1.5 Mathematical Structures
and Encyclopedic Narrative
Bourbaki’s three great structures, upon which their encyclopedic mathematical project is built, present a useful critical approach to considering
the role of mathematics in postmodern fiction, particularly the encyclopedic novels of U.S. authors prominent in the latter half of the twentieth
century. By using topological, algebraic, and ordered structures as models
of allusion, metaphor, and cognition, we can better understand the role of
mathematics in the encyclopedic narratives of Don DeLillo, Thomas
Pynchon, and David Foster Wallace, respectively.
Bourbaki describes how topological structures ‘provide an abstract
mathematical formulation of the intuitive concepts of neighborhoods, limits, and continuity’.211 These topological structures (also known as topologies) describe a set via arrangements of elements ‘called points, in analogy
Mashaal, p. 79.
Mashaal, p. 52.
210
Shapiro, p. 288.
211
Mashaal, p. 79. Leo Corry that ‘the history of the development of topology, at least
from 1935 to 1955, cannot be told without considering in detail the role played in it by both
Bourbaki as a group and its individual members’—Corry, p. 295.
208
209
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50
S. J. TAYLOR
with the points of space in geometry’.212 They allow the study of ‘the
properties of geometric or abstract mathematical objects that remain
unchanged when the object is deformed continuously, without pinching
or tearing it’. Thus ‘the concept of continuity (or, more precisely, that of
a continuous function) is the cornerstone of this branch of mathematics’.
213
Topological structures present a valuable approach to Don DeLillo’s
Ratner’s Star, a novel that Tom LeClair has shown to be systematically
constructed upon the history of mathematics and the mathematical fiction
of Lewis Carroll. In Chap. 2, I will explore how topological structures
model DeLillo’s allusive engagement with these interdisciplinary intertexts in Ratner’s Star, emphasising a kind of never-ending Möbian relationship between source and primary text. The Möbius strip is regularly
considered a figure—and, since John Barth, an exhaustive figure—
emblematic of postmodern fiction. Instead, my reading considers the
Möbius strip as a topological structure, in the Bourbakian sense, and thus
supports an exploration of Ratner’s Star’s postmodern system of allusion
whereby intertexts (mathematical and literary) are not exhausted but continually replenished.214
While topological structures can be conceived with analogies to space,
algebraic structures are less intuitive. We most readily understand the term
‘algebra’ as denoting the equations we had to solve in school. More generally, however, ‘algebra deals with the basic operations of addition and multiplication’, and any structure using such basic associative operations to
combine two things to obtain a third is algebraic.215 Algebraic structures,
then, involve an associative rule connecting ‘any pair of elements with a
third element’.216 As I will demonstrate in Chap. 3, such algebraic structures model Thomas Pynchon’s metaphorical strategies in Gravity’s
Rainbow. Drawing on N. Katherine Hayles’s interdisciplinary approach to
Mashaal, p. 90.
Mashaal, p. 88.
214
Alan Clinton’s Lefebvrean reading of the literary encyclopedias explores how ‘contemporary encyclopedic narratives enact an aesthetic of crowdedness that relates to their situation in the era of late capitalism’, specifically how such texts ‘address issues of space in relation
to subjectivity and commodification’—Clinton, 4–5. In my reading of Ratner’s Star I will
suggest that DeLillo overcomes Clinton’s ‘crowdedness’ with topological structures as models for allusion.
215
Tubbs, p. 39.
216
Mashaal, p. 79.
212
213
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1 INTRODUCTION
51
the scientific ‘field’ and nonlinear dynamics, these structures emphasise
Pynchon’s resistance to linear determinism in favour of nonlinear potential. Re-engaging with Gravity’s Rainbow’s problematic partial differential
equation, associated with the dynamics of the V-2 rocket’s flight, in terms
of algebraic structures, however, it becomes clear that Pynchon’s mathematical inscription illustrates the generative potential of his metaphors: a
close analysis of the algebraic structure itself suggests a process of resisting
determinant linear trajectories, a process which complements the rendering of lines throughout his fiction. Through the resistant process to which
the equation strongly alludes—a recurvation and revolution against a
deterministic trajectory—I will argue that Pynchon’s algebraic structures
can maintain multiple, not necessarily straight nor simple, lines of interpretation, yielding a complicated interpretative plurality which cannot be
collapsed into simple allegory.
Part of Pynchon’s influence on subsequent postmodernist or post-­
postmodernist writers, including David Foster Wallace, can be seen in
their narrative deployment of mathematical equations. While algebraic
notation features throughout Wallace’s encyclopedic writing, he also consistently utilises hierarchies of narrative containers that resemble ordered
structures. Ordered structures utilise ‘common tools for comparison like
“greater than or equal to” and “less than or equal to”’ for the purposes of
ordering or comparing all or some elements of a set.217 In his work on
mathematical infinity, Georg Cantor famously used an ordered structure—
his Diagonal Proof—to distinguish positive rational numbers from their
magnitude and thereby show that ‘different sizes of infinity’ exist and can
be mathematically compared.218 It was Cantor’s work in this area that
established set theory’s authority, and from which foundation Bourbaki
constructed their structural image of mathematics. In Chap. 4, I will detail
how Cantor’s achievements were repurposed for their aesthetic qualities
by Wallace, in whose popular mathematics book Everything & More: A
Compact History of Infinity, I argue, narrative analogues of ordered structures are presented as hierarchies of containment. Re-engaging with this
central, yet critically undervalued text in Wallace’s oeuvre, through an
analysis of his ordered structures, allows a re-evaluation of the role of
ordered structures in his most celebrated work Infinite Jest, suggesting
217
218
Mashaal, p. 82.
Tubbs, pp. 58–61.
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52
S. J. TAYLOR
that—in both texts—ordered structures model Wallace’s system of cognition as it represents consciousness. Infinite Jest is generally considered to
be a ‘mathematical novel’, but aside from general conceptions of the novel
as fractally structured—a conception that Wallace perpetuated in
­interviews—criticism has been generally unclear in exposing how mathematics influences his overtly hierarchical text, complexly split, as it is,
between main body and a peritextual appendix of endnotes.219 This anatomical hierarchy—superior body, inferior appendix—resembles the mathematical ordering strategies of sets and subsets. With reference to Gregory
Bateson’s ecological conceptions of mind, I will show how, by moving
from the corporeal to the abstract, Wallace’s hierarchies become more
representative of thought patterns: both internal, in terms of personal
choice, and external, communicating and connecting outside the cranial
container. By considering maintext-endnote relations in terms of mathematical ordered structures we not only gain a better understanding of how
Infinite Jest emphasises the importance of choice in negotiating the hierarchies of text, but can also reconcile how ordered structures (and the
important role they play in Cantor’s mathematics of infinity) offer a
through-line between Infinite Jest and Everything and More, and so reposition the latter more centrally within Wallace’s oeuvre. The concluding
chapter will consider the ongoing legacy of mathematical structures, both
within the later works of DeLillo, Pynchon, and Wallace, as well as beyond
them in representative works by other authors of integrating mathematics
and fiction, and the growing relevance of the developing interdisciplinary
field of literature and mathematics studies to contemporary cultural issues.
219
Roberto Natalini, ‘David Foster Wallace and the Mathematics of Infinity’, in A
Companion to David Foster Wallace Studies, ed. by Marshall Boswell and Stephen J. Burn
(London: Palgrave Macmillan, 2013), pp. 43–57 (p. 46).
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