Literary Infinities: Number and Narrative in Modern Fiction 9781501331466, 9781501331480, 9781501331459

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Literary Infinities

Literary Infinities Number and Narrative in Modern Fiction Baylee Brits

BLOOMSBURY ACADEMIC Bloomsbury Publishing Inc 1385 Broadway, New York, NY 10018, USA 50 Bedford Square, London, WC1B 3DP, UK BLOOMSBURY, BLOOMSBURY ACADEMIC and the Diana logo are trademarks of Bloomsbury Publishing Plc First published 2018 Paperback edition first published 2019 Copyright © Baylee Brits, 2018 For legal purposes the Acknowledgments on p. ix constitute an extension of this copyright page. Cover design: Daniel Benneworth-Gray Cover image © Hugo Black Muecke, 2017 All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage or retrieval system, without prior permission in writing from the publishers. Bloomsbury Publishing Inc does not have any control over, or responsibility for, any third-party websites referred to or in this book. All internet addresses given in this book were correct at the time of going to press. The author and publisher regret any inconvenience caused if addresses have changed or sites have ceased to exist, but can accept no responsibility for any such changes. Library of Congress Cataloging-in-Publication Data Names: Brits, Baylee, author. Title: Literary infinities : number and narrative in modern fiction / Baylee Brits. Description: New York : Bloomsbury Academic, 2017. | Includes bibliographical references and index. Identifiers: LCCN 2017015965 (print) | LCCN 2017029630 (ebook) | ISBN 9781501331473 (ePub) | ISBN 9781501331459 (ePDF) | ISBN 9781501331466 (hardcover : alk. paper) Subjects: LCSH: Numbers in literature. | Infinite in literature. | Fiction–20th century–History and criticism. | Modernism (Literature) | Mathematics in literature. Classification: LCC PN56.N86 (ebook) | LCC PN56.N86 B75 2017 (print) | DDC 809/.915–dc23 LC record available at https://lccn.loc.gov/2017015965 ISBN: HB: 978-1-5013-3146-6 PB: 978-1-5013-5259-1 ePDF: 978-1-5013-3145-9 eBook: 978-1-5013-3147-3 Typeset by Integra Software Services Pvt. Ltd. To find out more about our authors and books visit www.bloomsbury.com and sign up for our newsletters.

For Jayne and Marius

Contents Acknowledgements

ix

Introduction

1

1

2

3

4

Actual Infinities: Cantor’s Proofs and Modern Fiction The Missed Encounter Cantor’s Transfinite Mallarmé and Meillassoux: Fixing the Infinite Allegory and Enigma

23

The Aleph: Jorge Luis Borges and the Measure of Prose Supplanting the Symbol for the ‘Thing Itself ’: Borges’s Ultraist Beginnings ‘Funes, His Memory’: A Transfinite Technogenesis of Perception ‘The Library of Babel’ ‘The Lottery of Babylon’ What is a Transfinite Allegory?

49

The Lemniscate: Infinite Shapes in the Work of Samuel Beckett A Mania for Symmetry: Molloy and the Continuous Deformation of Language Permutation and Division in Watt All Strange Away and Imagination Dead Imagine: Imagination by Numbers

89

One: J.M. Coetzee and the Name of the Number ‘Literature in the Lap of Mathematics’: The Quantification of Style In the Heart of the Country: Freedom and Equality

24 27 36 46

55 58 69 78 83

94 105 120 139 141 145

viii

Contents

The Childhood of Jesus and Mathematical Nominalism Counting as One, Rather than Counting to One

160 176

Conclusion – X: Literary Infinities after Zeno and Cantor

181

Works Cited Index

194 205

Acknowledgements In 1883, Cantor wrote, ‘Apart from the journey which strives to be carried out in the imagination [Phantasie] or in dreams, I say that a solid ground and base as well as a smooth path are absolutely necessary for secure traveling or wandering, a path which never breaks off, but one which must be and remain passable wherever the journey leads.’1 Cantor elaborated his metaphor of the path in order to claim the necessity of transfinite numbers for mathematics: ‘Every potential infinity (the wandering limit) leads to a Transfinitum (the sure path for wandering), and cannot be thought of without the latter.’2 This was the path that Cantor formulated for mathematics: he was convinced that transfinite numbers were the sure road to the infinite. Here I would like to thank Jayne, Marius and Maegan Brits, who have given me just such a ‘sure path for wandering’. I am grateful to my doctoral supervisor, Julian Murphet, at the Centre for Modernism Studies in Australia, who supervised the project that would eventually become this book. Literary Infinities has been immeasurably enriched by advice from Helen Rydstrand, Penelope Hone, Kate Montague, Alys Moody, Louise Mayhew and the Sydney Scrags group. I am particularly indebted to Rónán McDonald for his perspicuous feedback at various stages of preparation of the manuscript and for his confidence in this work. I am also grateful to Hugo Black Muecke, who has allowed me to use his artwork for the cover image. Finally, I owe thanks to Haaris Naqvi and Katherine De Chant at Bloomsbury for guiding this book from draft to manuscript to publication.

1

2

Cantor quoted in Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princeton, NJ: Princeton University Press, 1990), 127. Ibid., 127.

Introduction

Numbers are drivers of innovation and experimentation in some of the most important prose works of the twentieth century. Examples might include the ‘Ithaca’ episode of Joyce’s Ulysses (‘Nought nowhere was never reached …’), the sucking stones sequence in Beckett’s Molloy (‘I had say sixteen stones, four in each of my four pockets …’) and the repeated invocations of numbers (‘One, two and one, two, nine, second and five and that’) in Gertrude Stein’s Tender Buttons. The famous ‘Oulipo’ gathering of writers and mathematicians in the 1960s, featuring Perec and Queneau, produced bizarre mathematical formalism. But they did not exist ex nihilo. In each of these instances, number facilitates a literary departure from the constraints of representation, facilitating experimentation with literary form delivered from the traditional trajectories of meaning, if not delivered from the very demand for meaning itself. In this book, I consider how the traditional displacement between mathematics and literature is overcome in literature in the twentieth century through an engagement with the peculiarities of number as a formal language, and, above all, an engagement with the mathematical approach to the infinite. In short, numbers are significant because they produce literary form subtracted from semiotic meaning. Although the act of counting has always been central to literature, in the sense that poetic measure relies on number and pattern, number as a tool for literary expression, rather than a device to measure literary temporality, has not been adequately acknowledged. Number is distinguished from other signs in language because it is ‘presentational’, not representational. Numbers are a formal language. This means that a number does not stand in for anything, but rather is the thing itself. In this sense numbers present an interesting relation between thing and name, a crucial issue for a certain type of avantgarde writer – Gertrude Stein, for instance, whose experimental poems came

2

Literary Infinities

to resemble, rather than represent, objects, or J.M. Coetzee, whose novels put under pressure the capacities of words to name objects, people or states of being. Because numbers are extra-semiotic, extra-representational ‘things’, they also constitute a formal language that is, with regard to creativity, generational rather than representational. Rather than representing quantities or relationships between times and spaces, they generate quantities and constitute relationships. It is for this reason that numbers are essential to the work of Jorge Luis Borges, who – if we agree with Anthony Cascardi – was interested in the form of storytelling that creates a world in the tradition of myth and epic, rather than represents it in the tradition of the modern European novel.1 Since numbers are presentational not representational, they are also generative. These are two key elements that make mathematics attractive to the writer confronting the limits of language and that offer scope for a literature not confined to semiotic suggestion. There is a long history of art, literature and philosophy plundering mathematics and number for metaphorical material. For instance, it is a cliché that the number one stands in as a figure of wholeness or unity. Similarly, infinity traditionally stands in for endlessness and for a certain mystical or sublime aura. This book is not concerned with these co-optations; they construct no significant link between mathematics and literature because they are exclusively literary operations, where mathematics is transformed under the effects of rhetoric to take on a tropological function. Instead, this book is concerned with a genuine link between the mathematical and the literary that is developed in the late nineteenth century and across the twentieth century, a link that is facilitated by the advent of a modernist mathematics in the work of European mathematicians, most importantly the German mathematician and inventor of the transfinite numbers and set theory, Georg Cantor. Cantor’s vision of an ‘actual infinity’, one that is either mathematically or philosophically outside of the grasp of human notation and reasoning, provides a paradigm for grasping the mathematical absolute without rendering it as a trope of the unattainable. 1

This argument is elaborated in: Anthony Cascardi, ‘Mimesis and Modernism: The Case of Jorge Luis Borges’, in Literary Philosophers: Borges, Calvino, Eco, ed. Jorge J.E. Garcia, Carolyn Korsmeyer, and Rodolphe Gasché (New York and London: Routledge, 2002).

Introduction

3

There were several transformative events in the discipline of mathematics around the turn of the century but none, perhaps, as singularly influential or imaginatively bold as Georg Cantor’s development of the transfinite numbers. Cantor’s transfinites are numbers that measure multiple ‘actual’ infinities of different sizes. Even today, the idea of measuring or speaking of the relative size of different infinities seems preposterous. By developing this notation, Cantor brought infinity, the apparent antithesis of human knowledge, into the rational sphere: into the grasp of human thought and manipulation. He chose the Hebrew letter ‘aleph’ (ℵ) as the symbol for the transfinite cardinals2 and thus fixed the sign by which infinity would be written, added and subtracted.3 The discovery of a number that measures the infinity of the numbers constitutes a transgressive reformulation of our ideas about the ‘actual’, the infinite and what is termed the ‘generic’: a word used to describe something that may have measure but not determination.4 The very notion of a measure of the infinite that has certain finite capacities (one can add or multiply transfinite numbers) constitutes a radical refutation of the laws of number and infinity, which demands an interrogation of the very foundations of mathematics. Cantor’s ‘transfinite’ revolution is a remarkable moment of epistemological reaching, one that I hope to show held signal attractions for twentieth-century writers who were grappling with the limits of expression and striving, to use Samuel Beckett’s words, to ‘eff the ineffable’.5 Joseph Warren Dauben, the pre-eminent Anglophone scholar of Cantor’s legacy, compares the advent of transfinite set theory to the Renaissance 2

3

4

5

Cardinality is the measure of objects in a set, whereas ordinality is the order type of the set. A set is a collection of distinguishable objects. Set theory arose from Cantor’s work on cardinality: the number of objects in an infinite set. Cantor managed to distinguish the cardinality of two infinite sets: the set of real numbers, and the set of natural numbers. Cantor proved that the set of real numbers could not be counted, but that the set of natural numbers could (and, as such, that one could establish a cardinality for the set of natural numbers). Whereas the natural numbers have a cardinality of ℵ0, the set of real numbers is greater than this set. Here lies the radical proposition of infinite sets of different ‘sizes’. It should be noted that the transfinite ordinals and the notation for the transfinite ordinals were developed prior to the transfinite cardinals. Here I am focusing on the transfinite cardinals and neglecting the transfinite ordinals due to their significance regarding the uncountability proofs. Here I am glossing Peter Hallward’s excellent definition. See: Peter Hallward, ‘Generic Sovereignty: The Philosophy of Alain Badiou’, Angelaki: Journal of the Theoretical Humanities 3, no. 3 (2008), 87. The line from Watt is as follows: ‘perhaps also because of what we know partakes in no small measure of the nature of what has so happily been called the unutterable or ineffable, so that any attempt to utter or eff it is doomed to fail, doomed, doomed to fail’. See: Samuel Beckett, Watt, ed. C.J. Ackerley (London: Faber and Faber, 2009), 52–53.

4

Literary Infinities

‘transition in advancing from the closed world of Aristotle’s universe to the infinite world of the post-Copernican era’ which was ‘in many respects a painful and traumatic one, but profound in its implications for the subsequent history of Western thought’.6 Cantor’s transfinite numbers would provoke such a shift in mathematics: a reorientation of the ground upon which mathematics is founded. The formal and intellectual connections between this paradigm shift in the foundations of mathematics and the tremors that reverberated through innovative literary work in the subsequent decades have not hitherto been widely acknowledged by scholars of literature and science. This book will show that Cantor’s development of ‘actual infinities’, which captures perhaps the most nebulous and contradictory mathematical idea in numerical notation, is achieved through the use of a form shared by both mathematics and literature. In this book, I will call this shared form ‘transfinite allegory’. In the first chapter of this book I will discuss Cantor in relation to the French poet Stéphane Mallarmé and his disciple and compatriot, Paul Valéry, and provide an initial account of transfinite allegory. In the chapters that follow I will analyse the use of mathematics and transfinite structures in the work of Argentine miniaturist Jorge Luis Borges, Irish novelist and playwright Samuel Beckett and South African novelist J.M. Coetzee. Each of these writers participates in a different instantiation of literary modernism in vastly different regional and linguistic contexts. They represent various tendencies of modernism, including post- or semi-colonial modernism and ‘late’ modernism, their lives spanning the nineteenth to the twenty-first centuries, their careers spanning Spanish modernismo, the Parisian avantgarde, and North American post-structuralism.7 What unites these four writers is the centrality of number and a modernist conception of infinity to the literary innovations that shape and define their work. It is number that will mediate for each of these writers one of the defining constraints of 6

7

Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princeton, NJ: Princeton University Press, 1990), 4. I am referring, here, to Tyrus Miller’s concept of ‘late modernism’. For Miller, late modernism is ‘a distinctly self-conscious manifestation of the ageing and decline of modernism, in both its institutional and ideological dimensions. More surprising, however, such writing also strongly anticipates future developments, so that without forcing, it might easily fit into a narrative of emergent postmodernism’. See: Tyrus Miller, Late Modernism: Politics, Fiction, and the Arts Between the World Wars (Berkeley and Los Angeles: University of California Press, 1999), 7.

Introduction

5

literary modernism. This constraint is a paradox. Modernist art veers between the hubristic ambition for the representative capacities of art and a lack of faith in precisely the elements and techniques from which that art is formed. The ambition for total expression coupled with a critical lack of faith in the substances available for expression is perhaps even the fundamental artistic contradiction in modernism. Into this breach between desire and circumstance comes the truth affording capacity of number. Each of these writers comes to literature wrestling with the limits and limitations of language. Jorge Luis Borges, in 1921, produced an ‘ultraist manifesto’ with a Spanish avant-garde group that called for a liberation of the symbol from ornamental language in favour of ‘primordial metaphor’.8 Samuel Beckett famously elaborates, in a 1937 letter to Axel Kaun, his exasperation with his medium. ‘Grammar and Style!’ Beckett writes, ‘To me they seem to have become as irrelevant as a Biedemeier bathing suit or the imperturbability of a gentleman.’9 And J.M. Coetzee condemned the effects of conventional literary voice for creating stultifying identities for readers and writers in a 1984 essay entitled ‘A Note on Writing’: ‘A is written by X’ comes to function as ‘a linguistic metaphor for a particular kind of writing […] where the machine runs the operator’.10 In each case, these authors find the received structures of language and expression to be hindrances to their artistic task. The argument that I develop over the following chapters claims that these four writers took number to be valuable precisely for its mathematical identity and difference to natural language. It is the status of the numeral as a ‘presentational mark’, capable of generating things and worlds rather than representing them, that makes it key to resolving limitations inherited from traditional literary modes of style and address. There is a rich and extensive engagement in nineteenth-century literature with the empirical sciences, despite the fact that this century

8

9

10

See Beatriz Sarlo’s discussion and translation (the most authoritative translation that I have come across) of the ultraist manifesto in her book: Beatriz Sarlo, Jorge Luis Borges: A Writer on the Edge, ed. John King, 2nd ed. (London and New York: Verso, 2006), 113. Samuel Beckett, The Letters of Samuel Beckett, Volume 1: 1929–1940, ed. George Craig and Dan Gunn (Cambridge and New York: Cambridge University Press, 2009), 518. J.M. Coetzee, ‘A Note on Writing’, in Doubling the Point: Essays and Interviews, ed. David Attwell (Cambridge: Harvard University Press, 1992), 95.

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Literary Infinities

also saw the consolidation and specialization of scholarly disciplines. Yet, pure mathematics, as a speculative or formal rather than empirical science did not have the same presence in literature or literary studies as biology, for instance.11 This is because literature and mathematics are at once inextricably intimate and implacably oppositional: poetry needs its metrics, but the values we ascribe to literature – humane, affective, figurative, narratological  – are far from the perceived abstractions and bloodless symmetries of mathematics. Unlike, say, biology, mathematics has traditionally been considered an art, aligned with the poetic or expressive, and yet the form of writing peculiar to mathematics – pure presentation – is excluded from what we understand to be relevant to literature or, even, to the exercise of language. In the work of Borges, Beckett and Coetzee, we will see a shift in this polarity, harnessing rather than negating that distance and embracing those paradoxes of the number: textual and not semantic, quotidian and radically abstract, the tool of empiricism and objectivity yet fundamentally grounded in a void or zero, or the irrational, or the infinite. In other words, in the writers considered in this book we 11

The novels written by the ‘realists’ and the ‘naturalists’ of the nineteenth century (as well as into the twentieth century) are the most evident testimony to this. Laura Otis, in her preface to her anthology, Literature and Science in the nineteenth century, explains that it was only in the 1830s that science took on a definition that separated it from general skill: At an 1833 meeting of the British Association, William Whewall proposed the term ‘scientist’ for investigators who until then had been known as natural philosophers. In the nineteenth century, ‘science’ came to signify the study of the natural and physical world. Until that time it had denoted any sort of knowledge or skill, including the ‘science’ of boxing. In contrast, the notion of a ‘split’ between literature and science, or a ‘gap’ to be ‘bridged’ between the two, was never a nineteenth century phenomenon. (xxv) Despite elaborating a distinction between science and skill, or science and literature, Otis’s history of the interaction between literature and science does not find a great separation between the two but rather suggests that science and literature overlapped in terms of subject matter. Otis justifies the separation of mathematics from science in her anthology on the basis of the fact that nineteenthcentury mathematics, along with technology, would have been considered an ‘art’ not a ‘science’ ‘… indicating the use of practical skill rather than systematic inquiry’ (xxvii). Where mathematics is included in the anthology, it is done so in the legacy of August Comte and the positivist circles around Comte, who ‘understood mathematics and physics as the source of rigorous laws and consequently the foundation on which other disciplines might be based’ (xxvii). Otis thus places mathematics at the start of her anthology to represent the way in which writers turned from these rigid confines to richer, non-positivistic sciences dealing with evolution and other empirical findings. See: Laura Otis, ‘Introduction’, in Literature and Science in the Nineteenth Century (Oxford and New York: Oxford University Press, 2002). Mark S. Morrison charts this trajectory into the twentieth century in his book Modernism, Science and Technology (New York: Bloomsbury Academic, 2017). Morrison notes that the twentieth century saw a broad move away from paradigms of the nineteenth century, and his work focuses on the shifts in understandings of thermodynamics and relativity.

Introduction

7

see mathematics become intimate with prose precisely on the basis of its irreducible difference from prose, not its relationality.12 The mathematician Jeremy Gray associates Cantor with what he calls a ‘mathematical modernism’. In Gray’s work, Cantor is only one of the actors in a wider transformation of mathematics that occurs at the end of the nineteenth and start of the twentieth centuries. In Plato’s Ghost: The Modernist Transformation of Mathematics, Gray argues that ‘the period from 1890 to 1930 saw mathematics go through a modernist transformation’.13 Gray’s book charts a shift in mathematics, arguing that the year 1900 saw a substantial shift in the way that mathematics was carried out: in terms of both its institutional status (the ‘autonomy’ of mathematicians from scientists, in particular) and its intellectual ambition (new visions of what constitutes a proof, for instance), nominating Hilbert’s Grundlagen der Geometrie as perhaps the ‘single exemplary work that ushered in modernism’.14 For Gray, this applies to the whole of mathematics, or at least its constituent sub-disciplines, rather than one particular event or proof: Taken together all the changes in mathematics here described and the connections to other intellectual disciplines that were then animated constitute a development that cannot be described adequately as progress in this or that branch of mathematics (logic and philosophy) but must be seen as a single cultural shift, which I call mathematical modernism.15

Crucially, Gray ties this mathematical ‘modernism’ to a new Platonist perspective in mathematics. Whilst Gray acknowledges that there is ‘little direct influence’ of cultural modernism upon his notion of mathematical modernism, he does construct a definite basis for the comparison.16 Most significantly for 12

13

14 15 16

Another juncture at which mathematics resembles or echoes artistic practice is in the controversy surrounding whether mathematical innovations constitute ‘creations’ or ‘discoveries’. This debate is particularly pertinent to Cantor, and Mary Tiles sums up the issue of creation (or in Tiles’s terms, ‘invention’) versus discovery succinctly: ‘Did Cantor discover the rich and strange world of transfinite sets (which Hilbert was to call Cantor’s Paradise) or did he (with a little help from his friends) create it? Are set theorists now discovering more about the universe to which Cantor showed them the way, or are they continuing the creative process?’ Mary Tiles, The Philosophy of Set Theory: An Historical Introduction to Cantor’s Paradise (Mineola, NY: Dover, 2004), 1. Jeremy Gray, Plato’s Ghost: The Modernist Transformation of Mathematics (Princeton and Woodstock, NJ: Princeton University Press, 2008), 1. Ibid., 5 Ibid., 4 Ibid., 7.

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my work, Gray constructs a link between these two modernisms that does not necessarily imply a similarity between the two, but rather uses a third, material metaphor to place the two domains in relation to each other: Some features of bats and birds, or ichthyosaurs and dolphins, are alike because the requirements of efficient flying or swimming promote them, but one species does not inherit them from the other. The common features in the present case are hard to discern, but the sheer size of society, its extensive diversification, the existence of cultural activities remote from immediate practical needs, and their high degree of cultural hegemony are certainly present in each.17

This is, in a certain sense, a means by which to articulate the fact that Cantor and the modernists emerged from the same zeitgeist: Gray points, here, at an affinity without necessarily establishing direct influence. This book cannot hope to address whether, broadly understood, the discipline of mathematics can be understood to have practised a form of modernism analogous, say, to what we mean by modernism in music. This would require an assessment of the wider late-nineteenth-century and early-twentieth-century developments in mathematics. Rather, I am considering quite the inverse of Gray’s topic: the presence of a modernist mathematics in the literature of the twentieth century. By articulating the points at which mathematics becomes relevant to literature, or to any of the arts, we can start to reflect on the possibility of a modernism in mathematics from the ‘other side’, finding literary affinities with a ‘modernist’ mathematics. This book maps Cantor’s ‘mathematical modernism’ to four writers who have idiosyncratic stakes in a twentieth-century literary modernism. As the choice of authors indicates, this is not a periodizing study which gathers all of the engagements with the infinite in literary modernism. I am not seeking to theorize or describe the approach to number of a broad historical movement (using, say, one of the usual historical boundaries of modernism, 1914–1939) or to catalogue an engagement with infinity or mathematics in the set of authors broadly described as modernist. Rather, my use of the terms ‘modernism’ and ‘modernist’ tend towards an inductive rather than deductive method. While 17

Ibid., 8.

Introduction

9

I begin with some general remarks, these are in order to situate my selected authors rather than to establish a theory of modernism and number of which they are instances. This is not to say that fruitful comparisons and conclusions are not available, but they are established from the ‘ground up’, as it were. This modernism is no longer bound to describing a period or delimited to a few decades at the beginning of the twentieth century, and it can encompass various instantiations.18 This is a selective foray as opposed to a comprehensive cartography. It is also a prolegomena to a different sort of literary criticism, one that tries to trace the contact points between two discrete spheres of knowing: narrative and number. The literary context for this operation must be situated within a broader centrality of number for biological and political life. There are two overarching narratives that focalize such a shift: the narrative of the Enlightenment dissolution of quasi-ecclesiastical hierarchies of being and the ascendency of a ‘regime’ of number through the biopolitical organization of life in the twentieth century. Enlightenment science and philosophy is often taken to be a process that severed the ‘great chain of being’ by virtue of eliminating a stable hierarchy of existence, posited in some absolute. The concept of the ‘great chain of being’ or scala naturae was originally developed by Aristotle (though analogous concepts were developed by other Ancient Greek philosophers, notably Plato) as a mode of hierarchic empirical classification of animals and humans. In Ray Brassier’s words, it was the Enlightenment ‘disenchantment of the world’ and the ‘coruscating potency of reason’ that ‘shattered the ‘great chain

18

This shift is well summed up in the editorial introduction to a recent special issue of Filozofski Vestnik on ‘modernism revisited’. Tyrus Miller and Aleš Erjavec note, ‘Since the 1970s and 1980s when the concept of postmodernism was advanced and hotly debated, the concept of “modernism” was not simply superseded, but also itself became a major object of criticism, questioning, negation and reinscription’ (9). The implication, here, is that modernism has breached its historical bounds and constitutes an aesthetic endeavour that has maintained its rigour and potency at various moments over the twentieth century, and now again holds relevance. This is ratified by Julian Murphet. In his introduction to the inaugural issue of Affirmations: Of the Modern, Murphet writes, ‘Modernism and modernity have now completed the second decade of a remarkable, some would have said unpredictable, revival of intellectual fortunes. […] In any event, this “Modernism” is different in kind from the variable emphases on organic form, Verfremdungseffecte, abstraction, and existentiality of an earlier moment’ (1–2). Modernism is protean without being relative, attached to a certain historical movement and locale but not bounded by it, diverse without being unrecognisable. See Tyrus Miller and Aleš Erjavec, ‘Editorial’, Filozofski Vestnik 35, no. 2 (2014), and Julian Murphet, ‘Introduction: On the Market and Uneven Development’, Affirmations: Of the Modern 1, no. 1 (2013).

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of being’.19 However, rather than a deformation of this sequence of existence, this philosophical transformation offered a substitute for this version of order and hierarchy in another hegemonic narrative: that of quantification. The Enlightenment ‘rupture’ of a sequence or narrative of existence such as the ‘great chain of being’ did not give rise to new non-linear or non-totalizing modes of thought and organization. Instead, the Enlightenment ushered in an era of quantification. Divine or transcendental orders of being were replaced by a new, secular absolute: the number line. In the words of J.L. Heilbron, editor of The Quantifying Spirit in the Eighteenth Century, ‘the later 18th century saw a rapid increase in the range and intensity of application of mathematical methods’.20 The proliferation of mathematics across the sciences of the eighteenth century saw the establishment of a single marker of intellectual vigour: the numerical measure of the object and, bound up with this, the quantification of those objects offering themselves for scientific measure.21 This would hold for population as much as for plants, animals and the heavenly bodies.22 It was this same predominance of quantification that would produce ‘reification’, the concept developed by Marx to capture the process by which human attributes or acts become identified and

19

20

21

Ray Brassier, Nihil Unbound: Enlightenment and Extinction (Hampshire and New York: Palgrave MacMillan, 2007), xi. J.L. Heilbron, ‘Introductory Essay’, in The Quantifying Spirit in the Eighteenth Century, ed. Tore Frängsmyr, J.L. Heilbron, and Robin E. Rider (Berkeley and Los Angeles: University of California Press, 1990), 2. Gunnar Broberg’s study of the significance of quantification to the natural science and history of the eighteenth century clearly establishes the dual nature of quantification for science: the potential objects of study proliferate with the new mechanisms of measure (what Broberg calls ‘the challenge of plenitude’) at the same time that the method of study is itself an act of quantification. Broberg writes: The word ‘quantitative’ applies to natural history during the second half of the eighteenthcentury in two distinct but related ways: as characteristic of the object and of the method of study. As for objects, the sheer number of known and estimated forms forced new approaches to the storage and retrieval of information; as for method, these new approaches were instrumentalist and, in the dominant system of Linnæus, mathematical. These features – the overwhelming flow of information and the determination to inventory and survey it for useful purposes – characterize much of the learned activity of the late Enlightenment. (45)

22

The clearest example of this is found in the quantification of animal species: the number of discoveries of new species in the eighteenth century triggered many estimates of the total number of species on the planet. As the century wore on, the estimates increased exponentially. See Gunnar Broberg, ‘The Broken Circle’, in The Quantifying Spirit in the Eighteenth Century, ed. Frängsmyr et al. For Theodore Porter, the adoption of quantification in the human sciences was not a one-directional import from the sciences. Instead, Porter views quantification as a fundamental communication tool that facilitates the impersonality necessary to meet institutional or bureaucratic demands. See Theodore Porter, Trust in Numbers: The Pursuit of Objectivity in Public Life (Princeton, NJ: Princeton University Press, 1995).

Introduction

11

evaluated on the same basis by which things and commodities are identified and evaluated. Reification is the extension of a quantificatory method and spirit into human life. It is the process by which the ‘spirit of quantification’ comes to be radicalized in the twentieth century, producing what French philosopher Alain Badiou calls the ‘despotism of number’.23 For Badiou, the ‘singular condition’ of thought in the twentieth century will be number: Having interred the pathologies of an unbridled will, the happy correlation of a Market without restrictions and a Democracy without shores would finally have established that the meaning of the century lies in pacification or in the wisdom of mediocrity. The century would thereby express the victory of the economy, in all senses of the term: the victory of Capital.24

The victory of the economy is consolidated by the quantification of life and norms constructed through numerical parameters. The ascendency of number in social and political life translates, for Badiou, into a century that sustains a regime of number but also, somewhat paradoxically, forgets the problems and possibilities entailed in the foundations of number. In other words, this is a century that embraces the count so closely as to completely naturalize it. In Badiou’s summation, number now ‘governs our conception of the political, with the currency […] of the majority’; it ‘governs the quasi-totality of the “human sciences”’ as well as the economy.25 Above all, for Badiou, ‘Number informs our souls. What is it to exist, if not to give a favourable account of oneself?’26 What is the relationship, then, between Cantor’s ‘actual infinities’, the Enlightenment and twentieth-century passions for quantification and the reification, and the numerical ‘despotism’ that Badiou refers to? Is Cantor’s ‘actualization’ of that which is deemed beyond human science and thought another instance of quantification, and, if we follow Badiou, a ‘forgetting’ of the foundations of mathematics? Cantor’s transfinite numbers occupy a liminal position as regards quantification, seeming to embrace it on one level, and utterly reject it on another. Cantor’s generation of a numerical infinite 23 24 25 26

Alain Badiou, Number and Numbers, trans. Robin Mackay (Cambridge: Polity Press, 2008), 1. Alain Badiou, The Century, trans. Alberto Toscano (Cambridge: Polity Press, 2007), 2–3. Badiou, Number and Numbers, 2–3. Ibid., 2–3.

12

Literary Infinities

that is not necessarily tied to a withdrawn ideal or divinity (which I will explain at length in the following chapter) at once renders an idea that was previously ‘transcendental’ or simply elusive ‘actual’, a task that may seem, on surface level, to be radically positivist or naively quantificatory. On the one hand, Cantor’s ‘actual infinities’ ratify the Lockean privileging of experience as a source of knowledge. It is precisely this unity of the actual (what we can see and work with: a concrete number form) with a revolution in noetic and metaphysical sequence that we find echoed in Cantor’s transfinites. And yet, simultaneously, Cantor’s work would disturb the central Enlightenment connection between the actual and a consistent, continuous number line that has the capacity to measure any of science’s objects. It is this affirmation of ‘actual’, material measure for which exceeds and confounds the finitude of experience that presents a novel mathematical discovery pertinent to the form and possibilities entailed in modernist literary infinities. This book investigates the formal, theoretical and artistic possibilities of an actual infinity as it is manifest in Cantor’s transfinite numbers. Through an analysis of number and the transfinite in fiction, I will forge a formal and conceptual connection between two domains regularly understood to be antithetical, without succumbing to the pitfall peculiar to this type of analysis: allowing one domain to co-opt the other. The mathematical or numerical literary criticism that does exist has often been characterized either by a reduction of mathematical form to simply a technique of abstraction or by a co-opting of mathematics as concept or analogy, negating the unmitigated difference between writing as presentation and writing as representation. For many literary critics, mathematical writing and signs occupy a domain that is irreconcilable with the subjective and phenomenal world of art, except when commuted to metaphor. This circumvents both the textual existence of the numeral and its distance from the identity of signs in natural language. Number is an instance of pure presentation distinct from representation, which nonetheless remains a mark or a ‘symbol’. It is this dual status of the mathematical mark (existing as a ‘presentation’ yet occupying a textual domain) that is comprehended and exploited in art, and must be realized by a criticism capable of retaining the same autonomy for mathematics. The near-ideogrammatic form of ∞ (the lemniscate; the sign for infinity) captures this inconsistent and contingent bridge

Introduction

13

between visual-semiotic implication and numerical referent, conjoining both quantity and quality in the signifier for the absolute infinite. The numeral remains a mark and susceptible to symbolic attribution yet properly – mathematically – must exist outside signification. There is a similar bridge between the experience of the numeral and its capacity for abstraction: numerical identity is at once both quotidian – governing the minutiae of everyday life – and transcendental. We use numbers all the time, without there being any clear signified origin for the numeral. We encounter number every day, define our actions and identities by number, and yet it remains ‘transcendental’ in the sense that it occupies precisely the definition of the ‘presupposed’: it exceeds the limits of language because it has no referent other than itself, and is fully and immanently available at all times, and for this very reason is rarely interrogated itself. What is a number? How do we establish the validity of the numerical stakes in continuity, consistency and the role of the number as the ultimate ‘unit’? Such questions pinpoint a problem that will in the twentieth century become literary as well as mathematical, social as well as scientific. Three literary theorists represent the possibilities of confronting the mathematical stakes of literature as well as the methodological pitfalls of integrating mathematics into literary theory. It is worth dwelling on the divergence of these methods for the study of number and literature in order to illuminate the essential element determining such studies: the definition of and philosophical stakes in number. Each of these theorists occupies a vastly different position in literary theory: J. Hillis Miller, Franco Moretti and Steven Connor have little in common other than a preoccupation with number. The vast methodological and critical difference between their studies can be traced back to irreconcilable concepts of number. J. Hillis Miller, for instance, does not develop a numerical methodology for literary studies but rather looks at a philosophical extension of number, which transforms number into a concept. In Zero Plus One, Miller engages with number as a concept, as an expressive ground that literature operates through and enquires into. In Miller’s gloss, infinity is ‘zero’s reciprocal’ and it ‘is a feature of zero that it is incommensurate with all other numbers, or rather, it is a number and not a number’.27 Crucially, 27

J. Hillis Miller, Zero Plus One (Valénzia: Biblioteca Javier Coy d’estudis nord-americans, 2003), 11.

14

Literary Infinities

this work on zero as a concept is part of a wider reflection on comparative literature. Miller’s work on zero is contextualized by the necessity to go beyond national literatures and their comparison to a genuine world literature that exceeds national peculiarities. His contentions about zero and comparative literature are contained as two parts in a single volume (Zero Plus One) linked because they are both bound up with the ‘untranslatable, idiomatic and obscure’.28 Indeed, Miller claims that the ‘zero dimension’ of literature corresponds to ‘the language crisis in comparative literature’.29 Miller uses zero to collect and describe acts or literary productions or modes of expression, and in this sense zero becomes simply a descriptor, a cipher of a cipher rather than a mathematical entity operating within a work. Here, zero is the partially semiotic sign for that which exceeds linguistic capture. In this study, number is in danger of being used to express something, functioning as a concept, metaphor or allegory of precisely that which is divergent from literary form. In other words, number is forced into participating in a connotative or representational scheme. The problem here is that in a mathematical sense zero is simply not idiomatic or obscure, and is only untranslatable insofar as it is not within a linguistic economy (and perhaps, then, nontranslatable rather than untranslatable). In order to understand precisely what mathematical difference, and the writing of ‘presentation’, may afford literature, one needs a revised notion of zero: one acutely aware of the shifts between zero as conceptual metaphor and zero as number, even whilst the identity of the latter allows the creation of the former. 28 29

Ibid. Ibid. Number has been persistently related to the idea of universal communication and Miller seems to be operating in something of a tradition of thought on number and forms of ‘Esperanto’. In the work of the writer Zalkind Hourwitz, for instance, we find a similar claim to number functioning as the antidote to the specificity and national identity and diversity of languages. Robin E. Rider summarizes Hourwitz’s project as follows: The idea that mathematical concepts, especially the concepts of number and their representation by Arabic numerals, were universally comprehensible, held persistent appeal to those who aimed at universal, rational languages. In 1801, for example, Zalkind Hourwitz proposed a Polygraphie that relied on the assignment of a number to each word in the basic polyglot dictionary. The same number thus served to designate words with the same meaning in several languages. For a discussion of this see Robin E. Rider, ‘Measure of Ideas, Rule of Language: Mathematics and Language in the Eighteenth-century’, in The Quantifying Spirit in the Eighteenth Century, ed. ToreFrängsmyer, et al., 125.

Introduction

15

On the other hand, Franco Moretti’s work on number and literature avoids the metaphoric traps of Miller’s analyses but collapses the significance of number for literature into quantitative data production: an empirical tool to pursue distributive tendencies in literary works.30 For the last decade or so Moretti’s critical work on the novel has revolved around developing a quantitative methodology, driven by the polemical claim that literary history will quickly become very different from what it is now: it will become ‘second hand’: a patchwork of other people’s research, without a single direct textual reading. Still ambitious, and actually even more so than before (world literature!); but the ambition is now directly proportional to the distance from the text: the more ambitious the project, the greater must the distance be.31

As in Miller’s work, this ‘numerical’ literary study addresses the viability of world literature, and approaches to texts that emerge from different times and national contexts. Like Miller, Moretti uses number in the context of a study of world literature, undertaken through a methodology appropriate to planetary scope: quantitative studies, which deal only in quantifiable data, usually but not necessarily numerical, formed through counts, measures or pinpoints. Once coordinates are established a diagram can be produced, and this secondary representation then analysed to establish fluctuations, topographies or evolutions. ‘Distance’ is the key predicate of valid world literature methodology, and the form of reading that that these studies should take. Moretti posits an 30

This is part of a wider shift towards quantitative methodologies in literary studies. Matthew Wickman sums up this turn to the quantitative particularly well: The incorporation of ‘big data’ into the humanities is largely characterized by a spirit of pragmatism – a surprising turn, perhaps, after so many decades of ‘big theory’ about the impact of new technologies. In literary studies electronic tools, processing massive amounts of information, make it possible to identify previously undetectable relationships between disparate corners of the canon or within the seams of individual texts, and the interest and anxiety inspired by this new wizardry seem evident in growing numbers of digital humanities symposia, data visualization websites, and conference panels on new media. In this environment familiar arguments concerning the impact of technology on everyday life and on our conception of the human (think of popular refrains involving the cyborg, the posthuman, and so on) are receding behind the growing conviction that we simply need to get on with this new work of inputting, encoding, and uploading. (1)

31

In the same article, Wickman provides a critique of Moretti that is also based on an accusation of naiveté regarding number. See Wickman, ‘Robert Burns and Big Data; Or, Pests of Quantity and Visualisation’, Modern Language Quarterly 75, no. 1 (2014). Franco Moretti, ‘Conjectures on World Literature’, New Left Review 1 (2000), n.p.,

16

Literary Infinities

untapped numerical resource that exists only between texts, and not only that, but between very many texts: the truth of texts being based on a resonance that occurs cumulatively. His distance from the texts is achieved through a naturalization – an absolute closeness – to numeracy, without critical relation to the social regime of numeracy that he participates in. In his direct connection between abstraction and ambition, Moretti misplaces the actual distance of his readings. In fact, this work is not distant at all. The supposed ‘distance’ is achieved by the ubiquity of digital technologies and computation: the work is only ‘distant’ to the texts by virtue of embracing an absolute intimacy with data modelling. This is an example of a study that ratifies number as the appropriate means by which to achieve scope, perspective and generality in order to produce greater ‘awareness’ of broad trends in literary history. Such an affirmation, however, loses all critical relation to the number; indeed, we can say that number in Moretti’s work is substantively ‘forgotten’. The third important instance of recent ‘numerical’ criticism comes much closer to retaining the specificity of mathematics without naturalizing number and quantification. Steven Connor’s important work on numbers in Carroll, Dickens, Huxley and Beckett also focuses on the crucial but problematic status that number occupies in mathematics and works to reclaim the shared project of literature and mathematics on the basis of a ‘differentiated indifference’ at the heart of numerical existence.32 This book shares many aspects of the task that Connor undertakes, but differs in perspective on the relation between modernism and a mathematical ‘actual’ infinity. Many writers as well as literary critics maintain a romantic conception of number even whilst participating in modernism in every other sense. Connor argues that ‘modernist writers […] mount a sustained assault against the real of number, determined to assert quality over quantity, determined to assert the hazy, nebular, indefinite or indistinct’, and whilst this holds true for many writers, it is also indicative of an artistic perspective that took the ‘actuality’ of mathematics as a threat rather than an opportunity.33 This scheme retains an opposition between 32

33

Steven Connor, ‘“What’s One and One and One and One and One and One and One and One and One and One?” Literature, Number and Death’, Steven Connor, December 2013, n.p., http:// stevenconnor.com/oneandone.html (accessed 2 December 2016). Ibid.

Introduction

17

mathematics and literature, in which one mode of writing must neutralize the other. What is compelling about Connor’s investigation into mathematics and modernism is his recognition of number not only as a tool of the critic, but as something deployed by writers and poets towards formal ends, and as such his work details an array of literary ‘attractions’ to number, including the graphic, material presence of number in print culture and in the cityscape. Connor makes much of the fact that definitions of number as well as processes of counting can inspire both ‘giddiness’ and ‘horror’.34 In this, Connor’s work is groundbreaking and approaches mathematics in literature from the perspective that undergirds Badiou’s criticism of the ‘tyranny’ of number in the twentieth century. Key to his work is an understanding of the difference between quotidian number and the foundations of those quotidian numbers, which lie in the further reaches of pure mathematics and, as he suggests, simply don’t have the habitual surety that we associate with numbers. He makes a crucial distinction between ‘monotonous numbering’ and what is properly numerical, in what he loosely terms a ‘Bad Infinity’ and a ‘Good Infinity’ of numbers, co-opting number as a hero in an imagined war against the quantitative.35 However, pitting ‘Good Infinity’ as the hero in the war against the quantitative neglects the ‘actuality’ of modern infinities in favour of a desired ‘relationality’ and ‘inexactitude’. This actual infinity sits in stark contrast to precisely this numerical sublimity that preoccupies Connor. In his haste to enlist mathematics for a pseudoethical position of an ever-receding horizon of thought and possibility, and to expound the heroic possibilities of number as a refutation of all exactitude and independent identity, Connor neglects what many take to be the most important advance in pure mathematics in the last century and a half: the founding of a new form of modern mathematics based precisely on a concretion of numerical identity rather than a reiteration of any ‘nebulousness’. Instead of a ‘Good Infinity’ or a ‘Bad Infinity’ in modern mathematics we are dealing with an ‘actual

34

35

Steven Connor, ‘Hilarious Arithmetic: Annual Churchill Lective at the University of Bristol’, Steven Connor, n.d., n.p., http://stevenconnor.com/hilarious.html (accessed 12 December 2014). Steven Connor, ‘“What’s One and One and One and One and One and One and One and One and One and One?” Literature, Number and Death’.

18

Literary Infinities

infinity’. If mathematics is the signifier of a regime of number whose foundations are remote or absent, then mathematics cannot also be the locus of an ethical remedy for this: mathematics has no space for the ‘ethical’ set, only the empty set. The demand, now, is to understand literary experimentation with a form of notation that is other to the word and that is more and other than a submissive, suggestible, evocative symbol that conforms to the metaphysics of the literary. It is this question that this book addresses. There are two instances of literary work on number that stand out, here, as realizing precisely this goal. In an important essay on ‘the pursuit of number’, Marjorie Perloff looks at mathematics in the work of Russian futurist Velimir Khlebnikov (1885– 1922) and Irish poet W.B. Yeats (1865–1939).36 Both writers were interested in numerical models of history and being: in Yeats’s case, the ‘Great Wheel of History’ or the ‘Four Faculties’ and, in Khlebnikov’s case, the geometric models that he took to underlie the progression of history. Perloff argues that the numerical preoccupations in the work of both of these poets is not a preservation of romanticism but is thoroughly modernist, drawing out the distinction between the ‘ultimate abstraction’ of number (using Alfred North Whitehead’s term) and the contrasting specificity, and indeed materiality, of numerical value. Perloff focuses on Khlebnikov’s vision of being and the world as a great vibrating string, which connects abstracted order that governs events to a resonant materiality and logical system. By demonstrating the material consequence of Khlebnikov’s mathematics, Perloff shows that the association of abstraction, determinism and materiality with resonance, if not reason, enacts the transition from a romantic idealism to a modernist vision of the world or universe. This vision of a modern mathematics, divorced from either a romantic or positivist relation to number and the infinite, is captured in another seminal work on literature and mathematics. In Beckett and Badiou: The Pathos of Intermittency, Andrew Gibson argues that romantic philosophy broke the integral link between mathematics and philosophy. Gibson notes: ‘Romantic philosophy more or less completely separates philosophy from mathematics, with Hegel playing the decisive role. The anti-philosophical stance of positivism 36

Marjorie Perloff, ‘The Pursuit of Number: Yeats, Khlebnikov, and the Mathematics of Modernism’, in Poetic License (Chicago, IL: Northwestern University Press, 1990).

Introduction

19

does no more than mirror the anti-mathematical stance of romanticism.’37 The romantic conception of infinity directly renders it as a metaphor for a boundless exteriority, an openness without end. […] By the same token, infinity becomes an object of insatiable yearning. It is placed within a structure that opposes it to transience, historical mortality, the birth and death of ideas.38

This is not a problem in itself, but it evacuates infinity of its peculiarly mathematical import and prevents a genuine literary relation to mathematics. Gibson identifies this romantic conception of the infinite as an ‘auratic conception of the infinite’ (which closely resembles the Burkean sublime), whereas the positivist notion of the infinite lies in antithesis to the generation of the aura – the other side, we might say, of the same coin and at an equal distance from any ‘actual infinite’.39 It is, then, unsurprising that we see a mathematics of the infinite return to art – if not philosophy – at the precise moment in which culture must confront the loss of aura: the moment at which art must adapt to ‘an age of technological reproducibility’.40 This moment occurs at the start of the twentieth century, with European and Anglophone modernism. For each of the writers that I am considering, here – Borges, Beckett and Coetzee – the relation to number is bound up with a relation to the mechanization of either the image or the poem: for Borges, the filmstrip; for Beckett, the photograph, sound recording and television screen; for Coetzee, computer programming. In these cases, the problem of an art without aura parallels the problem of an infinite without sublimity. Perloff ’s understanding of the necessity of materiality to a modernist mathematics and Gibson’s rendering of the romantic conception of the infinite as ‘auratic’ together articulate the necessity and the specificity of a vision of the infinite in an age of rapid technological innovation. 37

38 39 40

Andrew Gibson, Beckett and Badiou: The Pathos of Intermittency (Oxford and New York: Oxford University Press, 2006), 6–7. Ibid. Ibid., 7 Walter Benjamin, ‘The Work of Art in the Age of Its Technological Reproducability’, in The Work of Art in the Age of Its Technological Reproducibility and Other Writings on Media, ed. Michal W. Jennings, Brigid Doherty, and Thomas Y. Levin (Cambridge and London: The Belknap Press of Harvard University Press, 2008), 19.

20

Literary Infinities

Formally, this book maps the reciprocity between mathematics and literature along the lines of the lemniscate. This mark provides an image of the reciprocal structure that I map here, without properly conforming either to a presentational or to a representational scheme. The lemniscate maintains the autonomy of the tropological and semiotic domain of literature on one of its loops, and the presentational domain of mathematics on another, but it also allows the two domains meet at a certain point. This point is the ‘transfinite’. Where this point touches the mathematical domain, it takes the form of Cantor’s transfinite numbers; where it faces the literary domain, it constitutes a type of formal allegory, which I will call ‘transfinite allegory’. But this visual model should not be interpreted to imply a literature of the ‘infinite’, here. Cantorian set theory and proofs of the actual infinite do not lead to a literature of the infinite, just as Cantor’s ‘actual infinite’ is absolutely not collapsible into an unspecified ‘infinite’ or ‘infinity’. We might find a literature of the infinite precisely where Andrew Gibson locates the most vehement rejection of actual infinity: in romanticism, where there are many analogies for an absolute or an infinite, but where the infinite is ultimately immaterial, absent or intangible referent.41 In romanticism, ‘We are separated from infinity by an uncrossable frontier’ and that frontier is the ‘boundless exteriority, an openness without end’.42 Indeed, in Romanticism the infinite is not just intangible but too often it stands in for or is the intangible. I will argue that in the instances of modernist fiction at issue here we come to see a transfinite rather than infinite operation, an actual literary infinity rather than simply a literary infinity. This is what allows an embrace again of the link between mathematics and literature, and what distinguishes a modernist reciprocity between prose and mathematics from a romantic rejection of an actual infinite.43 41 42 43

Gibson, Beckett and Badiou, 6–7, 51–52. Ibid., 6. It may seem that the charge of a ‘rejection of the mathematical infinite’ is unfair, here, in particular because Cantor’s proofs most certainly come after the bulk of what can be considered literary romanticism (and the proofs were hardly accessible to a Anglophone literary community), but I do not think that this charge has to imply a failure to anticipate Cantor. The issue of the truth or falsity of infinity is one that is hardly new and, indeed, has persisted along the lines of an Aristotelian versus Platonic view of the world: a divergence that I will articulate, in my chapter on Samuel Beckett, as a distinction between viewing multiplicity as genera or as generic. So this avoidance of the actual infinite in favour of the ideal infinite is not anachronistic, and the embrace of an infinity that is ideal, and ephemeral, can be seen as an opposition to Cantorian thought without any direct engagement between these philosophies and Cantor’s mathematics.

Introduction

21

The five writers I select here for in-depth treatment are heterogeneous. I make no claim that they are representative or exemplary of a broad literary movement. Rather, the analyses that follow are comparable to drilling explanatory holes in an untapped well – the links between number and narrative established here offer analytic forays into five key oeuvres in what is no doubt a wider field of ‘mathematics and modernism’ as well as ‘mathematics and literature’ more generally. The first chapter addresses the theoretical and mathematical context for this book. In this chapter, I situate Cantor’s transfinite numbers, and his diagonalization proofs, in terms of the wider mathematical context of the nineteenth century and in terms of the impact of these proofs in the twentieth century. In this chapter I consider the possibility of Cantor’s proofs as a triumph of modernism and present two engagements with literature and mathematics that set the scene for the following analyses. The first of these is a short story: Paul Valéry’s short text on the poet Verlaine and the mathematician Poincaré passing each other on the street. The second of these is critical: French philosopher Quentin Meillassoux recently discovered a ‘code’ at the heart of Stéphane Mallarmé’s Un coup de dés jamais n’abolira le hazard, a code that he suggests is a poetic ‘concretion’ of the infinite. Following this, I will outline a model for comparison between Cantor’s ‘actual infinite’ and Mallarmé’s ‘actual infinite’: transfinite allegory. I will theorize ‘transfinite allegory’ as the model shared by Cantor’s proofs and the literary infinities of Mallarmé and of those who work in the legacy of the actual infinite. ‘Transfinite allegory’ will also provide the model by which literature can approach mathematical infinities without subsuming mathematics to a representational economy. The second chapter deals with a writer that exists at the crossroads between the nineteenth century and the twentieth century, European modernism and the South American periphery, indeed Symbolisme and modernism. In this chapter, entitled ‘The Aleph: Jorge Luis Borges and the Measure of Prose’, I will analyse the relation between number and total worlds, and number and the capacities and limits of prose representation in the short stories of Jorge Luis Borges. I will show how Borges inverts the traditions of prose description to create fictional worlds that defy the limits of representation. In the following chapter, ‘The Lemniscate: Infinite Shapes in the Work of Samuel Beckett’, I analyse two of Beckett’s novels and three short prose works. This chapter focuses on the transformation of naturalism in Molloy and

22

Literary Infinities

Watt through different numerical models, most significantly permutation. Beckett’s prose connects mathematical and literary crises of foundation by attempting what Beckett would call a literature of the ‘unword’ that is decidedly mathematical. The repeated negation of semantic reference and the radicalization of naturalist description both rely on a reworking of the numeracy of prose. Addressing Beckett’s later work, I analyse the numeral as a mark of representation that cannot be determined by ‘condition’ or ‘content’, focusing on All Strange Away, Imagination Dead Imagine and The Way. This status of the numeral allows Beckett to invert the naturalist ambition for fictional worlds in order to achieve a ‘generic literature’. The fourth chapter is titled ‘One: J.M. Coetzee and the Name of the Number’ and considers Coetzee’s investment in structuralism and the ‘quantification’ of style, and the role of number in two of Coetzee’s novels. Here I consider, in particular, Coetzee’s engagement with the uniqueness of the symbol, what counts as ‘I’ and, most importantly, what allows for the consistency of the count. I will argue that the site of formal experimentation in these novels is the generic operation of – to use Badiou’s terms – what is considered to ‘count-as-one’. Rather than the infinite being a ‘special interest’, it can be seen as a substantial influence in some of the key textual innovations of the past century. Collecting these diverse engagements with the infinite together under one conceptual and thematic investigation allows for an overarching theory of transfinite allegory, which is developed in the introduction and then more substantially discussed in the conclusion. Number has always been present as the logos alogos of language and poetics but – to echo Dan Mellamphy’s Beckettian rendering of this – comes to function as the essential point of non-being, the negated twin self, of twentieth-century prose through its difference from linguistic signification.44 In this regard, this book rescues the numerical for the literary, specifying a domain of literary production whose achievement is to decline the tropological in favour of the formal, to explore the aesthetic and affective heights of declining to ‘mean something’, to generate worlds as well as to represent them and to harness the links to another art too long divorced from expression: mathematics. 44

For a reading of Beckett’s ‘etre manqué assasine’ in relation to the logos alogos of fiction, see: Dan Mellamphy, ‘Alchemical Endgame: “Checkmate” in Beckett and Eliot’, in Alchemical Traditions from Antiquity to the Avant-Garde, ed. Aaron Cheak (Melbourne: Numen Books, 2013), 552.

1

Actual Infinities: Cantor’s Proofs and Modern Fiction

There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. But from this very fact the necessity of the actual infinite can be demonstrated. Georg Cantor, 1885 The unique number/which cannot be another. Stéphane Mallarmé, 1897 No one will expel us from the paradise that Cantor has created. David Hilbert, 1925 Is there a fruitful connection between literary infinities and mathematical ones? How might we conceptualize a relation that is mutually illuminating without the need to insist on similarity? And how might such a connection be drawn without eroding the specificity of each of these domains? By and large, literature and mathematics have been represented as two separate worlds that bypass each other without any interaction. This planetary metaphor is not without its own value: the relationship between these two worlds does not involve direct communication; these worlds are not co-constituted and do not resemble each other. What they share is relation to a third element: the source of gravity around which they both revolve. If we take literary and mathematical innovation to be something more than contingent cultural flux, we might claim that these two worlds inevitably fall into a relationship with each other via the conduit of a common orientation: the infinite. In this

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Literary Infinities

preliminary, contextualizing chapter I will address what can be called two ‘transfinite’ endeavours: Mallarmé’s coded poetry, and Cantor’s transfinite numbers, and opportunities for situating these endeavours in terms of an allegorical structure. But first, an excursus on distance.

The Missed Encounter The short text ‘Passage de Verlaine’ by Paul Valéry (1871–1945) is one of the rare works that address the distance between mathematics and literature. Valéry, friend and champion of Stéphane Mallarmé, is the inimitable poet who is also a polymath. Valéry read modern mathematics and was versed in the perennial problems posed by the numerical infinite, especially as it is figured in Ancient Greek thought.1 He is distinctive for privileging mathematics as the discipline in which the imagination is at its most adventurous: In comparison to the dream, the real is a convention. And the same applies for the pure imagination. […] The imagination is always curiously timid. It rarely risks combinations far from all probable use and reality. It is the mathematicians who are led furthest from this by the necessity to interpret or demonstrate their equations when they want to generalize or study the whole domain. They write with a greater generality than they can see. And afterwards, they attempt to see.2

Valéry existed between the worlds of science and poetry, a position that is perhaps best exemplified in the short text ‘Verlaine and Poincaré’. In this text he describes a scene at the Jardin du Luxembourg, where two profound intellects – the symbolist poet, Paul Verlaine (1844–1896), and the mathematician, Henri Poincaré (1854–1912) – very nearly cross paths on a daily basis, without encountering or even being aware of each other. Poincaré

1

2

Valéry’s poem ‘The Graveyard by the Sea’ is one of the best examples of Valéry’s practice of drawing connections between states of being and mathematical problems. See Paul Valéry, ‘The Graveyard by the Sea’, in Selected Writings of Paul Valéry, trans. Malcolm Cowley, C. Day Lewis, and Jackson Mathews (New York: New Directions Publishing, 1950), 41. Valéry quoted in Karin Krauthausen, ‘Paul Valéry and Geometry: Instrument, Writing Model, Practice’, Configurations 18 (2010), 243.

Actual Infinities: Cantor’s Proofs and Modern Fiction

25

frequently passes along the same street as Valéry, alone, hunched, and with a gaze that is ‘empty and fixed’.3 Minutes later, Verlaine heads down the same street to a tavern, ‘flanked by his friends, leaning on a woman’s arm… [he] would speak, pounding on the pavement, to his small devoted retinue’.4 At that time, Verlaine was France’s most significant symbolist poet, whose goal, as put most famously by Jean Moréas in the Le Symbolisme, was to ‘clothe the Ideal in a perceptible form’.5 The only link between Poincaré and Verlaine is Valéry, the silent observer who takes note of the scene from a distance. Valéry’s anecdote portrays two gods traversing the streets of Paris, both divine but separated by a ‘spiritual distance’ that is ‘immense’.6 What is particularly striking in Valéry’s anecdote is the fact that the poet speaks to his entourage, whereas the mathematician, solitary, does not. This allegorizes the means by which mathematics is distinguished from the empirical sciences as well as literature and the arts, a distinction that can be articulated in a variety of ways but which in essence pertains to the distinction between presentation and representation. Pure mathematics only became distinguished as a discipline distinct from other sciences in the eighteenth century, on the basis that it does not measure or record empirical data but produces proofs that are pursued not for the sake of utility but for the truth that these proofs would formalize.7 In Valéry’s anecdote, it is the divergence between expression and silence, and between sociality and isolation, that is the initial deciding factor in the mutual exclusivity between the figure of the mathematician and the figure of the poet. And yet these two figures are united by the same pursuit of truth rather than measurement. Valéry’s story describes a missed encounter that is perhaps synecdochic of the separation between mathematics and poetry. At the same time, through 3

4 5

6 7

Paul Valéry, ‘Verlaine and Poincaré’, in Mathematical Lives: Protagonists of the Twentieth Century, ed. Claudio Bartocci, Renato Betti, Angelo Geurraggio, Robert Lucchetti, and Kim Williams (Berlin and Heidelberg: Springer-Verlag, 2011), 25. Ibid. Jean Moreas, ‘The Symbolist Manifesto’, in Manifesto: A Century of Isms, ed. Mary Ann Caws (Lincoln and London: University of Nebraska Press, 2000), 50. Valéry, ‘Verlaine and Poincaré’, 26. This definition was popularized by the Bourbaki group. See Nicolas Bourbaki, Elements of Mathematics: Theory of Sets (Berlin and Heidelberg: Springer-Verlag, 1968), 6–7.

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Literary Infinities

an emphasis on purity, abstraction, and divinity, Valéry manages to gesture towards a shared sphere: What incomparably different effects the sight of even the same street could produce in those two systems that followed so quickly upon one another. In order to conceive of it, I had to choose between two admirable orders of things that were mutually exclusive in appearance, but that resembled one another in the purity and depth of their purpose.8

Valéry retained a lifelong obsession with the relation between the artistic imagination and geometry. His topological notions of the imagination find form within his poems, which echoes his own deformation of mathematical orthodoxy. Valéry was deeply concerned with the new discipline of topology and adopted the idea of the Riemann manifold as a scheme for the functioning of the imagination.9 For Valéry, just as topological forms are characterized not by content but by connectivity, so too was the imagination characterized by internal transformation and connectivity.10 In ‘Passage de Verlaine’, Valéry, the writer and theorist of the geometry of the imagination, is able to see the two magisteria passing each other. Valéry manages to form his own account of a relation between the two – finding analogues between mathematical entities and the phenomenal world that he develops in his Cahiers. Yet what is crucial to emphasize here, and what this short story shows us, is the difference between two worlds that nonetheless both privilege ‘purity’ and ‘depth of purpose’. In the following two sections I will present another ‘passage’ of two minds, albeit one that does not have the convenient scene of the Parisian street at which they intersect. Stéphane Mallarmé and Georg Cantor both developed forms of ‘actual infinity’ in the nineteenth century. These two figures would have been unaware of each other’s work, and certainly the poetry of Mallarmé and the mathematics of Cantor have rarely been read in terms of each other, save in the work of Badiou and, indirectly, Quentin Meillassoux. The link that I will hazard between Cantor’s transfinite numbers and the literary ‘actual’ infinities (for which Mallarmé’s work is the 8 9 10

Valéry, ‘Verlaine and Poincaré’, 26. Krauthausen, ‘Paul Valéry and Geometry’, 237. Ibid.

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27

representative, here) takes the form of allegory, which I will discuss in the concluding section of this chapter.

Cantor’s Transfinite What is the ‘paradise’ that Cantor created? This phrase comes from David Hilbert’s famous call for mathematicians to resolve issues in ‘set theory’ with the rallying declaration that no one could ‘expel’ the mathematicians from the ‘paradise’ that Cantor had created.11 ‘Paradise’, here, describes the radical new potential offered to mathematics by the ‘set theory’ that follows from Cantor’s proofs, as well as a radical formalization of the foundations of mathematics. It is also, of course, a biblical reference. Hilbert here has replaced the Garden of Eden with set theory: a godless paradise that only the fecklessness of mathematicians can ruin. There is no deity to expel the mathematicians from the paradise: once the proof is in existence, its offer of paradise is eternal, and the challenge merely has to be met by future generations. There are two crucial co-implicated issues that are resolved in Cantor’s proofs: the first is the numerical instantiation of the actual infinite; the second is the stabilization of the number line and, most importantly and controversially, the ‘proof ’ of the continuum. Whilst both of these profound steps in mathematics resolved many deeply troubling conundrums in mathematics, these steps also generated a whole new domain of paradoxes: paradise had its own problems. In this section I will summarize Cantor’s discoveries and the interpretation of these discoveries as ‘modern’ or ‘modernist’, beginning with Cantor’s proofs of the ‘uncountability’ of the real numbers. Proving the discrete existence of numbers is a key metamathematical problem that has two significant implications. All mathematics can be built once you have a coherent theory of the whole number. But the problems that prevented such a theory proved insoluble until the development of Cantorian set theory. The need for a theory of the discrete existence of numbers and their 11

This famous and now oft-quoted reference to paradise originates in ‘Über Das Unendliche’, a lecture delivered in 1925. For a discussion of Hilbert’s phrase and its implication for notions of mathematical creativity see Mary Tiles, The Philosophy of Set Theory: An Historical Introduction to Cantor’s Paradise (Mineola, NY: Dover, 2004), 1.

28

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placement in a continuum arose from the existence of irrational numbers. Irrationals pose a problem for number theory as a whole because they never reach a limit; these are numbers with decimal places that never stop. The irrational is in other words a crevasse without floor between two whole numbers that thwarts attempts to prove the stability and consistency of the numerical continuum. If these numbers get ever more refined and ever smaller, the idea of counting upwards from one whole number through the decimals to the next number is an impossibility: a destination that keeps receding (a problem most famously represented in Zeno’s Paradoxes). Potentially infinite quantities disrupt arithmetic because they follow different rules: this other and uncountable universe of irrational numbers (and their association with infinitesimal quantities) would be demarcated from the countable infinity of whole numbers by Cantor. The most important theory that attempted to stave off the problems that irrationals posed for arithmetic, prior to Cantor’s work, was written by the French mathematician Augustin-Louis Cauchy (1789–1857). Cauchy suggested that the endless exactitude of the infinitesimal could be ignored by virtue of more important criteria for whole numbers: the limit. Cauchy demonstrated that whole numbers can be differentiated by virtue of the fact that they constitute a limit for a series: ‘When the successive values attributed to a variable approach a fixed value indefinitely so as to end by differing from it as little as is wished, this fixed value is called the “limit” of all the others’, but a limit that was not, crucially, a limit that affected whole numbers.12 Cauchy would stipulate the problem with irrationals in terms of a definition of numerical limit: ‘Thus an irrational number is the limit of the various fractions which furnish more and more approximate values of it.’13 The whole number would then be a limit of a convergent series. However, this theory would not overcome the impasse of the irrational, as Cauchy’s most important interlocutor in number theory, Karl Weierstrass (1815–1897) showed. This deeply attractive theory of numerical identity requires a definition of a real number; as such it does not prove the existence of such numbers in a continuum. Cantor’s stabilization of the 12

13

Cauchy is quoted in: Julian Havil, The Irrationals: A Story of the Numbers You Can’t Count On (Princeton and Woodstock, NJ: Princeton University Press, 2012), 238. Ibid.

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number line comes from, on one level, actually affirming the distinct identity of the irrationals from the natural numbers. Cantor proved that irrational numbers (along with all the category of all the real numbers) are uncountable. An irrational is simply any number that cannot be expressed as a fraction, or, more simply, a number with a decimal string that does not end; it is one of the types of ‘real’ numbers. The best-known example of an irrational number is π. Douglas Hofstadter aptly describes Cantor’s diagonal argument, which demonstrated the ‘uncountability’ of the irrationals, as a ‘twist’.14 Hofstadter’s classic text Gödel, Escher, Bach contains one of the clearest descriptions of the uncountability proofs: ‘What Cantor wanted to show was that a “directory” of real numbers were made, it would inevitably leave some real numbers out – so that actually the notion of a complete directory of real numbers is a contradiction in terms.’15 The transfinite numbers are ‘directories’ of infinite sizes. Cantor proved this by demonstrating that there is always one number not included in a register of all real numbers between zero and one. He did this by constructing a hypothetical table that assigns a number to all the real numbers (placed in rows) as such: 1) 1.1438934578935096783758349654374632944343228938472389 … (row 1)

2) 1.422895347543589345748935734895748395834756348975847 … (row 2) 3) 1. n … (row n) Cantor proceeds by identifying a number ‘d’ (here I am following, for convenience, Hofstadter’s notation) from a diagonal extraction from the decimal extensions (here, d could be constructed from the numbers that appear in bold). Here things get strange, and this is where the proof is perhaps best approached as a thought experiment. Imagine that 1 is subtracted from each of the decimals in the number ‘d’. This would produce a totally new number – and surely this ‘new’ number should be somewhere in that complete directory of real numbers? The astounding answer is that it is not. Why? Because we have subtracted one digit from ‘d’ it is now not the same as the original ‘d’ and it is different from the number in row one, row two, and, indeed, all the 14

15

Douglas R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid, 20th-anniversary edition (London and New York: Penguin Books, 2000), 420. Ibid., 421.

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rows of the table. It is a real number and cannot possibly be on the list of real numbers. More importantly, to use Hofstadter’s words: ‘The set of integers is just not big enough to index the set of real numbers.’16 In other words, to again make use of Mary Tiles’s wonderful line, the question ‘how many points in a line’ is answerable for the natural numbers, but not the reals: the number of real numbers is, and will by definition always be, uncountable.17 This point in the history of mathematics identifies what will come to characterize a new sequence in mathematics, involving a break from both classicist and logicist praxis, with significant implications for the philosophy of novelty and contingency.18 It is worth taking note at this juncture of the form of Cantor’s intuitive proof. The claim for uncountability here rests on a number being used in two ‘senses’: as the index and as that which is indexed. This diagonal ‘method’ of Cantor’s would be replicated in at least two of the most significant mathematical proofs produced in the twentieth century: Kurt Gödel’s ‘Incompleteness Theorems’ and Alan Turing’s ‘On Computable Numbers’. Both Gödel and Turing relied on the problems that arise from numbers being used both as an index and as that which is indexed, or, in other words, self-reference. Essentially diagonalization is a process of applying a principle to that very principle, or, in the words of mathematician Haim Gaifman, ‘Diagonalization then consists in applying to an object the function it represents.’19 The essential importance of self reference in Cantor’s proofs, and in subsequent developments in set theory, will become clear in the final section of this sub-chapter, where I address a literary form of self-reference: allegory, and its possible relation to a transfinite poetics or prose. 16 17 18

Ibid. Tiles, The Philosophy of Set Theory, 10. Most significantly, this uprooted stable Kantian critique as the predominant philosophical point of departure (idealism or another weaker correlationism as the only possible engagement with the ‘real’) and the Cartesian order of the world (the proof of an unintelligible but existent God, the cogito as the site of novelty and thought). Quentin Meillassoux summarizes this very well: It is precisely this totalisation of the thinkable which can no longer be guaranteed a priori. For we now know – indeed, we have known it at least since Cantor’s revolutionary settheory  – that we have no grounds for maintaining that the conceivable is necessarily totalisable. For one of the fundamental components of this revolution was the detotalisation of number, a detotalisation also known as ‘the transfinite.’

19

See: Quentin Meillassoux, After Finitude: An Essay on the Necessity of Contingency, trans. Ray Brassier (London and New York: Continuum, 2008), 103. Haim Gaifman, Chapter 0: The Easy Way to Gödel’s Proof and Related Topics, 2007, 2 http://www. columbia.edu/~hg17/Inc07-chap0.pdf (accessed 30 November 2014).

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There are two inflexions of the infinite, as Cantor in his Contributions to the Founding of the Theory of Transfinite Numbers (two texts published in 1895 and 1897) recognizes: the destination of the number line and the infinite quality of the irrational number. The former constitutes a countable infinity, whereas the latter does not. These countable infinities can be represented by a numerical notation: the transfinite numbers. A transfinite number is an actual infinity, without being the absolute infinity. Cantor represented the transfinite cardinal numbers with aleph letters, which linked the finite numbers that we use to count with their infinite potentialities, ‘fix[ing] the infinite’ as Quentin Meillassoux would say.20 A transfinite number is particularly difficult to understand, because, whilst it is an infinite number (in the sense that there is a transfinite number for any infinite set), Cantor wanted to reserve the absolute infinite for God. Transfinite numbers can be determined through the one-toone correspondence principle. In Morris Kline’s words, ‘Cantor’s greatness lies in his perception of the importance of the one-to-one correspondence principle and in his courage to pursue its consequences. If two infinite classes can be put into one-to-one correspondence then, according to Cantor, they have the same Number of objects in them.’21 By establishing a one-to-one correspondence between two sets (matching the objects of one set to the objects of another), one can tell the number of objects in the set (or whether one is greater or lesser). For instance (in an example used by Kline), the set a) 1 2 3 4 5 6 7 …

can be said to have the same number of objects as the set b) 6 7 8 9 10 11 12 …

simply because one can show that there is a 1–1 correspondence between these sets. By showing that some sets of numbers do not have a 1–1 correspondence, Cantor can prove that some sets are ‘larger’ than others. For instance, completely 20

21

Quentin Meillassoux, The Number and the Siren: A Decipherment of Mallarmé’s Coup de Dés, trans. Robin Mackay (Falmouth: Urbanomic, 2012), 99. Morris Kline, Mathematical Thought From Ancient to Modern Times, vol. 1 (Oxford and New York: Oxford University Press, 1990), 398.

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counter-intuitively, the number of squares and the number of natural numbers are the same, because for each natural number there is a square and hence both sets have a cardinality of ℵ0. However, ordinal numbers cannot be put in a one-to-one correspondence with natural numbers or squares and thus comprise a ‘larger’ infinity: ℵ1. Thus, Cantor demonstrates the existence of two different sizes of infinities, and thereafter extrapolating the possibility of not one infinite, but infinite countable infinities, of different ‘sizes’. Cantor numbered these different-sized infinities with what he first called ‘symbols of infinity’ and would later term the ‘transfinites’.22 Of the transfinite numbers, Cantor writes: Every aggregate of distinct things can be regarded as a unitary thing in which the things first mentioned are constitutive elements. If we abstract both from the nature of the things given and the order in which they are given, we get the ‘cardinal number’ or ‘power’ of the aggregate, a general concept in which the elements, as so-called units, have so grown organically into one another to make a unitary whole that none of them ranks above the others.23

In addition to cardinal numbers, which measure the size of these aggregates or ‘sets’, Cantor devised ‘ordinal’ numbers, which measure the ‘order type’ of sets. Infinite sets – say, for instance, the set of all even numbers – are represented by transfinite ordinal and cardinal numbers, which measure or ‘count’ the sets according to the 1–1 correspondence principle. Transfinite numbers are exempt from the problem that infinitesimals cause for whole numbers. Take, for example, ω (omega – the order type of natural numbers), which is the lowest transfinite ordinal number. Where ω is ‘the limit to which the variable finite whole number v tends’, then ‘ω is the least transfinite ordinal number 22

Akihiro Kanamori, ‘Cohen and Set Theory’, The Bulletin of Symbolic Logic 12, no. 3 (2008), 353. ‘1-1 correspondence’ has been known as ‘bijection’ since the Bourbaki account of set theory. Bijection is simply the existence of corresponding pairs in two sets. There is no bijection if two sets are of different sizes. Bijections can only apply to finite sets; for infinite sets one replaces such a bijection with a cardinal number. In the case of finite sets, a cardinal would just be a number counting the elements of a set, but in infinite sets, a transfinite number is used. The use of transfinite cardinals to describe infinite sets implies different ‘sizes’ of infinity, and hence implies different infinities.

23

Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers (Mineola, NY: Dover, 1955), 74.

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which is greater than all finite numbers; exactly in the same way that √2 is the limit of certain variable, increasing, rational numbers, with this difference: the difference between √2 and these approximating fractions becomes as small as we wish, whereas ω – v is always equal to ω’.24 This is thus not a matter of saying that whole numbers approach ω but rather that these numbers are all an equal distance from ω. The attribution of a transfinite number to the set of natural numbers (but not to the set of reals) stabilizes a continuum (a countable infinity) and solves the arithmetical problems associated with infinitesimals and numerical identity generally. In this, Cantor has introduced a positive account of the infinite – a numerical value or ‘actualization’ of the infinite – into mathematics.25 In David Foster Wallace’s words, after Cantor, the world ‘spins, now, in a new kind of all-formal Void’.26 This void cuts right to the heart of what can be considered true about mathematics, and the connection between the mathematical proof and the real. Cantor occupied a liminal position between romanticism and an emergent modernism: he was convinced, on the one hand, that his proofs were delivered to him by God, and went to significant lengths to involve the church in his work. But for the Catholic clergy of the nineteenth century Cantor’s proofs appeared heretical by virtue of what can be described as their ‘Platonism’: they posit actual infinities accessible to abstract mathematics, whereas theologically, the ‘actual infinite’ is supposed to be reserved for God. In this, Cantor appears as an Adam figure, grasping at forbidden knowledge in the Garden of Eden. It is this echo of the Platonic that divided the mathematical response to his theories. For Gauss, this Platonism was heretical in itself: infinity was only a manner of speaking and should not be formalized as an entity beyond a vague figurative suggestion. Whereas for Hilbert and others who sought, in the early decades of the twenty-first century, to axiomatize mathematics in a way consistent with Cantor’s proofs, this Platonism would make Cantor’s proofs heroic. 24 25

26

Ibid., 77. A Platonist would take all numbers to be actually existent, and this would necessarily include the transfinite numbers. As such, Platonist Cantor discovers rather than creates the transfinites. A formalist, on the other hand, would take Cantor to have created the transfinite numbers. Cantor believed squarely in the former: the numbers were actually existent, and this fact was made indisputable by his proofs. David Foster Wallace, Everything and More: A Compact History of Infinity, 2nd editio (New York: W. W. Norton and Company, 2010), 305.

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The contemporary French philosopher Alain Badiou has made the strongest case for Cantor’s breakthrough constituting both a triumph of modernity and a reintroduction of Platonism into thought. Unlike Jeremy Gray, Badiou does not argue for a modernism internal to mathematics, so much as a broad achievement of a ‘modernity’ in mathematics that echoes similar achievements in art and politics (the most notable comparison that Badiou makes is between Cantor and Mallarmé). As Andrew Gibson points out, although Badiou’s work fundamentally revolves around a philosophy of truth and event, he implicitly also provides a theory of modernity in the domains of art, love, science, and politics. In Gibson’s words, for Badiou, ‘The aesthetic domain catches up with the political domain with the first great modern writer, Mallarmé, with the emergence of Cantor, the scientific domain catches up too.’27 Badiou goes so far as to claim that the radical implications of Cantor’s theory were played out in literature independently of any direct engagement with mathematics. For Badiou, Mallarmé is the example par excellence here.28 Badiou argues that every age holds a certain conception of infinity and its place regarding divinity, destiny, materiality, and desire and Cantor’s ‘pure interruption’ in numerical succession constituted a radical break with those conceptions of his age.29 The infinite is no longer a ‘law without limit’ or a completed totality, but a measure outside of the limits of numeric repetition.30 In the light of the refutation of wholeness, the mathematicians of the early twentieth century developed an understanding of types of numbers, each with their own apparatuses by which one works with these numbers or arrives at these numbers. As such, Badiou claims that it was not mathematics but literature that first assumed the challenge of the modern: although mathematicians were bestowed with Cantor’s revolutionary truths, they suppressed the radical consequences of these truths in order to develop types of numbers. For Badiou, then, the transfinite numbers are a direct model of the achievement of the modern. But does the transfinite directly offer anything to literary innovation? Is there any bridge of connection between this mathematics and experimental language? There are two forms of ‘doubling’ 27

28 29 30

Andrew Gibson, Beckett and Badiou: The Pathos of Intermittency (Oxford and New York: Oxford University Press, 2006), 257. See, for instance, ibid., 13. Ibid. Ibid.

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and paradox in Cantor’s work that I take to be pertinent to other modes of regarding infinities, particularly the literary mode. The first instance of a doubling is this creation of an actual but not absolute infinity: just one mode in which infinities are manifest, according to Cantor.31 The mathematical transfinite doubles absolute infinity in an actual and countable realm, formalizing a paradox that constitutes a logos alogos.32 In Cantor’s work, mathematics is simply one version of the logos alogos: one means by which to regard infinities. This is a kind of allegorical structure: the ‘in abstracto’ is the mathematical other of a theological presence. We have an ‘actualization’ of that which – by traditional terms – is supposed to formalize the antithesis of actualization. Cantor’s proofs thus offer perhaps the most important model from which to explore other means by which to ‘regard’ infinities; inscribed in his very own proof is the notion of a finite supplement of an infinite in Deo. The second form of doubling and paradox occurs within the proof itself: Cantor uses numbers to index numbers, producing an impasse in the process of indexing and demonstrating that the real numbers are not countable. This

31

Cantor was convinced that the transfinite numbers were one form of the infinite: a mathematical form. He thus left room for a divine form of the infinite separate from this mathematical form. Dauben summarizes Cantor’s position on this succinctly: For example, there were a number of different ways in which Cantor felt the concept of infinity could be regarded. One of these was the form it assumed ‘in Deo extramundano alterno omnipotenti sive natura naturante’, or the Absolute. When the infinite served in this capacity he regarded it as capable of no change or increase, and he said that it was therefore to be thought of as mathematically indeterminable. Were it determinable, then it would have been limited in some manner.

32

See: Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princeton, NJ: Princeton University Press, 1990), 245. The term logos alogos seems to have been created by the Pythagoreans, according to Simóne Weil. In Weil’s words, One of the first problems they [the Pythagoreans] ran up against was that the diagonal of a square could not be expressed as a rational number. The proof of this is to be found in an appendix to Book X of Euclid’s Elements. The Pythagoreans used the Greek word logos to refer to a number, and seem to have coined the phrase logos alogos (a logos that is not a logos) to refer to the incommensurability of the diagonal of a square, which we now refer to as an irrational number. [Simóne Weil, Lectures on Philosophy, trans. Hugh Price (Cambridge and New York: Cambridge University Press, 1978), 116 ]. The term logos alogos has been used outside of this direct mathematical reference to the irrational numbers, however. In Dan Mellamphy’s work, it is used to refer to a ‘wordless language’ and to the ‘word unheard’ in Beckett’s work [Dan Mellamphy, ‘Alchemical Endgame: “Checkmate” in Beckett and Eliot’, in Alchemical Traditions from Antiquity to the AvantGarde, ed. Aaron Cheak (Melbourne: Numen Books, 2013), 552 ]. This extended meaning allows the term to refer to something coexisting in or as its antithesis.

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mode of self-reference is the same form that produces antinomies of all sorts, including semantic antinomies (most famously, the Liar’s Paradox: ‘all Cretans are liars’). The transfinite thus shows us a connection between certain forms of doubling (which occur by pairing an actual instantiation with an ‘uncountable’ or ‘transcendental’ pair; and the formalization of paradoxes within a proof by virtue of the ‘doubling’ of sense or self-reference) and the discovery of ‘actual’ infinities. I will begin to show, below, that literary ‘actual infinities’ share this model with Cantor’s mathematical transfinites: in this formal echo lies the most important link between the literary and the mathematical infinite.

Mallarmé and Meillassoux: Fixing the Infinite So what might a peculiarly literary transfinite look like? And what are the precedents for a theory of the literary transfinite? The most important precedent for the articulation of a literary transfinite, which I will assess here, emerges from a ‘discovery’ of a code in Mallarmé’s masterpiece Un Coup de Dés Jamais N’Abolira le Hasard (hereafter Coup de dés) by French philosopher Quentin Meillassoux. Meillassoux’s reading is both shocking and controversial, but also extends a substantial record of reading Mallarmé in terms of the infinite, of which Badiou’s interpretation is perhaps the most important recent instance. Meillassoux’s ‘deciphering’ of the poem is unique by virtue of the actual number he finds hidden in the poem, but also his explicit connection between infinity and chance. The Coup de dés has traditionally been read in terms of making the ‘invisible’ visible. Valéry, one of the poem’s first readers, recalled the experience of reading in terms of a concretion of the ephemeral: It seemed to me that I was looking at the form and pattern of a thought, placed for the first time in finite space. Here space itself truly spoke, dreamed, and gave birth to temporal forms. Expectancy, doubt, concentration, all were visible things.33

Meillassoux’s reading deals with this oscillation between what can and cannot be apprehended in terms of the poetic infinite. Un Coup de dés has 33

Paul Valéry, ‘Mallarme’s “Coup de Dés”’, in Selected Writings of Paul Valéry, trans. Cowley et al., 218.

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always been recognized as a profound break in poetry, but this new reading of the poem in terms of number suggests that the proto-modernist break with traditional poetic graphology and metrics, and the poem’s focus on the generic (the ‘only the place will have taken place’) is achieved via a poetic use of number. Meillassoux finds an endogenous code (located within the text) in the Coup de dés, rather than a kind of exogenous code (located in some external meaning).34 He thus uncovers simplistic, cryptographical ‘solution’ that gathers together every formal aspect of the poem under a single number, ‘707’. Mallarmé wrote the Coup de dés in response to a crisis in French verse: the introduction of vers libre that presented a threat to the traditional alexandrine. The alexandrine (or dodecasyllable) is a twelve-syllable line frequently split by a caesura after six syllables. The significance of the alexandrine in French poetry is roughly commensurate with the significance of pentameter in English verse.35 Mallarmé is perhaps most famous for transforming the poetic capacities of the line by severing the alexandrine and constructing nonlinear, non-sequential metre through resonance across words and sounds exploded from lines on the page, making the poem generate meaning and resonance across linear, diagonal, and graphological axes. However, this is not to say that Mallarmé was straightforwardly opposed to the alexandrine. He had no naïve notion of vers libre necessarily surpassing the systematic strictures of the past. Indeed, certain significant lines of the Coup de dés are rendered in alexandrine. Mallarmé understood vers libre to be central for poetic individuation and the alexandrine to be central to ceremonial purpose.36 There was, nonetheless, a crisis that Mallarmé thought poetry faced. ‘Literature here is undergoing an exquisite crisis, a fundamental crisis’, wrote Mallarmé in 1897, The faithful supporters of the alexandrine, our hexameter, are loosening from within the rigid and puerile mechanism of its beat; the ear, set free from an artificial counter, discovers delight in discerning on its own all the possible combinations that twelve timbres can make amongst themselves. 34 35

36

Meillassoux, The Number and the Siren, 5. See: Clive Scott, French Verse-Art: A Study (Cambridge and New York: Cambridge University Press, 1980), xii. Meillassoux, The Number and the Siren, 22.

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It’s a taste we should consider very modern.37 The Coup de dés manifests this freeing of the line from an ‘artificial counter’ most clearly. The poem consists of various syntactic threads but is equally dependent on a meticulous graphology. The Coup de dés is overtly governed by number: it begins and ends with dice being thrown. The captain of the ship, holding the dice, senses and anticipates the appearance of a ‘unique number unlike every other number’ as he oscillates over whether to throw the dice as the waters surge around him.38 The poem ends with the master engulfed by the whirlpool, with just the feather of his cap sitting above the water. A siren arises from the waves and beats her tail against the rock that destroyed the master’s ship, destroying it. We never find out whether the master throws the dice or not: the numerical image that we are left with, at the close of the poem, is of the Septentrion constellation, which is rendered in terms of a count: A CONSTELLATION cold with neglect and desuetude not so much that it fails to number on some vacant and higher surface the successive impact starrily of a total count in the making39

The poem concludes with the line: ‘Every Thought Emits a Throw of Dice.’ Despite the appearance of a radical form of liberated verse, the Coup De dés is in fact structured by a single number that is not the number twelve, as in the alexandrine, but a ‘unique number’ that Meillassoux claims is seven. The poem revolves around number in a more fundamental way than the simple use of a metrical arrangement. Meillassoux has, through a process of laborious, multidirectional counting, extracted a number that essentially structures the 37

38

39

Stéphane Mallarmé, ‘Crise de Vers’, in Mallarmé: The Poet and His Circle, ed. and trans. Rosemary Lloyd (Ithaca, NY; and London: Cornell University Press, 1999), 227–228. Stéphane Mallarmé, ‘A Throw of the Dice Will Never Abolish Chance’, in The Number and the Siren, by Meillassoux. Ibid., 273.

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Coup de dés. Meillassoux reads the poem numerically on a number of different levels: he counts the words in the poem (707), the last line (7), analyses the symbolism of the number seven in (the Septentrion) and the significance of ‘si’, the seventh musical note in the poem (amongst other significant suggestions). Meillassoux enumerates or deciphers several hidden references to the number, reading backwards and diagonally across the poem. Simultaneously, Meillassoux reads the poem as a reflection upon the task of poetry itself: the shipwreck is an allegory for the crisis of verse in France, the feather from the master’s cap is simultaneously the symbol of the writer’s quill, and the unique number that the poem produces is one that can replace the alexandrine metric: it is a poem about and for the future of poetry. Meillassoux’s claim that the Coup de dés possesses a secret numerical key is both ratified and complicated by the fact that – as Clive Scott in French Verse Art: A Study – puts it, ‘The French line is syllabic before it is accentual. That is to say, its unity is mathematical, and the rhythmic units or measures resulting from the accents which occur in the line are significant as fractions of the total number of syllables the line has.’40 This ‘mathesis’ of verse peculiar to French poetry goes a long way to establishing a precedent for precisely Mallarmé’s mathematical revision of the alexandrine. Mallarmé’s insistence on a crisis, in the above quote, is not so much a departure from implicit French metrical tradition as an insistence upon its very fundamentals, and this constitutes a radicalization from the inside rather than the outside.41 Whilst traditionally, in Scott’s words, it is important to remember that ‘accents are, of course, created by language’ not number, the mathematical organization of the poem still provides ‘customary sizes of measure’ and so ‘to know that one is reading alexandrines or octosyllables is already to locate oneself in a certain area of expectations, and to exclude certain possibilities’.42 There are two parts to Meillassoux’s discovery, however: the deciphering of the code and, then, the potential refutation of that deciphering. He locates a potential code in the poem and subsequently finds a string of ambiguities that call that code into question. In Meillassoux’s final analysis, the code is ‘undecideable’ 40 41 42

Scott, French Verse-Art: A Study, 17. Ibid., 17. Ibid.

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or, more accurately, it is both posited and effaced in the construction of the poem. The number, in Meillassoux’s terms, ‘quavers’.43 And it is this quavering that is the most significant trait of Mallarmé’s code; indeed, it is this element of contingency in the code itself that renders it part of an infinite poetic process. For Meillassoux, this ‘quavering’ is a realization of Mallarmé’s own convictions that any insertion of the divine into poetry must be through diffusion not representation. A ‘throw’ that properly diffuses chance into the poem would ‘produce a Number that presents the hesitation’, that ‘would be at one and the same time this Number premeditated by the count of the Poem – 707 – and not this Number at all, becoming […] a noncoded total’ or a ‘negative proof ’.44 And indeed the poem ratifies this: ‘The signs or the metaphors of Chance […] are not Chance itself.’45 And the code itself ‘quavers’ – for Meillassoux – precisely by virtue of Mallarmé’s choice to include it. By making the decision to include the code, a ‘bequest he made of his memory’, Mallarmé constructed a poem whose centrepiece was an irreducibly finite gesture.46 Meillassoux argues that this is precisely the nature of a ‘unique number’: ‘If the code was entirely unambiguous, the imperfect subjectives through which the Number is characterized (“were it to exist”, “were it to be ciphered”, “were it to be illuminated”, etc.) would cease to be pertinent once the code was decrypted.’47 In other words, a code that can properly instantiate a literature that makes the ‘invisible visible’ would fail at its task if, once decrypted, it simply made its secret visible once and for all. The code has to be perpetually hypothetical, subject to speculation without ever allowing itself to be defined once and for all. This ‘quavering’ of the code allows it to perpetually remain an instantiation of chance, just as, in the poem, we never know whether the master throws the dice.

43 44 45 46

47

Meillassoux, The Number and the Siren, 138. Ibid., 139. Ibid., 141. As such, the Number would indeed bring together all three properties of Chance: (a) it would contain two equally determinate opposites (707) and another number close to it, but without any relation to the code); (b) it would be eternal (the uncertainty is forever inscribed in the meaning of the Poem, since we can never determine the ‘correct’ solution); (c) it would be real (since it would refer to the act – perhaps effectuated, but undecidably so – of the man Mallamé coding the Coup de dés). (Ibid., 144) Ibid., 159.

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41

Mallarmé’s poetry is distinctive within symbolism in that his preoccupation with irreducibility was not wedded to the ideal as a whole pure form that poetry was the conduit of. Instead, for Mallarmé, the ideal is subtly replaced with a concept of universalism devoid of attributes or particularities, or to use Badiou’s words, a recognition of the truth that the One is ‘Not’. In other words, the ideal is not manifest as a totality but is instead a material order that is never fully determined or delimitable by language. This is summed up by Peter Broome and Graham Chesters, who note Mallarmé’s extension of Baudelaire’s idea of correspondences: ‘[Mallarmé’s] work takes to their most absolute and purified conclusion […] the belief that behind the phenomena of the material world lies an anterior, immutable Reality; that material forms, imprisoned in the plane of the contingent and accidental, testify nevertheless to a pattern of meaning and a mysterious order beyond themselves.’48 What is crucial in this description is recourse to contingency, which distinguishes Mallarmé’s preoccupation with the relation between the poetic surface and more fundamental orders. In order to fully understand Mallarmé’s relation to contingency, however, this quote needs to paired with the famous statement that forms the title of the poem we are concerned with, here: Un Coup de dés Jamais N’Abolira Le Hasard: a throw of the dice will never abolish chance. Symbolist literature, in Arthur Symons’s words, was a task of rendering the invisible world real: ‘After the world has starved its soul long enough in the contemplation and the re-arrangement of material things, comes the turn of the soul; and with it comes the literature in which the visible world is no longer a reality, and the unseen world no longer a dream’.49 In Mallarmé’s poem, emphasis on the unseen quickly finds its place in the silence inherent to verse, and the vagaries of style and the fixities of language suited to material world become a hindrance to the work of the poet. Hugh Kenner calls silence ‘the arch-Symbolist’s best-known preoccupation’, figuring language as a wrench preventing intimacy with the object: ‘To name is to destroy, thought Mallarmé.’50 48

49

50

Peter Broome and Graham Chesters, An Anthology of Modern French Poetry 1850–1950 (Cambridge and New York: Cambridge University Press, 1976), 35. Arthur Symons, The Symbolist Movement in Literature (New York: E.P. Dutton and Company, 1958), 4. Hugh Kenner, The Pound Era (Berkeley and Los Angeles: University of California Press, 1971), 136.

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This vision of contingency and ‘genericity’ must supersede that arbitrariness of naming for the contingency of a singular future. But how can a poem inscribe the contingency of the real that is necessarily beyond it? Simply through a suggested, but not fully secure, code posited at the heart of the poem? How can one produce a literature of the ‘perhaps’ – of contingency – rather than a literature of what ‘is’? This is one of the fundamental aesthetic questions of the nineteenth and twentieth centuries, and Mallarmé’s ‘quavering’ code is one poetic attempt to answer it. A literature of contingency offers a formal approach to an infinite art: one that can actualize the infinite ‘perfection beyond perfection’ rather than alluding to a temporal experience of eternity. In other words, can there be an art that is infinite or that thinks infinity rather than describing infinity or thinking about the infinite? For Mallarmé, any art capable of thinking infinity must also be an art that was based on chance. In this poem, it is the contingency of the code rather than the experience of endlessness that is the formal cipher for the infinite: chance, as perpetually indeterminate, offers a kind of poetic or textual ‘actualization’ of the infinite without having recourse to (finite) sentiment. For Mallarmé and in turn for Meillassoux, literary infinity would have to be realized through a combination of the absurd and the hypothetical, perpetually insecure idea: In short, in an act where chance is in play, it is always chance that accomplishes its proper Idea in affirming or denying itself. Before its existence, negation and affirmation are exhausted. It contains the Absurd – implies it, but in the latent state, preventing it from existing: which permits Infinity to be.51

Meillassoux would claim, thus, that the ‘unique number’ is at once literally manifest in the poem and, simultaneously, that it can never be directly ‘extracted’ – that it will always be suspended in chance. It is this double logic of the code that allows it to straddle a contradiction: the code will perpetually instantiate chance. Key to this is the relation between reader and the poem. Although Meillassoux does not put it in these terms, we might claim that the code manages to present an instance of ‘infinite chance’ and ‘unique number’ because the reader’s apprehension of the number will always exceed poetic evidence, and the 51

Mallarmé quoted in: Meillassoux, The Number and the Siren: A Decipherment of Mallarmé’s Coup de Dés, 29–30.

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poem’s dimensions will always exceed a single reading. It is the instability and disjunction between the reader and poem that Meillassoux is preoccupied with: the infinite chance is rendered only through a speculative reading of the poem. If we were to return Meillassoux’s reading of the Coup de dés to literary interpretation, rather than the ambiguous detective work of ‘deciphering’, we could recuperate this ‘ambiguity’ of the code in terms of the necessary dynamic between text and reader that is essential to the definition of allegory. This is not to say that the code of the Coup de dés is purely a product of reading, but that Mallarmé understood that the only way to compose a literary infinity is via allegorical form, and Meillassoux, without ever having recourse to allegory, is in fact rediscovering this form in his work. The definition of allegory that I refer to here is not the hackneyed, one-dimensional ‘resolution’ of a text via a single authoritative process but rather something quite different, a perpetual dialectic of reader and writer. Frederic Jameson sums up this vision of allegory best: our traditional concept of allegory […] is that of an elaborate set of figures and personifications to be read against some one-to-one table of equivalence: this is, so to speak, a one-dimensional view of this signifying process, which might only be set in motion and complexified were we willing to entertain the more alarming notion that such equivalencies are themselves in constant change and transformation at each perpetual present of the text.52

This broader definition of allegory and allegorical interpretation, which takes into account the contingency and variability of signification, is relevant to Meillassoux’s insistence on the unique number necessarily being ambiguous: the reading of the text oscillates between the clear presence of a number and the abolishment of the code (just as the Siren abolishes the rock upon which the master’s ship crashed). The ‘oscillation’ resembles precisely the polysemy and ‘constant change and transformation’ that Jameson designates as allegorical. It is clear that the poem directly addresses the crisis in French verse through the metaphor of the sinking ship, and suggests a solution through the figures of the siren and the Septentrion. In this sense the allegory is a ‘formal’ because it allegorizes the very processes of its own composition. 52

Frederic Jameson, ‘Third World Literature in the Era of Multinational Capitalism’, Social Text no. 15 (1986), 73.

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Formal allegory was most famously theorized by Paul de Man. For de Man interpretation is always an act of reading another meaning into the text and ‘any narrative is primarily the allegory of its own reading’.53 This rather complex circuitry of innate textual allegory happens via what de Man calls the ‘rhetorical model of the trope’.54 In its structure of deferred or displaced reference, the model of trope mirrors the model of reading itself. In de Man’s theory of literature and meaning all language involves a displacement between referent and significance. A symbol of an olive branch, for instance, refers to ‘reconciliation’, but this meaning is entirely independent of the olive branch itself. The referent – the olive branch – is independent of its significance, which occurs by virtue of interpretation. This displacement is essential to all language: its medium always skirts what it purports to capture. In de Man’s powerful rereading of this essential feature of language, all figural form in fact allegorizes its own reading: the very construction of the trope, which involves a divergence and path between referent and significance, is that same as the process that occurs between text and reader, words on a page and the instability of their interpretation. This is ‘formal allegory’: an instance where the text allegorizes its own formal processes. On one level, this occurs through Mallarmé’s direct engagement in this poem with a crisis in French verse: the entire ‘plot’ of the Coup de dés, if we can call it a plot, revolves around a complex allegory of the destruction of a poetic limitation (the rock) and the ascension to a celestial rather than earthly alternative (the Septentrion). On another level, however, a formal allegory occurs precisely in Meillassoux’s reading of the code in the text. Meillassoux discovers this anew when he shows that the presence of a code can never be ‘finalized’ – that the suggestion of the code will oscillate between certainty and uncertainty. The code inscribes, above all, the poem’s own limitations in establishing semantic meaning through reference to a number, to composition, that ultimately undergirds any ‘pure’ symbolism. Equally, the poem stages its own limitation: it cannot produce a ‘unique number that cannot be another’ but

53

54

Paul de Man, Allegories of Reading: Figural Language in Rousseau, Nietzsche, Rilke and Proust (New Haven, CT; and London: Yale University Press, 1979), 76. Ibid., 15.

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can allegorize this, through the ‘quavering’ placement of the number 707 in the text. Likewise, the throw of the dice so central to Meillassoux’s reading allegorizes the process of simultaneous chance and determination found in the very production of a poem: the inevitably unstable dialectic between writer and reader, intention and reception, composition and distribution. Meillassoux’s code, then, is ultimately allegorical in the same sense that de Man speaks of allegory. On one level, Meillassoux does recognize this in Mallarmé’s text (claiming it for eternity and infinity by virtue of its self-referential delineation of its own poetic task), but he falls into the trap of understanding this to be peculiar to a text with a code, rather than to poetic form that embraces a certain form of allegorical doubling. My reading of Meillassoux’s code as fundamentally a mode of allegorical composition and reading (more properly than a deciphering) does not refute his reading necessarily, but rather places it in a literary context that I take to be necessary to understanding a peculiarly poetic relation to the ‘unique number’. Understanding the unique number – 707 – in the Coup de dés as allegorical on the levels of both composition and interpretation allows us to recognize that the ‘literary infinity’ in Mallarmé’s poem is achieved through formal allegory. So, to ask the question again, then: What might a peculiarly literary transfinite look like? The hypothesis that I will test over the following chapters is this: a literary transfinite, a properly modern concept of the infinite in literature, will operate according to that ‘doubling’ process crucial to both Cantor’s uncountability proofs and Mallarmé’s poem; in the case of the former, it is the application of a notation system to itself, and in the case of the latter the application of a process of signification to itself. Of course, this same doubling process is found in de Man’s theory of allegory: when one applies linguistic bifurcation to itself on a different level (the phenomenology of reading rather than the formal processes of the text), we find an allegorical structure between the creation of meaning intra-text and extra-text. This doubling, here, will involve the allegorical structure that we see in Mallarmé’s work, and which Meillassoux elucidates through what he calls a ‘deciphering’. This formal realization of a literary transfinite will occur through what can be named ‘transfinite allegory’.

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Allegory and Enigma What unites the analyses by Meillassoux and Badiou is a conviction that the innovations and implications of Cantor’s work are not exclusive to mathematics. This is not to say, however, that the mathematical theory of multiplicity and the grasp of infinite extension are the same as forms or concepts of this in other domains. These two thinkers do not even make the claim that it is analogous: this would involve a ‘similarity’ between literature and mathematics that is not visible other than with extensive speculation. For Badiou, art and mathematics are irreconcilable but share a truth, and become modern at different times and through the procedures that follow from different events. And for Meillassoux, there is no direct link, only Mallarmé’s singular conviction that a ‘unique number’ could contribute to a poetic infinite. I will argue here that from a literary perspective the right form of comparison or linkage between these two domains is allegorical. Rita Copeland and Paul Struck provide a basic definition of allegory that indicates its origins in displaced speech: ‘The term itself has Greek origins: allos (other) and agoreuein (to speak in public), produce a sense of “other speaking.” In its most common usage it refers to two related procedures, a manner of composing and a method of interpreting.’55 The descriptor ‘allegorical’ can refer to both as a form of reading, which seeks the meaning of (or in) an ‘other speech’ as well as a mode of composition. But indeed, this double implication of the word is perhaps not double at all: the method of composing an allegory originates in a kind of creative reading of one ‘speech’ into another. Key to the entire process of both allegorical reading and allegorical composition is the consistency and autonomy of each narrative. Allegorical form rests on certain vision of composition, which anticipates a reader who can locate a meaning, a tale or a significance that is not fully manifest in the content of the allegory but is located outside of it. It is this aspect of allegory that was so important to Paul de Man, and which he ‘doubles’ by reading allegorical structure into the very form of language itself.

55

Rita Copeland and Peter Struck, The Cambridge Companion to Allegory (Cambridge and New York: Cambridge University Press, 2010), 2.

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Allegory has not always been privileged as a literary form: for Coleridge, famously, allegory was associated with ‘translation’ whereas symbolism was a form of ‘translucence’. Indeed, if we follow the narrative of Copeland and Struck, the nineteenth-century privileged symbolism over allegory based on the perceived aesthetic immediacy of the former and the deductive, conceptual requirements of the latter. Even in the criticism of the last quarter century allegory has remained a suspect category because, in Imre Szeman’s words, it is in danger of creating ‘a presumed passage from text to context that is epistemologically and politically suspect’ because of ‘the naive mode of one-to-one mapping that it seems to imply’.56 And yet the definitions of allegory produced by Jameson and de Man discussed above suggest a different potential in allegorical form and reading. In these definitions, what becomes interesting about allegory is precisely the absence of an immanent meaning contained within one form: it is the mutually exclusive coexistence of two ‘speeches’, two ‘narratives’ that are co-implicated precisely by being the ‘other’ of the pair. This ‘other’ is not a measure of difference (this story claims one ending, the other claims a divergent one) between the two texts, but a different existence of the text, perhaps on a different plane or as a different book. This structure befits the relationship between literature and mathematics: two utterly autonomous worlds that are bound together in their form and implication. At the same time, this structure befits the way that mathematics is used in literature; fiction requires an allegorical ‘doubling’ in order to introduce another logos into the representational scheme of fiction and properly produce its own infinity. Allegory has been related to both revelation and the pursuit of enigma. Here, number functions as the enigma of language for these three writers working within and after an era of modernism. A literary transfinite relies, as the mathematical one does, on a ‘doubling’ of its measures. Cantor’s diagonal proofs relied on a problem created by using numbers in ‘two senses’: on the one hand, as the index, and, on the other hand, as that which is indexed. My analyses over the following three chapters will show that where we find a 56

Imre Szeman, ‘Who’s Afraid of National Allegory? Jameson, Literary Criticism, Globalization’, The South Atlantic Quarterly 100, no. 3 (2001), 806.

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literary ‘transfinite’ (or, ‘actual infinity’), we also find just such a ‘doubling’ process. This doubling takes the form of an allegory, most importantly the peculiarly literary form whereby a text allegorizes its own formal processes. Just as allegory involves both a method of reading and a method of interpreting, so too does allegory occur in both directions in this study. This allegorical process is not only a theoretical model for thinking the interconnectedness between mathematics and literature. I’d like to suggest, here, that it is also the mode by which literature approaches mathematics. Here we have the Boromean knot between the transfinite in literature, mathematics, and the model that unites the two in literary theory: allegory. Allegory, and in particular formal allegory, is key to each of the writers that I analyse in the following chapters. Most importantly, this allegorical process enables them to produce prose that includes or approaches precisely that which exceeds it: totality, being outside of language, and above all the processes of counting and quantification that structure their prose. Cantor relies on a doubling of number in his uncountability proofs in order to prove the uncountability of real numbers and to show that the infinity of real numbers is larger than the infinity of natural numbers. It is the conjunction (and disjunction) of these two forms of counting that produces something radically new. In Mallarmé, the poet ‘doubles’ the metrical essence of the poem to produce a ‘quavering’ between chance and determination, finitude and eternity, a literary crisis and its potential but unfulfilled solution. In Borges, Beckett, and Coetzee, we will see a doubling of the task of prose literature through ‘transfinite’ allegory, which indexes the enumeration of the prose as it proceeds. This formal doubling produces a literature that revolves around its own enigma: number and the revelatory capacities of the transfinite.

2

The Aleph: Jorge Luis Borges and the Measure of Prose

There is a concept which corrupts and upsets all others. I refer not to Evil, whose limited realm is that of ethics; I refer to the infinite. Borges, ‘Avatars of the Tortoise’ For the great Argentine miniaturist, Jorge Luis Borges, numbers were essential to an artistic revivification of what he took to be the turgid prose of the early twentieth century. Indeed, the connection between numeracy and fictional worlds is key to his later prose and delivers the singular style of allegory that is perhaps his unique formal invention. On a poetic level, the young Borges felt he needed to break with the symbolism of the nineteenth century, and in terms of prose he needed to break with the other dominant literary paradigm: naturalism. His perspectives on late nineteenth-century poetry are apparent in the manifestos and experiments that came out of his participation in avantgarde groups in Spain the early 1920s. Whilst he and his family were stranded in Europe during the First World War, the young Borges became involved with the Ultraists in Seville and Madrid, returning to Argentina in the early 1920s convinced that primordial metaphor would rescue verse from poetic ornamentation. The Ultraists were modernists whose attempts to create spare, evocative verse that celebrated machinery and urban life echoed the work of the Imagists, who preceded them by less than a decade. Borges eventually disowned this ‘Ultraist’ enterprise, but it would provide the foundation for his persistent preoccupation with fictional worlds created without the suffocating overlay of symbolism, and his commitment to the construction of fictional worlds through systems of abstraction, suggestion and – most importantly – a

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mathematized imagination. Borges’s perspective on the saturation of the novel form is made explicit in his essay entitled ‘A Defense of Bouvard and Pécuchet’. In this essay, Borges argues that Flaubert, creator of the realist novel, is also the first to ‘shatter’ it.1 Borges also offers a connection between numerals in Ulysses and the death of the novel: Flaubert instinctively sensed that death [of the novel], which is indeed taking place (is not Ulysses, with its maps and timetables and exactitudes the magnificent death throes of a genre?), and in the fifth chapter of the work, he condemned the ‘statistical or ethnographic’ novels of Balzac and, by extension, Zola.2

Numbers, here, are the harbingers of the death of a genre. The statistical quality of Balzac and Zola’s narratives (an account of a reality) perhaps give a sense of the death of the genre, but it is Joyce’s self-conscious parody of these statistics in Ulysses that we see this most clearly. James Ramey, in a study of Borges and Joyce, agrees with Cesar Augusto Sagado that Borges’s relationship to the novel is ‘eschatological’.3 The performance and drama of the death of the genre, which he takes to be reflected in Ulysses (a book that Borges had, elsewhere, called a ‘miracle’) is signaled by the numeral.4 In Borges’s eyes, Joyce is attuned to the failure of the novelistic relation to the ‘actual’ instituted by Defoe, Fielding and others, which established characters as subjects of a measurable world, facilitating the immediacy that would make the novel such a cultural force over the following centuries. Borges needed a new relation between prose and number: one that did not enumerate the reality it sets out to describe or reduce realism to the staid presence of what Flaubert recognizes, in Bouvard and Pécuchet, as the ‘statistical’ or ‘ethnographic’ novels of Balzac.

1

2 3

4

Jorge Luis Borges, ‘A Defense of Bouvard and Pécuchet’, in Selected Non-Fictions, ed. Eliot Weinberger, trans. Esther Allen (New York: Penguin, 1999), 389. Ibid. James Ramey, ‘Synecdoche and Literary Parasitism in Borges and Joyce’, Comparative Literature 61, no. 2 (2009), 142. Borges reviewed Ulysses in superlative terms in 1925. See: Jorge Luis Borges, ‘Joyce’s “Ulysses”’, in Selected Non-Fictions, ed. Eliot Weinberger, trans. Suzanne Jill Levine (New York: Penguin, 1999), 13.

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Crucially – and this is where Borges is both bold and original – this did not require a flight from the matheme but an embrace of it. Fiction and mathematics needed to be reciprocally liberated, rather than wrenched apart. Borges’s grasp of the reciprocal relation between the two disciplines would be resolved in his short stories through an unprecedented use of transfinite allegory, a mathesis of fiction.5 The exactitudes of realism, which for Borges had beached the leviathan genre, were to be alleviated by a new, modern vision of that which compromised the novel in the first place: the number. Borges was a reader of mathematical texts, including Russell and Whitehead’s Principia Mathematica and Kasner and Newman’s Mathematics and the Imagination, and he engaged directly with the issue of an actual infinity in his essays and stories. He had an explicit and sustained engagement with Cantor’s transfinite numbers, naming one of his most important collections of stories The Aleph, after the symbol for the transfinite.6 In his 1940 book review of Mathematics and the Imagination, he predicts that, in addition to ‘Mauthner’s Dictionary of Philosophy, Lewes’s Biographical History of Philosophy, Liddell Hart’s History of the War of 1914–1918, Boswell’s Life of Samuel Johnson and Gustav Spiller’s psychological study The Mind of Man’ this volume was likely to become one of the works that he has ‘most reread and scribbled with notes’.7 Borges was particularly well versed in the consequences of Cantor’s theory of the infinite, and its differentiation from vaguer, non-mathematical visions like Nietzsche’s ‘Eternal Return’. In an essay entitled ‘The Doctrine of Cycles’, Borges pits Cantor’s transfinites against the Eternal Recurrence of Nietzsche’s Zarathustra. ‘[Cantor] asserts the perfect infinity of the number of points in the universe, and even in one meter of the universe, or a fraction of that meter.

5

6

7

For Michel Foucault, a ‘mathesis’ was the ‘general science of order; a theory of signs analyzing representation; the arrangement of identities and differences into ordered tables’. A ‘universal method’ of mathesis is, for Foucault, ‘algebra’. See: Michel Foucault, The Order of Things: An Archeology of the Human Sciences (London and New York: Routledge Classics, 2002), 79. Floyd Merrell suggests that there is a crucial difference between Borges and Cantor: Cantor hoped that the paradoxes of his set theory would be resolved, making a total system. Borges, on the other hand, had no such hope, and revelled in the paradoxes. See Floyd Merrell, Unthinking Thinking: Jorge Luis Borges, Mathematics and the New Physics (West Lafayette, Indiana: Purdue University Press, 1991), 61. In this review, Borges also refers to Bertrand Russell’s classic Principia Mathematica, from which he took copious notes. See: Jorge Luis Borges, ‘From Allegories to Novels’, in Selected Non-Fictions, ed. Eliot Weinberger, trans. Esther Allen (New York: Penguin Books, 1999), 337.

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The operation of counting is, for him, nothing else than that of comparing two series,’8 Borges explains. His illustration, as ever, is vivid: For example, if the first-born sons of all the houses of Egypt were killed by the Angel, except for those who lived in a house that had a red mark on the door, it is clear that as many sons were saved as there were red marks, and an enumeration of precisely how many of these there were does not matter. […] The set of natural numbers is infinite, but it is possible to demonstrate that, within it, there are as many odd numbers as even.9

In Borges’s argument, Nietzsche’s eternal return (or ‘eternal recurrence’) is impoverished because it falls into the trap of assuming that an infinite number of particles cannot be presented in an infinite number of ways due to the existence of finite rather than infinite force. Eternal Recurrence is a product of Nietzsche’s attempt to rid the world of a type of thought that presumes that the world resembles a god, or appears through the will and intervention of a god. For Nietzsche, novelty was not infinite – and external recurrence hence inevitable – because a godless universe does not contain infinite force. If it did, the universe would still be ruled by a deity. What is attractive for Borges in Cantor’s theories of the infinite is that the profusion of forms of counting negate Nietzsche’s single count. For Cantor, the infinity of natural numbers and the infinity of points in space belong to different sets and are incommensurable. These counts give way, for Borges, to an infinite generation; an intellectual prospect that Nietzsche’s Eternal Recurrence is devoid of because he only accounts for one infinity, not multiple infinities (or perhaps ‘transfinities’) of different sizes. This modern vision of infinite generation is more than a simple intellectual intrigue for Borges. Instead, these Cantorian proofs of ‘actual infinites’ and generic difference will provide Borges with a prose form that possesses the vitality that he felt the novel and poetry had lost. Echoing the formal allegories used in the uncountability proofs, in this chapter I will call Borges’s formal solution to the problem of literary finitude transfinite allegory. 8

9

Jorge Luis Borges, ‘The Doctrine of the Cycles’, in Selected Non-Fictions, ed. Eliot Weinberger, trans. Esther Allen (New York: Penguin, 1999), 116. Ibid., 116–117.

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Transfinite allegory allows for exactly what Cantor’s portmanteau suggests: the presentation of that which is beyond finitude but not – like God – absolutely exclusive of any finite representation. Literature itself cannot be infinite: it cannot create or be a total world because it is bound to the finitude of reading.10 Anthony Cascardi has argued that rather than replicate worlds, Borges’s approach would be to create worlds. ‘The work of art,’ Cascardi writes, can ‘reassert its claim to be something more or other than a mimesis of the world, in part by reflecting on the impossibility of it ever being a full and complete mimesis of the world’.11 In this vein, Cascardi claims that the ‘imperfections’ worked in to Borges’s worlds ‘suggest how art remembers what it was like to be a world, not just to be like the world’.12 This is a kind of refusal to ‘clone’ the world through description, one that entails a refusal of broader representational completion. These imperfections, in Cascardi’s argument, speak to art’s ‘memory’ of the time in which it did create a world: the time of myth and epic that preceded the Romance and the novel. This mimesis becomes the product of another mimesis: a world contingently created from another. This is an act not of linear progression but ‘permutations and combinations’ and especially, I would add, transfinite numbers.13 The question of mathematics in Borges’s work has been explored from a variety of angles. William Goldbloom Bloch’s monograph, The Unimaginable Mathematics of the Library of Babel, is the most important and wideranging extended study of mathematics in Borges’s oeuvre.14 Bloch’s work on mathematics in Borges’s stories pursues not a coherent narrative of mathematics and literary form but rather embraces and describes the eclectic mathematical forms relevant to Borges’s stories. The second full-length study of mathematics

10

11

12 13 14

There are works that contradict this, though, most notably John Cage’s Organ/As Slow as Possible. For an analysis of this work in terms of the finitude of the experience of the artwork, see Lendl Barcelos, ‘The Nuclear Sonic: Listening to Millennial Matter’, in Aesthetics After Finitude, ed. Baylee Brits, Prudence Gibson, and Amy Ireland (Melbourne: Re-Press, 2016). Anthony Cascardi, ‘Mimesis and Modernism: The Case of Jorge Luis Borges’, in Literary Philosophers: Borges, Calvino, Eco, ed. Jorge J.E. Garcia, Carolyn Korsmeyer, and Rodolphe Gasché (New York and London: Routledge, 2002), 116. Ibid., 116 Ibid., 113. William Goldbloom Bloch, The Unimaginable Mathematics of Borges’ Library of Babel (Oxford: Oxford University Press, 2008).

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in Borges’s work takes a similar approach. Guillermo Martínez’s Borges and Mathematics provides a full index of the range of mathematical themes in Borges’s work and, like Bloch’s study, investigates eclectic manifestations of mathematics in the short stories. The themes that Martinez investigates include infinity, abstraction and concretion, and artificial intelligence. Martinez concludes by developing a direct connection between literary form and mathematics: the short story, for Martinez, is a ‘logical system’; the link between the fiction and mathematics resides in the function of the author as a ‘manipulator of systems’.15 In this chapter, I will not attempt a catalogue of the diverse mathematical forms in Borges’s oeuvre. Instead, my focus is on a particularly mathematical phenomenon: the existence of transfinite form. Floyd Merrell links ‘the demise of totalising narratives’ that Cascardi is speaking of directly to Cantor’s infinities, citing Cantor’s remark that ‘the least particle contains a world full of an infinity of creations’.16 There are two problems, here: a literary one and a mathematical one. The first is the problem of the literary creation of novelty and the state of Barthesian ‘exhaustion’ that Cascardi cites: the state of cultural saturation whereby literature comes to take other literature as its subject matter. The second is the mathematical paradox of completing the ‘set of all sets’. Merell defines this perfectly in his gloss of Cantor: ‘Everything is contained in everything else.’17 Looking at this through the combined perspective of literature and mathematics, this appears, then, not as recollection of a lost mimesis (as Cascardi feels it does) so much as a mathesis of literature. This is not a restitution of a descriptive totality, but a modern awareness of the mathematical presence of multiple totalities, each of different measure, and the representational demands made by this new concept of infinite. Here, the problem of literary creation and the prospect of mathematical modernism come together. In each case what is required is the transfinite operation upon the totality of writing, or, to follow Cantor, the totality of sets.

15

16 17

Guillermo Martinez, Borges and Mathematics, trans. Andrea Labinger (West Lafayette, IN: Purdue University Press, 2012), 69–70. Merrell, Unthinking Thinking: Jorge Luis Borges, Mathematics and the New Physics, 61. Ibid.

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Supplanting the Symbol for the ‘Thing Itself ’: Borges’s Ultraist Beginnings Borges occupies an enigmatic position on the fringes of modernism. He was an eclectic and highly idiosyncratic reader, and his influences and allusions render his work anomalous in terms of larger literary movements of the early twentieth century. Nonetheless, he was also an active and formative member of several literary collectives and journals in Spain and Argentina, and his early years as a student were formed by collective endeavours to move away from the limitations of nineteenth-century symbolism and initiate a new form of poetry focused on a generative rather than evocative practice of symbolism. Borges was associated, in 1921, with Ultraist writers of Madrid and Seville who opposed Modernismo, the aesthetic ideological cluster that Spanish poetry had orbited around since the 1890s, whose most famous practitioner was the Nicaraguan poet Rubén Darío (1867–1916), who was responsible for inspiring a wave of Modernismo writing in Latin American writers including Jose Marti, Julian del Casal, Salvador Díaz Mirón, José Asunción Silva and Manuel Gutiérrez Nájera.18 The term ‘Modernismo’ is not the Spanish version of ‘modernism’; in fact, Modernismo, although broadly linked to European modernism, describes an earlier phenomenon in Spanish literature characterized by reaction against the brute materiality of quotidian modern life, and hence an attraction to the transcendental possibilities that the Symbolists placed in poetry and the Parnassian refutation of positivism.19 The Ultraists were a self-styled avant-garde group whose manifesto for poetry also harked back to the Parnassians, revolving around the poet’s work as creation rather than representation, and a poetry that had the capacity to generate a certain image, experience or state, above and beyond any mimetic endeavour. They gathered at the Café Colonial in Madrid, their conversations and publications revolved around renewing

18

19

Kelly Washbourne, ‘Introduction’, in After-Dinner Conversation: The Diary of a Decadent, ed. José Asunción Silva (Austin: University of Texas Press, 2005), 8. See Gwen Kirkpatrick’s discussion of this in: Gwen Kirkpatrick, The Dissonant Legacy of Modernismo (Berkeley and Los Angeles: University of California Press, 1998), 38–43.

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a symbolism that they took to have become ornate and rigid within Modernismo. In Borges’s own words: Ultraism in Seville and Madrid was a desire for renewal; it was a desire to define a new cycle in the arts; it was a poetry written as if with big red letters on the leaves of a calendar and whose proudest emblems – airplanes, antennae, and propellers – plainly state a chronological newness.20

This renewal involved a shift to ‘modern’ images and content, in part, but also involved a direct reformation of the symbol. The Ultraists sought to wage a literary war against the excesses of sentiment that they saw committed in Modernismo, and did this through an attempt to generate a primordial sort of metaphor through poetic conjunctions: ‘The Ultraist movement’s […] programme affirmed the image as the fundamental element in poetry, the abolition of logical and syntactical links, and the brevity of the poem as a formal proof of the condensation of meaning.’21 For the Ultraists, primordial metaphor was to replace stock symbols to generate a new, revitalized poetic idealism. Each instance of primordial metaphor would produce a moment of transcendental unity, very much in line with symbolist visions of the infinite, or the ideal ‘One’. The influence of the Symbolists and the Parnassians upon Modernismo and Ultraism is complicated: there is no broad base for the influence of either of these groups. Ultraism was a reaction against Modernismo, which was deeply influenced by the French Symbolists and Parnassians, and yet Ultraism would develop its own relation to the symbol and the ideal that also profits from the legacy of those same groups. Borges took the movement back to Argentina with him, publishing an Ultraist manifesto in a literary journal that he initiated. This journal was called Prisma: Revista Mural and was printed on big poster-sized cards of brown paper, the poems flanked by woodcuts by his sister, Norah. There were only two issues of Prisma: one from December 1921 and the other a few months later, in March 1922. Upon seeing the posters 20

21

This quote was published in Inquisitions in 1925, and the translation cited is from Emir Rodiguez Monegal See: Emir Rodiguez Monegal, Jorge Luis Borges: A Literary Biography (New York: Paragon House Publishers, 1988), 173. Beatriz Sarlo, Jorge Luis Borges: A Writer on the Edge, ed. John King, 2nd ed. (London and New York: Verso, 2006), 113.

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Alfredo Bianchi, editor of the established literary magazine Nosotros, invited Borges and his collective to publish a special issue of Nosotros on Ultraism. Borges published another manifesto in this commissioned issue of Nosotros entitled ‘Ultraismo’, in which he outlined the tenets of the movement. Beatriz Sarlo paraphrases the three key elements of this manifesto: ‘Its programme affirmed the image as the fundamental element in poetry, the abolition of logical and syntactical links, and the brevity of the poem as a formal proof of the condensation of meaning.’22 Ultraism involved an approach to poetry that focused on the internal coherence of the poem (rather than external evocation) and the use of archetypes which solved the problem of tropological language in terms of hierarchy. The archetype was a primordial signifier, and thus not an efflorescent or ornamental use of language, but one purportedly connected to the very origins of subjectivity. This at once aligns Borges with certain more successful strands of European literary modernism and at the same time differentiates him from these. For instance, Ezra Pound, in the essay ‘Imagisme,’ notes that the first rule of the Imagists is the ‘direct treatment of the “thing”, whether subjective or objective’: a rule that went along with the embrace of free verse under the banner of poetic necessity, as opposed to poetic ‘dilettantism’.23 It is this simultaneity of treatment of the thing itself and the emphasis on exactitude that resonates with and predates the Ultraist manifestos. And yet Ultraism premised this form of direct verse upon an idealism antithetical to most modernism: the idealism of ‘primordial signification’ and ‘pure style’. It is in this ambiguous status of ultraism that we see an early Platonism in Borges, albeit one still concerned with a sublime absolute. Ultraism, seeped in a still romantic idealism, would be preoccupied with only one number: one. In what follows I will show that Borges does not entirely reject the fantasy of pure style after his Ultraist phase, though he holds it to be no longer exclusive of erudition, history and a certain baroque effect. Rather, Borges uses number to buttress a version of pure style that is not puritan, anti-intellectual or, indeed, totalitarian. His move away from Ultraist ‘symbolism’ will allow him to retain a preoccupation with the internal coherence of the text and the inclusion of 22 23

Ibid., 113. F.S. Flint, ‘Imagisme’, Poetry: A Magazine of Verse 1, no. 6 (1913), 199, 200.

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the generative principles of fiction within the text itself. The drive for textual autonomy in ultraism constitutes Borges’s first steps towards an art that – to again use Cascardi’s phrase – asks, ‘What it is like to be a world?’ In his later work, this project comes to involve a fictional incorporation of both an origin and the world it produces instead of being couched in a still-romantic idealism that seeks a link to a transcendental real where metaphors or allegories are secured. Allegory here becomes relevant not because there is a connection to a primordial meaning or realm of narrative but because it performs a formal doubling that incorporates a reflection of the processes of composition in the text.

‘Funes, His Memory’: A Transfinite Technogenesis of Perception ‘Funes, His Memory’ (hereafter ‘Funes’), first published in La Nacion on the 7 May 1942 and subsequently in the collection Artifices in 1944,24 presents a brief, exquisitely crafted fictional exploration of the phenomenal and representational significance of number.25 The story takes the form of a memorial to a dead Uruguayan who was subject to an extraordinary mental transformation as a result of a horse riding accident. The story is narrated in the first person by an unnamed narrator who is never entirely dissociated from Borges himself. This narrator describes his two trips to the Uruguayan town of Fray Bentos and his three encounters there with Ireneo Funes, an adolescent at their first meeting, and later a young man. The first encounter between our narrator and Funes, a gaucho who has ‘the delicate fingers of the plainsman who can braid leather’, is brief but haunting.26 24

25

26

Here I use Andrew Hurley’s translation. This story has also been translated by James Irby as ‘Funes, the Memorius’, which bears a more immediate resonance with the Spanish title ‘Funes el memorioso’ (spelt as either ‘memorius’ or ‘memorious’ depending on the translator). Andrew Hurley decided to depart from the Spanish title in order not to introduce a strange or idiosyncratic title that might suggest more or otherwise than Borges had intended with his Spanish title. Although Hurley’s translation is a superior translation, his title loses the evocation and mystery of Irby’s ‘memorius’. In Spanish, ‘memorious’, as Hurley notes, is not an uncommon word and signifies a ‘powerful memory’. Whilst this is true, I feel it is imperative to include the appellation for Funes in the title (even if the English word is perhaps rather too suggestive). Here I use the version in the Collected Fictions. Jorge Luis Borges, ‘Funes, His Memory’, in Collected Fictions, trans. Andrew Hurley (New York: Penguin Books, 1998). Monegal, Jorge Luis Borges: A Literary Biography, 56, 137.

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In 1884, the narrator visits Fray Bentos and goes out riding with his cousin, Bernardo. Storm clouds are gathering, ‘the wind flailed the trees wildly, and I was filled with the fear (the hope) that we would be surprised in the open countryside by the elemental water. We ran a kind of race against the storm’.27 Against this backdrop, they encounter a boy running along a wall. Borges’s cousin knows the child, and asks him the time; ‘Unexpectedly, Bernardo shouted out to [the boy] – Whats the time, Ireneo? Without consulting the sky, without a second’s pause, the boy replied, Four minutes till eight, young Bernardo Juan Francisco.’28 The boy is described as incredibly haughty, his voice ‘shrill and mocking’, and through this brief exchange he is marked as an uncanny figure with extraordinary capabilities for the abstract measure of time.29 When the narrator, years later, returns to the town, it is this strange and brief encounter that prompts him to ask after ‘chronometric Funes’.30 The appellation ‘chronometric Funes’ registers Funes’s capacity to count time as an attribute or a qualifier, and not yet a proper epithet as in ‘Funes, the Memorius’. Here, ‘chronometic’ implies a function, the measure of time, where metric is: ‘a binary function of topological space that gives, for any two points of the space, a value equal to the distance between them, or to a value treated as analogous to the distance for the purpose of analysis’.31 This definition is helpful in emphasizing the way in which Funes operates in Borges’s and the readers’ first encounter with him: he traverses an abstract logic of space through a binary function. A ‘binary function’ is derived from the distance between a two-part structure comprising of two points separated on a topological map, here hours, minutes or seconds (3.58 am to 3.59 am, or x to y). In this case, the function is the analogue of the progression of time according to a clock. The young Funes manages – without visible effort or transition – to abstract from the situation he is in to tell the time. Here, he is effectively able to tie continuous phenomenal existence to inorganic measure in a profound and extraordinary way, forcing a literal connection between phenomenality and its measure, or,

27 28 29 30 31

Borges, ‘Funes, His Memory’, 132. Ibid. Ibid. Ibid. Oxford English Dictionary, 2nd ed., s.v. ‘metric’.

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in other words, presence and presentation. This is a pure, isomorphic analogy, unlike the type of analogy usually found in fiction and poetics: this metric for temporal experience is perhaps the purest form of symbolism, formalizing a perfect coherence between a mode of measure and temporal existence. At the time of the narrator’s second visit to Fray Bentos, Funes is crippled and bedridden after a horse riding accident. But this physical debility pales in the light of the profound cognitive transformation that the accident has triggered in him. After being knocked unconscious in the accident, Funes awakes to find ‘the present was so rich, so clear, that it was almost unbearable, as were his oldest and even his most trivial memories’.32 Our narrator informs us that Funes remains incredibly proud, yet also experiences a fundamental transformation of self. If Funes is still proud and haughty, it could only conceivably be as a defence mechanism, or a relic of his former self: Funes’s memories are so acute that regardless of any physical impairment he is immobilized by the perfection and acuity of each of these memories. His transformation is a sort of ‘second birth’, but a curious one that does not fully inaugurate a new life: He had lived, he said, for nineteen years as though in a dream: he looked without seeing, heard without listening, forgot everything, or virtually everything. When he fell, he’d been knocked unconscious: when he came to again, the present was so rich, so clear, that it was almost unbearable, as were his oldest and even his most trivial memories.33

This characterization of Funes’s earlier life is less an explanation of a peculiarly amnesic youth than a comparison with his new state: in contrast to this new total and unrelenting awareness of detail, Funes did forget virtually everything. Now, he struggles to bear the acuity of the present. Indeed, we might summarize this simply by observing the catechrestic movement between the minute (as in the measure of time) and the minute (as in the profoundly intricate or tiny) that goes on between the two encounters with Funes. The capacity for counting that Funes possessed before his accident is an achievement in abstract and continuous thought: the boy running along the 32 33

Ibid., 135. Ibid., 134–135.

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wall seemed to be able to maintain a remarkably exact track of time (a form of counting) that was not dependent on any external factors other than a constant measure. In Funes’s post-accident existence, he is subject to an extraordinary eidetic memory, to a constant awareness of the minute: With one quick look, you and I perceive three wineglasses on a table; Funes perceived every grape that had been pressed into the wine and all the stalks and tendrils of its vineyard. He knew the forms of the clouds in the southern sky on the morning of April 30, 1882, and he could compare them in his memory with the veins in the marbled binding of a book he had seen only once, or with the feathers of spray lifted by an oar on the Rio Negro on the eve of the Battle of Quebracho.34

Fascinatingly, the narrator delivers the sense of randomness through specificity: the inclusion of a date in the above quote emphasizes allencompassing memory, where even the most arbitrary day or scene is subject to an extraordinary, or apparently infinite, retentional exactitude. This exactitude is not only retentional, though: Funes also has an exquisite sense of detail that exceeds memory; he is aware of the nuances of time and space that memory passes over: ‘His perception and his memory were perfect.’35 Of course, this sense of arbitrariness is conveyed through number; the use of digits in the text emphasizes both the apparently arbitrary and precise nature of Funes’s memory. This strategic detail in the memorial brings into sharp relief the extremity of Funes’s experience. Every object appears as a veritable barrage of other events, objects and forms; it cannot be determined by a single idea or name. This inability to phenomenally arrest the stable, singular thing is represented by number and – where number does not suffice – through the rhetorical use of ‘every’ and ‘all’. Funes remembers not just a book he once encountered but all the veins in the binding: detail too inconsequential, miniscule and undifferentiated for normal human perception. The narrator compares this state of perception to our own geometric intuitions: ‘A circle drawn on a blackboard, a right triangle, a rhombus – all these forms we can fully intuit; Ireneo could do the same with the stormy mane of a young colt.’36 This radical intuition becomes a 34 35 36

Ibid., 135. Ibid., italics mine. Ibid.

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form of phenomenal nominalism, where perception is so acute it disqualifies abstraction. In this sense, there is also radical equality of objects: the detail of an apparently random cloud formation on a certain afternoon many years ago is as acute as any other memory or event in Funes’s past, regardless of significance. This might seem to be an exacerbated awareness of singularity, in the sense that the singularity of each object actually precludes an abstract grasp of the quality of singularity as such. Yet Funes, immune to abstract thought, has no concept of the singular but purely an unrelenting perception of it. The distinction between Funes’s two conditions is at once considerable and barely distinguishable: although there is a radical difference in the perception of detail, in both cases Funes is not so much capable of an abstract traversal of space and time, but rather is such a traversal. In the first instance, however, the function that accompanies Funes’s existence is tied to a calculation system; in the second instance, he must submit to the anarchy of pure memory. The simple difference here is the presence of a continuum in the former and a plenum in the latter. ‘Funes’ thus revolves around the moment of correlation between mathematical order and phenomenal awareness. Abstract thought is lost to Funes, and as such he is suspended by what we might call – to modify French philosopher Bernard Stiegler’s term somewhat – a ‘retentional infinitude’ that leaves him at the mercy of his extraordinary memory. ‘When life becomes technical,’ Stiegler writes, ‘it is also to be understood as “retentional finitude”.’37 In Stiegler’s theory, retentional finitude refers to the finitude of memory, or the technical boundaries of memory; it refers to the fact that memories cannot be contained by the wider entity of memory but are mutable and subject to loss.38 The notion of retentional finitude is closely related to technics. For Stiegler, the human experience of time is supplemented by technics: by inorganic measure. Measured time is a technical division of reality into regular sections: it is the insertion of a metric into rhythm (and in this sense, despite the fact that the metric is inserted into rhythm, the conceptual externalization of rhythm succeeds the metric, in Stiegler’s terms, ‘techno-genesis structurally precedes socio-genesis’39). The 37

38 39

Bernard Stiegler, Technics and Time, 1: The Fault of Epimetheus, trans. Richard Beardsworth and George Collins (Stanford, CA: Stanford University Press, 1998), 17. Ibid. Ibid.

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younger Funes, running along the wall, has an extraordinary technical cognitive facility: he is able to maintain a perfect equivalence between his phenomenal existence and the ordering abstraction of time. The older Funes is paralysed in a cognitive world where the overwhelming beauty and acuity of each lived scene possesses him, rather than Funes being able – through abstraction and categorization – to possess the memories. Funes does retain aspects of his former self: he can still determine correspondences between phenomena in time, but can now ‘rewind’ or pause this: he can move back and forwards in memory, zoom in or out and is not subject to the movement of the clock. This is not an antinomial transformation but a developmental one. The significant words, here, are ‘rewind’ and ‘pause’. ‘Funes’ developmental arc is also the arc of the new ‘technics of time’ and aesthetic duration found in the filmstrip. (And, notably, the other name for an eidetic memory like that of Funes is, of course, a ‘photographic’ memory.) It is here that we begin to see the tripartite arrangement between literature, mathematics and other representational technologies. ‘Just as the entire mode of existence of human collectives changes over long historical periods, so too does their mode of perception’, writes Walter Benjamin in ‘The Work of Art in the Age of its Technological Reproducibility’.40 There are two key changes in perception at work in this scenario: the first, as I have suggested, is mathematical, the second is cinematographic. These two are bound up together at the intersection between perception and calculation. At one point in this story, the narrator bemoans the fact that in Fray Bentos there were no phonographers and no cinematographers to capture Funes’s abilities. Funes thus exists, literally in his two conditions and figuratively in his rural life, on the border between two regimes of technicity: the cinematographers were not in Fray Bentos for Funes yet, but the future of retentional finitude would be forever altered by the new perceptive regime of the filmstrip; in this Funes’s condition is prophetic. Subject to retentional infinitude, Funes is cognitively – not only physically – paralysed. His state of exquisite memory has made him ‘mechanised’. This may

40

Walter Benjamin, ‘The Work of Art in the Age of Its Technological Reproducability’, in The Work of Art in the Age of Its Technological Reproducibility and Other Writings on Media, ed. Michal W. Jennings, Brigid Doherty, and Thomas Y. Levin (Cambridge and London: The Belknap Press of Harvard University Press, 2008), 23.

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seem counter-intuitive: Is Funes’s mind not in fact reduced to some pure, not mechanized state by virtue of its transcendence from any metric? If we pursue the impact of calculation upon phenomenality and the existence of organic life, this is in fact not the case. For Stiegler, the technicization of science [which is also the numeration of science] constitutes its eidetic blinding. […] Technicization is what produces a loss of memory, as was already the case in Plato’s Phaedrus. […] With the advent of calculation, which will come to determine the essence of modernity, the memory of originary eidetic intuitions, upon which all apodictic processes and meaning are founded, is lost.41

Stiegler’s narrative of modernity suggests an idealized pre-modern past where eidetic intuitions are unsullied by calculation. This imagined premodern condition appears in Borges’s criticism too. Gabriel Riera diagnoses two tendencies in Borges’s criticism: on the one hand, those who read Borges as ‘textualist’ and postmodern, and, on the other hand, those who see in Borges’s work ‘as stand-ins for the restitution of an auratic reality – or, as if his staging of the “fables of the One” could be equated with a mystical temptation to which he finally succumbs’.42 In this diagnosis, Riera is referring in particular to readings of Borges’s dense intertextuality and his engagement with pre-modern or preEnlightenment texts, where recourse to texts that might retain a conception of the sacred, or some divine or transcendental totality, is seen to imply that Borges himself sought an originary or pre-modern sacred state. Applying Stiegler’s narrative of pre-modern eidetic intuitions to Borges’s stories seems entirely susceptible to this. And yet, what remains fundamentally useful here – beyond any fantasy of Borges recuperating a pre-modern unity of perception and its objects – is the connection between the ‘advent of calculation’ and eidetic experience. Stiegler’s theory of technicization, memory and counting provides an important lens by which to understand the change that Funes undergoes. The only way to modulate the acuity of Funes’s memory is through adopting a form of technicization – numeration – as a means of facilitating 41 42

Stiegler, Technics and Time, 1: The Fault of Epimetheus, 3. Gabriel Riera, ‘“The One Does Not Exist”: Borges and Modernity’s Predicament’, Romance Studies 24, no. 1 (2006), 56.

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a retentional finitude, as a means by which to impose a mode of eidetic blinding for Funes’s mind. Using Lamarck’s distinction between living and non-living beings, Stiegler notes that the living beings are necessarily subject to organization: biology is organizational but anti-mechanical. Here, it is the inorganic, not the organic, that attends the experience of pure singularity. This ‘inorganic’ measure is precisely inorganic because not tied to finite (human) experience and subject to organization. It is, rather, an avatar of an infinite filmstrip, constituted by an innumerable rate of frames per second, each containing every detail at every time, independent of the ‘action’ or ‘point’ of the frame in a wider sequence. And, as would befit the new visual capacities of the filmstrip, the infinity, here, is not in the fact that the filmstrip is ‘neverending’ but that each still has an equally denumerable infinity of points. Funes tells the narrator that at one time after his accident he attempts to develop a sort of number system, which he completes to over 24,000 numerals, and that he also attempts to categorize his days. Funes ‘resolved to reduce every one of his past days to some seventy thousand recollections, which he would then define by numbers’ but he gives up, realizing (with a quite astounding naivety) that the project is ‘interminable’.43 In this sense, his number system is redundant because it lacks a continuum: a consistent form of both similarity and difference between numbers, as well as some regulated system that could found the spaces in between numbers. In other words, Funes’s numbers have no clear relationship to a number line; there is no proper metric. Despite the fact that Funes’s numbers appear to be independent of any numerical continuum, these numbers are not entirely foreign or absolutely anomalous to mathematics. Just as Funes’s memory does not constitute an annihilation of memory, so too are these numbers not the antithesis of number. Rather, this other continuum corresponds to a type of magnitude that does not require quantification: the irrational number. In the first chapter, I detailed Cantor’s discovery of transfinite numbers. Transfinite numbers are always cardinal or ordinal numbers, which measure different infinities (for instance, the infinity of natural numbers). Transfinite numbers are closely connected to irrational numbers. Indeed, Cantor called 43

Borges, ‘Funes, His Memory’, 136.

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his transfinites the ‘new irrationals’, because, like the irrationals, transfinite numbers are also ‘delineated forms or modifications of the actual infinite’.44 Irrational numbers (‘pi’ is the most famous) cannot be integrated into logorithmic sensory experience or conceptual abstract knowledge in the same way that Funes’s eidetic experience (memory alogos) cannot be retained as normal, finite memory. The Ancient Greeks would term magnitudes that were irrational numbers alogos: without reason. Funes’s memory exceeds finitude – including the limits of the human senses and the bounds of temporal continuity – but is not absolute, and it constitutes a measure of a totality without indexing or containing that totality (a transfinite number is not the end of the numerical continuum). It is thus that any attempt at restoring a chronometric or categorical function to Funes’s memories – he is said to have reconstructed whole days but this process itself takes whole days – is in fact a reliving, an irreducible and incompressible rerun of the day, a number with no cipher or concept. This overwhelming minutiae is an irrational phenomenal experience. Borges here presents a form of experience excluded from prose precisely because it operates without any metric, or is, in set theoretical terms ‘uncountable’. The technogenesis of Funes’s later consciousness is uncountable in contrast to, say, the metric of the watch that defined his earlier self. Rather than reflecting the function x◊y, his phenomenal existence is tied with an infinite extensity. This is not an absolute (Funes’s memory must depart from his own existence; it is not the memory of God) but is, rather, uncountable. And yet Funes’s uncountable memory does not make this an irrational story, or a story that itself contains a certain uncountability. Where Funes’s mind is associated with the irrational number, the story is more broadly associated with a transfinite process; the type of process that can, without having to exhaust all the digits of the irrational number, show or contain that which is uncountable. Upon his second trip to Fray Bentos, the narrator had been learning Latin, and had brought several books with him: ‘In my suitcase I had

44

This remark comes from a Cantor’s ‘Mitteilungen zur Lehre vom Transfinite’ written in 1887. This is translated and cited in: Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princeton, NJ: Princeton University Press, 1990), 128.

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brought with me Lhomond’s De viris illustribus, Quicherat’s Thesaurus, Julius Caesar’s commentaries, and an odd-numbered volume of Pliny’s Naturalis historia.’45 Word gets back to Funes that he is in possession of these books, which prompts him to send a ‘flowery sententious letter’ requesting loan of the Latin books and a dictionary ‘for a full understanding of the text, since I must plead ignorance of Latin’.46 Days later, upon arriving at Funes’s home to collect his two books – the Quicherat and the Pliny – Borges hears Funes reciting Latin, specifically the ‘twenty-fourth chapter of the seventh book of Pliny’s Naturalis historia’, whose subject is memory.47 Given the exactitude of Funes’s memory, it is even questionable as to whether he is properly reciting the text or whether he is remembering the moment of reading, raising one of Borges’s favoured problems: What is the nature of the minimal difference that constitutes repetition? Such a memory would not simply recall the text but relive it. The narrator explains that following are the last words of this chapter in the Pliny text: ‘so that nothing is restored to the hearing of the same words’.48 The narrator, who had sent Funes the books and dictionary spitefully, to demonstrate the absurdity of Funes’s pretension to learn Latin with a text and the mere aid of a dictionary, is quite stunned by Funes’s ability, and indeed the ability to speak in and follow sentences seems to be the only kind of continuum that Funes can muster. This last phrase from Pliny resounds as a precaution in the context of Funes’s predicament, in that it seems to take heed of the function of memory as a bastion (so that nothing is restored) against irrational retentional capacity  – a cinematographic consciousness – that Pliny seems to be warning against. The true node of singularity in this story can be found in Borges’s narration of Funes himself; it is Funes who appears to us as singular, and we are necessarily barred from his experience. It would seem that this is a consciousness that can only be reported, not experienced, by Borges’s readers themselves. Yet, this is too hasty a conclusion, because we find the singularity embodied by Funes in 45 46 47 48

Borges, ‘Funes, His Memory’, 133. Ibid. Ibid., 133. Ibid., 134.

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the composition of the short story. The text opens with a performance of the narrator’s own recollection: I recall him (although I have no right to speak that sacred verb – only one man on earth did, and that man is dead) holding a dark passionflower in his hand, seeing it as it had never been seen, even had it been stared at from the first light of dawn till the last light of evening for an entire life-time.49

Here Funes is constructed from the ‘outside’, and we see Funes as experiencing an end of the continuum, rather than the end of the continuum, which is necessarily barred from representation and phenomenality and must be embedded in the story through the narrator’s voice. Funes is always at one narrative removed from the text (his voice is mediated by the narrator), and his form of experience is never replicated, only reported. The repetition of ‘I recall him’ invokes Funes as a singularity, but a singularity that remains fixed in time for the narrator who retains a finite existence in time and space, a finite perception and recall. The narrator’s ‘memorial’ becomes a memory of an encounter with Funes’s memory, a narrative testimony: ‘I recall him – his taciturn face, its Indian features, its extraordinary remoteness – behind the cigarette.’50 The kernel of an ‘uncountable’ consciousness is presented without being represented, and the fidelity to that aconceptuality is never resolved, lending an enigmatic quality to the story. Borges has thus, in the first instance, managed to embed existence-outside-of-representation through narrative recursion, an ‘uncountable’ consciousness within a ‘countable’ one. This intersection between number and the transition from an abstract to a phenomenological absolute has been overwhelmingly read in the critical literature as moral fables that are preoccupied with human finitude. Edmond Wright’s claim that the story strips numbers of some ‘outward’ identity to reveal their ‘nakedness’ does not seem quite right: Remove the decimal, the binary (based on 2), the duodecimal (based on 12), the ‘undevigintal’ (based on 19, which is perfectly possible) – if truth be told, remove a system of mnemonics based on any number whatsoever, and we

49 50

Ibid., 131. Ibid., 131, italics mine.

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are left with the numbers in all their naked glory, each an abstract singularity worthy of the proper name.51

The notion of a ‘naked glory’ of numbers, whilst initially appealing, neglects the most fundamental device that allows numerical presentation to structure a prose story here: narrative recursion conditioning the experience of ‘singularity’. Funes’s attempts to construct his own number system and his paralysis all contribute to a situation which is defined by the possibility of presentation whilst never conflating mathematical writing with the writing of prose, or uncountability with the necessarily ‘countable’ and finite consciousness of the narrator and, even, of representation. Irrational numbers are formally revealed whilst being simultaneously concealed; the one who utters that ‘sacred verb’ ‘remember’ has no right to. The story incorporates its own generic, that which exists prior to (or even primordial to) the narrative. In this sense, there is an allegorical incorporation of the story’s own technicity in a dual form. We move from a chronometric existence – the watch as the symbol of organized life – to an irrational one; to a vision of ‘infinite’ detail held in the photograph, the film still. This is a vision that is replicated rather than represented. Of course, then, this presentation of ‘uncountability’ renders Funes’s memory the existential equivalent of ‘ᒀ’ in Cantor’s notation; the cardinality of the continuum. It is this fictional ‘proof ’ of uncountable existence that brings into relief the measurable infinity of countable numbers, the world that the narrator exists in.

‘The Library of Babel’ Borges was obsessed with the labyrinth as a combination of both order and disorder, and as a predicament that requires some mysterious key to unlock. In Emir Rodiguez Monegal’s words, labyrinths are ‘according to tradition, the representation of ordered chaos, a chaos submitted to human intelligence, a deliberate disorder that contains its own code’.52 The ‘code’, here, is also 51

52

Edmond Wright, ‘Jorge Luis Borges’s “Funes: The Memorious”: A Philosophical Narrative’, Partial Answers: Journal of LIterature and the History of Ideas 5, no. 1 (2007), 44. Monegal, Jorge Luis Borges: A Literary Biography, 42.

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the ‘clue’, a word that originates with the Greek term for the thread given to Daedelus by Ariadne, to help him find his way out of the labyrinth that he had built to house the Minotaur. Borges’s most famous labyrinthine tale is perhaps ‘The Library of Babel’, which is a narrative account of a universe that takes the form of a library ordered by regular geometric cells. ‘The Library of Babel’ is a labyrinth because it contains all possible books and thus, most importantly, the ‘book of Books’ that forms the code for all others. The existence of this book is a rumour. It has not been found by any of the ‘librarians’ but provides the most important justification for the existence of the library, as well as an impetus for the librarians’ search through the books. This rumour is derived from speculation that emerges from the logic of the library itself; the speculation that the library contains every possible permutation of the alphabet of Babel. This book in many ways stands in for the mythological minotaur: it is both the source of the library and that which the library keeps hidden. The library is thus a labyrinth in the sense that its nature presents a conundrum: on the one hand, a quest to find this essential book, and, on the other hand, a metaphysical problem concerning the order and contents of the library and the lives lived in the library. In both of these cases, we are dealing with the mystery of the thought of the library, an unspoken divine that drives the questions of the narrator or a clue that unravels the genesis and laws of the library. The labyrinth is thus a double bind in the sense that the appearance of the question is also the grain, the method, of the search. The Library of Babel houses the books that the inhabitants – the librarians – spend their lives combing. The narrator is an ageing librarian whose sight is failing and who seems to be reaching the end of his or her life: ‘Now that my eyes can hardly decipher what I write, I am preparing to die just a few leagues from the hexagon in which I was born.’53 The narrator’s brief and strange account seems to be at once an outline of the only world in which he has lived his life, as well as an explanation of the world for one who lives outside the library. In this sense, the short story is fundamentally contradictory: the

53

Jorge Luis Borges, ‘The Library of Babel’, in Collected Fictions, trans. Andrew Hurley (New York: Penguin Books, 1998), 112.

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representative account of a total universe is presented as if for the foreigner (the various details of the uniform structure of the library are noted) or, in other words, a world that supposedly contains no outside is presented for an outside. The story opens with the essential structures of the library: The universe (which others call the Library) is composed of an indefinite and perhaps infinite number of hexagonal galleries, with vast air shafts between, surrounded by very low railings. From any of the hexagons one can see, interminably, the upper and lower floors. The distribution of the galleries is invariable. Twenty shelves, five long shelves per side, cover all the sides except two; their height, which is the distance from floor to ceiling, scarcely exceeds that of a normal bookcase.54

How the narrator knows the height of a ‘normal bookcase’ is a mystery: he has lived his whole life in the library. The self-conscious estrangement that occurs here is carried right through the story and is particularly significant, given that the production of texts has such consequences for the constitution of the library. The ‘significance’ of this is a kind of anti-significance: this discrepancy in the form of the narrative signals the contradiction in positing any totality at the same time that it constructs the ‘total’ library. The epigraph to ‘The Library’ is taken from Robert Burton’s Anatomy of Melancholy (Pt 2., Sec. II, Mem. IV) and foregrounds the issue of textual production as a process of combinatorics: ‘By this art you may contemplate the variation of the 23 letters.’55 Though Borges only provides a snippet of this section from Burton’s masterpiece, the passage that it falls within is fascinating in terms of its use of numbers. Burton sets out, here, to make the point that through the simple variant combinations of twenty-three letters of the alphabet (during Burton’s time the Classical Latin alphabet contained twentythree letters, not yet including j, u and w) one is afforded an exceptional imaginative range. Within a wider section on the value of literature and discourse as a solution to melancholy, Burton cites numerical combinations of the letters of the alphabet to give a sense of the vast possibility that emerges

54 55

Ibid. Ibid.

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from language alone. What is intriguing is Burton’s exact delineation of this realm of possibility: Ten words may be varied 40,320 several ways: by this art you may examine how many men may stand one by another in the whole superficies of the earth, some say 148,456,800,000,000, assignando singulis passum quadratum (assigning a square foot to each), how many men, supposing all the world as habitable as France, as fruitful and so long-lived, may be born in 60,000 years, and so may you demonstrate with Archimedes how many sands the mass of the whole world might contain if all sandy, if you did but first know how much a small cube as big as a mustard-seed might hold, with infinite such.56

Here, in order to imagine the full expressive range afforded by the humble alphabet, Burton in fact turns away from the alphabet and starts listing very large numerical figures. It is numbers that ascribe, and describe, the expressive capacities of the alphabet. Of the very large numbers in the theory of eternal recurrence, Borges writes: ‘This chaste, painless squandering of enormous numbers undoubtedly yields the peculiar pleasure of all excesses.’57 Burton seems to be applying this very technique, here. There would, conceivably, be a number to describe the extent of variation of the twenty-three letters: a number that sits outside of the alphabetic universe that Burton opens for his reader. The task of the librarians, the contemplation of the books, the limits of their universe, and the chances of their finding a ‘book of books’, happens in the context of just such numbers as Burton deploys. One might say, in other words, that the capacities of language to describe and imagine vast possibilities – even the simple, random combinatorial possibilities of the alphabet – are undergirded by a number because whilst the permutative capacities of language go beyond the finite imagination, one can still index this infinitude with a kind of abstract shorthand: mathematics. Of course, wherever we are dealing with an index of an infinite sum larger than that which is expressible, we are dealing with a transfinite quantity. Indeed, here, the ‘truth’ of the alphabet, and so of literature, appears transfinite. The narrator is careful in systematically describing the library and the means by which the librarians understand its potential totality. There are only 56 57

Robert Burton, The Anatomy of Melancholy (New York: The New York Review of Books, 2001), 95. Borges, ‘The Doctrine of the Cycles’, 116.

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two ‘narrow provisions’ – two small spaces in the cells – in which the librarians may sleep or relieve themselves. The books in the library do not necessarily make sense: rather, they are arrangements of letters that do not necessarily produce words, sentences or meaning of any kind and, in fact, statistically will not do so. The ‘meaning’ is instead produced in instantiation or permutation. Each book contains another possible arrangement of the letters or characters of the twenty-five-letter alphabet, even if it is only the most minimal change of just one character. Thus, the librarians presume that the library must contain a ‘written’ account of every life, and of all future events in the library, and in this sense the library is infinite to its inhabitants because it expresses, in its very form, all possibility.58 However, the random arrangement of letters in the books means that ‘for every rational line or forthright statement, there are leagues of senseless cacophony, verbal nonsense, and incoherency’.59 Upon introducing the library, our ageing librarian relates the basic axioms of his world: the first principle being that ‘the Library exists has existed ab aeternitate’.60 The narrator repeats what he or she calls ‘the classic dictum’, supposedly a common belief about the nature of the library, which is in fact a rephrasing of Giordano Bruno’s description of God: ‘The Library is a sphere whose exact center is any hexagon and whose circumference is unattainable.’61 This is, of course, also a version of infinity severed from the number line, the version reflected in Cantor’s uncountability proofs (where, as Borges himself notes, all you need is a 1–1 correspondence to construct a measure of infinity). This ‘classic dictum’ is less a statement about a cosmic distribution impossible to imagine than a claim about the library as instantiation. Reminiscent

58

59 60 61

The number for all possible combinations of the twenty-five orthographic symbols in the library is of course not infinite. It is merely, as William Goldbloom Bloch points out, ‘unimaginably vast’. The number that Bloch comes up with is 251,312,000. This of course, pertains only to a sense of the infinite rather than the infinite as such. It also pertains, however, to some certainty that the library never repeats a volume: a fact which is necessarily unverifiable because any full index of the library would be an entire replication of the library. As such, the safest ‘size’ to posit for the Library of Babel is somewhere in between 251,312,000 and ∞. This number, however, does not include the possible arrangements of letters on the spines of the books. Current research, as Bloch notes, posits the size of our observable universe as 1.5 × 1026 m across. Goldbloom calculates this in terms of cubic metres, and demonstrates – astonishingly – that our universe ‘doesn’t make the slightest dent in the Library’ in terms of cubic size. See: Bloch, The Unimaginable Mathematics of Borges’ Library of Babel, 18–19. Borges, ‘The Library of Babel’, 114. Ibid., 113. Ibid.

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of Asimov’s Nine Billion Names of God, the ‘meaning’ of the library is the inclusion of every possible arrangement of letters, and in this inclusion the library is complete and is manifest as a labyrinth. The library presents the form and limits of everything the librarians can know, and it is precisely these limits that form the basis for theories of the being of the library. The second principle of the library stipulates the number of letters: There are twenty-five orthographic symbols. That discovery enabled mankind, three hundred years ago, to formulate a general theory of the Library and thereby satisfactorily solve the riddle that no conjecture had been able to divine – the formless and chaotic nature of virtually all books.62

The librarian relates an instance where a book his father ‘saw in a hexagon in circuit 15-94, consisted of the letters M C V perversely repeated from the first line to the last. Another (much consulted in this zone) is a mere labyrinth of letters whose penultimate page contains the phrase O Time thy pyramids’.63 The letters M C V demonstrate this ‘chaotic nature of all books’, and the significance of this text is found in the absence rather than in the presence of any semiotic important.64 This nonsensical, repetitive book proves that this alphabet and the dialects, languages and various forms of meaning that it produces also contain its antithesis: the capacity for meaninglessness. What is deeply unsettling, regarding this story, is the presence of a footnote that implies that the work we are reading is a copy: ‘The original manuscript has neither numbers nor capital letters’.65 The version of ‘The Library of Babel’ that we are reading does include numbers – for instance, it refers to a hexagon in circuit ‘15-94’ – and so there must be an ‘original’ copy, then, where this would have been spelt out: fifteen-ninety four. Once again, what is behind the distribution of the hexagons is a notation system separate from the alphabet, and what signals the status of the text as a copy is the supervenience of numeracy in a universe totalized by an alphabet.

62 63 64 65

Ibid., 113. Ibid., 113–114. Ibid. Ibid., 113.

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Regardless of the limit of the orthographic symbols, we know that if the number line is potentially infinite, then so too are the books of the library. Our language is indeed inflected with the mathematical infinite by virtue of the fact that all numbers also have words: 5 is also ‘five’. If every number has a corresponding word, then language can also become an infinite set. The conditions for being and the conditions for knowing both revolve around the number of letters and the possible combinations of these letters. The twenty-five orthographic symbols are also a notation for an unimaginably large number of combinations. It is in the number twenty-five that we see the simultaneity of the ontic and the epistemic (the being of the universe, and the limits of what can be known in that universe), but, curiously, without the implication of a totality. This is not ‘god’ or some other philosophical absolute (consciousness, perhaps) but simply the number twenty-five.66 But, insofar as these letters can be combined to form number words, any calculation of combinations becomes redundant, precisely insofar as numbers can continue infinitely, even when they are made from words rather than mathematical notation. All one could produce is a key for an endless creative process: those simple twenty-five orthographic symbols that are vastly more generative of permutational combinations than the decimal system. These problems of the finite and the infinite, so stirring to the librarians, are the source of the Vindications, the Crimson Hexagon, and the Man of the Book. The notion that the library contains all possible narratives, including those that will occur in the future and those that have occurred in the past (‘the archangel’s autobiographies’), inspires religious fervour: When it was announced that the Library contained all books, the first reaction was unbounded joy. […] There was no personal, no world problem, whose eloquent solution did not exist – somewhere in some hexagon. […] At that period there was much talk of The Vindications – books of apologiæ and prophecies that would vindicate for all time the actions of every person in the universe and that held wondrous arcana for men’s futures. Thousands of the greedy individuals abandoned their sweet native hexagons and rushed downstairs, upstairs, spurred by the vain desire to find their Vindication.67 66 67

Ibid. Ibid., 115.

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This self-fulfilling fervour comes from the most interesting (and horrifying) possibility that the library materializes, which is validation simply through existence. In the Library of Babel, the realm of possibility is already delimited through the record of all possibility. Each vindication is an individual manifestation of the necessity and reason for existence. This passion for knowledge is reversed in another superstition held at one time by some of the inhabitants of the library: ‘the Book-Man. On some shelf in some hexagon, it was argued, there must exist a book that is the cipher and perfect compendium of all other books, and some librarian must have examined that book; this librarian is analogous to a god’.68 Another form of this speculation is the presence of a total book with a circular spine: ‘Mystics claim that their ecstasies reveal to them a circular chamber containing an enormous circular book with a continuous spine that goes completely around the walls. But their testimony is suspect, their words obscure. That cyclical book is God.’69 The library is the measure of time and life because it contains all possibility at once, already rolled out and specified, and the vindications or the great circular book are merely the most explicit, extended version of this. The simple possibility of an essential book that provides the key to the unravelling of the rest of the library, the codex for the universe that proceeds from it, is the most condensed version of the same impulse. This labyrinthine morphology is distinctive because it posits a conflation of ontology and epistemology at its core: the key that lets Daedalus escape his labyrinth. The possibilities for the ‘art’ of twenty-five orthographic symbols is at once the scope of all knowledge of the library (epistemology), insofar as it is mediated by language, but equally the conditions for being in the library, insofar as the world exists for housing the books and, more minimally, the librarians who attend to the books (ontology). Alfred North Whitehead theorizes the distinction between the modern and the pre- or non-modern on the basis of a conflation, or severing, of the link between ontology and epistemology. For Whitehead, the modern severing between ontology and epistemology contradicts the foundations from which modern science developed, which are rooted (in his account in Science and the Modern World) in medieval Christianity as much as Ancient 68 69

Ibid., 116. Ibid., 113.

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Greek science.70 For Whitehead, Greek ‘science’ was never truly a science, but a necessary extension of metaphysics. Whitehead contends that it is ‘scholastic divinity’ that forms the necessary ground for the Enlightenment flowering of empiricism.71 Crucially, innovations within science emerging from scholastic divinity relied on what Whitehead calls the ‘instinctive belief ’ in the ‘secret’ and the metaphysical consistency behind natural order: ‘I mean the inexpugnable belief that every detailed occurrence can be correlated with its antecedents in a perfectly definite manner, exemplifying general principles.’72 The forgotten foundation of modern science is in the presumption of completion (the presumption of a total rationality in the universe) and, by virtue of this completion, the presumption of consistency. In other words, the flowering of naturalism emerged from the medieval Christian requirement for an invisible order (a secret), that underlies and generalizes the most minute natural phenomena, creating a necessary link between ontology and epistemology. In Whitehead’s history and theory of the Enlightenment, the connection between representation and presentation is severed by the secularization of modern science, the disavowal of the foundational connection (the Christian connection that is also the Parmenidean connection) between ontology and epistemology that produces the conditions for modern science in the first place. There are two points regarding ‘The Library of Babel’ that are significant here: the first concerns a mode of formal allegory that emerges from this conflation of ontology and epistemology, and the second regards the consequence of this, which is the production of an ‘in-significant totality’. The library revolves around a book that provides a foundational node where ontology

70

Murray Code provides an excellent summary of Whitehead’s perspective on the conflation of ontology and epistemology. Even more significantly, he frames this modern ‘denial’ of its foundations in terms of a failure to be properly modern: It would be better to describe Whitehead as attempting to frame a thoroughly nonmodern naturalism in the sense outlined by Bruno Latour, who charges the moderns with never having been truly modern since they never tried to bring all the explanatory resources of nature, culture and discourse under one roof. They instead opened up a chasm between epistemology and ontology by trying the former to a sensationalist theory of perception and the latter to the doctrine of mechanistic materialism.

71 72

See: Murray Code, Process, Reality and the Power of Symbols: Thinking with A.N. Whitehead (Hampshire and New York: Palgrave MacMillan, 2007), 62. Alfred North Whitehead, Science and the Modern World (New York: The Free Press, 1925), 12. Ibid., 15.

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and epistemology are conflated: it is the key for the labyrinth, providing the source and extent of possibility of the material universe of the library. In other words, the library revolves around a moment of pure presentation, not yet an instantiation or permutation, and not a representation of the twenty-five orthographic symbols. This ‘book’ would be entirely future oriented: not referring to or representing anything, but providing the molecular form of the world that will surround it. This book provides the same function for the librarians as what Whitehead calls the ‘secret’ that animated the Christian scholastics, and conflates the ontic and the epistemic in the same way. Just as the library revolves around some necessary microcosm that is the total book, The Library is reciprocally constructed around a total book: the universe of Babel, and all of its contents. The Library is only an allegory of a total world, the extensity of language and the function of presentation in our world insofar as the library occupies a similar distance to its potential ‘total book’. The short story here revolves around an imagined total book that justifies and provides the reason for the universe that the narrator lives in, just as the world that we encounter in the story orbits around the same structure (which provides a rationale for the estranging contradictions of the story, whereby the narrator explains his world as if to someone from another world, as if there were an outside). In the case of The Library (the story), the total book is of course ‘The Library of Babel’, and in The Library (the world) the total book is of course The Golden Book, the Vindications, the circular book or whatever other figure of an essential or total book the librarians might formulate. This reciprocity allows this story to become a perfect formal allegory, whereby the contents of the story reflect the narrative’s own creation and explanation of a total universe.

‘The Lottery of Babylon’ The possibility of a kind of ‘ontic book’ is again explored in the story ‘The Lottery of Babylon’. This story – as the title makes obvious – is closely connected to the ‘Library of Babel’. The word ‘Babel’ can mean ‘confusion’ but also – when lengthened to Babylon – becomes the word for the ‘gateway to God’. The ancient city of Babylon, whose ruins are in contemporary Iraq, is the location for the biblical myth of the

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‘Tower of Babel’. The inhabitants of the tower were said to have one language, a story that became the source for the myth of ‘Babel’ as the universally understood language, the universally shared language: here complete comprehension is also the ‘gateway’ to an absolute, to God. In ‘The Lottery in Babylon’, we have a twisted form of the universal language. This ‘language’ is not something universally comprehended by the inhabitants, but rather concerns what they are universally subjected to: the radical contingency of fate, and the absence (even if it is a potentiality of absence) of free will. In Borges’s Babylon, the contingencies of fate are controlled by a lottery that – much like a municipal organization – evolves over the years to ‘better serve’ the people of Babylon. The ‘atrocious variety’ of the narrator’s identities and roles in his society are a direct result of outcomes of this lottery. ‘Like all men in Babylon, I have been proconsul; like all, a slave’, he reports, ‘I have also known omnipotence, opprobrium, imprisonment.’73 Babylon, the universal language, is but an ‘infinite game of chance’, where life happens according to combinatorics, just as in ‘The Library of Babel’. Unlike The Library, however, there are only three letters of consequence in Babylon: Look here, through this gash in my cape you can see on my stomach a crimson tattoo – it is the second letter, Beth. On nights when the moon is full, this symbol gives me power over men with the mark of Gimel, but it subjects me to those with the Aleph, who on nights when there is no moon owe obedience to those marked with the Gimel.74

Aleph, Beth and Gimel, the first three letters of the Hebrew alphabet, assign classes in Babylon and enforce the arbitrariness of power and submission. The initial, by implication primitive, lotteries consisted of draws ‘in broad daylight’ and resembled traditional lotteries in that the winners received coins.75 The problem with this system was that the ‘moral force’ of the lottery was lacking.76 In this sense, the story becomes testimony to a certain attachment to a moral necessity in chance, and to some inherent virtue in lives modulated by numbers or arbitrary signs. In its second variation, the lottery involves

73 74 75 76

Jorge Luis Borges, ‘The Lottery in Babylon’, in Collected Fictions (New York: Penguin, 1998), 101. Ibid. Ibid. Ibid.

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lucky and unlucky draws, and one could either end up winning or owing a sum of money. The implication became that those who did not buy tickets were cowardly, and those who drew ‘unlucky’ numbers and were fined became subjects of contempt, and thus there was a successful moral scourge to be had. The final version of the lottery perfects this. After a revolution inspired by the unfair drawing of multiple lots by the upper classes, the lottery is transformed, and every citizen of Babylon is automatically a participant, creating a true democracy of the lottery. The Company (the body that administers the lottery) becomes the all-powerful force in Babylon and fates are decided – many times during each citizen’s life – based on lots drawn. Radical contingency – the capacity for freedom and wealth one day, and slavery or death the next – comes to govern and normalize the lives of the inhabitants of Babel: ‘Mine is a dizzying country in which the Lottery is a major element of reality; until this day, I have thought as little about it as about the conduct of indecipherable gods or of my heart.’77 The narrator goes on to recount elements of his life and how the lotteries of Babylon have come to eventually modulate the social order of the city: Once initiated in the mysteries of Baal, every free man automatically participated in the sacred drawings, which took place in the labyrinths of the god every sixty nights and which determined his destiny until the next drawing. The consequences were incalculable. A fortunate play could bring about his promotion to the council of wise men or the imprisonment of an enemy (public or private)… A bad play: mutilation, different kinds of infamy, death.78

The lottery is administered by ‘the Company’, which ensures – and here we see the key paradox in this story – the social efficacy of the lottery. There is frequently social dissatisfaction with the form of the lottery, usually of an existential nature, which the Company then seeks to remedy. Strangely, the very source of the lottery – the exhilaration and submission of chance – will make it seem inadequate to the Babylonians: In many cases the knowledge that certain happinesses were the simple product of chance would have diminished their virtue. To avoid that

77 78

Ibid., 101. Ibid., 103.

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obstacle, the agents of the Company made use of the power of suggestion and magic. Their steps, their manoeuvrings, were secret. To find out about the intimate hopes and terrors of each individual, they had astrologists and spies.79

The solution, then, will be to obscure the sources and ‘reasons’ for chance. This reality of Babel is peculiar because it bears a strange relation to the real. This is not quite as paradoxical as it sounds: everyday life is structured here by what is precisely antithetical to the continuation of that life: utter contingency. Instead of a theory, however, this is a surrender to what the Babylonians take to be the real: However unlikely it might seem, no one had tried out before then a general theory of chance. Babylonians are not very speculative. They revere the judgments of fate, they deliver to them their lives, their hopes, their panic, but it does not occur to them to investigate fate’s labyrinthine laws nor the gyratory spheres which reveal it.80

What the Babylonians are after, here, is the sensation of submission to fate, which is existentially comforting, rather than any revealed truth regarding fate. Here, we find another transformation of presentation: brute happening without reason, matter without causality and an ontology that seems to ‘appear’ by virtue of the constant exposure of the citizens to the vagaries of fate, the return to generic being in the moment of suspension in which the lots are drawn. Here, ontology appears as an affect wrought by subjective exposure and the perpetual ‘year zeros’ that the lottery repeatedly inaugurates. The numbers of the lottery are thus meant to provoke the ‘sensation’ of number.81 In this story

79 80 81

Ibid., 104. Ibid., 104. One of Cantor’s precursor’s, the Italian mathematician Bolzano, developed a distinction between two types of infinities that would be crucial to the development of the transfinite several decades later. Bolzano defined syncategorematic and categorematic infinities, the first pertaining to potential infinities, and the second pertaining to actual infinities. It is this distinction, primitive to the full realization of the transfinite as a measure of actual infinity, which is operative in Babylon: the use of a categorematic infinity (the rules of the game and chance) to obscure or render impotent a syncategorematic infinity. It is the ever-increasing branches of chance in the lottery that shows the increasing desperation of The Company to stave off the syncategorematic, imitating its omnipotence in ever-greater degrees of detail yet still retaining control.

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we again see an instance where the kernel of presentation is again suggested (the ‘appearance’ of determinism in the casting of lots) but ultimately hidden (the replacement of municipal governance with a lottery). The lottery in Babylon is thus not a speculative enterprise, but rather a normative one. The lottery attempts to shift the fundamentally contingent process of cause and effect to a randomized process that both originates in and dictates human agency. The reintegration of fate into the workings of the ‘Company’ perversely subsumes the human perception of causes behind effects into a well-founded and predictable system. The lottery creates a norm of radical contingency and unpredictability in the lives of the citizens: unpredictability becomes what is expected, but the form of this unpredictability must remain a mystery. So, then, the primitive versions of the lottery are said to ‘appeal not to all a man’s faculties but only to his hopefulness’, thus creating ‘public indifference’.82 The successful elaborate later versions evoke ‘all the vicissitudes of terror and hope’.83 Here, we again find Borgesian narrative wit: the story is finely constructed in order to conceal what it claims to present. Just as the Library of Babel conceals, and Funes only appears to suffer, so too does the world of Babylon revolve around what it necessarily cannot avow: the kernel of the real, the nature of presentation and the true possibility of a unique number. The lottery, as our narrator describes it, is the ‘insertion of chaos into cosmos’.84 This is the annihilation of novelty through numeracy and the exertion of a finite randomness that can show a definite link between cause and effect, assuring the citizens of Babylon of the presence of a continuum, at the same time allowing them to be seized by sublime contingency. As the Company becomes more and more complex, so too do the branches of contingency, and each prospect is never quite assured or quite what it seems. Once again, the lottery here realizes a kind of total book staked in numerical chance (the buying of lots), whereby all the prospects for being and the appearance of change and novelty are rendered by one, secretive authority and the shady rules by which that authority proceeds.

82 83 84

Ibid., 102. Ibid. Ibid., 104.

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What is a Transfinite Allegory? Borges begins his 1949 article, ‘From Allegories to Novels’, with a polemical claim about the redundancy of allegorical form: ‘For all of us, allegory is an aesthetic mistake. (I first wrote, “Is nothing but an error of aesthetics”, but then I noticed that my sentence involved an allegory.)’85 In this article, Borges traces literary development out of allegory – which he associates with prenovelistic epic and Romance – to the novel form, and articulates why allegory in its traditional sense is a literary impossibility in his time. The entire problem with allegory, in this essay, revolves around a loss of belief, a loss of some ephemerality that renders the symbolic inscription ‘inseparable from artistic intuition’.86 These are Benedetto Croce’s words, and Borges quotes him at length to illustrate the poverty of allegory: If the symbol is conceived as separable [from artistic intuition], if the symbol can be expressed on the one hand, and the thing symbolised can be expressed on the other, we fall back into the intellectualist error; the supposed symbol is the exposition of an abstract concept; it is an allegory; it is science, or an art that apes science.87

Whilst Croce appends a number of qualifications to this, the essence is clear for Borges: if allegory is not secure on an intuitive and spiritual level, it is merely, to use Croce’s word again, a kind of ‘cryptography’.88 On the other hand, allegory is, for G.K. Chesterton, a supplement to language, which is necessarily crude and inadequate in its representation of the world. In this case, any perspective on allegory as a ‘justifiable’ art form depends on a prior perspective on the representational capacities of language. Borges’s own conclusion is that allegory is ‘a fable of abstractions’: of Gods and destinies, and in this sense he implies that that allegory is associated with pre-modern forms: epic and prenovelistic Romance (his example, notably, is Roman de la Rose). The novel on the other hand is a ‘fable of individuals’.89 Only insofar as ‘the individuals that 85 86 87 88 89

Borges, ‘From Allegories to Novels’, 337. Ibid. Ibid. This quote, from Benedetto Croce, is reproduced for Borges’s purposes in: Ibid., 338. Ibid., 339.

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novelists present aspire to be generic’ is there an element of allegory in the novel.90 Although Borges does not claim this explicitly, the latent conclusion appears to be that that allegorical form becomes ‘unjustifiable’ if there is no longer a belief in or sense of abstraction. However, as we saw in the essay on Flaubert, ‘A Defense of Bouvard and Pécuchet’ (written five years later), Borges does not see the novelistic form as defensible either. What form, then, must be forged from the death of the novel and the loss of a ‘justification’ for allegory? I would like to suggest that it is a modified form of allegory that Borges finds as a formal replacement: one that does not require a belief in abstractions, and one whose generic is no longer has a name, or a divinity. Many of Borges’ most successful stories revolve around the capacities of fiction to suggest worlds beyond itself, and the source and meaning of the extensity of fiction. But to look at this only at the level of content misses precisely the mode in which the ‘other world’ comes into being, which is through number. Like Mallarmé, Borges is interested in a ‘total book’: an essential structure or principle that underlies the possibilities available to fiction, a simultaneously genetic and generative kernel. I have argued above that Borges creates a genuinely generative fiction: one that produces other worlds without the necessity of their completion or coherence. This neither relies on any spiritual foundation for allegory nor is it limited to that ‘intellectualist error’ or inadequate form, an awkward science of the symbol (though it replicates this form closely). This form is not rooted in any genericity of individual experience, either, and takes no coherent subject as its point of origin. Rather, this new form rests not on divinity but on ℵ0 and can be called ‘transfinite allegory’. In each of the stories analysed above, we see the following mechanism at work: the stories revolve around either an actual infinite (as in the ‘Lottery in Babylon’ or the ‘The Library of Babel’) or two irreconcilable forms of numerical continuum (in ‘Funes, His Memory’) that are never represented – determined – within the story. In each case, number is embedded in the fiction but unseen: we find the ontic code in the ‘Library of Babel’ in the rumour of the Golden Book, and in ‘Funes’ we have the subjective apprehension of transfinite number locked in the paralysed body 90

Ibid.

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of the character. Of course, this is in turn defied by the fact that Borges has generated a fictional instantiation – or ‘measure’ – of a ‘total world’ through the reciprocity of formal allegory, just as our relation to the character Funes is mediated by a narrator (and the memory of that narrator). In this sense, each of these stories contain and allude to their own origins or the origins and structures of the lives of their characters: actual infinites that – like the Company in Babylon – bring worlds into being without providing the rationale for these worlds or patently contradicting the credibility of these worlds. In other words, these infinities remain an alogos – a point of non-rationality – at the heart of the stories. Within each of the stories there is thus a double inclusion of number, which ultimately produces a structural allegory internal to each story. In Funes, for example, we see number twice over: first, in terms of Funes’s capacity to track time, and thus his ability to attach number to a chronos, to an independent, abstract and arbitrary movement, and secondly in the numbers deployed by the narrator to convince us of the extensity of Funes’s perception. Here, Funes has a consciousness that is brought into being through mechanical number; later, this will be reversed. Funes’s second ‘birth’ constructs a different epistemology, and he acquires an apparently infinite perception: exceeding finitude without positing an absolute perception. Crucially, there is a structural gap between the double presence of number in these texts, which brings into relief the distinction between a transfinite number and ordinary counting in the arrangement of presentation and its necessary effacement by representation. This structural gap is in fact another instantiation of what Paul de Man argues is the internal allegory essential to fictional writing. De Man’s work in Allegories of Reading and Blindness and Insight, which I discussed in the previous chapter, looks at the ways that literary form constructs linkages between objects and events and creates images through tropological uses of language, simultaneously collapsing that very content to the abstract machinations of language itself. This is exactly the form that Borges gives literary historical weight to in the essay ‘From Allegories to Novels’, where the operations of the symbol subtracted from intuition and spiritual guarantee inevitably must, in Croce’s words, ‘ape science’. De Man constructs this argument through a method of reading that straddles both literature and criticism, which brings

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out the process of linguistic doubling essential to fiction. In his critique of formalist criticism de Man notes: On the one hand literature cannot merely be received as a definite unit of referential meaning that can be decoded without leaving a residue. The code is unusually conspicuous, complex, and enigmatic; it attracts an inordinate amount of attention to itself, and this attention has to acquire the rigour of a method. The structural movement of concentration on the code for its own sake cannot be avoided, and literature necessarily breeds its own formalism.91

Despite the fact that the critics ‘cry out for the fresh air of referential meaning’ in their mappings of poetic language, the creative work of tropology undermines the possibility of precisely the unified work that the formalist critic seeks, always leaving a residue in its outward and endless referentiality. Writing on Rilke, de Man shows how the celebration and content of the poems is achieved through language devices. This is a traditional analytic reading until de Man doubles his analysis over to show how the very content of the poems is subsumed to a dexterity with language, indeed, what is expressed pertains not to God, the topic of the poems, but rather to the movement of language. De Man shows through his analysis of the poetry of Rilke and others that the axis of content – the story, the gesture or the sentiment of the poems – is inextricable from form, but also that form, curiously, has a life of its own. De Man starts from the basic assertion that themes – when they are put under close analysis – are seen to be indelibly linked to modes of figuration. The most forceful of images or claims do not rest on their own constitution but, rather, some other story, other logos beyond that theme or referent, that accompany the act of image creation or staking a claim. This autonomy of form means that the content of the poems also functions as an allegory of what happens in the language of the poem. This claim for internal allegory operates according to the same principle whereby we see the two forms of number in Borges’s texts, which are placed in an irresolvable contradiction: the formal rendering of the transfinite number points outside of the text, to an autonomous existence, which is embedded but not contained by the boundaries of the stories. It is this discovery of the 91

Paul de Man, Allegories of Reading: Figural Language in Rousseau, Nietzsche, Rilke and Proust (New Haven, CT; and London: Yale University Press, 1979), 4.

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singularity of fictional language that is anticipated and worked with in Borges fiction: not only do we have de Man’s thesis ratified in the ‘doubling’ of language and number in Borges’s stories, but we in fact have a self-conscious engagement with this gap, made explicit in the use of number in these stories. Here I am pointing to a simultaneity between the ‘gap’ that characterizes the unique discourse of fiction in de Man’s understanding – in that all fictional texts necessarily allegorize their own formal properties – and the ‘gap’ between the two forms of number in Borges. Borges’s stories allegorize their own processes of composition by bringing into relief their own ‘enigma’: an infinite presentation rather than a finite representation. This is not, however, a clear parallel. It would seem that the double forms of number in the stories allegorize the organization of expression and tropological movement of the very fictional work. However, there is something additional that disrupts this: the presence of number in the texts. The autonomous and material aspects of language that necessarily defy both the formalist critics and any pretension to unified poetic works are rendered in Borges’s work as the essential collapse of ontology and epistemology in substance: ‘The Library of Babel’ must posit the library in order to posit, within that world, the superstition of a ‘Golden Book’, for instance. In other words, what we have with Borges is the replacement of a stable relationship between language and the world with number. Number here comes to stand in for the link between sign and referent, and fascinatingly the restitution of a foundation for language in Borges’s work is not found in natural language at all, but the other form of language: mathematical writing or the ontic mark. What this modified version of de Man’s theory of formal allegory affords us is a concept of allegory that is not a reduction to content, or an addition upon content, nor a mythological or humanist ‘resolution’ to the stories. This allows us to approach the numerical aspects of the text that generate the tendency to allegorize and narrativize ‘out of ’ the stories, and allow us to start to approach and theorize the formal components of a modernist mathesis of fiction. The internal allegory necessary to fiction is restaged in Borges’s work by the double presence of number, producing an allegory that does not demonstrate the sheer autonomy of language from content, and the separation of the sign from the signifier, but posits a generic and genetic device at the heart of the machinations of language: number.

3

The Lemniscate: Infinite Shapes in the Work of Samuel Beckett

The art of Samuel Beckett has become an art of zero, as we all know. J.M. Coetzee, Doubling the Point That’s how he reasons, wide of the mark, but wide of what mark, answer us that. Samuel Beckett, Texts for Nothing It is a commonplace to identify ‘bareness’, or ‘minimalism’, or ‘lessening’ as the governing aesthetic principle of the prose, theatre and television works of Samuel Beckett. This attribution of ‘bareness’ rends Beckett’s artistic achievement in the terms of subtraction: creativity is valued here for what it takes away from ‘reality’, ‘scene’ or ‘thought’.1 The barren stage sets of Endgame and Waiting for Godot, and aphasic narrators of Molloy, Malone Dies and The Unnamable exemplify this visual or linguistic ‘minimum’.2 This aesthetic is linked to mathematics in two senses. First, it relies on forms of stylistic subtraction to get ‘behind’ language, to remove the obfuscation from language, or, rather, the obfuscation that is language. Beckett’s texts share with mathematics, in Ingo Berensmeyer’s words,

1

2

French philosopher Alain Badiou has written the most substantial account of ‘subtraction’ in Samuel Beckett’s work. See Alain Badiou, On Beckett, trans. Nina Power and Alberto Toscano (Manchester: Clinamen Press, 2003). Power and Toscano provide a succinct summary of Badiou’s notion of subtraction in Beckett: ‘Beckett’s writing draws its force and urgency precisely from the way that it subtracts itself from our impressions and intuitions; in other words, from the manner in which it evacuates our muddled and spontaneous phenomenologies to reveal a sparse but essential set of invariant functions that determine our “generic humanity”’ (xiii–xiv). For an account of the link between aphasic speech and modernist aesthetics in Beckett, see Laura Salisbury, ‘What Is the Word: Beckett’s Aphasic Modernism’, Journal of Beckett Studies 17, no. 1–2 (2008), 80–128.

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‘the formal principle of reducing complexity, both linguistic and experiential, in order to gain a higher degree of (structured) complexity’.3 The subtraction from language, as a process of formalism, cleaves to both the process of mathematical work and the type of language that we find in mathematics. Secondly, Beckett’s ‘minimal’ undertaking is implicated in a renovation of the novel form in the wake of the shift in the foundations of counting, the most basic system of order underpinning reality. Molloy claims that he has become ‘a little less’ than ‘the creature [he was] in the beginning, and the middle [of his tale]’, a process he exults as the most he can hope for.4 In this reflection, Molloy unites life, chronology and narrative in an arc of beginning, middle and end, yet reverses the traditional numerical assumption behind the arc of the novel: maturation, complication, consequence. These stages are defined, in the case of the traditional novel, by becoming a little more or a little different than what was before. Beckett’s ‘lessening’ or ‘subtraction’ emerges from an artistic endeavour preoccupied with foundations; preoccupied with the absence of a solid linguistic, semiotic and symbolic ground upon which the fictional world is constructed.5 In Beckett’s words, this came from a general ‘rupture in the lines of communication’.6 In this chapter, I will consider the way that this rupture in communication is intimately bound up with a rupture in the capacity to count. Beckett’s minimum is a search for a new foundation in the wake of the disintegration of various atomic elements of literature: the individual, the word, the progress of time. J.M. Coetzee’s statement – that ‘Beckett’s art is

3

4 5

6

Ingo Berensmeyer, ‘“Twofold Vibration”: Samuel Beckett’s Laws of Form’, Poetics Today 25, no. 3 (2004), 466. Samuel Beckett, Molloy, ed. Shane Weller (London: Faber and Faber, 2009), 28. This is the antithesis, as Ruby Cohn points out, of Joyce’s ‘apotheosis of the word’. See: Ruby Cohn, A Beckett Canon (Ann Arbor: University of Michigan Press, 2001), 89. Beckett opens his 1934 review, entitled ‘Recent Irish Poetry’, with the polemical paragraph that contains this phrase: I propose, as rough principle of individuation in this essay, the degree in which the younger Irish poets evince awareness of the new thing that has happened, or the old thing that has happened again, namely the breakdown of the object, whether current, historical, mythical or spook. The thermolaters – and they pullulate in Ireland – adoring the stuff of song as incorruptible, uninjurable and unchangeable, never at a loss to know when they are in the Presence, would no doubt like this amended to breakdown of the subject. It comes to the same thing – rupture in the lines of communication. Samuel Beckett, ‘Recent Irish Poetry’, in Disjecta: Miscellaneous Writings and a Dramatic Fragment, ed. Ruby Cohn (London: John Calder, 1983), 79.

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an art of zero’ – incorporates the critical fatigue that this necessary but now clichéd aesthetic diagnosis involves (‘as we all know’). But he also signals a crucial, and less frequently remarked upon aspect of this aesthetic: the fact that an aesthetic problem is also profoundly a mathematical problem. Coetzee elaborates his initial remark on zero precisely in terms of the mathematical quest for foundations: If we can justify an initial segmentation of a set into classes X and not-X, said the mathematician Richard Dedekind, the whole structure of mathematics will follow as a gigantic footnote. Beckett is mathematician enough to appreciate this lesson: make a single sure affirmation, and from it the whole contingent world of bicycles and greatcoats can, with a little patience, a little diligence, be deduced.7

Here, Coetzee echoes Beckett’s famous claim about the ‘unword’. In the oftquoted ‘Letter to Axel Kaun’, Beckett articulates his own criticism of the ‘real of language’, claiming that words obscure more than they reveal. In this letter, Beckett imagines an alternative literature that circumvents such limitations. This is the literature of the ‘unword’.8 The Beckett of 1939 could not find ‘any reason why that terrible materiality of the word surface should not be capable of being dissolved’ and thus saw the task for literature as one which worked to reveal what is prior to or behind language, a ground of genuine pre-linguistic expression.9 From this, the capacity to secure a single adequate affirmation – the security of the first word – should follow.10 This starting point for literary creation results in an attempt to induce literature rather than create it. It is this aspect of Beckett’s work in which he already seems to be a late modernist.

7

8

9 10

J.M. Coetzee, ‘Samuel Beckett and the Temptations of Style’, in Doubling the Point (Cambridge, MA: Harvard University Press, 1992), 43. Samuel Beckett, The Letters of Samuel Beckett, Volume 1: 1929–1940, ed. George Craig and Dan Gunn (Cambridge and New York: Cambridge University Press, 2009), 518. Ibid., 518. Coetzee’s ‘single, sure affirmation’ is echoed by Hugh Culik is the idea of a ‘descriptive sufficiency’: ‘In his aesthetic agenda we see […] a reformulation [of the modern] through a strategy that relies upon the metaphoric power of non-literary fields such as neurology, aphasiology, and mathematics to represent two related ideas: first, the descriptive (in)sufficiency of language, and second, the (in)ability of a formal system to comprehend itself.’ More than an analogy between one formal system and another, mathematics is the alogos of literature, essential, rather than supplementary, to Beckett’s modernism. Hugh Culik, ‘Mathematics as Metaphor: Samuel Beckett and the Esthetics of Incompleteness’, Papers on Language and Literature 29, no. 2 (1993), 132.

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There are multiple short studies that have collectively established the centrality of mathematics in Samuel Beckett’s work. Together, these studies provide the most important precedent for work on mathematics in Beckett’s oeuvre. The work of Chris Ackerley, Hugh Culik and Brian Macaskill has illuminated varied mathematical elements in Beckett’s work and catalogued mathematical episodes in his oeuvre; elaborated the literary significance of mathematical modes of negation and incompleteness; and theorized the connection between literary and mathematical uncertainty, respectively.11 The present chapter explores the foundations of number and the significance of infinite shapes in Beckett’s novels and his short prose works. Through an analysis of two of Beckett’s novels, Molloy and Watt, and a consideration of the unpublished short text entitled The Way, I will argue that the radicalization of naturalist prose form in Beckett’s novels occurs through a replacement of geometry with topology, and a replacement of the requirement for ‘typicality’ with the requirement for ‘genericity’. In both cases, this is a replacement of a finite mathematics with infinite form. I will argue that the replacement of one with the other is less a refutation of novelistic form so much as a fuller realization of it. ‘Typicality’ and ‘types’ echo the Aristotelian category of genera: a unified identity of an animal, plant or object that differentiates it

11

Macaskill’s essay ‘The Logic of Coprophilia: Mathematics and Beckett’s Molloy’ analyses ‘an affinity between Beckett’s style and numerical practice’ but also rejects a ‘naïve belief in mathematical certainty’ (14). Indeed, it is mathematical uncertainty that Macaskill focuses on, looking at Molloy’s ‘logic of […] being and the logic of literature which delivers his being’ through Aristotelian logic, Gödel’s incompleteness theorem and Russell’s paradoxes. Mackaskill produces a masterful analysis of Molloy’s relation to zero, in particular the ‘zero’ of the anus, the ungenerative orifice that so preoccupies and disturbs Molloy. Hugh Culik’s ‘Mathematics as Metaphor: Samuel Beckett and the Esthetic of Incompleteness’ has a similar focus to Macaskill, looking at the necessary incompleteness of ‘descriptive sufficiency’ (132). Culik works with the ‘Pythagorean struggle with irrational numbers’ (132) and Beckett’s development of Pythagorean metaphors. Culik focuses largely on the novel Murphy, the mind–body problem, and the ‘Pythagorean ambition to mathematize the world’ (143), in order to elaborate the significance of mathematical models for the very possibility of Beckett’s art. Culik’s essay is particularly significant, here, because he connects Beckett’s formal solutions to representational issues with mathematics and – briefly – his modernist context. Ackerley’s entry on ‘mathematics’ in the Grove Companion to Samuel Beckett details instances and engagements with mathematics across the entire oeuvre of Beckett. This brief article is encyclopaedic in nature and does not possess the scope or intention for detailed analysis. See Culik, ‘Mathematics as Metaphor: Samuel Beckett and the Esthetics of Incompleteness’, 131–151; Brian Macaskill, ‘The Logic of Coprophilia: Mathematics and Beckett’s “Molloy”’, SubStance 17, no. 3 (1988), 13–21; C.J. Ackerley and S.E. Gontarski, The Grove Companion to Samuel Beckett (New York: Grove Press, 2004), 347–358.

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from others. Despite the morphological resemblance to ‘genera’, the generic, on the other hand, is the antithesis of the ‘typical’: it is a placeholder for a multiplicity that may have ‘measure but not determination’.12 I will argue that this ‘generic literature’ involves a form of counting that does not proceed in a cumulative and linear fashion, but rather through a continuous topological process. This installs a new naturalist numeracy that befits generic rather than ‘typical’ prose fiction. Beckett’s late short fiction works also pursue the task of creating an art of the ‘unword’: a word that betrays the ‘grammar and style’ or the ‘surface’ of language and presents itself as an opening to what is outside or beneath the phenomenal fabric of dynamic speech and writing.13 In the second half of this chapter, I will demonstrate how two key texts in Beckett’s late prose – All Strange Away and Imagination Dead Imagine – again realize this sort of ‘generic literature’ by evacuating traditional elements of the phenomenal, including narrative experience, the production or representation of a world and vivid imagery. In these three late works, Beckett is concerned with the two most basic predicates of experience: space and time, which, in the Kantian tradition, are considered to be conditions of phenomenality, rather than content of phenomenality.14 In these texts it is explicitly number, and, in particular, magnitude rather than measure, that allows Beckett to pursue these ‘conditions’ through a medium that is traditionally associated with the representation of the phenomenal world. As in his novels, in Beckett’s short prose he creates a fiction that possesses measure without totality, or measure without completion or determination. This ‘literature without measure’ is also, then, a literature of infinitude, existing as creation-by-magnitude outside of experience and even imagination, insofar as imagination is connected to the finitude of experience. This is the contradiction that Beckett manages to inhabit in this later work: he has produced a literature of x: a literature of that which we take to stand outside of phenomenal experience, after the death of the imagination. 12

13 14

Once again, here I am glossing Hallward’s definition of the generic. See: Peter Hallward, ‘Generic Sovereignty: The Philosophy of Alain Badiou’, Angelaki: Journal of the Theoretical Humanities 3, no. 3 (2008), 87. Beckett, The Letters of Samuel Beckett, Volume 1: 1929–1940, 518. Immanuel Kant, Critique of Pure Reason, trans. Marcus Weigelt (London: Penguin Classics, 2007), 65.

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A Mania for Symmetry: Molloy and the Continuous Deformation of Language The two novels that Beckett is perhaps best known for, Molloy and Watt, stage various processes of enumeration that divert rather than facilitate narrative progress. In Molloy, the famous ‘sucking stones sequence’ is merely one instance of the eponymous character’s attempts to resolve a stable relation between body and world, pleasure and identity, through a numerical system. Molloy is one of three novels that are now grouped as a ‘trilogy’, which consists of Molloy and Malone Dies, both published in French in 1951 and then translated in 1956, and The Unnamable, published in French in 1953. Molloy lives in what he thinks is his mother’s room – although this is not verifiable for either Molloy or the reader. Molloy does not know how he came to reside in this room, but occupies himself with writing, and tolerates the daily intrusion of a woman who opens the door and places his food beside it and takes out the chamber pot, as well as the weekly intrusion of a man who comes every Sunday – Molloy thinks it is a Sunday – to collect his papers (Molloy seems to spend some of his time writing), return last week’s papers (which appear with a series of editorial marks on them) and give him money. Molloy had previously taken a rambling journey – mostly on a bicycle but eventually walking – to find his mother, but presumably has not succeeded, and has somehow ended up incapacitated in her room. It is the tale of this journey that Molloy will narrate for his readers. Molloy broadly does not know why he writes or why he is in the place he is in. His account constitutes the limits of testimony in the sense that there is no necessity to his extended autobiography: he is a point on a narrative map without coordinates rather than a character with place and history. There is, of course, a reciprocal relation between the ‘point’ and zero. The point is the precursor sign of zero, just as zero properly functions as a placeholder (there are ‘zero’ hundreds in 7056) in large numbers. Molloy appears less as a character and more as a placeholder; his narrative is pointless; his journey, and – as we will see – even his use of language, is circular. This substitution of character with placeholder is exemplified early in the novel, where Molloy relates his observation of two figures on the road. Molloy

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names them ‘A’ and ‘C’ and observes that they were ‘going slowly towards each other, unconscious of what they were doing. […] At first a wide space lay between them. They couldn’t have seen each other, even had they raised their heads and looked about’.15 Like Molloy and Moran, A and C walk to meet each other, until they ‘stop breast to breast’ and then ‘turned towards the sea which, far in the east, beyond the fields, loomed high in the waning sky, and exchanged a few words. Then each went on his way’.16 A and C are not characters: they are generic placeholders, and hence they can be without proper names, but are instead indicated by algebraic placeholders. Dan Mellamphy construes this same arbitrariness found in Endgame in terms of figuration. Mother Pegg is, for Mellamphy, a ‘non-persona [who] functions as the focal point not only of that play in particular (Endgame), but figuratively – or rather, figurelessly: that is, as a function (functionally) rather than a figure per se – more generally in Beckett’s work’.17 In this sense, A and C do not properly achieve the status of the figure: they retain the figure whilst evacuating it of its substance, thereby ‘functioning figurelessly’ in Mellamphy’s formulation. What Beckett presents, here, is not entirely alien to the novel form, so much as a radicalization of a crucial element of the novel. Novelists have long relied upon the arbitrariness of the numerical figure to ground their narratives in fact; indeed, facts are required to support the apparatus of fictionality. The requirement for facts or a sense of ‘facticity’ is, of course, one part of the paradox of fictionality. The novel has traditionally been defined against any straightforward depiction of ‘real life’, society and the world, in order to embrace the narrative freedom that comes through avoiding literal description of one’s surroundings (and avoid the charge of libel). In Catherine Gallagher’s words, ‘because the novel defined itself against the scandalous libel, it used fiction as the diacritical mark of its differentiation’ and yet simultaneously, ‘its fictionality also differentiated itself from previous incredible forms’.18 This is the double bind of fictionality: the novel must at once assert its fictional status in order to 15 16 17

18

Beckett, Molloy, 4–5. Ibid., 5. Dan Mellamphy, ‘Alchemical Endgame: “Checkmate” in Beckett and Eliot’, in Alchemical Traditions from Antiquity to the Avant-Garde, ed. Aaron Cheak (Melbourne: Numen Books, 2013), 491. Catherine Gallagher, ‘The Rise of Fictionality’, in The Novel, Volume 1, ed. Franco Moretti (New Haven, CT; and London: Princeton University Press, 2006), 340.

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have purchase on the world, and yet simultaneously disavow the outer reaches of fiction – the ‘incredible’ or unbelievable. In Gallagher’s terms, this requires both an embrace and a disavowal of fictionality: ‘the novel slowly opens the conceptual space of fictionality in the process of seeming to narrow its practice’.19 This novelistic reliance on fact is thus not straightforward: fictional facts must be neither incredible nor true. As such, characters must embody a paradox that Beckett would have enjoyed: they must be typically singular. The character must be far enough from the world to be a distinct persona yet recognizable enough to evoke sympathy and empathy. This paradox is embraced too closely in Molloy: rather than character being typical through empathy and recognition, they are typical by virtue of mathesis. The implication of a ‘mathematical typicality’ is that the object is not elaborated or filled with predicates but rather evacuated of content: made generic – a placeholder for any content, thus in effect exceeding and undoing any typicality whatsoever. This is novelistic naturalism taken to an extreme, where typicality is realized through an algebraic sign (replacing character) or subordination to seriality (replacing verisimilitude). These algebraic figures and the circularity of the narrative are accompanied by a pervasive narrative exhaustion. Molloy is virtually immobilized and seems to be waiting for his death, although he doesn’t quite have the energy to perform any action that might hurry it along. Molloy is – aside from his narrative activities – the exemplar of extreme passivity: All grows dim. A little more and you’ll go blind. It’s in the head. It doesn’t work any more, it says, I don’t work any more. You go dumb as well and sounds fade. The threshold scarcely crossed that’s how it is.20

These reflections on his world are shaped by his awareness of the contingency of perception and, as such, constitute an alienated analytic relation to his own journey: But now he knows these hills, that is to say he knows them better, and if ever again he sees them from afar it will be I think with other eyes, and not only that but the within, all that inner space one never sees, the brain and heart

19 20

Ibid., 340. Beckett, Molloy, 4.

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and other caverns where thought and feeling dance their sabbath, all that too quite differently disposed.21

Molloy’s sense of his own stagnating journey rends his perception at once radically without purchase and wedded to a cold logic of probability. This strange ‘scholastic autism’,22 to use Anthony Cordingley’s words, is exemplified when Molloy kills a man in a completely bizarre fashion. After hitting him on the head with one of his crutches, he kicks him to death by swinging over him using his crutches as supports. Slightly stranger is the geometric impulse that operates behind this: ‘I rested a moment, then got up, picked up my crutches, took up my position on the other side of the body and applied myself with method to the same exercise. I always had a mania for symmetry.’23 This ‘mania for symmetry’ constitutes a wider aesthetic inversion that swaps the structure of passion for the passion for structure in the same way that Molloy swaps the figurative (a character, or a potential one) for the placeholder (A or C). A reading of symmetry and journey, or symmetry and act, in Molloy necessarily becomes a reading of identity and symmetry as well, which is exemplified in the unusual naming. The fact that Molloy and Moran share an initial – indeed, a first syllable – is perhaps unremarkable (although less so when the prevalence of the letter ‘M’ – the perfectly symmetrical letter halfway through the alphabet – across Beckett’s oeuvre is considered, including ‘Martha’, Moran’s housekeeper, ‘Murphy’, ‘Malone’ and ‘Mercier’). However, when considered in terms of Molloy’s personal theory of language, and indeed personal application of language, this coincidence does become significant as a symmetry. Unlike the symbol, this symmetry exists without any ‘ground’ for significance, forcing the reader into replicating Molloy’s ‘mania for symmetry’. Molloy’s mother has a particular name for him, although he claims her choice is misguided, and he in turn has his name for her, which seems to be equally idiosyncratic: She never called me son, fortunately, but Dan, I don’t know why, my name is not Dan. Dan was my father’s name perhaps, yes, perhaps she took me

21 22

23

Ibid., 6. Anthony Cordingley, ‘Samuel Beckett’s Debt to Aristotle: Cosmology, Syllogism, Space, Time’, Samuel Beckett Today/Aujourd’hui 22 (2010), 184. Beckett, Molloy, 79.

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for my father […] I called her Mag, when I had to call her something. And I called her Mag because for me, without my knowing why, the letter g abolished the syllable Ma, and as it were spat on it, better than any other letter would have done.24

Molloy’s rejection of his mother is symbolized here for the reader through a satirical compliance with the appellation he is meant to use for her. Molloy’s half-resistance presents the uniquely Beckettian structure of the ‘unword’, a word which retains its identity as a lexeme but is evacuated of content, whereby he retains the word ‘Ma’ but manages to negate it by adding a velar plosive ‘g’.25 This addition of a letter makes Molloy’s idiosyncratic language synthetic. A synthetic language adds or clusters individual morphemes to create more complex ones.26 Molloy’s construction of ‘Mag’, however, is derivational: out of a word and a letter, a new word is produced. Derivational synthetic constructions are usually produced by combining two words to create a new word, in opposition to relational construction whereby a new word is created to express a grammatical construction. Here we see a kind of negative synthetic creation: Molloy does not properly create a new word, but generates a version of it in order to negate that word. There is, as such, never synthesis of two words, or a proper replacement of the word, so much as a continuous negation: presenting the appropriate appellation whilst also, each time, denaturing it in a circular fashion. 24 25

Ibid., 13 Moran has the same synthetic approach to language as Molloy, referring to Molloy’s mother as ‘Mother Molloy, or Mollose’ and musing over the reasons he might add the suffix ‘ose’ to her name: Of these two names, Molloy and Mollose, the second seemed to me perhaps the more correct. But barely. What I heard, in my soul I suppose, where the acoustics are so bad, was a first syllable, Mol, very clear, followed almost at once by a second, very thick, as though gobbled by the first, and which might have been oy as it might have been ose, or ne, or even oc. (ibid., 107)

26

Here, Moran splits the Molloy that exists in his mind versus the one presented to him by Gaber through a suffix, to make ‘Molloy’ and ‘Mollose’. He nonetheless must relinquish ‘Mollose’ ‘since Gaber had said Molloy, not once but several times…. I was compelled to admit that I too should have said Molloy and that in saying Mollose I was at fault’ (ibid., 107). Jeanne-Sarah de Larquier pushes this even further, using Molloy’s own ‘mania for symmetry’ to read into his name a graphic and alphabetic symmetry, one can ‘think of “M-ollo-Y”, where the ‘ollo’ not only is symmetric, but also graphically looks like a bicycle, Molloy’s main means of transportation’, an exemplary instance of speculative symbolism. Jeanne-Sarah de Larquier, ‘Beckett’s Molloy: Inscribing Molloy in a Metalanguage Story’, French Forum 29, no. 4 (2004), 52. Where isolated languages have one morpheme per word (or, with a less rigorous definition, few morphemes per word), synthetic languages will have multiple morphemes per word (the hyper version of this is called polysynthetic language; synthesis upon synthesis).

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This produces a constant but not final negation: a kind of circle of existences and erasure. This circularity both reveals and disrupts the geometric form that structures the efficacy of the sign, just as narrative circularity reveals the necessity of forms of mensuration to the wider plot. This is a first instance of grammatical topology replacing typology, or deformation replacing shape. Where linguistic typology is concerned with structure of language and construction of words through morpheme, topology – a branch of geometry – studies continuous form, uniting the fields of set theory and geometry to focus on formation and deformation in place. In this deformation of the juvenile idiom for mother, content is replaced with structure. Saussure’s image of language as a chain of signification does not apply here. Rather, we have a circular process of negating whilst preserving: a linguistic replacement of geometry with continuous deformation. This formal circularity recalls Beckett’s participation in Eugene Jolas’s manifesto entitled Poetry is Vertical (co-authored with Hans Arp, Thomas McGreevy and others). This famed 1932 modernist manifesto was signed by nine modernist writers and artists involved with the journal transition, including Beckett. Jolas’s manifesto records the modernist impulses towards essential depths and near-mystic truths in art: ‘Poetry builds a nexus between the “I” and the “you” by leading the emotions of the sunken, telluric depths upward toward the illumination of a collective reality and a totalistic universe.’27 In addition to his participation in Jolas’s vertical manifesto, Beckett gave a ‘parodic lecture […] in November 1930 titled “Le Concentrisme”’, in which he invents a French literary avant-garde movement that he calls ‘les concentristes’ (homophonous with ‘les cons sont tristes’). A concentriste is a ‘biconvex Buddha’, a ‘prism on the staircase’, at once ‘excluant et exclu’.28 ‘Concentrisme’ realizes in style what the vertical manifesto does not: the pointlessness, the satirical tone of the manifesto and the chaotic polysemy of the title contributes significantly to a genuine artistic call for formal divestment from distinct and unproblematic categories of self and other, linear rather than vertical intention. ‘Concentrisme’ hints at the radical possibilities of a topological rather than 27

28

Hans Arp et al., ‘Poetry Is Vertical’, in Manifesto: A Century of Isms, ed. Mary Ann Caws (Lincoln and London: University of Nebraska Press, 2001), 529. Beckett is quoted in: Berensmeyer, ‘“Twofold Vibration”: Samuel Beckett’s Laws of Form’, 471.

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geometric approach to fiction, one which involves negation, satire and insincerity in order to achieve a multiplication of the geometric form inherent to fiction, rather than a rigid ‘verticisation’ of processes of composition. ‘Concentrisme’ is Beckett’s first reorganization of the circle of signification, a reverberating series of circles that will, in these novels, be replaced with another more subtle mode of continuity: topology. This ‘circular deformation’ beneath the names in Molloy is complemented by events in the narrative that can be called ‘manias for geometry’. The nowfamous episode in Molloy, the ‘sucking stones sequence’, revolves around a circular obsession of Molloy’s. He has collected sixteen pebbles, which he sucks for some sort of comfort, whether that is physical or mental, or both: I had say sixteen stones, four in each of my four pockets these being the two pockets of my trousers and the two pockets of my greatcoat. Taking a stone from the right pocket of my greatcoat, and putting it in my mouth, I replaced it in the right pocket of my greatcoat by a stone from the right pocket of my trousers, which I replaced by a stone from the left pocket of my trousers, which I replaced by a stone from the left pocket of my greatcoat, which I replaced by the stone which was in my mouth, as soon as I had finished sucking it.29

The stones thus circulate in a clockwise direction from pocket to pocket, Molloy sucking each stone in turn. However, unless the stones manage to form a line in Molloy’s pockets, which would only happen by fantastic coincidence, Molloy cannot really know how long any of these stones has been in his pocket. The stones more or less – or at least often – move in a clockwise direction, although Molloy never knows if a complete rotation is performed. Eventually, Molloy must come to terms with this: The possibility nevertheless remained of my always chancing on the same stone, within each group of four, and consequently of my sucking, not the sixteen turn and turn about as I wished, but in fact four only, always the same, turn and turn about. So I had to seek elsewhere than in the mode of circulation.30 29 30

Beckett, Molloy, 64. Ibid., 65.

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This model of circulation leaks constantly by virtue of – like every other fact in Molloy’s life – an absence of verifiability. The system, here, is flawed because it is partial and based on comfort: it is only a creeping sense of the inadequacy of his system, an experiential doubt that prompts Molloy to ‘confess’ that he must revise his system. The physical comfort here is only in part the alleviation of hunger: it is equally the ‘bodily need’ to ‘suck the stones in the way that I have described, not haphazard, but with method’.31 The ‘reason’ behind the sucking stones is both psychological and physiological and models, for Molloy, a coherent relation between mind and object, language and organization of the world. Together with Watt, Molloy has been called an ‘attack upon Cartesian rationality’ by C.J. Ackerley, who claims that the book ‘probe[s] the soft centres of the rationalist enterprise’.32 For Ackerley, the ‘soft centre’ of reason implies a criticism of the Cartesian project: Cartesian rationality relies on a dualism of body and mind and takes scientific method to be the privileged route to the attainment of truth, all of which Ackerley holds to be deeply suspect. Molloy’s sequence certainly models some of the most urgent issues at stake in Cartesian philosophy. Most importantly, the sucking stones illustrate Molloy’s psychic investment in what is called ‘numerical materialism’, a term that refers to a formalization of bodily extension that originated with Cartesian geometry. In Robin Mackay’s somewhat broader conception, numerical materialism is ‘an inquiry into the extent and nature of number’s dominion over any philosophy calling itself a materialism; but also an inquiry into the materiality of number and numerical practices’.33 Yet the Cartesian stakes in the sucking stones sequence by no means produce some direct criticism of Cartesian rationality, so much as an amendment to it. Descartes (1596–1650) is most famous for the ‘Cartesian coordinate system’ that forms the basic graph with four axes. Undergirding this coordinate system are two theoretical positions. First, Descartes allows arithmetic – the manipulation of points on the number line – to be mapped onto a geometric system, and on the other hand Descartes formalizes a concept of identity 31 32 33

Ibid., 68. C.J. Ackerley, ‘Preface’, in Watt, ed. Samuel Beckett (London: Faber and Faber, 2009), i. Robin Mackay, ‘Editorial Introduction’, in Collapse, vol. 1 (Oxford: Urbanomic, 2007), 6.

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that makes mind and body essentially separate entities that are nonetheless also sutured to each other. Descartes distinguished numerical and qualitative identity: numerical identity is the persistence of identity despite some material modification. In Cartesian terms, if I have a leg amputated, I still retain the same ‘numerical identity’ despite the loss of a limb. Here Descartes’s famous res cogitans (non-dimensional matter) and res extensa (three-dimensional matter) demarcate two concepts of number that exempt the human from the law of numerical materialism. It is the res cogitans, the soul – the point of intersection between vertical and horizontal axes, the 0, 0 coordinate – which remains consistent despite any greater or lesser material measure. This 0,0 point is the source for numerical extension, and the principle that maintains identity despite changes to this extension. This is also the point in the graph that has no horizontal coordinates and no vertical coordinates. Molloy’s ‘sucking stones’ is in one way a perfect material model for Descartes’s theory: a numerical and material system (even though it takes Molloy a couple of tries to make this system ‘count’) that instantiates a geometric extension of Molloy’s soul, which is given consistency by Molloy’s ‘system’ and which, in turn, alleviates both a psychological and a physical hunger. Here we have a geometric mapping of Molloy qua rational being, and the sucking stones are thus an image of extension: of the consistent, coherent ‘numbers’ of Molloy’s identity. As in Descartes’s own theory, this system perpetuates the ‘sensory contradiction’, whereby the integrity of identity – here the individuality of each of the stones, the uniqueness and consistency of their numbers that would ensure a proper circulation – is based in the recognition of each of the stones. Descartes did not derive this profound origin of Western geometry in metaphysics but rather grounded it in sense experience. Vlad Alexandrescu reminds us that ‘Descartes relativizes knowledge of the world through the senses’, simultaneously establishing a ‘modus operandi that is geometrical’.34 Algebraic geometry originates, here, in a model of proprioception rather than some conceptual system divorced from human embodiment. Proprioception is the generic origin of perception itself, referring not to perception of the

34

Vlad Alexandrescu, ‘Y a-T-Il Un Critère d’Individuation Des Corps Physiques Chez Descartes’, ARCHES: Revue Internationale Des Sciences Humaines 5 (2003), 436.

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world around us so much as our sense of the extensity of our bodies. Like the Cartesian coordinate system, Molloy’s geometrical modus operandi is reliant upon his senses, and in order to complete his system he must enable some form of recognition that the stones move from pocket to pocket as they should. This is less a critique of rationality than a recognition of the proprioception – the ‘soft’ centre – essential to reason. Just as Molloy transforms the circulation system of language through his lexical construction ‘Mag’, so too does he transform the circulation of this proxy for bodily extension. This transformation is facilitated by the word ‘trim’, which acts as a sort of conceptual hinge between habit and a better way of doing things: ‘One day suddenly it dawned on me, dimly, that I might perhaps achieve my purpose without increasing the number of my pockets, or reducing the number of my stones, but simply by sacrificing the principle of trim’, Molloy announces.35 Both the origin and the substance of ‘trim’ are a mystery to Molloy: The meaning of this illumination, which suddenly began to sing within me, like a verse of Isaiah, or of Jeremiah, I did not penetrate at once… Finally I seemed to grasp that this word trim could not here mean anything else, anything better, than the distribution of the sixteen stones in four groups of four, one group in each pocket, and that it was my refusal to consider any distribution other than this that had vitiated my calculations until then and rendered the problem literally insoluble.36

The word ‘trim’ is radically vague, here, because it is overdetermined. Trim can mean ‘the condition of being properly balanced’, ‘in good condition or order’, ‘to strengthen’, ‘to comfort’ as well as ‘to become pregnant’.37 Here, ‘trim’ cannot itself enact that which it refers to, showing up the impossibility of language to fortify a perfect circular system (the sucking stones). Like Molloy’s ‘partial negation’ of the word ‘Ma’, ‘trim’ allegorizes the very process of linguistic expression and its incapacity for true performative equality with acts or systems.

35 36 37

Beckett, Molloy, 66. Ibid. Oxford English Dictionary, 2nd ed., s.v. ‘trim’.

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Molloy’s stones are for sucking, and thus circulate from his pockets to his mouth and back, a kind of reversal of the way that words are supposed to emanate from the mouth in a semiotic rather than abstract circulation. This brings into relief the flipside of natural language, its unacknowledged mathematical inverse: the patterns that resemble counting beneath and within language. Here there is an indistinction between a phrase and an actuality that belies the synthetic form of language. This is astounding simply for the fact that structure, not semiotics, and reason, not the subject, comes first: it is the pattern of points that precedes the content. Likewise, it is the ironic contradiction sustained in ‘trim’ that demonstrates the impossibility of a descriptive adequacy (a circular reciprocity between word and referent), resulting in, perhaps, the kind of broken totality whereby number, numerical extension and quality are confused. Mathematics in Molloy is, then, not an allegory of some intended meaning or a diagnostic tool: it is the expression of the non-semiotic jump between reason and material (partial, recurrent negation), mind and extension (proprioception), language and referent. This is where the organization of sign and referent defaults from a geometric analogy to a topological one. This ‘continuous deformation’ is perhaps most clearly realized in a short, unpublished text written by Beckett in 1981. Beckett produced seven manuscript versions of a text provisionally entitled ‘The Way’ which was finally split into two sections under the subheadings ‘8’ and ‘∞’. Each of these sections record a journey which charts the shape of the lemniscate. This diagram is the first draft of a short, two-part text that traces movement without locale and figure without subjectivity. In the subsequent drafts, this diagram will be rendered in prose: ‘Forth and back across a barren same winding one-way way. […] Through emptiness the beaten ways as fixed as if enclosed. Were the eye to look unending void.’38 In ‘The Way’, and in the prose texts, Beckett constructs a reciprocal relation between matheme and word, soul and extension, the mute mark and its semiotic other. He does this with a substantial dose of humour: the text traces the path of a mathematical figure, describing a journey around a figure eight and then a lemniscate. The development, affective import and ultimately conclusion of this

38

Samuel Beckett, ‘The Way’ (Carlton Lake Collection, Harry Ransom Humanities Research Center, The University of Texas at Austin, Box 17, Folder 3, 1981).

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short and beautiful prose work are all based on a predetermined structure: the shape of a mathematical sign. The prose creates the mathematical mark, just as the mathematical mark creates the prose: this reciprocity between the matheme and the text presents an obscure allegorical relation between the two domains: one ‘language’ (either the prose or the matheme) is the implicit ‘other’, structuring our access to the other ‘language’ (whether prose or matheme). Beckett’s work involves circular form that – by virtue of geometric inevitability of the curve that will eventually meet its origin – negates what is usually considered to be narrative change or transformation that must be, necessarily, linear or helical. This is not, however, an issue of simple substitution of one narrative geometric model with another. Beckett replaces linear narrative not only with circular form but with something at once more subtle and more disruptive to a literary naturalism: circular, continuous deformation.

Permutation and Division in Watt In the novel Watt, various characters engage in forms of enumeration as compulsions towards order, practices that defy their purpose and negate any narrative ‘account’ rather than stabilize it. The events that carry the plot of Watt are distinctive for their absence of meaningful outcome. The purposes or ends of the activities that Watt undertakes is, often, not so much productive as establishing an order for the sake of order itself: It fell to Watt to weigh, to measure and to count, with the utmost exactness, the ingredients that composed this dish […] and to mix them thoroughly together without loss, so that not one could be distinguished from another, and to put them on to boil, and when boiling to keep them on the boil…. This was a task that taxed Watt’s powers, both of mind and of body, to the utmost, it was so delicate, and rude.39

This activity does not create substantial difference but, rather, both material and psychological chaos. The urgency of exactitude weighs on Watt just as it weighs on the reader waiting for telos or substance to be attributed to the 39

Samuel Beckett, Watt, ed. C.J. Ackerley (London: Faber and Faber, 2009), 73.

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process of ordering. In other words, what is missing from these endeavours of precision is, precisely, what or, the equivalent in this tale, Watt: a character bearing goal, intention and disposition. Watt was written between 1942 and 1944 whilst Beckett was in hiding in France during the Second World War, but it would only be published in English in 1953, and only much later translated into French by Beckett himself and published in 1968. Watt features two key figures named ‘Watt’ (What) and ‘Knott’ (Not). In this novel, issues of matter, identity and negation are explored through processes characteristic of mathematical enquiry: demonstration, refutation, compatibility and completion. Much has been written characterizing this text as an attempt to assert rationality in the irrational world of France controlled by the Vichy regime and the violence of the Second World War. John J. Mood has called Watt ‘[Beckett’s] most devastating depiction of the cul-de-sac of modern Western rationalistic philosophy’.40 The connection between rationalism and mathematics here revolves around the possibility of the stability of the category, and in the symbolic markers that mathematics provides to mark the gait of reason. This subversion is enacted, on one level, through replacing novelistic conventions with logical or mathematical claims that stand in contradiction to their context, a process that allows Beckett to bring out the contradictions entailed in exercises of objective rationality. The plot of Watt follows an unusual count: As Watt told the beginning of his story, not first, but second, so not fourth, but third, now he told its end. Two, one, four, three, that was the order in which Watt told his story. Heroic quatrains are not otherwise elaborated.41

This emphasis on (arbitrary) structure is coupled with an emphasis on – as J.M. Coetzee puts it – the ‘fictiveness of the fiction’,42 through the inclusion of remarks on the plausibility of the narrative. (Coetzee’s example pinpoints this: ‘Haemophilia is […] an enlargement of the prostate. But not in this work.’43)

40 41 42

43

John J. Mood, ‘“The Personal System” – Samuel Beckett’s Watt’, PMLA 86, no. 2 (1971), 255. Beckett, Watt, 186. J.M. Coetzee, ‘The Comedy of Point of View in “Murphy”’, in Doubling the Point: Essays and Interviews, ed. David Attwell (Cambridge: Harvard University Press, 1992), 37. Beckett, Watt, 102.

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This stress on plausibility serves to foreclose both the development of character and any representation of change through the constant reference to its own fictionality. Here I will look at how this rupture in novelistic bounds or discourse is inextricably bound up with the processes of counting in this novel. Describing these two narrative markers as a negation or undoing of the discourse of the novel is an attempt to recognize the dual affirmation and negation that repeatedly surfaces in Beckett’s texts. Beckett’s subversion of the novel form is thus produced through retaining certain of the conventions of narration, whilst subsequently exposing and undermining such conventions, whereby ‘the aesthetic goal is one of an immanent presentation of states of affairs that, in not changing, nevertheless expose their voids’.44 In Watt we see an attempt to renounce the conventions of the novel to reveal the generic capacities of language, or, in other words, to evacuate the predicates of the novel that rely on the evocation of transformation and verisimilitude in order to expose precisely the ‘gaps’ or ‘perforations’ (to use Beckett’s term) of language. By migrating mathematical forms – notably forms of permutation and divisibility – into the novel, Beckett simultaneously reveals and undermines the enumeration of naturalist and realist forms of narrative, presenting a third way that again disperses coherent character and situates both voice and identity outside of the speaking subject. This ‘third way’ takes the form of ‘narrative permutation’, a numerical term which migrates to the literary in  order to describe enumeration without progression, or an attempt at completion or totalization of some system that, by its very nature, cannot be either completed or contained. The short poem in the Addenda is part reflection on Watt’s narrative, part parody of the whole category of addenda, footnotes or other textual miscellany, and part poetic comedy. The poem seems to enquire after the entitlement to tell the story of the ‘old man’, presumably Watt, here, which, through reiterations of the question, creates an equivalence between this act and ‘nothingness’: who may tell the tale of the old man?

44

Julian Murphet, ‘The Mortification of Novelistic Discourse in Beckett’s “Trilogy”’, Paper presented at Beyond Historicism: Resituating Samuel Beckett, University of New South Wales, Sydney, Australia, 7–8 December 2012.

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weigh absence in a scale? mete want with a span? the sum assess of the world’s woes? nothingness in words enclose?45

Here, the asyndeton creates a sense of accumulation, and the question that (graphically and literally) pile on top of one another produce iterations rather than poetic or syntactic progression or culmination. Each of these questions is oriented towards the limits of linguistic and aesthetic possibility. As Mood rightly points out, the mathematics of this novel demonstrate a ‘meticulously flawed form’ of Watt: ‘Not just in the content (the helplessness of Watt as developed in characterizations and plot, and the use of philosophical and literary allusions) but also in the form is this impotence and ignorance portrayed.’46 These repetitions, and the negation of meaning that they effect, occur frequently throughout the text. The most significant instances of this, outside of this exemplary poem in the Addenda, are the permutative lists that Watt and others undertake. There is a circulation of servants at Mr Knott’s house. One servant works upstairs, and the other downstairs. When a new servant arrives at the house  – always out of the blue, it seems – then the servant on the upper floor must leave immediately. The servant on the lower floor subsequently moves upstairs, and the new arrival begins work downstairs. We learn of this process through the events that follow Watt’s arrival at the house – namely, the departure of Arsene, the servant on the upper floor – and, later, through the arrival of Arthur and the departure of Erskine. After Watt’s arrival at the house, Arsene gathers his belongings together and embarks upon a bizarre monologue before leaving. This monologue is an exemplary surge of ‘weigh[ing] absence in a scale’, ‘met[ing] want with a span’, and ‘the sum assess[ment] of the worlds woes’. Arsene compares the shifts in his

45 46

Beckett, Watt, 247. Mood, ‘“The Personal System” – Samuel Beckett’s Watt’, 263.

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capacities for reason to the barely perceptible, sudden shift in the particles of reality: The change. In what did it consist? It is hard to say. Something slipped. There I was, warm and bright, smoking my tobacco-pipe, watching the warm bright wall, when suddenly somewhere some little thing slipped, some little tiny thing. Gliss – iss – iss – STOP!47

Here, there is an allusion to the ground shifting; all the component parts of reality are maintained, but something, inexplicably, and without event, had changed. The short sentences indicate the confounding of Arsene’s narrative voice: the sentences are scattato until – for a moment – he relates his state of being prior to the change and temporarily takes up a conventional narrative voice: ‘There I was, warm and bright, smoking my tobacco-pipe.’ The immediate departure from this confident narration is a transition away from imagery and into sound; this ‘Gliss-iss-iss’ is the culmination of those short sentences in a final stammer between sound and word. In Arsene’s experience of a shift in reality, which he compares to a shift in sand on a dune, we see a kind of inverse or forbidden anagnorisis: a discovery is indeed made, a variation in the world apprehended, but the content of this is subtracted from the experience. Everything is different and the absolute hold of reality undermined, but there is no positive content to this: there is an involuntary and unspecific sense of transition. Watt has a very similar impression as regards the story of the Galls. The Galls are a father–son outfit who arrive at Mr Knott’s to tune the piano. What perplexes Watt about the event is precisely that a palpable experience of ‘nothing’ seems to occur, persist or grate at a normal sense of reality: What distressed Watt in this incident of the Galls father and son, and in subsequent similar incidents, was not so much that he did not know what had happened, for he did not care what had happened, as that nothing had happened, that a thing that was nothing had happened, with the utmost formal distinctness, and that it continued to happen, in his mind, he supposed, though he did not know exactly what that meant.48

47 48

Beckett, Watt, 43. Ibid., 76.

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This sense of the ‘utmost formal distinctness’ of nothing is in fact formally realized in the cataloguing, or enumeration, that Arsene undertakes, and which will be replicated by Watt. In a tirade seemingly on the general state of things, Arsene bemoans: The Tuesday scowls, the Wednesday growls, the Thursday curses, the Friday howls, the Saturday snotes, the Sunday yams, the Monday morns, the Monday morns. The whacks, the moans, the cracks, the groans, the welts, the squeaks, the belts, the shrieks, the pricks, the prayers, the kicks, the tears, the skelps, and the yelps.49

Arsene’s lists are virtually unreadable due to their excessive accumulation and a sense of the madness of permutation: a form of counting that at once enumerates new and distinctive elements yet equally seems not to progress or forward motion. Whole pages in Watt are given to this form of permutative list, and, as such, reading happens at the level of page as opposed to the level of the line. Although in Arsene’s monologue we have an array of different events, sounds, or gestures, each different from the next, the grammatical repetition and the parataxis homogenizes the various elements of the list, collapsing into a group of ‘one thing after the next’. Yet another list is genealogical. This list painfully enumerates familial lines: ‘And the poor old lousy old earth, my earth and my father’s and my mother’s and my father’s father’s and my mother’s mother’s and my father’s mothers and my mother’s fathers and my father’s mother’s father’s’ and so on.50 The list concludes with ‘An excrement’, a feature that becomes a refrain when Arsene starts another list and ends it with ‘a turd’.51 In other words, these permutations constitute a means towards no end, or towards an end that is only waste or filth. This permutation ends, ultimately, with a corporealization of the count: the non-progressive enumerative cycle (a partial, but not full, negation of any other enumerative act) runs until its stops, or is stopped, by a return to the body. Watt himself will enumerate lists very similar to those of Arsene, for instance in regard to the affair with the Lynch family, where he meticulously processes all of the Lynch generations. This is made even more explicit when 49 50 51

Ibid., 46. Ibid. Ibid., 47.

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these lists become outlines of various possible states of affairs; the solution to the problem of Mr Knott’s leftovers, for instance, is dealt with by enumerating possible situations with the number of objections listed next to them: 1. Mr Knott was responsible for this arrangement, and knew that such an arrangement existed, and was content. […] 12. Mr Knott was not responsible for the arrangement, but knew that he was responsible for the arrangement, but did not know that any arrangement existed, and was content.52

This enumeration is preoccupied with potential situations and their extensions, rather than events that might directly affect Watt. As such, for Watt, enumeration exorcizes a logical problem without solving it, precisely because of the absence of any finality to that enumerative procedure. These iterations become meaningless excrement for Arsene, for Watt and for the reader, because the evocation of meaningful difference is replaced by the assault of permutation. Beckett’s ‘permutation’ constitutes a distinct narrative ‘count’ separate from the numerical stakes of other modes of representation, including, to use Georg Lukács’s scheme, ‘narration’ and ‘description’. Lukács’s essay ‘Narrate or Describe’, published in 1936, holds two different forms of writing, or ‘creative method’ in opposition.53 The two forms – narration and description –occur in literary naturalism, and Lukács uses examples from Tolstoy and Zola to illustrate the respective paradigms. Naturalism is the literary movement with conviction in the capacities of realism to accurately and adequately represent the everyday. For Lukács, naturalism constitutes an aesthetic procedure that exceeds the historical bounds of late nineteenth- and early twentiethcentury French tendency in prose fiction (as exemplified by Zola) and, as such, Lukács considers Beckett a naturalist because he is above all concerned with the ‘triviality of everyday existence’ and his plots do ‘not articulate the essential relevance in an event and in the reaction to event’.54 Lukács explains 52 53

54

Ibid., 74–75. Georg Lukács, ‘Narrate or Describe?’, in Writer and Critic and Other Essays, ed. Arthur Kahn (London: The Merlin Press, 1978), 110. Georg Lukács, ‘Preface’, in Writer and Critic and Other Essays, ed. Arthur Kahn (London: The Merlin Press, 1978), 14.

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the separation between realism and naturalism using the analogy of visual art: ‘It is ultimately the writer’s approach to reality that determines whether he produces a painting or a photograph, an articulate statement or a mute babbling.’55 The photograph, here, is a mute babbling – the new technics of image subordinated to ‘painting’ because it relies on ‘technical virtuosity’ rather than the rich gestural ambiguity of realism. ‘Narrate or Describe’ extends this opposition between realism and naturalism by discussing two naturalist forms: description and narration. In Lukács’s examples, there is an external viewpoint in the case of description, but in narration events are related from the perspective of a character. The opposition between the two aesthetic regimes is thus a division between stasis and dynamism. Narration manages to integrate an event into the work, thereby allowing the reader to experience the event, whereas description objectifies an event, facilitating a much more limited reading experience. The key difference between these two modes, for Lukács, is the narrative temporality that they facilitate, and the sense (or lack) of vitalism that arises from this temporality. Description renders an event outside of temporality, privileging elements over experience, and creating a kind of narrative stasis: a scene or event total in its constitution and suspended outside of time.56 In each form we have a totality: in description, we have a total world, with as many elements as possible included, whereas in narration we have a total experience. Each form has a different unit of measure: description attempts totality by including as many elements of a scene as possible (each element contributing to quantity), whereas narration achieves totality by subsuming all these elements to the privileged literary end: readerly experience. In other words, Lukács could frame these two modes of fiction in terms of two different definitions of count: description relies on counting as determining a number or reciting the order of numerals, whereas narration relies on ‘counting’ as bearing significance or particular import. More importantly, every literary work, despite its final attribution, will inevitably contain elements of both narration and description: whilst for Lukács the former should take precedence, moments of description 55 56

Ibid. Lukács, ‘Narrate or Describe?’, 111. Gérard Gennette theorized this in terms of duration, naming it ‘descriptive pause’.

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are nonetheless necessary. Here, we see the unique numerical literary purview, whereby a quantitative count and qualitative count are interchangeable and co-implicated. These two literary forms – naturalism and realism, description and narration – are also ‘two basically divergent approaches to language’, both of which rely on a certain cataloguing of plausible objects to achieve their literary, and ideological, effect.57 Just as words, rather than events, can be the sources of causality in Molloy, so too in Watt logic and enumeration impose themselves on the speakers producing the spinning out of a voice that confoundingly combines both of Lukács’s aesthetic regimes: both the stasis of description (the content of the lists) but equally the dynamism of narration (we witness these episodes of permutation happen to Watt) are present in a third form of narrative: permutation. What is the significance of permutation, then, for narrative vitality, and for the relation between narration and object, and narration and experience? In Watt, narration and description are given over to the duration of the fit of enumeration – literally, in the stasis experienced by both character and reader. These permutative passages are ‘fixations’: rather than conveying the dynamism of experience, or creating a world, these passages instead – and paradoxically in Lukács’s terms – create an experience of stasis. In other words, ‘permutation’ combines the antithetical elements of narration and description and confounds them both. In Watt, the principle of novelistic description is forced into temporary suspension when there is no economy, when the act of enumeration cannot stop or become accumulation. For Watt, these enumerations are not purely exercises of rationality, to attain an objective scrutiny upon his world, but are both compulsive and cathartic episodes. Arsene comments on his own ‘personal system’, and the desire to keep it contained and complete: my personal system was so distended at the period of which I speak that the distinction between what was inside it and what was outside it was not at all easy to draw. Everything that happened happened inside it, and at the same time everything that happened happened outside it.58

57 58

Ibid., 120. Beckett, Watt, 43.

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This description, which encompasses a combination of compulsion and catharsis, is also a departure from either narration or description to a third mechanism that presents a very different view of the human and language. This mechanism is ‘permutation’. Watt’s enumerative activity formalizes and exorcizes the varied suspicions and confusions that haunt him: ‘For Watt considered, with reason, that he was successful, in this enterprise, when he could evolve, from the meticulous phantoms that beset him, a hypothesis proper to disperse them, as often as this might be found necessary.’59 Here, rationality follows a cathartic function: ‘for to explain had always been to exorcize, for Watt’.60 In this experience of stasis, facilitated by a numerical regime that integrates both ‘senses’ of the verb ‘to count’ Beckett produces not only the numerical stakes inherent to literature but, equally, the humour inherent to mathematics. This ‘humour’ is well summed up in Gilles Chatelet’s reflection on the mathematical figure: ‘Why do figures fascinate so many simple souls, and the impatient, always so fond of references and certainties? Almost by definition, a figure is not open to discussion; there is indeed an imbecilic virility to the number, stubborn and always ready to hide behind a kind of scientific immunity.’61 This ‘imbecilic virility’ of the number fittingly describes the paradoxical status of Beckett’s formal permutation. This permutative organization achieves an experience of narrative stasis. The lists effect, for Watt, a catharsis by virtue of an exhaustion of reason, the experience of having let something that should, properly, be antithetical to experience, run its course. In this, the text suspends literary duration. Unlike words, the ‘universe of number’ is unproblematically coextensive with itself, each and all integers immediately present, without dependence on the temporal process of articulation. It is through this default from description to permutation that Beckett achieves, in Watt, the evocation of a timeless realm. The episodes of counting, and their distinctly permutative nature, can be read, together with Watt’s theory about the croaking frogs, in terms of a triptych of unorthodox systems of counting. What is astounding about Watt is 59 60 61

Ibid., 77. Ibid., 78. Gilles Chatelet, To Live and Think Like Pigs, trans. Robin Mackay (New York: Sequence Press, 2014), 51.

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precisely the eclecticism of different counting forms, each, however, included by virtue of its capacity to contradict the naturalist count, the stoppages that these counts put on the possibility of flowing, conventional narrative. This final form of prohibitive count harks back to the initial, onomatopoeic stumbling that replaced Arsene’s monologue, the: ‘Gliss-iss-iss’, where we saw a first turn away from semiotics to signal a content-less experience in the only way possible: through structure, here by the reproduction of a rhythm through sound. Later in the novel, the arrivals and departures of servants at Mr Knott’s house are represented as the three different voices in a chorus of croaking frogs. Watt had earlier, on his way to Mr Knott’s house, encountered another kind of numerical singing, quite distinct for its utter randomness in the text, whilst also conforming to the other modes of permutative counting, which end with excrement or void. The song runs as follows: ‘Fifty-two point one/four two eight five seven one/four two eight five seven one/oh a bun a big fat bun’, concluding with ‘and everyone is gone home to oblivion’.62 The singing of the frogs presents a development from this. Each of the counting systems is reduced to a different sound, non-human and non-linguistic and, emphatically, non-symbolic. After leaving Mr Knott’s, Watt considers how much time he spent there, and attempts to work out a logic behind the different times spent at the house by different servants, and whether these correspond to each other (each servant’s period in the house being dependent on the arrival of new servants, which triggers the departure of the servant on the top floor). Whilst preoccupied with this, Watt recalls an experience from a long time ago, when he was young, sober and also lying in a ditch, and heard the chorus of three frogs: Krak! – – – – – – – Krek! – – – – Krek! – – Krik! – – Krik! – – Krik! – 63

Each enumeration here takes on a non-linguistic form of numbering (Krik, Krak or Krek) that differs from the other two. Here, algebraic marks are replaced by sonic variations. In the case of Watt’s theory about the croaking,

62 63

Beckett, Watt, 28. Ibid., 117.

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each numbering form has a sonic domain, even if this is an arbitrary attribution, and thus has a continuity that is based on rhythm and the breaking of silence (aurally permutative). This corresponds exactly to Watt’s own musings about the abstract laws of the movement of servants – hypothetically ‘Tom’, ‘Dick’ and ‘Harry’ – whereby the narrator concludes that it was not the Tomness of Tom, the Dickness of Dick, the Harryness of Harry, however remarkable in themselves that preoccupied Watt, for the moment, but their Tomness, their Dickness, their Harryness then, their then-Tomness, then-Dickness, then-Harryness; nor the ordaining of a being to come […] as in a musical composition bar a hundred say by say bar ten […] the time taken to have been true, the time taken to be proved true, whatever that is.64

These rhythmic sonic bursts that order possible states of affairs (regarding ‘Tom’, ‘Dick’ and ‘Harry’) crystallize the process of permutation in an aural form. This is a final, additional algebraic instance of permutation which is certainly a ‘worsening’ of numeracy in that the elements of permutation are no longer logical possibilities but amphibian cries. The hermeticism of such a number system, and an assumption of a sonic algebra to stand in for the musical notation, parodies mathematics by revealing ‘imbecilic’ limits of numeracy. Reading the croaks as counts, and indeed as ordering counting systems, presents counting in a non-linear and non-textual fashion, a nonhuman modality, even. This number-as-animal-sound is the culmination of the ‘virile imbecility’ described by Chatelet: each croak is complete and self-contained in its meaning, like a number, and its power and totality rests precisely in this solipsism.65 The tabulated ‘croaking’ system, combined with other permutation systems in Watt reads pictorially as well as textually, and entirely in keeping with Lukács’s diagnosis of naturalism as closer to a photograph than the painting.66 Like the impression of light on photosensitive paper, naturalist description is immune to content, or, in other words, ‘blind’ to the meaning and significance of the 64 65

66

Ibid., 127. Croak, of course, is a synonym for death, and this completes, on a symbolic level, this negation of a narrative count. Ibid., 117–118.

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overall picture. The episodes of permutation construct a type of difference that is distinct from the establishment of traditional descriptive differentiation. The type of difference and specificity created by permutation can be read as a radicalization of the naturalist form along the same lines that Lukács sees a radicalized pictorial naturalism in the photograph. These lists are logically and aesthetically unnecessary because they are reducible to sets, and reducible to a continuing and consistent principle. The futility of permutation and the fact that it confounds narrative occurs because it refutes both definitions of count inherent to narration and description: the quantitative count and the qualitative count. Instead, permutation, lacking meaning in itself, gestures towards a metacount: one native to the cardinal and ordinal numbers. In other words, what these permutations reveal are not their component parts (there is no meaningful or rich multiplicity) but rather a principle of division. The shift here is from the substance of the count to the logical exhaustion of the set; the structural laws of the fictional world start to become an issue of the integrity of the set when the naturalist project is taken to its limit. In other words, these lists realize, quite perfectly, answers to those contradictory questions of the poem from the Addenda: here we have the effect of nothingness measured. No content is produced but we wind up with a measure nonetheless: the measure that we find in the repeated form of each notch in the permutation; a measure that makes the rest of the permutation redundant. In these lists we have the first and clear illustration of the limits of novelistic prose. These lists fail aesthetically by virtue of their radicalization of a naturalism: their attempt to achieve a complete description divorces these passages, and by extension the plot of the novel, from significance or importance or progressive development. This then isolates the mode of description alien to the novel form: generic description, which revolves around a world that spins inward or outward to the null set, as Beckett desires his fiction to do, the silenced accompaniment to description that instigates ‘genera’, the various discrepancies in the attributes of a differentiated and vivid world. More importantly, the anti-aesthetic generated by the mathematics of the text relies on an abstract difference in the implicit metaphysical foundations of regimes of description and narration. Lukács evaluates the regimes of naturalism and realism according to a concept of difference that echoes

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Aristotelian difference, which is based in ‘genera’ or types, rather than Platonic difference, which is based in division. This differentiation between the genera and the mathematical sets are best described by Albert Lautman, who construes this divergence as one that replicates the distinction between the Aristotelian work on species versus Platonic division. Lautman explains the ordering processes of axiomatic set theory as pertaining to division rather than genera: It is therefore not Aristotelian logic, that of genera and species, that plays a part here, but the Platonic method of division, as taught in the Sophist and the Philebus, for which the unity of Being is a unit of composition and a starting point for the search for principles that are united in the Ideas.67

Lautman sees a fundamental Platonism reflected in the processes of arithmetic, algebra and multiplication. Regarding the ‘philosophical importance of the activity of dissociation in mathematics’ (essentially the activity of forming sets) Lautman notes that certain notions of elementary arithmetic and algebra, which seem simple and primitive, envelop a plurality of logical or mathematical notions, delicate to specify but in all cases clearly distinguishable from one another. It is in this way that arithmetic equality is the only equivalence relation such that the countability of the individuals of a set is conflated with the countability of classes of equivalent individuals as defined by this relation.68

What this mathematical explanation is referring to is the philosophical coordinates that exist but are perhaps not recognized when undertaking simple or seemingly self-evident mathematical activities like arithmetic. Here, Lautman is pointing to the ‘countability’ of individuals in a set and the countability of classes, which are equivalent in arithmetic. One can also apply arithmetic principles across classes. This is an instance of a wider phenomenon that Lautman is trying to specify: the division of classes and individuals does not sacrifice the generality of arithmetic. Here we have

67

68

Albert Lautman, Mathematics, Ideas and the Physical Real, trans. Simón B. Duffy (London and New York: Continuum, 2011), 41. Ibid., 40.

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division without genera, which, to add a potentially confusing claim to this, allows for the generic integrity of arithmetic (genera and generic being utterly distinct terms in this discursive framework). Axiomatic set theory relies, thus, on a conceptual scheme of division as opposed to genre. Set theory does not rely on sets to demarcate different genres, or species, from each other to avoid applying improper axioms or functions to those sets. Rather, the reliance on sets exists for the purpose of sustaining division rather than ‘extended types’.69 In Watt, permutation replaces normal narrative counting, thus substituting a usual regime of difference with another one: one which enacts a kind of hyper-naturalism, resulting in a halting of narration, a suspension (for both reader and character) of plot and fictional world, as the lists run their course. But this is not only an act of taking naturalism to its extreme, not only a matter of amplifying it, but also indicating a fundamentally different form of difference (and the grouping of sets). This partial negation of the dominant aesthetic enumerative procedure here reveals the distinction between this form of difference and the meaningful difference created in description and narration: this difference is based on set theory, and on the division of the worlds into classes and genitive structures, whereas the latter mode of difference is based on meaningful difference between elements. In the former, we are concerned with a principle that elaborates its way down a certain set: that of the Lynch family. The division of that is enacted here is not a matter of meaningful difference – the individuality of the members of the Lynch family, a recognition of their existence – so much as an attempt to apply the genitive principle and elaborate the parameters of a set. This ‘difference in difference’ is one way that the novel is comedic: in Watt we find an inappropriate form of difference in the place of another. Yet more fundamentally it is this form of radicalized description and Platonic discourse that makes Watt a ‘mortification of novelistic discourse’70 or an ‘anti-novel’71: a prose event that achieves its subversion of the novel form through elements that are mathematical in

69 70 71

Ibid., 42. Murphet, ‘The Mortification of Novelistic Discourse in Beckett’s “Trilogy”’. n.p. John Bolin, Beckett and the Modern Novel (Cambridge and New York: Cambridge University Press, 2013), 5, 71, 164.

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register, and relate to the metaphysical consequences of mathematics, in other words, to the enumerative organization of the world.

All Strange Away and Imagination Dead Imagine: Imagination by Numbers In the preface to Writer and Critic, Lukács writes, ‘With its forms of organisation, its science, and its techniques of manipulation, modern life moves relentlessly toward reducing the word to the mechanical simplicity of a mere sign.’72 The sign, here, is the word without depth or warmth: it is language dissociated from its animation by human beings. For Lukács, the transmutation of word to sign implies a radical departure from life, for the dynamism of everyday language derives precisely from its always being either less or more in vocabulary and syntax than mere signs: less in that in its ambiguity it skirts the essence of the object being discussed, more in that in its very imprecision it articulates the concrete essence of an entire concrete complex.73

Beckett’s late short fiction, theatrical and television works continue his task of creating an art of the ‘unword’: a word that betrays the ‘grammar and style’ or the ‘surface’ of language and presents itself as an opening to what is outside or beneath the phenomenal fabric of dynamic speech and writing.74 In this chapter, I will demonstrate how this work – whether textual or televisual – realizes this shift from the word to the sign by creating prose that attempts to evacuate traditional elements of the phenomenal: narrative experience, production or representation of a world and vivid imagery. In these late works, Beckett is concerned with the two most basic predicates of experience: space and time, which, the Kantian tradition, are considered to be conditions of phenomenality, rather than content of phenomenality.75

72 73 74 75

Lukács, ‘Preface’, 11. Ibid., 11. Beckett, The Letters of Samuel Beckett, Volume 1: 1929–1940, 518. Kant, Critique of Pure Reason, 65.

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S.E. Gontarski frames Beckett’s later artistic concerns in terms of abstract boundaries, claiming that ‘Despite his early insistence on “keeping our genres more or less distinct”, Beckett seemed in this later phase of his work to have stretched beyond such limitations, beyond the generic boundaries to examine the diaphanous membrane separating inside from outside, perception from imagination, self from others, narrative from experience.’76 I take the most important of these ‘generic boundaries’ to be this distinction between conditions of phenomena (space and time) and content of phenomena (the predicate). The formal device that enables this transition exists between these two pillars of experience: the numeral, the mark of presentation that is neither condition nor content, but which exists in the paradoxical and nebulous space of measuring conditions and potential conditions. In this chapter, I will analyse the numerical presentation of space and time – topos and duration – in these works, as well as the transformation of syntax into a technique of measure. In the previous chapter, I looked at the numerical stakes in Lukács’s definition of naturalism, whereby the realist achieves an expressive narrative immersion in the scene but the naturalist, on the other hand, enumerates the scene.77 The naturalist, in Lukács rendering, stands outside of the fictional scene that he or she presents and as such is associated with the mechanization and objectivity of the photograph, whereas the realist retains the gestural quality of the painting. Here, I again will look at Beckett’s naturalism in numerical terms, exploring the reformulation of spatiality in prose and television work from a domain of measure to a domain of magnitude. It is this transition, from measure to magnitude, and from content of phenomena to conditions for phenomena, that will enable Beckett to produce a ‘generic’ rather than a ‘general’ literature, radicalizing the tenets of naturalism against itself. This is, again, a realization of both Beckett’s early commitment to the ‘unword’ as well as an inversion of the naturalist task: the writer stands outside of the scene without the illusion of the veridical, indeed, the writer stands outside of the fictional scene precisely to divest himself or herself of the task of the veridical and claim, for art, what is on the ‘other side’ of the membrane of textual representation. 76

77

S.E. Gontarski, ‘Introduction, From Unabandoned Works: Beckett’s Short Prose’, in Samuel Beckett The Complete Short Prose 1929–1989 (New York: Grove Press, 1995), xxix. See Lukács, ‘Preface’, 14.

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The 1950s were a transitional decade for Beckett, and the production of theatrical and prose works of this period are characterized by subordination of the word to a particular kind of ‘sign’: the geometric coordinate. Beckett’s first work for the theatre was En Attendant Godot, written in 1952 and translated into English in 1953. Fin de Partie, Actes sans Paroles I and Actes sans Paroles II and Krapp’s Last Tape would all follow in the extraordinarily prolific output of the next few years. The short prose works All Strange Away and Imagination Dead Imagine are exemplary transitional pieces in that the prose is given over to instructions for the creation or direction of a spatial scene, with narrative attempts to both ‘deaden’ the imagination and to produce the barest generation of an abstract, cognitive spatial product. Ruby Cohn claims that ‘Beckett came close to painting still lives in movement’, an idea that captures the various contradictions of Beckett’s later short prose and theatre, in particular the concern with duration and extension, in the form of contortion in or by time and space.78 In All Strange Away and Imagination Dead Imagine, we see a move away from a preoccupation with the circulation system of words and the developments of idiosyncratic or subversive logics. Instead, we see a consideration of the spatial stakes of numeracy, or an engagement with number and image that attempts to reveal a material grain that exists behind language, which, in these works, is a topos that precedes and shapes duration. Appropriately, Beckett is not interested in representing space and time (which would contradict a genuine definition of conditions of phenomena) so much as generating it. Once again, number fulfils the function of a modernist generic, a formalization ‘behind’ form, radicalizing the modes of counting within naturalism. All Strange Away (1964) is both a chronologically and formally transitional work, appearing between Beckett’s prose and his theatre or television productions. The first line of All Strange Away forms the title of Imagination Dead Imagine, which was initially written as a shorter version of All Strange Away before being developed into a separate piece. This latter text still retains much of the content of All Strange Away but is devoid of the more substantial pornographic passages of the earlier text. All Strange Away has at best a tenuous narrative thread. A protagonist, who we know nothing of, produces a 78

Cohn, A Beckett Canon, 31.

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one-paragraph monologue that is set in what Cohn identifies as a ‘hemicycle’. A ‘hemicycle’ is, Cohn explains, the same as a semicircle, but the latter word ‘lacks the resonances of enclosure – hem – and of repetition – cycle’.79 The hemicycle is described as ‘bonewhite’ and the ‘floor like bleached dirt’, and thus the text, like the play Endgame, gives the impression of being set in a skull.80 All Strange Away is divided into two sections, the first without a title and the second entitled ‘Diagram’. The prose describes a scene initially populated by a man, who is observed by the two onlookers Jolly and Draeger Praeger Draeger, and is subjected to images and memories almost as a patient in a psychiatric or psychoanalytic clinic might occupy a world of fantasy watched by distant, observant eyes. Later in the monologue, these figures will disappear and the space will be filled only with Emma, her face, ‘arse’, knees and feet each aligned to one of the coordinates of the space. All Strange Away is narrated in a stream of consciousness in an instructive register (‘take his coat off, no, naked, all right, leave it for the moment’81), and, as in Imagination Dead Imagine, the reader is enjoined to imagine the elements of the scene, thereby literalizing the demand that the traditional fictive work places upon its audience. The scene is described using coordinates – ‘Five foot square, six high, no way in, none out’ – privileging an efficacy of description and a breathless crowding of clauses that befit the entrapment of the closed space.82 We are told that ‘someone’ in the ‘place’ is talking to himself ‘in the last person’, which suggests a conflation of the narrator and the man in the space, although this conclusion remains speculative.83 The narrator breathlessly stipulates the predicates of the scene, the contortions of the body in the scene and the shifts in light and colour, producing a monologue that is formally closer to stage directions (or an author’s notebook) than to a work in its own right. As such, these two texts constantly configure themselves as a supplement to fiction rather than an original; fictionality is first and foremost the concern of the text rather than fiction per se. The constant pauses and the listing structure of the sentences 79 80

81 82 83

Ibid., 286. Samuel Beckett, ‘All Strange Away’, in Samuel Beckett The Complete Short Prose 1929–1989, ed. S.E. Gontarski (New York: Grove Press, n.d.), 173. Ibid., 170. Ibid., 169. Ibid.

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give the sense of an abbreviated speech and the text comes to resemble an instruction manual, dense with cues for the reader to abandon imagery: ‘Islands, waters, azure, verdure, one glimpse and vanished, endlessly, omit.’84 Staples of the exotic (islands) or vivid (azure) are evoked only for the purpose of exorcism (‘one glimpse and vanished’), a process familiar from the cathartic lists in Watt. Death pervades both Imagination Dead Imagine and All Strange Away. The narrator of Imagination Dead Imagine must ‘crawl out of the frowsy deathbed and drag it to a place to die in’ and the numbers of that begin, from the first page, to structure the monologue strike one as symptomatic of narratorial exhaustion.85 The refrain that closes so many of the narrator’s sentences – ‘that again’ – instils a weariness in the monologue compounded only by the numbers which demarcate the closed scene in which the narrator finds himself. Imagination Dead Imagine opens by evoking an imagination after life or, an imagination after some sort of finitude: ‘No trace anywhere of life, you say, pah, no difficulty there, imagination not dead yet, yes, dead, good, imagination dead imagine.’86 There is a curious narrative logic to this. The imagination is, initially not dead and then dies, and this is affirmed (‘yes, dead, good’). Yet this quickly becomes a cycle: the imagination ‘kills itself ’ in order to imagine itself dead again in order, precisely, to resurrect itself. The reader must imagine a dead imagination, an imagination that is almost but not entirely negated. This imagination exists but is obsolete, or – like Lukács’s sign – lifeless, without breath or animation. The injunction to imagine a dead imagination is an injunction to imagine thought outside of the human or beyond a certain finitude and thus presents a paradox that reveals the limits of literary representation. These lines ask us to evoke fossilized mental worlds, or, in other words, forms of thought devoid of content and hence outside of human experience or existence, reminiscent of what Quentin Meillassoux calls the ‘glacial world’ of primary qualities.87 This evocation of the obsolete or dead imagination is aided by the use of coordinates to map the space and movement in the space. The body 84

85 86 87

Samuel Beckett, ‘Imagination Dead Imagine’, in Samuel Beckett The Complete Short Prose 1929– 1989, ed. S.E. Gontarski (New York: Grove Press, 1995), 182. Beckett, ‘All Strange Away’, 169. Beckett, ‘Imagination Dead Imagine’, 182. Quentin Meillassoux, After Finitude: An Essay on the Necessity of Contingency, trans. Ray Brassier (London and New York: Continuum, 2008), 115.

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in the text contorts according to four angles of the quadrilateral, rendered as the algebraic points A, B, C and D (and e, f, g and h that set the bounds of the ceiling), which contract to reduce the size of the space they demarcate. Just as in Watt, where the descriptive, evocative and narrative conventions of the novel are doubled back in the fiction to create a kind of anti-novel, in All Strange Away and Imagination Dead Imagine these same conventions are rejected. The rejection, this time, comes not in the form of permutation but in a narrative that is structured around magnitude, rather than traditional forms of measure. This use of magnitude rather than measure echoes the pre-Enlightenment subject-centred measure famously depicted by Da Vinci’s ‘Vitruvian Man’, the Renaissance exemplar of the connection between art and mathematics. In All Strange Away and Imagination Dead Imagine, as in the ‘Vitruvian Man’, we see images structured by a mathematics that proceeds from the subject rather than a countable system; a mathematics against objective measure. All Strange Away begins with the two conceptual coordinates that will govern the piece, imagination and place: A place, that again. Never another question. A place, then someone in it, that again. […] Five foot square, six high, no way in, none out, try for him there. Stool, bare walls when the light comes on, women’s faces on the walls when the light comes on. […] Light off and let him be, on the stool, talking to himself in the last person, murmuring, no sound, Now where is he, no, Now he is here.88

One of the defining features of these two works is the use of coordinates to map the scene and provide a substitute for images of contorting bodies. In the previous chapter, in Molloy and Watt, I argued that Beckett doubles the traits of the novel to stage, indicate or inflate these traits, working within and against the form at the same time. The two short texts under consideration here replicate this structure. All Strange Away literally commands the reader to imagine scenes whilst simultaneously evacuating their imagination of imagery. In this text imagery is replaced by coordinates that are abstract and mathematical and thus do not produce what we might associate with an image or an imaginative experience. This is compounded by the question of who the 88

Beckett, ‘All Strange Away’, 169.

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narrator is addressing. There is often little to suggest that the narrator of All Strange Away is recounting anything other than an imaginative scene within his or her own head: a ‘fancy’, to use the narrator’s word, and indeed the white rotunda literally resembles the inside of a skull.89 But injunctions like ‘see how he crouches down and back to see’ suggest that his vision is also either observation or communication.90 There are barely perceptible changes in speaker or address; initially, the narrative voice seems to address the man in the room or whoever is directing him, but then, with a self-reflexive remark turns outwards to address the reader: ‘Light flows, eyes close, stay closed till it ebbs, no, can’t do that, eyes stay open, all right, look at that later.’91 This oscillation between a kind of stream of consciousness and the compulsive injunction or instruction that accompanies it becomes an increasingly confounded series of positions with the introduction of geometric coordinates: Call floor angles deasil a, b, c and d and ceiling likewise e, f, g and Beckett, say Jolly at b and Draeger at d, lean him for rest with feet at a and head at g, in dark and light, eyes glaring, murmuring He’s not here, no sound, Fancy is his only hope.92

The hemicycle thus contains a cube and here we have a radicalization of naturalism as Lukács defined it: the fictional scene is mapped utterly without the limitations of perspective within the scene but from a thought outside the scene. This is then not the depiction of a scene witnessed or experienced by the narrator (as would befit realism, in Lukács’s terms) so much as a scene that extends from the narrator’s imagination, from a cognitive or imaginative position outside of the narrative scene. If indeed these scenes are set in a skull, or in the mind of the narrator, then there is a contradiction formalized here: the imagination is both the subject from which these visions emerge and being its object. The positions of the figures in this scene become increasingly obscene and visions or views of Emma capture not a whole being but – like a series 89

90 91 92

Both James Knowlson and Graham Fraser note that the rotunda appears to be the inside of a skull. See: James Knowlson, Damned to Fame: The Life of Samuel Beckett (London: Bloomsbury, 1996), 531 and Graham Fraser, ‘The Pornographic Imagination in All Strange Away’, Modern Fiction Studies 41, no. 3 (1995), 516. Beckett, ‘All Strange Away’, 171. Ibid., 170. Ibid., 171.

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of photographs rather than a film or a painting – fragments of her: ‘First face alone, lovely beyond words, leave it at that, then deasil breasts alone, then thighs and cunt alone, then arse and hole alone.’93 Emma is split into eight images, each of which is focused on individually. The coordinates that map Emma’s contortions are increasingly obsessive. The exactitude of the positions eliminates the sensuality that might convincingly accompany fantasy or pornography, suspending imagery and producing an imagination of utter abstraction: ‘For nine and nine eighteen that is four feet and more across in which to kneel, arse on heels, hands on thighs, trunk best bowed.’94 It is notable that these are not Cartesian coordinates, where the body might be mapped out on a two-dimensional graph in a stable representation. Instead, the body parts are assigned to points or generic placeholders. This form of measure requires only the notion of distance between A, B, C and D. In other words, this is no longer Cartesian measure, but topology, the study of place under a continuous ‘deformation’ (no longer a geo-metry, measure of the earth, but a topo-logy, a study of place). Topology emerged from a combination of geometry and set theory and focuses on properties of spaces undergoing a change, such as the emergence of one object from another. Emma’s ‘measures’ are algebraic but these signs do not stand in for numbers that can be derived from an equation. Instead, we have coordinates without stable measure that are put in flux by the narrative progression, which – in a diminution perhaps typical of Beckett’s prose – retracts the cube that Emma is contained within. In Beckett’s early publication on Marcel Proust, he introduces the idea of a literary geometry that describes its object without measuring it: It will be impossible to prepare the hundreds of masks that rightly belong to the objects of even his most disinterested scrutiny. He accepts regretfully the sacred ruler and compass of literary geometry. But he will refuse to extend his submission to spatial scales, he will refuse to measure the length and weight of man in terms of his body instead of in terms of his years.95 93 94 95

Ibid., 171. Ibid., 172. Samuel Beckett, Proust and Three Dialogues with Georges Duthuit (London: Calder and Boyars, 1987), 12.

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Here, in All Strange Away, we have Beckett’s own realization of this literary principle. Where Proust might measure a man in terms of time, Beckett retains a spatial form and substitutes measure for magnitude. The second section of All Strange Away, entitled ‘Diagram’, is preoccupied with Emma instead of the man of the earlier section, and Emma’s bizarre interaction with either a grey rubber ball, or a ‘sprayer bulb’. ‘Diagram’ consists of a fantasy revolving around Emma’s contorting body, although whether Emma and the positions we are told to imagine her in are products of the man’s fantasies or an external narrator is unclear. This ambiguity is maintained by the injunctions to imagine, the instructive register and the physical impossibility of the contortions, rendering the scenes as potential rather than actual. ‘Diagram’ quickly turns into a kind of detached pornographic scene: ‘Imagine him kissing, caressing, licking, sucking, fucking and buggering all this stuff, no sound.’96 This scene is again rendered in points a, b, c and d, evoking an abstracted sexual combinatorics that can potentially be plotted or re-enacted by virtue of a graph, but, most interestingly, a graph with no necessary measure: Rotunda then as before no change for the moment in dark and light no visible source spread even no shadow slow on third seconds to full same off to black two foot high at highest six and a half round good measure […] floor bleached dirt or similar, head wedged against wall at a with blank face on left cheek and the rest the only way that is arse wedged against wall at c and knees wedged against wall ab a few inches from face. 97

Both duration and imperial measure are cited in the descriptions, but these are contained with a space that is essentially measured by four letters: a, b, c and d. Here, the alphabet is being used ‘numerically’, hijacked for use in a second order that is outside of the order of natural language. At the same time, we have a kind of literalization of the process of composition: the letters of the alphabet construct an imaginary scene and delimit the movement within that scene.

96 97

Beckett, ‘All Strange Away’, 171. Ibid., 178.

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The figure is placed in fantasy, in the imagination not the world, through this move to geometry, which renders the scene abstract in the same gesture that it describes movement and position. The contortions of Emma’s body in the space recall a failed, de-eroticized peep show. The mathematical presentation of the scene recalls Jean Baudrillard’s theorization of the relationship between ever more ‘real’ – exact and high definition – images and their purchase on intelligibility and visibility. Writing on the distance between virtual reality and the ‘real world’, artificial reality and the artificial, and, pornography and desire, Baudrillard claims: The highest definition of sex (in pornography) corresponds to the lowest definition of desire. The highest definition of language (as computer coding) corresponds to the lowest definition of sense. Everywhere high definition corresponds to a world where referential substance is scarcely to be found any more.98

This pairing – sex and language – echoes the dynamic in All Strange Away between passages that enumerate sexual acts literally and figuratively within cognitive space (the skull) with reference to a hypothetical, mathematized scene.99 Just as numbers provide ever greater exactitude for the scene, so too

98 99

Jean Baudrillard, Art and Artefact, ed. Nicholas Zurburgg (London: Sage, 1997), 26–27. The Marquis de Sade produced exquisitely numerical erotic scenes. Gaëtan Brulotte notes, ‘All this is rendered with a strong penchant for mathematics, according to which to count and to measure have an erotic value all of their own […] For Sade, eroticism is the opposite of wild abandon’ (57). This association between mathematics and the erotic emerges precisely from what would seem to be their antithesis. For Brulotte Sade inverts the formalism in ‘monastic, military and industrial traditions’ (57), transforming the strictures of these domains into the capacity for jouissance. For a broader discussion of this, see Gaëtan Brulotte, ‘Sade and Erotic Discourse’, Paragraph 23, no. 1 (2000), 57. The inversion of rules to produce jouissance involves commuting precisely these rules to the domain not of reason, where they purport to emanate from and influence, but the domain of the senses, the ‘logic’ of here emerging not from reason but desire. The erotic potential of the number is, much more frequently in Sade, simply an issue of quantification. More broadly, Sade’s ‘calculations’ emerge from desire and the impetus for pleasure: ‘The faculty given me by Nature whereby I may dispose myself in a favorable sense toward such-and-such an object and against some other, depending upon the amount of pleasure or pain I derive from these objects: a calculation governed absolutely by my senses […]’ (34). Numbers, here, are calculations of the senses: the exact inversion of their usual function in discourse, scholarship, society. This is a perennial artistic truth: the dry objectivity of the rule barely obscures the debauchery implicit in its very stringency. See Marquis de Sade, Juliette, trans. Austryn Wainhouse (New York: Grove Press, 1968).

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is there a loss of imagery, exactly the task of a text that seeks to ‘imagine dead imagination’.100 In the Critique of Pure Reason Kant theorizes the a priori conditions for experience in an argument that – if read in an imperative rather than declarative mode – echoes the tone and task of All Strange Away: Remove from your empirical concept of a body everything that stems from experience, one by one: the colour, the hardness or softness, the weight, and even the impenetrability, and there still remains the space which the body (now entirely vanished) occupied, and this you cannot remove.101

In Kant’s figuring, mathematics is only a priori insofar as mathematical statements contain their concept in the predicate, as opposed to requiring experience to associate the concept with the predicate. For Kant, space–time is the condition of phenomenality, and it is the intuitive a priori notion of space that produces one aspect of mathematics – geometry, and the intuitive a priori notion of time that produces another aspect of mathematics – counting and analytical mathematics Such a priori propositions are ‘indispensib[le] […] for the possibility of experience itself ’.102 Beckett was familiar with Kant’s work and, indeed, as Gontarski and Ackerley point out in their companion to Beckett, the narrator of Company summarizes one of Kant’s central claims ‘Pure reason? Beyond experience’.103 A version of this statement – ‘Pure reason? Beyond imagination’ – seems to

100

Such an idea is echoed by Ihab Hassan in his model of the relationship between Joyce and Beckett. Hassan uses Elizabeth Sewell’s opposition between nightmare and number to describe the distinction between Joyce and Beckett. While Joyce is the writer of ‘nightmare’, Beckett is the writer of ‘number’: ‘The language of Nightmare is that of confusion and multiple reference; it creates a world in which all is necessary, all significant; everything is there at once. But the language of Number empties the mind of reference; it creates a world of pure and arbitrary order; nothing there is out of place’ (67). Here, Joyce’s ‘maximal’ prose stands in for proliferation in and of language, which is opposed to the ‘language of Number’ of Beckett, whose overwhelming feature is to ‘empty’ rather than ‘fill’. Hassan goes on to note that ‘the structure of Beckett’s work is miraculously empty – anything can be made to fill it – as the structure of Joyce’s is ineluctable. There is profound parody in this; the parody of archetypes of numbers’ (67–68). If numbers, indeed, serve to ‘empty the imagination’, they would occupy the paradoxical space that the line ‘imagine dead imagination’ evokes. See Ihab Hassan, ‘Joyce-Beckett: A Scenario in Eight Scenes and a Voice’, in Paracriticisms: Seven Speculations of the Times (Champaign: Illini Books, 1984). 101 Kant, Critique of Pure Reason, 40. 102 Ibid., 40. 103 Ackerley and Gontarski, The Grove Companion to Samuel Beckett, 295.

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be at issue in these two texts. The coordinates – A, B, C, D – of All Strange Away and Imagine Dead Imagination are not simply alphabetic marks divorced from words but are, here, marks of the partial a priori, what Kant called the ‘synthetic a priori’ in his Critique of Pure Reason. The concept of the a priori is relevant here in that it refracts the significance of topological marks in All Strange Away in terms of a broader artistic project of a ‘generic’ literature. For Kant the structures that a priori knowledge follows are informed not by experience (‘synthetic’ knowledge), but by reason, hence being termed ‘analytic’. An ‘analytic’ proposition is true by virtue of the terms it contains (the famous example: ‘all bachelors are unmarried’) as opposed to some reference to the world (‘all bachelors are unhappy’). However, a priori knowledge, whilst separate from empirical evidence, is not completely separate from experience. In Kant’s formulation, the a priori has a restricted relation to experience: it can contain its predicates and, as such, not require any additional information from experience, even if the definition of identity of those predicates requires some experience in the world.104 From this, Kant concludes that knowledge a priori is either pure or impure. Pure knowledge a priori is that with which no empirical element is mixed up, it is knowledge that is the pure product of cognition. For example, the proposition, ‘Every change has a cause,’ is a proposition a priori, but impure, because change is a conception which can only be derived from experience.105

It is only time and space that are pure a priori. This does not, crucially, mean that space and time are intrusions of the noumenal into perception, but instead that they are pure products of human cognition. Space and time are, in other words, the limits of reason, the ‘rose-tinted glasses’ that human cognition colours reality with anterior to the accumulation of experience. For Kant, analytic a priori statements are ‘pure’ a priori statements (and, will constitute ‘transcendental logic’106) that relate to time and space, whereas synthetic a priori statements require understanding of a concept or definition but not necessarily experience. Mathematical propositions assume the status of ‘synthetic a priori’  – which is 104

See Kant, Critique of Pure Reason, 38. Ibid., 2. 106 Ibid., 97. 105

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a qualified a priori that requires a degree of experience to understand the terms but not the truth of a proposition. Such cognitions are not verified by observation or experience (they are not objective moments of understanding) but emerge from the domain of transcendental logic.107 Despite the restrictions put on mathematical language by Kant, there is a crucial revelation regarding non-semiotic language here. Language stripped of its semiotic capacity is reduced to either propositions of possession or relation: space and time are the ultimate subtraction from experience to pure cognition. Beckett rejects the ‘dust’ of words and imagery in order to work with the conditions of experience, which, in these terms, are spatial: the flux of A, B, C and D. Lukács’s sign, a word without life or breath, is here perfectly correlated with the topological point, the terms by which spatial or temporal possession or relation are articulated. The sign, then, is that which indexes conditions of experience rather than content of experience. The a priori is thereby also the form of proposition that exists behind the ‘veil’ of world experience that Beckett refers to in his formulation of the ‘unword’: ‘More and more my language appears to me like a veil which one has to tear apart in order to get to those things (or the nothingness) lying behind it.’108 All Strange Away and Imagine Dead Imagine outline their fictional worlds not through evocation or propositions that might recall the world but rather through points. Like All Strange Away, Imagination Dead Imagine also moves quickly into passages of measurement, which outline the points that demarcate the scene and the positions that the figures take up. The rotunda is expressed in terms of four points: ‘Two diameters at right angles AB CD divide the white ground into two semicircles ACB BDA.’109 Crucially, a point is – in the Euclidean definition – ‘that which has no part’.110 Here, we have a strange instance of the a priori: it is not the statement here that reflects its definition, but the component parts which themselves are a priori, outside of empirical verification and both transcendental and material. These geometric points are

107

Kant, Critique of Pure Reason, 39. Beckett, The Letters of Samuel Beckett, Volume 1: 1929–1940, 518. 109 Beckett, ‘Imagination Dead Imagine’, 182. 110 Euclid, The Thirteen Books of the Elements, ed. trans. Sir Thomas L. Heath, 2nd ed. (New York: Dover Publications, 1956), 155. 108

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marks which have no content or extension and are symbols that exist evacuated of substance or import because they themselves occupy no space or time (the point is literally ‘that which has no part’); these points generate rather than represent time and space. Take, for instance, one of the lines from the opening of Imagination Dead Imagine: ‘Till all white in the whiteness the rotunda’.111 What this absence of punctuation suggests is a passage: one moves through the sentence on the premise of the ‘till’, which is a contraction from ‘until’, a preposition. We start, then, in this sentence, with an axiom of linkage: we are held, pre-posed or pre-posited, even, before all becomes white. The object of the preposition, the ‘white’, is doubled over. All becomes white in the whiteness, a transformation happens against minimal difference. We start with a temporal preposition and are moved from this, by the modification from white into whiteness, into a locus: the rotunda. This sentence moves from preposition to position through merely five words, a shift from time to space. Here the absence of conjunctions indicates a lack of logical relation and equally a lack of possession or subjectivity. This is abetted by the absence of ‘I’ or ‘my’ in the piece, which renders such a description abstract. Likewise, even the rotunda cannot unambiguously have any attributes. The whiteness is connected to the rotunda only through incomplete movement but is not clearly a quality of it. Similarly, the narrator of this piece speaks of a process of ‘fall’ and ‘rise’ based on certain variables and according to certain parameters. The shift from a rise to a fall, or a fall to a rise, may be sparked by ‘pauses of varying length, from the fraction of the second to what would have seemed, in other times, other places, an eternity’.112 Such a flux of temporal experience can only arise from a process that equally has no objective referent. The language – ‘the rise now fall, the fall rise’ – gives us verbs of incomplete movement that are without object and occur only through the deformation of one modality of change into another (the rise will turn to fall). The grammar here is constructed, oddly, through the removal of punctuation and clauses or words that facilitate logical relations, moving away from syntactic implication and inference to transition purely

111 112

Beckett, ‘Imagination Dead Imagine’, 182. Ibid., 183.

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across points in a domain or scene. The minimal content is only of time and space, two factors that Kant identified as pure cognitive products, independent of human experience and, as such, a priori. Of course, Kant’s understanding of mathematics and the a priori cannot quite account for this ‘mathematics of magnitude’ rather than measure. A precursor to this ‘measure without count’ exists: it is a pre-Enlightenment mathematics iconicized in Da Vinci’s ‘Vitruvian Man’. The Vitruvian Man and Beckett’s ‘Emma’ are comparable by virtue of a shared anatomical magnitudes. The ‘Vitruvian Man’ details the proportions of a man in terms of a circle and a square, and purports to illustrate the way in which man’s proportions echo the geometric essence of the universe: symmetrical, smaller units of the body exist as units of measure for the rest of the body (the length of the fingers equals that of the palm, the length of a foot is equal to four palms, etc.), the body possesses a central point (the navel) when mapped within a circle. For Da Vinci, the Vitruvian Man is a topological exercise, where we take topology to be an exercise of generating number in domain. Contrary to the simplistic assumption that the Vitruvian man provides or represents empirical measure, and hence some logo- or phallocentric norm – however godly – Da Vinci’s work is in fact a radically anti-empirical piece (albeit avant la lettre). This Renaissance ideal man is notably also a man whose ideality is determined not by measure but by magnitude. Here the hand is the base unit that determines proportions: the human is a microcosm of a wider and universal proportion, and there is no unit, no mechanism or constant that would indicate or allow objective measure. In All Strange Away, Beckett has relinquished sensible proportion but, like Da Vinci, places the figure in a square and animates it, moving Emma into positions with coordinates, turning a question of harmony and proportion into an exercise in pornography: For nine and nine eighteen that is four feet and more across in which to kneel, arse on heels, hands on thighs, trunk best bowed and crown on ground. And even sit, knees drawn up, trunk best bowed, bed between knees, arms round knees to hold all together. And even lie, arse to knees say diagonal ac, feet say at d, head on left cheek at b.113 113

Beckett, ‘All Strange Away’, 172.

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Beckett’s Anti-Vitruvian figures are positioned in terms of various magnitudes, defying harmonic mathematics for obscene mathematics, and debauching perfect existence in a circle or square through the contradiction between eroticism and mathematics. This might, indeed, also be a riff on the notion of ‘obscene’, a term which is thought to come from the Latin ‘onto’ – ob – and the Latin for filth, caenum, but also, in English (and French) becomes ‘onto-scene’ or a ‘putting in scene’. This use of the topological transition away from measure to magnitude must be recognized in terms of a mathematical tradition that exists on either side of the Enlightenment, in twentieth-century innovations in mathematics, as well as early-modern mathematics. Pre-Enlightenment mathematics mapped the world according to magnitudes, as we see in the use of the hand to measure and construct a cosmic geometry. In turn, the use of ideal shapes and biological units to express geometrical correspondence rather than measure installs a topological universality sundered from objectivity. Topology defies objectivity in the same sense that Renaissance mathematics does by virtue of the combination of arithmetic and geometry in this branch of mathematics. The division between arithmetic and geometry echoes, as the French philosopher and mathematician Albert Lautman puts it, the philosophical opposition of the relation between rule (analysis) and domain (geometry).114 Where geometry was concerned with actual place, the practice of arithmetic was not bound to ‘reality’ in the same way. Topology combines the study of domain with the arithmetical practice of what would, in philosophical terms, be considered speculative rules: rules that do not necessarily have to exist or be ratified in the empirical world, or based on experience. In other words, topology does not require an object, but, rather, a spatial possibility; it is a mathematics capable of generation, without the requirement for representation. Of course, the subject of these two short texts echoes Kant’s famous transcendental subject, which is always an x: ‘By this “I,” or “He,” or “It,” who or which thinks, nothing more is represented than a transcendental subject of thought; x, which is cognized only by means of the thoughts that are its predicates.’115 Appropriately x moves only in the skull, and talks to himself in the

114 115

Lautman, Mathematics, Ideas and the Physical Real, 9. Kant, Critique of Pure Reason, 142.

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‘fourth’ person, a voice that takes the self as both subject and object at the same time (the most famous example of the rarely used ‘fourth person’ comes from The Strange Case of Dr Jekyll and Mr Hyde). This formulation resembles the subject Beckett has at work in All Strange Away. In pursuit of a possible literature that can – like mathematics – write the a priori, or the unword, Beckett has replaced character with the transcendental subject, with x, achieving a form of algebraic art with no central substance, only ‘the thoughts that are its predicates’.116 This is the second incarnation of Beckett’s replacement of character with a point. In the first part of this chapter on Beckett, we saw how in Molloy the character is ‘situated’ in terms of Cartesian coordinates: at the 0, 0 point allocated to the soul on the graph that delineates numerical extension. In this case, the cogito is the medium for rationality, through which reason moves and extends itself, making it another point, another x, on the graph for which there is no centre. Here, we have the same process of replacing character with x but this occurs through a different technique. In this situation, the subject becomes x by virtue of its existence as thinking subject without identity. The subject’s thoughts constitute its predicates, distributing alphabetic markers in a space that emerges from the mind. This is a literature of the dead imagination. The points describe a scene outside of the world, outside, even, of what we know sensually and experientially to be possible. In All Strange Away and Imagination Dead Imagine, Beckett contorts rather than ‘fleshes out’ the figures that appear in both short texts, replacing evocative languages that might generate imagery with topological coordinates and spatial distribution. The replacement of imagery with image-coordinates circumvents imagination (precisely what these texts claim to take up, but rather defy) and allows the a priori construction of image to intrude upon the fictional expectation of imagery. This provides a kind of reverse and perverse enactment of the generic drive for a primitive signifier or for the ‘unword’ that Beckett seeks, producing a priori coordinates which at once generate a literary symbolism and negate it with an absence of referentiality. However, this is not a case of simple presentation and then negation. This is a hortatory negation that is effected, repeatedly, through a textual pulsation. Beckett’s use of topological and a priori coordinates in Imagination Dead Imagine and All 116

Ibid., 319.

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Strange Away serve to foreground the stakes in his frustration with language; the problem of communicating pure cognitive products, unmuddied by the word, but also the human centrality to any desire or drive for such an a priori. The internal undoing of the unword and the surrendering of objectivity in mathematics both seek a universality that is not found within phenomena. ‘Words are dangerous tools,’ writes the mathematician Hermann Weyl, ‘The scientist must thrust through the fog of abstract words to reach the concrete rock of reality.’117 Weyl’s complaint becomes strangely reminiscent of Beckett’s frustration at the porousness of words – his frustration at the fact that language acts like a fog rather than rock. Does the writer of the ‘unword’ then become a scientist here, piercing through the fog of abstraction that is language? Not in the way that one would expect. Weyl talks about a mathematics of the possible as a means to get at the ‘concrete’; inverting one’s expectations regarding the nature of the concrete. ‘Mathematical thinking’ is thinking in terms of functions and variables, and its principles extend ‘over all possible, rather than over all actually existing, specifications’.118 This ‘mathematical thinking’ might be the name for the form literary work that Beckett produces in Molloy and Watt, whereby the geometric organization of words is replaced by topological form and generic prose. Part of this mathematical thinking, for Weyl, is then the ‘step of abstraction where intuitive ideas are replaced by purely symbolic construction’.119 He illustrates this ‘step of abstraction’ (and its necessary link to possibility) with a mathematical example of time and space that elides the words (which muddy perception) ‘past, present and future’: A world point is represented by a point in this picture, the motion of a small body by a world line, the propagation of light with its velocity c radiating from a light signal at the world point O by a vertical straight circular cone with vertex at O (light cone). The active future of a given world point O, here-now, contains all those events which can still be influenced by what happens at O, while its passive past consists of all those world points from which any influence, any message, can reach O.120 117

Hermann Weyl, ‘The Mathematical Way of Thinking’, in The World of Mathematics: Volume Three, ed. James R. Newman (Mineola, NY: Dover Publications, 1956), 1836. 118 Ibid., 1834. 119 Ibid. 120 Ibid., 1836.

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What is interesting here is the insistence upon an evacuation of content in mathematics whilst forgoing an objective viewpoint (that of measure, of external reference) and hence privileging a grammar of the domain. Here lies the inversion of naturalism. On the one hand this work features a world represented by a narrator that still stands outside of this world, the narrator that does not participate in his scene, is not possessed by his scene, but at the same time relinquishes all objectivity. Of equal interest is the fact that here the concretion afforded by the use of mathematical symbols pertains less to the reference of each of the symbols so much as the grammar that they construct together. This is not necessarily a grammar in the sense of a linguistic grammar, but rather one of world-points, space-times, extension and signals, axes, bodies and motion. This is a grammar that Beckett achieves in these two short prose works: All Strange Away and Imagine Dead Imagination. This is grammar of topology; in other words, a grammar of a world that is no longer composed of axes which determine the movement of time, but one is which the propagation of light determines the ‘still to be’ and the darkness the ‘what has been’, without any clear designation of day or night, without any clear measure, only a magnitude.

4

One: J.M. Coetzee and the Name of the Number

All in all, this patchy imitation of Oxford English studies had proved a dull mistress from whom I had been thankful to turn to the embrace of mathematics; but now, after four years in the computer industry during which even my sleeping hours had been invaded by picayune problems in logic, I was ready to have another try. J.M. Coetzee, ‘Remembering Texas’ I know all the numbers. Do you want to hear them? I know 134 and I know 7 and I know’ – he draws a deep breath – ‘4623551 and I know 888 and I know 92 and I know – J.M. Coetzee, The Childhood of Jesus John Maxwell Coetzee’s novels have frequently been accused of a lack of worldly engagement, circumventing the immediate realities of apartheid for engagements with European modernism. This accusation is one of form as much as content. The novels are seen to miss the important ‘ethical work’ done by literary confrontations with the ‘other’ by allegorizing worlds, situations and encounters with the foreign.1 Coetzee, when he is condemned, is done so 1

Nadine Gordimer’s classic review of Coetzee’s The Life and Times of Michael K notes that: J.M. Coetzee, a writer with an imagination that soars like a lark and sees from up there like an eagle, chose allegory for his first few novels. It seemed he did so out of a kind of opposing desire to hold himself clear of events and their daily, grubby, tragic consequences in which, like everyone else living in South Africa, he is up to the neck, and about which he had an inner compulsion to write. (n.p.) This otherwise laudatory review confronts impoverishment of the abstractions involved in allegorical writing. See Gordimer, ‘The Idea of Gardening’, The New York Review of Books, 2 February (1984): n.p. http://www.nybooks.com/articles/1984/02/02/the-idea-of-gardening/ (accessed 2 December 2016). The split in aesthetic ideologies between Gordimer and Coetzee typifies the varied interpretations of the political and social stakes of allegory.

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as the allegorical novelist, the writer who places his characters allos agoreuo, outside the assembly. This outsider quality has persisted, in spite of his international acclaim and canonization. This status is as much a question of his restaging of the aesthetic problems of European modernism as it is a question of that which is other to representational language: mathematical language, the presentational sign and the hermetic quality of the number. Apart from Coetzee’s university education in mathematics, and his years in London working in computer programming, there is a formal use of mathematics in Coetzee’s fiction that I will argue is bound together with the socially and ethically oblique aspects of his work by the knot of allegory. The mathematics in his novels is often central to the elements of formal innovation in his work, as well as the Borromean imbrication of epistemology, language and modern subjectivity that his work confronts. The young Coetzee left Cape Town in 1962 for London to pursue the promise of a literary life, as well as a respectable career associated with mathematics that might support the bohemian existence of a writer. He had taken honours in two undergraduate degrees from the University of Cape Town: in English (awarded in 1960) and mathematics (awarded in 1961). As the reminiscence in the epigraph suggests, it was in fact mathematics that captured the imagination of the young Coetzee in the academic realm, whereas his work in literary criticism (though not, it seems, the promise of a life as a poet) left much to be desired. In London Coetzee would work as a computer programmer and write a master’s thesis on Ford Madox Ford, whose novel, The Good Soldier, he would describe as ‘probably the finest example of literary pure mathematics in English’.2 Here, Coetzee echoes Pound’s quip that ‘poetry is a sort of inspired mathematics’.3 Although Coetzee offers a tantalizing conjunction here between mathematics and literature, he did not, in his analysis of Ford, carry out any extended investigation into a literary mathematics, and the two domains are awkwardly joined here: literature is identified as a version of mathematics and mathematics becomes a metaphor

2

3

Peter Johnston, ‘“Presences of the Infinite”: J.M. Coetzee and Mathematics’ (Royal Holloway, University of London, 2013), 63. Ezra Pound, The Spirit of Romance (London: J.M. Dent and Sons, 1910), 14.

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for a formal achievement. The ‘literary pure mathematics’ that Coetzee would achieve in his novels of subsequent decades would achieve a far more subtle reciprocity between the two domains. In this chapter I will look at numbers and infinity in two novels that bookend Coetzee’s career: his second novel, In the Heart of the Country, published in 1977, and one of his most recent works, The Childhood of Jesus, published in 2013 (as well as reflecting more briefly on the 2016 sequel to this novel, The Schooldays of Jesus, which has the same thematic concerns). In particular, I will consider Coetzee’s preoccupation with the mathematical and linguistic issue of what can be counted as ‘one’ and the communal stakes that undergird this problem. In both of these novels, Coetzee investigates what comes to count as ‘one’ and the processes of measuring continuity that this counting entails. Mathematics will thus afford Coetzee a particular fictional mode, one that allegorizes its own processes of composition by revealing those linguistic mechanisms of making something ‘count’. This chapter concludes with a reflection on the literary significance of what Alain Badiou calls the ‘count-as-one’. The count-as-one, as the ability to count as an individual unit, is the act that links the individual and the social, the specific and the generic, representation and presentation. This foundational numerical act is what makes the use of mathematics in Coetzee’s work ultimately political.

‘Literature in the Lap of Mathematics’: The Quantification of Style During his time at IBM, whilst also writing on Ford, Coetzee started to experiment with an applied form of literary–mathematical exchange: computer poems. Access to the IBM1401 and 7090, provided by his programming job, allowed him to run experimental programmes that would produce poems that contained a themed vocabulary.4 The programme involved inputting a semantic field and verse form into the computer, which would then arrange 4

See: J.M. Coetzee, Youth (London: Vintage, 2003), 81.

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the vocabulary to form the skeleton of a poem ready for editing by the programmer.5 Coetzee was interested in the extent to which a programme – a machinic intelligence that reproduces forms rather than creates them – could come close to the production of poetry that would pass for successful literary work. This earliest work was forged out of a scepticism towards the authority of the novel. Much of this fiction would revolve around the extent to which languages are not spoken but rather ‘speak us’ and the possibilities of a linguistic capacity to generate worlds through genuine, rather than repetitive, creativity. These questions are, of course, yet another instance of the modernist question of art ‘in the age of its technological reproducibility’.6 In this instance, the mechanization refers not to film, photography, copying or printing techniques, but the capacity of computation to contribute to authorship. Coetzee gives a fictionalized account of his work as a computer programmer in the autobiographical novel Youth: Scenes from a Provincial Life II: He has heard of computer programming but has no clear idea of what it is. He has never laid eyes on a computer, except in cartoons, where computers appear as box-like objects spitting out scrolls of paper. There are no computers in South Africa that he knows of.7

Coetzee came to computer science early: at a time prior to the ‘PC’ and mainstream computation. This field pushed his proclivity for pure mathematics into an applied domain of work and administration concerned 5

6

7

Johnston has provided the best documentation and analysis of Coetzee’s computer poetry. See, in particular, Johnston, ‘“Presences of the Infinite”: J.M. Coetzee and Mathematics’, 42–43 for an account of the ‘composition’ of the computer poems and an example of one of the poems. Details of the publication of these poems, and their significance can also be found in: J.C. Kannemeyer, JM Coetzee: A Life in Writing, trans. Michiel Heyns (Melbourne and London: Scribe, 2012). The accounts of these poems in Coetzee’s fictionalized autobiography and Kannemeyer’s biography differ greatly save on the issue of literary significance. Kannemeyer notes: In Youth Coetzee refers dismissively to the poems written in London, and none of these were published in magazines or otherwise preserved. He did, however, in The Lion and the Impala, II: 1, March–April 1963, a publication of the Dramatic Society of the University of Cape Town, publish an interesting experiment that suggests that his years as a computer programmer did at least yield some engrossing results. The poem, also published in the Cape Times, bears the title ‘Computer poem’ (123). Walter Benjamin, ‘The Work of Art in the Age of Its Technological Reproducability’, in The Work of Art in the Age of Its Technological Reproducibility and Other Writings on Media, ed. Michal W. Jennings, Brigid Doherty, and Thomas Y. Levin (Cambridge and London: The Belknap Press of Harvard University Press, 2008), 19. Coetzee, Youth, 44.

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with computational products with which he was not always comfortable. In Youth, the fictionalized version of this qualm is framed in terms of simple outflow processes and – overarching this – in relation to capital: There is something about programming that flummoxes him, yet that even the businessmen in the class have no trouble with. In his naiveté he had imagined that computer programming would be about ways of translating symbolic logic and set theory into digital codes. Instead the talk is all about inventories and outflows, about Customer A and Customer B. What are inventories and outflows, and what do they have to do with mathematics?8

The usurpation of pure logic by the simple inventories of accumulation irritates the autobiographical John. The set theoretical foundations of computation are obscured for retail operation. It is this distinction between what the fictional John refers to here as ‘mathematics’ (symbolic logic and set theory) and its transmutation and deformation in inventories and outflow that will become significant decades later in his fiction. The capacity of the computer for networked and systemic ‘reading’ of symbols resembles the machinic visions of language common to the literary theory of this time: the grooves and networks of grammar, be they linguistic convention and prediction (Chomsky, generative grammar9) or the psychic structures of the soul (Lacan, the unconscious is structured like a language10). Coetzee would, in his doctoral work in English Literature, draw a direct connection between the computer and language, undertaking a ‘stylostatistic’ analysis of Beckett’s work that would knit together statistical mathematics, computation and literary theory. In the review of a volume of poetry, Strange Attractors, Coetzee notes a ‘general enthusiasm […] for structuralism, that is, for systems that seemed to run themselves without need for intervention’.11 This emphasis on autonomous systems would even extend to the theories of authorship that he 8 9

10

11

Ibid., 45. Noam Chomsky, Aspects of the Theory of Syntax (Boston: The Massachusetts Institute of Technology, 1965). Lacan makes this claim, in several different versions, throughout his career. It is perhaps most famously put in Four Fundamental Concepts of Psychoanalysis. See: Jacques Lacan, The Seminar of Jacques Lacan: Book XI The Four Fundamental Concepts of Psychoanalysis, ed. Jacques-Alain Miller, trans. Alan Sheridan (New York and London: W. W. Norton and Company, 1998), 20. Coetzee quoted in: Johnston, ‘“Presences of the Infinite”: J.M. Coetzee and Mathematics’, 71.

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would wrestle with, in the coming decades, in his fiction. In the 1984 essay ‘A Note on Writing’, Coetzee summarizes such a notion of authorship, a notion redundant by the time of his writing: ‘One might also want to think,’ he writes, ‘of A is-written-by X (passive) as a linguistic metaphor for a particular kind of writing, writing in stereotyped forms and genres and characterological systems and narrative orderings, where the machine runs the operator.’12 It is this figuring of writing as an output system, and the structuralist notion of language as a system capable of running itself, that would form the heart of Coetzee’s fictional enquiries: each novel enquires into its status as just such an output system, as well as the status of linguistic expression as a kind of autonomous rather than agential system. Stylostatistics gathers numerical data from texts and uses this to draw conclusions about literature, mathematizing literary style to draw conclusions about it. Coetzee’s doctoral project sought to evaluate the success of Beckett’s modernist attempt to ‘evacuate’ style from his work, to write without and against style. In the oft-quoted 1937 letter to Axel Kaun, Beckett displays his exasperation with the mannerisms of literary form: ‘Grammar and Style!’ Beckett writes, ‘To me they seem to have become as irrelevant as a Biedemeier bathing suit or the imperturbability of a gentleman.’13 The young Coetzee – preoccupied with the same question at the dawn of the age of computing, wanted to know exactly how irrelevant style had become in Beckett’s experimental work. Coetzee’s project, here, echoes the linguistic instantiation of a wider crisis: how to speak some kind of truth when the conditions of one’s language condemn one to a fate of misrepresentation. The legacy of stylostatistics is something of a caricature of its beginnings. Like European structuralism, stylostatistics could have been open to a range of numerical formulations that were not necessarily blandly deterministic or quantitative. Indeed, Coetzee, with his knowledge of Gödel and Dedekind, would hardly

12

13

J.M. Coetzee, ‘A Note on Writing’, in Doubling the Point: Essays and Interviews, ed. David Attwell (Cambridge: Harvard University Press, 1992), 95. Samuel Beckett, The Letters of Samuel Beckett, Volume 1: 1929–1940, ed. George Craig and Dan Gunn (Cambridge and New York: Cambridge University Press, 2009), 518.

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have adopted quite so naïve a view. David Atwell notes that this pursuit was in fact also an enquiry into ‘the ontology of fictional discourse’, a topic that would remain a fascination for Coetzee and one which is clearly manifest in the two novels dealt with here.14 For Attwell, Coetzee’s ‘reflexive narrative’ (the fact that his novels typically enquire into the conditions of their own composition) emerges directly out of these early mathematical enquiries. In Attwell’s words, in his time at IBM, his ‘professional pursuit before his decision in the mid-1960’s to return to literature’ paved the way for ‘the later interest in the rule-conditioned character of discourse’.15 The combination of the early computer programming and poems and the research in stylistics suggest, here, that the ultimate ‘rule’ conditioning language for Coetzee might just be the numerical rule: what counts, how to give an account and what constitutes an adequate enumeration, a position that subjects stylostatistics and associated methodologies to their own modes of analysis, in true Cantorian fashion.

In the Heart of the Country: Freedom and Equality In The Heart of the Country is perhaps the most clearly ‘mathematized’ text in Coetzee’s oeuvre. What is immediately striking about this novel, written in response to the genre of the South African plaasroman, are the numbers that accompany the paragraphs. The plaasroman was chiefly an Afrikaans endeavour that celebrated personal connection with the land. Coetzee’s plaasroman constitutes an interrogation of the genre’s linguistic conditions through a careful manipulation of the grammar within that form. The novel consists of only one voice: a monologue by Magda, a young woman who lives on a farm with her widowed father and two labourers, Hendrik and Jakob, and Jakob’s wife, Anna, who works in the house. Magda identifies herself by virtue of her self-ascribed ‘exceptionality’, as a figure whose method of withdrawing

14

15

David Attwell, ‘Editor’s Introduction’, in Doubling the Point: Essays and Interviews, ed. J.M. Coetzee (Cambridge: Harvard University Press, 1992), 1. Ibid., 5.

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from the world, whose mode of affliction, is common to a literary tradition of the isolated and mad woman: I was in my room, in the emerald semi-dark of the shuttered late afternoon, reading a book or, more likely, supine with a damp towel over my eyes fighting a migraine. I am the one who stays in her room reading or writing or fighting migraines. The colonies are full of girls like that, but none, I think, so extreme as I. My father is the one who paces the floorboards back and forth, back and forth in his slow black boots. And then, for a third, there is the new wife, who lies late abed. Those are the antagonists.16

Alongside this simple, matter-of-fact outline of Magda’s situation, there is both a graphological and narrative count. Each of the paragraphs is numbered; this paragraph, for instance, is number 1 in a count that will run to 269. The ‘matter-of-fact outline’ presenting the antagonists takes the form of a count too: Magda coming first, her father second (although he too is ‘the one’) and the new wife being the third. After the first person opening of the paragraph, in the following sentences the first person singular becomes an object. In this minor manoeuvre, Coetzee signals the perversion of the plaasroman to come: the ‘I’, here, will transition back and forth from subject to object, inserting a disruptive, implicit self-reflexivity into Magda’s testimonial as well as initiating the linguistic demands of Magda’s unstable consciousness. In this monologue, Magda’s own status as subject or object of and in language, and the status of the other characters, is never secure. This brief presentation of both Magda’s own extremity of character and the introduction of the other two antagonists is quickly compromised. We learn that Magda’s lucid account of her existence is far exceeded by a near-hallucinatory relation to the world. Whilst her multiple narratives and interpretations of her world do become increasingly disturbed as she encounters opposition and difficulty throughout the novel, from the outset she possesses a bitter, warped relation to reality. Magda’s voice only realizes a single coherent narrative for a few pages before she departs from a single account of events and the world. This forking of narrative events is initiated by an uncertainty surrounding the identity of the new wife. Either Magda speaks of the bride of Hendrik – the 16

J.M. Coetzee, In The Heart of the Country (London: Vintage, 1999), 1.

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farm labourer – who eventually comes to work as a maid in the house, or she speaks of her father’s new wife, who also intrudes upon Magda’s domestic sphere. These two narratives are never simultaneous and quickly become a confusion that suggests that Magda is not narrating an actuality, but processing a trauma: her lonely, withdrawn father has used his power over Hendrik to start a sexual relationship with his young wife. After this initial scene setting, Magda plunges into an introverted world that quickly overturns any factuality or order this enumeration might have established. The novel is thus modified by the notation that accompanies it, which comes to serve a function of contrast. In no way is the monologic narrative consistent, objective or progressive the way that the numbers that follow the paragraphs are. The numbers thus seem to dutifully enumerate a narrative progression that never occurs. This omission thus departs from the literary convention of numbering chapters, evoking instead the conventions of witness statements and legal documents, which fits with Magda’s monologic account of events, her ‘witnessing’ of what goes on at the farm recorded for purposes of verification. This graphological suggestion of legal convention, as well as the implied logic of the consistent numerical stream, serves to bring into relief the contradictions and reformulations in Magda’s narrative that obscure rather than order a coherent narrative. As the narrative turns back upon itself what is illuminated are not events but the processes of repression and sublation that produce Magda’s reality. In other words, the annotated paragraphs create a tension between number and narrative sequence; the consistent and completed order of the numbers exists in contrast to the narrative failure to ‘count’. Whilst this numerical accompaniment to the monologue seems to be isolated – the numbers that are supposed to track progress cannot hope to do so – it also reflects a process of exchange, distance and reciprocity between the words and numbers that will preoccupy Magda throughout the novel. There are several different forms of numeracy that preoccupy Magda, all of which involve counting, but very different sorts of counting to that which indexes the paragraphs. Like the numbers alongside the paragraphs, these forms of counting are also mechanisms of ordering language. In the first instance, Magda’s conception of her own identity and alienation from the world presents itself as an overarching count. This invokes both a clear self/other distinction

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that might remedy her isolation from others, as well as an emphasis on rationality or explication as the means by which to remedy this. She identifies herself with zero: To my father, I have been an absence all my life. Therefore instead of being the womanly warmth at the heart of this house I have been zero, null, a vacuum towards which all collapses inward, a turbulence, muffled, grey, like a chill draft eddying through the corridors, neglected, vengeful.17

This identification with zero is an absent or negative identification, an identification that is a contradiction, indeed a negation of identity. It is from this negative identity that many of Magda’s problems with language stem. If Magda is not a one but a zero, how is she to address and communicate with others, who are cancelled out grammatically and socially by this negative identity? The question of whether zero is a number has been a persistent conundrum in philosophy and mathematics. If zero has no value, then can it still be considered to be a number? In Fibonacci’s work on Arabic numerals, for instance, zero is listed not as a number but as a sign.18 In the Liber Abaci: ‘These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1.’19 In this text, these signs, taken together with the ‘sign 0, which in Arabic is called zephirum, any number can be represented, as will be demonstrated’.20 Zero’s unique standing predicates a set of relations that would not be possible with any other number, for instance in the equation x + y = x, y must be zero. As zero, Magda does not ‘add’ anything to the world she lives in, she is the marker of insubstantiality. She has identity here, but her identity is without value; similarly, she is not a figure but a sign. And, as a sign, Magda’s function is to supplement the figures (1, 2, 3). ‘Zero’ is the numerical placeholder, in Magda’s metaphoric system, for the first-person pronoun, revealing one of the key numerical contradictions in her capacities for narrative.

17 18 19

20

Ibid., 2. Carl B. Boyer, History of Analytic Geometry (Mineola, NY: Dover Publications, 2004), 44. L.E. Sigler, Fibonacci’s Liber Abaci/Leonardo Pisano’s Book of Calculation (New York: SpringerVerlag, 2003), 17. Ibid.

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Magda craves intimacy, which she believes would solve the breach between her absence and the presence of others: If my father had been a weaker man he would have had a better daughter. But he has never needed anything. Enthralled by my need to be needed, I circle him like a moon. Such is my sole risible venture into the psychology of our debacle. To explain is to forgive, to be explained is to be forgiven, but I, I hope and fear, am inexplicable, unforgivable.21

Closing the distance between herself and others (curiously coupled with the salvation of her soul) would be achieved by the explicable, by an ‘unfolding’ of a situation. This unfolding is also numerical. With reasoned explication Magda could travel from zero upwards into numerical existence though, notably, only by addition, not multiplication. This echoes Emile Beneviste’s formulation where a first numerical count is set up in the polarity created by language – the I and the you – the latter being ‘other’ to the subject as well as inherently plural. Magda’s monologue in its entirety can be read as an attempt at explication and this is where the numerical progression can be seen to have a function, even if the narrative that accompanies this fails to ‘unfold’ anything. The numbers that accompany the paragraphs thus seem to allegorize Magda’s own desires for what language can achieve. The process of explanation is thwarted by Magda’s own personal theories of language. Magda’s reflections on language continually run up against what she takes to be its tautological, captivating form: words function to create the speaker, who in turn creates these words. In one gloss, she responds to the problem of equality that is at the heart of this tautology: ‘I create myself in words that create me, I who living among the downcast have never beheld myself in the equal regard of another’s eye, have never held another in equal regard of mine.’22 Here, Magda offers a pure reciprocity between performative capacities of words and the subject that uses or produces those words. This is Magda’s affliction: she feels trapped in a system that completes itself; she is mired in a condition where language creates creation. This compounds the

21 22

Coetzee, In The Heart of the Country, 6. Ibid., 8.

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isolation that Magda experiences in identifying with zero. As zero, Magda cannot be equal to another, nor can another be an equal to her. Her status as zero means that no equation can be correct with a positive numeracy on one side and zero on the other: 1+1≠0, and so on. The metaphor of Magda as zero carries through to semiotic numerical coimplication: Magda cannot balance equations socially; and thus she cannot lend a meaningful existence insofar as she cannot participate in operations of numerical equality. The theme of equality in language, and in relation to language, is taken up broadly and explicitly by Magda several times: ‘Of course, the truth is that I am equal to anything.’23 On this initial presumption about the personal pronoun, she repeatedly evokes either an equality or inequality between herself, others, words or objects. ‘I am spoken to not in words, which come to me quaint and veiled,’ Magda tells us, but in signs, in conformations, of face and hands, in postures of shoulders and feet, in nuances of tune and tone, in gaps and absences whose grammar has never been recorded. Reading the brown folk I grope, as they grope reading me: for they too hear my words only dully, listening for those overtones of the voice, those subtleties of the eyebrows that tell them my true meaning: ‘Beware, do not cross me.’

The communication that she describes here again happens not in words, or figures, but in ‘sings’, echoing the relation between zero and other numbers. Her linguistic alienation from those around her – in particular her black servants – is coupled with her increasingly acute and debilitating recognition of the absence of a foundation for language and, hence, the stable individual identity of the personal pronoun. Magda’s account exists in contrast to the progressive count that accompanies her words. And we have seen that the use of words within the monologue presents a problem of equality; nothing (precisely, zero) is equal to Magda and, as such, Magda is also ‘equal to anything’, negating all meaningful distance. This is a linguistic problem that straddles the social at its most concrete and quotidian and the philosophical at its most abstract. Of course, an equation is fundamentally a balancing act. Any legitimate equation has two numerical or algebraic formulations that 23

Ibid., 17.

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are, when placed on either side of the equals sign, equivalent. Magda can be said to be engaged in a process of literary equations, which resonate with the mathematical equation insofar as both have stakes in a balancing system regulated by an intransitive count, be it through words or numbers. When Magda understands herself to embody zero – thus grammatically cancelling whatever she may encounter in a radical ‘forbidding’ of equation, she is both equal to anything and equal to nothing.24 In 1982, five years after the publication of In the Heart of the Country, Italo Calvino published an essay entitled ‘Cybernetics and Ghosts’, which explores the same problems with language in terms of the advent of a cybernetic age. Calvino, reflecting the broader concerns of the French avant-garde group Oulipo, writes, ‘Literature as I knew it was a constant series of attempts to make one word stay put after another by following certain definite rules; or, more often, rules that were neither definite nor definable, but that might be extracted from a series of examples, or rules made up for the occasion’ (15). This definition of literature, prevalent across the modernist and postmodernist attempts to create an automated literature (from Charcot and Janet’s writing therapies to the work of Stein and Tzara), has particular consequences for the use of the personal pronoun. ‘And in these operations the person “I,” whether explicit or implicit, splits into a number of different figures,’ Calvino writes, ‘into an “I” who is writing and an “I” who is written, into an empirical “I” who looks over the shoulder of the “I” who is writing into a mythical “I” who serves as a model for the “I” who is written. The “I” of the author is dissolved in the writing’ (15). For Calvino, it is the capacity of the computer for rulebased composition that ultimately erodes the status of the author and hence the capacity for the ‘I’ of and in writing. For Coetzee, writing in In The Heart of the Country, this is as much a colonial condition as a computational one. Magda’s ‘I’ is dissolved by the absence of connection with those around her – which climaxes in the confusion and manipulation of her father’s liaison with a black woman – as well as an acute awareness of the rule-based foundation of language, or, in other words, its primacy as an immaterial formal system; she is 24

Peter Johnston has also noted Magda’s paradoxical relation to infinity: she is both nothing and the infinite at the same time. See Johnston, ‘“Presences of the Infinite”: J.M. Coetzee and Mathematics’, 200.

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zero and cannot ‘count’ in any communicative structures. The two processes – one formal, one social – are intertwined. The problem of an absence of a coherent personal pronoun, or a figure or a word rather than a ‘sign’, is simultaneously a problem with numeracy: the ability for things and people to count as units that would establish difference and hence the possibility of genuine communication. The ‘signs’ that Magda deals in are, crucially, not differentiated and unique units of meaning but, again echoing her own identification as zero, identical points. Magda expresses this in a numerical register: ‘Seated here I hold the goats and stones, the entire farm and even its environs, as far as I know them, suspended in this cool, alienating medium of mine, exchanging them item by item for my word counters.’25 This process, it seems, is ostensibly infinite: The world is full of people who want to make their own lives, but to few outside the desert is such freedom granted. Here in the middle of nowhere I can expand to infinity just as I can shrivel to the size of an ant. Many things I lack, but freedom is not one of them.26

The absence of any identity or stability for the sign, here, rests on the conflict between the infinite and the finite. Magda’s ‘infinite’ operations, whereby all is equal to all, contradict a world and a language that requires finitude to enable difference, communion and communication. It is precisely the oscillation of Magda’s equations that indicates this lack of finitude. She can never settle on a form of communication because there is no necessity to the equality. Either she could be equal to all else she encounters on the farm, or traumatically separated from everything because there is no numerical identity in language (‘Seated here … I exchange [all things] item by item for my word counters’). Magda’s coruscating subjectivity, caught in the circle of tautology (‘I create myself in words that create me’), erases any identity or specificity that numbers may possess through a process of frictionless exchange (‘I am equal to anything’). Magda thus exists in the unbounded latitude of the imagination, where all numbers are zero, because these acts of thought cannot modify the actuality 25 26

Ibid., 28. Ibid., 55.

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of what is. Her language skims the surface of the world, without being able to possess or bore into its material invariance. The excessive intertextuality in In the Heart of the Country continues this rolling process of equivalence. At the same time as Magda’s personal and solipsistic language excludes her from civilization, it also inscribes her in literary traditions. In spite of herself, Magda’s vision of herself outside of language is peculiarly culturally refined. She goes on: While I am free to be I, nothing is impossible. In the cloister of my room I am the mad hag I am destined to be. My clothes cackle with dribble, I hunch and twist, my feet blossom with horny callouses, this prim voice, spinning out sentences without occasion, gaping with boredom because nothing ever happens on the farm, cracks and oozes with the peevish loony sentiments that belong to the dead of the night when the censor snores, to the crazy hornpipe I dance with myself. 27

The last image invokes Pan, the Greek god associated with hornpipes and mad dances. The hunched, twisted figure and the calloused feet are typical of witches. Here again we see the problematic of ‘equation’ standing in for communication and authenticity; Magda’s own self-descriptions are littered with images that are not her own. These images are taken from the canon of Western art and inscribe her in representational convention if not cliché. In this intertextuality Magda is both created and creator, as well as both alienated from the world and inscribed in it, dispossessed and possessed, simultaneously. The sheer variety of allusion, which appears with increasing density at the end of the novel, generates something of a maelstrom of influence, where the references accrue meaning not through their specificity but through their number. The book ends with a slew of intertextuality even more explicit and intense than these first signals. Towards the end of the novel, when Magda lives isolated and mad, on her own on the farm, she begins to hear voices: It is my commerce with the voices that has kept me from becoming a beast. For I am sure that if the voices did not speak to me I would long ago have given up this articulated chip-chop and begun to howl or bellow or squawk.

27

Ibid., 8.

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[…] The voices speak to me out of machines that fly in the sky. They speak to me in Spanish. I know no Spanish whatsoever. However, it is characteristic of the Spanish that is spoken to me out of the flying machines that I find it immediately comprehensible.28

These flying machines (which seem to be planes flying overhead) send Magda messages, to which she responds by using stones to create words. These voices are not only completely comprehensible to Magda but, more than this, use words that are ‘tied to universal meanings’.29 The flying machines don’t produce their own language but – appropriate perhaps to their machinic nature – issue facsimiles, quotes that come, mostly, from canonical texts in literary history. The first message that she reports is a line by Novalis: ‘When we dream that we are dreaming, the moment of awakening is at hand.’30 The phrases address a variety of issues, sentiments and positions, and come from a variety of authors. The majority of the quotes sent by the machines overhead come from well-known European and American writers: G.W.F. Hegel, Jacques Lacan (‘it is a world of words that creates a world of things’), Simone Weil, Friedrich Nietzsche, Jean-Jacques Rousseau (‘every man born in slavery is born for slavery’) and Luis Cernuda. The sole exception to this is the sentence ‘A blind man dancing seems not to observe his period of mourning’, which comes from a book entitled Southwest Native American Oral Literature, which anthologizes a selection of oral stories. Magda attempts to ignore or disregard the messages, but seems not to succeed, eventually sending her own messages back. She gathers stones from the veld and ‘painted [them], one by one, with whitewash left over from the old days […] Forming the stones into letters twelve feet high I began to spell out messages to my saviors: CINDRLA ES MI; and the next day: VENE AL TERRA; and: QUIERO UN AUTR; and again: SON ISOLADO’.31 When Magda’s ‘word counters’ finally cluster to make words, they are snatches of meaning, which respond to the catchphrases from literary and philosophical classics. Just as Samuel Beckett used stones to

28 29 30

31

Ibid., 136. Ibid., 137. Ibid., 138. This quote is wrongly attributed, in the Oxford Dictionary of Quotations, to Coetzee rather than Novalis. Ibid., 144.

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function as points and ciphers of words in Molloy, so too do we see Coetzee’s Magda construct an idiosyncratic linguistic system using pebbles which she circulates to create different, truncated messages. In this action we might say that Coetzee ‘doubles the point’ – to reference the title of one of his first books of collected essays. Here the counters have become points or pixels that are not words themselves – unlike Beckett’s sucking stones sequence – but parts of adjectives and pronouns in various languages or shades of languages. Many of Magda’s phrases are not quite translatable – seemingly a mix of Spanish, Latin and Esperanto – but appeal to the objects in the sky to remedy her isolation. Even Magda, despite her preposterous access to this universal language, must admit a lack of understanding: How I cursed my lot on the sixth of these days for denying me what of all things I needed most, a lexicon of the true Spanish language! […] Why will no one speak to me in the true language of the heart?32

The ‘true language of the heart’ seeks yet another form of equality: a pure language with no need for words, one in which communication is a pure transaction without medium. Magda has failed at intersubjectivity, so she now seems to have commuted into an intertextual communication. If she cannot speak between the I and the you, she will speak between texts, and address possibly benevolent machines. She does so, naturally, in a version of a (purported) ‘universal’ language. Where the planes speak to Magda with lines from the ‘universal’ exchange of the Western canon, she enters into dialogue with a version of Esperanto: another language based in a potentially universal communicability. The universal communicability is, for the reader of In the Heart of the Country, only signalled by the intrusion of the classic and the mangled Esperanto. We are told, by Magda, that this facilitates a pure exchange, but we never see the textual evidence of this. Although we do not witness this universal language described as ‘Spanish’ there is, of course, another ‘universal’ language that is present right beside Magda’s monologue: the number system. Like ‘Spanish’, its presence is also silent: it accompanies the narrative but is not expressed within the narrative. 32

Ibid., 145.

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In the Heart of the Country is thus a story about agreement in language and the numerical relationship to this. Magda is preoccupied with the pragmatics of language, of how communication is constituted in terms of the abstractions of number that silently accompany the narrative. How does one resolve the discrepancy between person pronouns and the generic, universal pronouns, between ‘Hendrik,’ ‘you’ and ‘one’? What numerical revolution in her own identity is required for Magda to regard another as ‘equal’? She ‘knows’ brown people only through the dictionary, or some memory of contact, but has no immediate experience of them (she alludes to ‘the smoky sourness of brown people, I know that by heart. I must have had a brown nurse though I cannot recall her’33). There is a striking kinship between numbers, then, and pronouns – the complex of distance and intimacy, violence and vulnerability, in Magda’s situation mean that she remains as a zero and cannot progress to name, or embody, a one. Her pragmatics are formulated using numbers and forms of equivalence mediated through ciphers of objects, most notably the pebbles. Each of the objects that she gives a word to is made equal with other objects, is turned into a placeholder, a reshuffling of whitewashed stones. Magda’s ‘counters’ are abstractions of words stripped of their ‘identity’ or particular signification. In this sense, each thing is exchangeable for all else, as she claims. This is where Magda’s efforts become problematic, and indeed, this is where Magda inadvertently comes up against a persistent and galling problem in the theory of numbers, which is perhaps not uncoincidentally a problem that involves an intersection between language and numerical identity. Numbers themselves are not immune from the problems of identity, and for numerical equations to succeed there must be a provable base for the identity of numbers and the spaces between them. Take, for example, the number eight. Does this number ‘contain’ all the numbers that precede it in the number line? Or does this rather signify something distinct? Moreover, what exactly is eight? We agree that the number has a specificity – to use it to describe seven objects or no objects would be incorrect. This is precisely the problem that Magda confronts with words: radical equality exists, and one word counter could easily

33

Ibid., 29.

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replace another, stymieing progression and understanding. A classic essay in number theory that considers precisely this problem is Paul Benaceraff ’s ‘What Numbers Could Not Be’. The title alludes to both the dilemma – the identity, explication and reduction of numbers – and the solution. Benacerraf concludes, after a thorough logical interrogation of what a number might be, that ‘if truth be known, there are no such things as numbers; which is not to say that there are not at least two prime numbers between 15 and 20’.34 Benacerraf shows how the problem of numerical identity actually arises from a problem internal to the set theoretical understanding of numbers. Benacerraf begins with a thought experiment, one whereby the ‘pedagogical order’ becomes the ‘epistemological order’.35 Rather than learning to count objects, two children – Ernie and Johnny – are taught set theory. Ernie is taught about a set of numbers N with a ‘less than’ relation R. For Ernie, ‘the assumptions made by ordinary mortals about numbers were in fact theorems’.36 Ernie learns that there are two types of counting: intransitive and transitive, where the former ‘admits of a direct object’, whereas the latter does not.37 Transitive and intransitive counting are distinguished in part by their relation to the objects counted: There are two kinds of counting, corresponding to transitive and intransitive uses of the verb ‘to count’. In one, ‘counting’ admits of a direct object, as in ‘counting the marbles’; in the other it does not. The case I have in mind is that of the preoperative patient being prepared for the operating room. The ether mask is placed over his face and he is told to count, as far as he can. He has not been instructed to count anything at all.38

All goes well with the educational experiment until Ernie and Johnny discuss their respective proofs and discover a problem: ‘Comparing notes, they soon became aware that something was wrong, for a dispute immediately ensued about whether or not 3 belonged to 17.’39 This dispute arises from

34 35 36 37 38 39

Paul Benacerraf, ‘What Numbers Could Not Be’, The Philosophical Review 74, no. 1 (1965), 73. Ibid., 48. Ibid., 49. Ibid., 49. Ibid., 49–50. Ibid., 54.

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different conceptions of the identity of numbers as and within sets. For Ernie, sets of numbers look like this: [Ø], [Ø,[Ø]], [Ø,[Ø],[Ø,[Ø]]],…

The ‘number’ that succeeds the previous number in the set N under the recursive law R, here, is a set consisting of that number and all the numbers it contained (thereby making for alarmingly complex notation). For Ernie, as is illustrated in the above notation, two is clearly a member of three, and three a member of other numbers that it is less than. Whereas for Johnny, a set of numbers will look like this: [Ø], [[Ø]], [[[Ø]]],…40

This dispute is set along the divergent forms of the identity of numbers, and whether identity can be expressed in terms of one number’s relation to or possession of other numbers; for Ernie it is the latter, for Johnny the former. Each mode of arriving at numbers is correct: ‘The two accounts agree in over-all structure. They disagree when it comes to fixing the referents for the terms in question.’41 Benacerraf ’s question revolves around which set has the privileged relation to number. Where exactly do we find the things that are numbers in Ernie and Johnny’s models? What constitutes a number, and is its identity singular or does it consist of component parts? Can things individuate as numbers? Or, are numbers individuated as things? Benacerraf, ultimately, concludes that numbers do not exist. Numbers are not entities nor are they predicators or quantifiers. They cannot be individuated but must be understood not as objects but as an abstract structure and arithmetic must be understood as a ‘science that elaborates the abstract structure that all progressions have in common merely in virtue of being progressions’.42 This conundrum is perfectly true of Magda’s predicament. Numbers, ultimately, sit outside or beside the progressions that she attempts or perceives. The numbers accompany but do not intrude upon her paragraphs, enumerating the novel without providing any semblance of structure or order to Magda’s 40 41 42

Ibid., 55. Ibid., 56. Ibid., 70.

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increasingly disordered tale. In this, her narrative overturns Valéry’s idea that mathematics is ‘a model of acts of the mind’.43 Numbers are that which we cannot know other than by abstract progression: we are excluded, as Magda’s paragraphs are, from the actual count, from knowing the being of numbers, which accompany but are not included in the narrative. One could claim that the count Magda does evoke successfully is a count of literature, peppering her observations with artworks and texts from the canon, but it is precisely this excessive evocation which dissolves the relation between these classics, and the possibility of either Magda’s monologue or In the Heart of the Country responding formally rather than indexically to these works. It is in Magda’s failings to articulate herself as an individual and assert a proper numerical identity (identifying herself as ‘one’) that we find a literary allegory for Benacerraf ’s problem: Is Magda, the zero, even a number? Or is she relegated to belong only beside the numbers, in the paragraphs full of signs, thwarted universal languages, and quotes copied from canonical culture, issued by machines from overhead? The connection between these machines that emit symbols from overhead and the universal languages that Magda perceives and attempts to communicate in makes the machines seem to be more than simply planes. They also strike one as computers, akin to the ‘head’ on a Turing machine, printing symbols on the blank strip below. The words that come from Magda, from zero, from the sign that is not yet a signifier, appropriately exist beside the numbers but never with them. This is not so much an Oulippean ‘combinatory literature’ that revels in the freedom of some literary Rubik’s cube, but a literature of the crises of continuous and intelligible signification that revolves around the ability to identify as a ‘one’ and the problems of equality that stem from this. Coetzee’s interest in stylistics and his work in mathematics had indeed anticipated this fictional work: this novel is formed as an enquiry into the role of language as a rule bound system. This text allegorizes its own numerical processes: the capacity of Magda to use the pronoun, to assert her own voice through which narrative events take shape, to name and communicate with others and to count herself and others in language. In other words, the formal 43

Paul Valéry, The Collected Words of Paul Valéry, Volume 15: Moi trans. Marthiel and Jackson Mathews (Princeton, NJ: Princeton University Press, 1975).

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allegory that occurs here is fundamentally an allegory of the numerical processes of the text and the novel. These numerical processes that constitute the most fundamental rules of the linguistic system are staged in the reciprocity between number – the notation next to each paragraph – and its other: the attempts in prose to count and to make oneself and others count.

The Childhood of Jesus and Mathematical Nominalism Coetzee’s recent novel, The Childhood of Jesus (2013), is preoccupied with the problems of initiating a sequence where there is no origin story for identity, or, in other words, the capacity to continue to count when the first, founding number has been forgotten. The novel is a kind of analogue of the childhood of the biblical Jesus, and as such tells a story about a finite boy with divine, or properly infinite, qualities. The novel revolves around an unusual child, David, and the man who cares for him, Simón. David and Simón are refugees in a city whose inhabitants have been ‘washed clean’ of their memories and desires, given new identities and a new language, Spanish.44 I will argue here that David’s ‘divinity’ emerges in this novel out of his unwittingly nominalist mind. David can comprehend singularity but not sequence, probability or linearity. It is in David’s simultaneous embodiment of and sense of singularity that Coetzee produces a profound intersection between name and number: in this novel David embodies both the number without name and the name without number. It is through this reciprocal exchange between name and number that Coetzee achieves a broader reciprocity between mathematics and literature, indeed, this novel realizes what I will call a ‘transfinite exchange’ between literature and mathematics. The novel opens with a boy, David, and a man, Simón, arriving at what appears to be a welfare centre in a city named Novilla. It transpires that these new arrivals have come to this city having undertaken intensive Spanish lessons for several weeks at a processing camp called ‘Belstar’, where they were also assigned names and ages. They had arrived at this camp by boat, 44

Ibid., 20.

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having no memories of a past, having been ‘washed clean’. They are thus in the literal sense new arrivals in a new place, without origin stories and only the vaguest sense of a life before Novilla. It is this erasure of history and identity that precipitates the strange search for the boy’s mother. According to Simón, the boy’s mother had travelled by ship to Novilla ahead of the boy, expecting to be reunited with David at a later stage. David had papers relating to his mother but he lost them on the ship, an unusual predicament given that once his mother had undergone the rehabilitation process at Belstar, and been given a new name, an approximate age, and allocated housing, the papers would not have been useful. There are many subtle but intricate connections between The Childhood of Jesus and the biblical infancy gospels. Simón and David have arrived at Novilla (which shares its first letter with Nazareth), and, when there is no accommodation available at the relocation centre, they are forced to reside not quite in a stable but in an improvised shack at the back of a house. Like the infant Jesus, David only has a mother and his father figure is a ‘god-father’ or a kind of uncle, as Simón describes his own role. But of course, David does not remember or resemble his mother (once she has been found) and she does not know him: she will be a mother by nomination rather than biology or even proper adoption (and, perhaps appropriately, she is always referred to by her name, Inés, rather than directly as ‘mother’). Simón supposes from the outset that he and the boy will recognize her as soon as they see her, relying on an intuitive notion that mother and child have some form of transcendental bond, and foregoing the usual biological or legal necessity that would determine motherhood. The common situation shared by Jesus’s mother Mary and Inés is that they become mothers through nomination, and it is this nomination that will be the origin story for a child who has no traceable origins or, in another set of terms, no (earthly) father. Inés is thus, potentially, a virgin mother and indeed Saint Inés, or Saint Agnes, is the patron saint of virgins.45

45

The names, under the title, are suggestive without establishing allegory. David is a king and is suggested to be an ancestor of Jesus. Before becoming king he also triumphs over Goliath, defying the giant’s might by killing him with five small stones. ‘Simón’ was the first name of St Peter, and ‘Simón the Zealot’ was also an apostle of Jesus; Jesus also had a brother named Simón.

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In this novel, there is no message from the heavens, or any other world outside of Novilla, and the society is one based on pragmatic management without any apparent transcendental, religious or spiritual foundation: without any foundation based on another world. The absence of necessity or verifiability for such an origin story has arisen several times in Coetzee’s recent fiction, including in his Nobel Prize acceptance speech, which raises the connection between the recognition of a miracle and the naming of an allegory. In He and His Man – yet another text that revolves around the foundations of personal and the generic pronoun – ‘his man’ comes across a scene where a form (a cloudy one, at that) is nominated as an allegory: I came upon a crowd in the street, he writes, and a woman in their midst pointing to the heavens. See, she cries, an angel in white brandishing a flaming sword! And the crowd all nod among themselves, Indeed it is so, they say: an angel with a sword! But he, the saddler, can see no angel, no sword. All he can see is a strange-shaped cloud brighter on the one side than the other, from the shining of the sun. It is an allegory! cries the woman in the street, but he can see no allegory for the life of him. Thus is his report.46

The angel with the flaming sword appears again in The Childhood of Jesus as a statue in the garden of the housing estate that Inés originally lives in. The nomination of Inés as the child’s mother occurs in the same way that the designation of the miracle occurs in the above passage: a recognition of an allegory without a necessary link to the attributed meaning. In the short scene from He and His Man, the woman’s vision and the surrounding crowd’s acceptance of this vision is founded upon the naming of an allegory: another act of nomination that traverses a transcendental gap which presents no assurance of its existence or character. This is not to presume that there exists any allegory that is properly without a name or exists prior to nomination. Indeed, any proper allegory exists without immediate dependency on its ‘other’ tale. Rather, what is emphasized here is very

46

J.M. Coetzee. ‘He and His Man: The 2003 Nobel Lecture’, World Literature Today 78, 2 (May–August 2004), 19.

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creation of allegory as a process of symbolic attribution. The suggestion in this scene is that the divine communication is delivered by an appearance that must be named, an appearance that presents without offering its own name. This process of nomination also reflects the key tenet of allegory: the form or text or image deemed allegorical is named for something other than itself. This process of nomination does not involve a description of the cloudy form by those that deem it an allegory (only ‘his man’ witnessing but not participating in the scene stops to describe the shape of the clouds). Instead, it depicts the giving of another name to the text or shape and the recognition that the other name applies to or resonates with this once amorphous event. Simón’s process of ‘nominating’ Inés as David’s mother is not an act of pure chance or a refutation of reason. It is an appeal to a different reason that resides in intuition, a form of reason that follows not logic but allegory. From the moment of his arrival, Simón resists the absence of sexual desire and passionate attachment in Novilla, as well as more generally the absence of sensual pleasure. In Novilla food is nutritious and basic (bean paste and crackers), and Simón craves meat. Simón’s appeals to the sexual passions fall equally flat in the face of a world where these are unsolicited and absurd. His insistent attempts at becoming involved with women appear misguided and pathetic. Simón finds work as a stevedore, even though loading and unloading cargo everyday is physically demanding on his ageing body. His fellow stevedores are not unfriendly, but rather seem immune to the question of origins or the necessity of situations, precisely that which Simón – in part because of his obligation to find David’s mother – seems preoccupied by. When Simón questions whether their world is the best possible world, a fellow dock worker, Álvaro, assures him that this is not a possible world, it is the only world. Simón notes that to Álvaro there is no irony in that statement. Simón finds a similar problem in Elena, the mother of one of David’s friends: ‘Elena is an intelligent woman but she does not see any doubleness in the world, any difference between the way things seem and the way things are.’47 For Simón,

47

Ibid., 64.

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this means that the world ‘lacks weight’. ‘The music we hear lacks weight. Our lovemaking lacks weight,’ he complains, The food we eat, our dreary diet of bread, lacks substance – lacks the substantiality of animal flesh, with all the gravity of bloodletting and sacrifice behind it. Our very words lack weight, these Spanish words that do not come from our heart.48

For Simón, irony, desire and an interrogation of the necessity of labour and habit lend ‘weight’ to the components of that world. Of course, this notion of ‘weight’ is irremediably, maddeningly vague because it operates on a certain logical structure that involves neither induction nor deduction but tropology. Here, Simón seeks a ‘weight’ that emerges from the doubleness of words that can associate flesh (his heart) and gravity (a sincere use of language). This ‘doubling’ of words adds a currency to language that exists properly outside of the linguistic system: an association, a resonance that comes not from the mind at all but from the prelinguistic, somatic beating heart. In this sense, Simón seems to crave art – the creation of ‘weight’ outside of utility, function, causality. It is not insignificant that art, here, is aligned with religion; before language, Simón alludes to the substantiality of religious ritual. Simón’s desire to lend weight and value to his world lies in stark contradiction to the organization of Novilla and the disposition of its inhabitants. A society like Novilla, one that involves an exceptional consistency of social organization, has a peculiar relation to value: a numerical one. Novilla seems to be a society that, in providing all basic necessary services for its citizens, has mostly conquered the desire that leads to inequality and competition. Work is readily available, goods seem cheap, housing and education are free, and people are generally open-minded and amiable if incurious and somewhat passive. This is a society of consolidated biopolitical control. Biopolitics works with a curious numerical form: the population, which is a ‘global mass that is affected by the overall characteristics specific to life […] like birth, death, production, illness, and so on’, a number indistinct in its numerical form.49 This is a society 48 49

Ibid., 64–65. This is Foucault’s later definition of biopolitics. His earlier definition referenced ‘a generalised disciplinary society’. For a discussion of this distinction, see: Michell Senelart, ‘Course Context’, in Security, Territory, Population by Michel Foucault, trans. Graham Burchill (New York: Picador, 2007), 378.

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whose founding myth lies in the consistency of the count. The intransitive progression of numbers continues unbroken: there is no new number, or a number without a name; rather there is a sequence of numbers whose identity is their name, a sequence that continues unbroken for no one knows how long, but certainly long enough to generously provide for the calculus of human life. This is, more generally, also the numerical regime of the ‘camp’, which, in Giorgio Agamben’s words is ‘the fundamental biopolitical paradigm of the West’.50 The camp is, in Agamben’s definition, a ‘state of exception’ from normal law that nonetheless, despite this status, becomes the enduring norm. This is an apt description of Novilla. As a state of refugees, who reside under its ‘protective custody’, Novilla is a perfect example of a normalized state of exception.51 The most important aspect of Agamben’s definition of the camp is the claim that ‘the camp is a hybrid of law and fact in which the two terms have become indistinguishable’.52 This is clearly echoed in Alvaro’s notion of ‘the only world’, where possibility is delimited and articulated by the state of things. Here, in Novilla, the relation to number formalizes this: number exists only insofar as it institutes a norm, a law of order and emphatically not the appearance of singularity and novelty. What Simón seems to be looking for is an aspect to life in Novilla which will lend value that exceeds economic terms, a value that does not already have a place in the economy of number, a transcendental necessity, or a transcendental ideal. Anthony Uhlmann has suggested that this book may be a response to Coetzee’s Nobel Prize win, which recognizes authors based on ‘the most outstanding work in an ideal direction’.53 The question of idealism is most certainly under scrutiny, here, and Coetzee takes one of the dominant idealistic socio-economic models of the twentieth century, welfare state socialism, and recreates a version of it. This society may be described as ideal where ‘ideal’ is taken to imply the ‘best situation’ in biopolitical terms, or in terms of the successful management of life. Rather than being an ideally managed society, 50

51 52 53

Giorgio Agamben, Homo Sacer: Sovereign Power and Bare Life, trans. Daniel Heller-Roazen (Stanford, CA: Stanford University Press, 1998), 181. Ibid., 167. Ibid., 170. Anthony Uhlmann, ‘Signs for the Soul’, Sydney Review of Books, 9 July 2013, http://www. sydneyreviewofbooks.com/signs-for-the-soul/ (accessed 12 December 2014).

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then, Novilla seems to be a place that has no ideals left, no ideals beyond the affirmation of quotidian reality, which might exceed or transcend the successful management of life. Simón is looking for a necessity or an ideal that is qualitatively rather than quantitatively different from the Novillan conception of these terms. Simón uses the same words as his companions in Novilla but means the exact opposite. In Novilla, necessity implies the minimum requirement, whereas for Simón necessity implies imperative; and whereas ‘ideal’ implies the ‘suitable or practical’ in Novilla, so ‘ideal’ for Simón is an elusive standard of perfection. In each case, we see the split between a pragmatic and a transcendental side of the coin. Likewise, for Simón meaning is delivered not through a pragmatic rationality of cause and effect but through the ‘doubling’ necessary to a consciousness of the transcendental: through the aporetic leaps of metaphor and symbol. This is exemplified by the traversal of signs that leads Simón to Inés. Simón and David are on a walk, following a path that will take them to a ‘scenic spot’ that is signified on their map by a starburst. With the title of the book in the back of readers’ minds, this alludes to the Star of Bethlehem. Simón stumbles upon Inés playing tennis with her brothers at the end of the walk, when they have reached the point on the map that was supposed to be ‘scenic’. Simón, suspecting Inés is David’s mother, introduces himself and implores her to consider mothering David. He will later reflect on this failed conversation using a metaphor of the star: ‘But alas, it came too suddenly for her, this great moment, as it had come too suddenly for him. It had burst on him like a star, and he had failed it.’54 This evokes the famous line from T.S. Eliot’s Gerontion: ‘Signs are taken for wonders.’55 For Simón, the signs on the map indicate wonders indeed. As in the unreasonable and unverifiable ‘recognition’ of Inés as David’s mother, we see Simón link otherwise unconnected signs together associatively, joining together the three separate stars and their significations together to lead to a sort of miracle: the discovery of Inés. Whilst Simón struggles with his new life in Novilla, and the absence of ‘doubleness’ in this newfound home, it is David who will most fully – and 54 55

J.M. Coetzee, The Childhood of Jesus (London: Harvill Secker, 2013), 77. T.S. Eliot, ‘Gerontion’, in Collected Poems 1909–1962 (Orlando, FL: Harcourt, Brace and Company, 1991).

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unintentionally – resist the founding elements of that society. Like Simón, David’s resistance is also rooted in his use of language, in particular his capacity to name. David cannot think in terms of either universals or economics, but only singularity. David is unwittingly a nominalist and in his struggles in school with numbers and stories will undermine the notion of ‘possible worlds’ held by Simón as well as the conviction in ‘one world’ held by other members of Novilla. David is acutely aware of the singular and oblivious to the natural or habitual law or accepted interpretation of ‘the way things are’. This is most directly illustrated through a conversation between David and Simón as they are walking home one evening: ‘Come on, hurry up,’ [Simón] says irritably. ‘Keep your game for another day.’ ‘No. I don’t want to fall into a crack.’ ‘That’s nonsense. How can a big boy like you fall down a little crack like that?’ ‘Not that crack. Another crack.’ ‘Which crack? Point to the crack.’ ‘I don’t know! I don’t know which crack. Nobody knows.’ ‘Nobody knows because nobody can fall through a crack in the paving. Now hurry up.’ ‘I can! You can! Anyone can! You don’t know!’56

Whilst it initially appears that David is being facetious here, this behaviour will come to appear genuine. David’s way of seeing the world repeatedly flaunts probability for what others take to be contingency. Above all, David’s sense of possibility operates according to an approach to situations and objects as singular rather than general. Where Simón occupies what might be described as a ‘Platonic’ position, aware of the discrepancy between appearance and reality, the imminent and the ideal, so David is mired cognitively and behaviourally in a nominalism that recognizes only singularities. It is true on one level that David does recognize some form of law here. Somewhere there is a crack that will indeed swallow him up. But this law is that of contingency, possibility and singularity, rather than the law of norms or ‘reasonableness’. David’s disruptive behaviour will come to a head in his inability to learn to read and write. There are other laws of numeracy – and indeed of literacy – that this nominalist child struggles with. Long before his time has come to go to school, Simón has already begun to introduce David to books and counting. He acquires a copy of Don Quixote from a library and reads David the story. 56

Coetzee, The Childhood of Jesus, 35.

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But David refuses – or is unable – to accept the ‘doubleness’ necessary for a traditional reading of Don Quixote. He takes Don Quixote’s perspective, not Sancho’s, convinced that Quixote’s observation of a giant is correct and Sancho’s view of a windmill is incorrect.57 David’s problems with literature seem to be an extension of his problems with numeracy. His grasp on the narrative is not checked by any necessity of natural law, just as his experience with the cracks in the pavement was not checked by any attribute of size, gravity or the finitude of the crack. Giants and windmills have equal purchase in David’s mind because he attributes identity to objects on the basis of possibility rather than probability, putting Quixote on the side of singularity and Sancho on the side of probability. Further signs of David’s numerical dissonance appear in his interpretations of the stories that Inés tells him. His interpretations link the strange blank origin narratives of all the citizens of Novilla – which disrupt what we consider to be the natural law of progeny – with a wider problem of order. Inés tells David a story of three brothers, who leave home in turn to seek out a remedy for their mother’s illness from the ‘Wise Woman who guards the precious herb of cure’.58 Each of the first two brothers meets a different animal along the way, who offers to show each of them the way to the Wise Woman in exchange for food. Both of them dismiss the animals they meet and are never heard of again. The third brother meets a bear on his journey, and agrees to give the bear food. The bear asks for his heart, and he assents, following which the bear leads him to the Wise Woman where he procures the necessary herb and returns home. The herb heals the mother, but the third son turns into a star (he cannot continue to live with no heart), leaving his mother alone. David has decided that he wants to be like the third brother, which is significant in the first instance because it reinforces David’s idiosyncratic relation to the world. The third brother does not operate according to types or probability (that the bear will attack him) but rather takes his situation at face value, giving the bear food. Again, this exchange happens within the terms of a certain law, but the very conditions of the exchange exceed what we would consider to be a ‘law’ that operates within and enforces the stability of norms. When Simón 57 58

Ibid., 152–153. Ibid., 146.

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disputes the possibility of David becoming like the third brother, claiming that he can only be the first, because he has no siblings, David objects and tells him Inés will give him more brothers. The boy is demanding and uncompromising, and whilst he understands that numbers occur, he has no comprehension of sequence, no comprehension that, even if two brothers are born, he will still be first in the familial sequence. David is typically unresponsive to the requests and reasoning of his parents and others and counters Simón with intensity: ‘I want to be the third son! She promised me!’ ‘One comes before two, David, and two before three,’ Simón retorts, ‘Inés can make promises until she is blue in the face but she can’t change that. One-two-three. It’s a law even stronger than a law of nature. It is called the law of numbers.’59 Despite Simón’s reasoning, David will not, or cannot, be convinced. David effectively identifies himself as that singular number, a number not yet named ‘one’, or, to use the words of Christ (or God) from Revelation 22:13: ‘I am the Alpha and the Omega.’ The Alpha and the Omega: a beginning and an end, a number outside of sequence. The only number in mathematical history that may be akin to this ‘singular number’ is Cantor’s transfinite ‘aleph-null’ (ℵ0): the cardinal number that measures the infinity of all definable numbers. This, of course, echoes Magda’s identification with ‘zero’: the singular number that is the locus of the personal pronoun, which cancels the other numbers when they are multiplied or divided by it. The position in mathematical epistemology that exists in antithesis to the numerical regime of Novilla is nominalism: a philosophical position – espoused most famously in the twentieth century by Willard Quine – that enforces a system of what can and cannot be known, and seeks to clarify language and thought by injecting such a distinction into linguistics and the sciences. Nominalism is the philosophical position whereby universal entities are denied existence, though the existence of particular or singular entities is affirmed. Whilst nominalism is an operative version of metaphysics, in The Childhood of Jesus we encounter nominalism as epistemological condition. A nominalist mathematics would deny the existence of mathematical objects or entities; these are, rather, merely tools to assist in the description of particulars. 59

Ibid., 148.

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This is opposed to Platonism, which posits that such mathematical objects do have a positive existence in an ideal realm. On one level, nominalism can involve only a minor amendment to mathematics, which involves the clause that, although mathematics involves conceptual entities, any claim to the actual existence of a universal entity such as a number would be a transgression against the proper metaphysical implication of mathematics. On another level, nominalism can imply a much greater transformation in the way that objects in the worlds are recognized. If numbers are taken as objects, they cannot exist within a sequence but must be encountered individually, as proper nouns or names. David does not choose nominalism as a philosophical position but cognitively cannot comprehend universals, only particulars. In a society that runs on sequence, and identity that emerges from sequence, this contrary numerical cognition will be a disability. The nominalism that we encounter in the character of David will require a much more radical renunciation of mathematical tenets, with enormous upheavals in social and personal existence. The problem that precedes a successful nominalism is that of individuation, which, as defined by Ray Brassier, is the question of ‘how it is that something comes to be counted as one’.60 In Brassier’s account, Quinean individuation is linguistic: it bears not on a reality but on what is deemed a ‘one’ in language. The initial means of clarifying this is through grammatical significance: ‘We shall not forego all use of predicates and other words that are often taken to name abstract objects. We may still write “x is a dog”, or “x is between y and z”; for here “is a dog” and “is between … and” can be construed as syncategorematic: significant in context but naming nothing.’61 In this sense, universals or abstractions can be qualifiers but not the variables that establish meaning. Where Simón’s conception of the value of numbers differs from the narrative that structures Novilla (and hence has a resistance that is largely affective), David’s dissent arises from a different conception of the identity of numbers (and hence a

60 61

Ray Brassier, ‘Behold the Non-Rabbit: Kant, Quine, Laurelle’, Pli 12 (2001), 50. Nelson Goodman and Willard Van Orman Quine, ‘Steps Toward a Constructive Nominalism’, The Journal of Symbolic Logic 12, no. 4 (1947), 105.

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resistance that is cognitive). Simón understands the disjunction between forms of measure and the definition and import or objects of measure. As such, his view of numbers is tempered by a consciousness of that which exceeds quantification. David appears as kind of radical nominalist because he takes numbers to have an existence and identity that is not sequential. In other words, for David, each number is a ‘one’ in language and thought: his nominalism takes abstractions as existent, reversing the Quinean allocation of abstraction and singularity. When David explains to Simón that Inés believes he should not have to go to school because he is clever (and thus will be bored by school), Simón asks David what it is that he thinks makes him so clever. David answers that he is clever because he knows all numbers: ‘I know all the numbers. Do you want to hear them? I know 134 and I know 7 and I know’ – he draws a deep breath – ‘4623551 and I know 888 and I know 92 and I know -’ ‘Stop! That’s not knowing numbers, David. Knowing numbers means being able to count. It means knowing the order of the numbers.’62

When Simón challenges the boy to tell him the next number after 888, and corrects the boy when he produces a number nowhere near 888, David retorts that Simón cannot know which number comes after ‘888’ because he has never ‘been there’. David’s response to Simón suggests that his idea of knowing numbers differs very significantly from Simón’s: that numbers are encountered or arrived at rather than simply extracted from their sequence. Moreover, there is an implicit nod to the infinite in Simón’s choice of arbitrarily large number. The number 888 presents the upright (and finite) mark of the lemnsicate three times over. If David is reading for the singularity of the marks, rather than sequential numerical identity, his claim that he cannot know what comes ‘after’ 888 seems surprisingly astute. It is notable that, given the title of the book, David, here, reveals himself as ‘unfallen’ in biblical terms because he cannot comprehend sequence.

62

Coetzee, The Childhood of Jesus, 149.

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Eugenio, who is also a stevedore, and a friend of Simón’s, has his own point of view on David’s predicament. ‘An apple is an apple is an apple,’ explains Eugenio: An apple and another apple make two apples. One Simón and one Eugenio make two passengers in a car. A child doesn’t find statements like that hard to accept – an ordinary child. He doesn’t find them hard because they are true, because from birth we are, so to speak, attuned to their truth. As for being afraid of the empty spaces between numbers, have you ever pointed out to David that the number of numbers is infinite?63

Eugenio’s characterization of David seems to be correct, here. David is not attuned to the basic truth that guides others. And yet Eugenio reveals himself as also fundamentally incapable of truly comprehending David’s singularity. Eugenio seems to have no conception that the number of infinities is infinite, and that infinities can take on numerical value and be different ‘sizes’. Eugenio’s point regarding the infinity of numbers suggests a ‘whole’ infinity that exists as a single continuous totality without gaps. Eugenio does not dwell on what the perils of infinity might exactly be, though his conception here is revealing enough. The danger of infinity, it seems, is that any concept of matter never really stops: the ideal chair will always withdraw from the grasp of thought and knowledge. Eugenio then goes on to present a novel theory of the infinite: There are good infinities and bad infinities, Simón. […] A bad infinity is like finding yourself in a dream within a dream within yet another dream, and so forth endlessly. […] But the numbers aren’t like that. The numbers constitute a good infinity. Why? Because, like being infinite in number, they fill all the spaces in the universe, packed one against another tight as bricks. So we are all safe. There is nowhere to fall. Point that out to the boy. It will reassure him.64

Again, Eugenio’s analytic philosophy misses the point and he seems oblivious to any theory of numbers that may include the gaps between numbers or an infinity that does not resemble a totality. There is no conceptual

63 64

Ibid., 250. Ibid., 250.

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or philosophical acknowledgement of an entity like an irrational number, here. Eugenio describes a totality that is infinite, consistent and whole.65 He envisages numbers as bricks: each standardized, solid and strong but ultimately identical, distinguishable only in their existence in a sequence. This, it would seem, is the mathematics of ‘one world’ rather than possible worlds or worlds of singularities. And this mathematics of ‘one world’ is, equally, a mathematics of the camp, the place, to echo Agamben, where ‘law and fact’ are ‘indistinguishable’. David’s notion does not so much comprehend the infinite but rather is itself inflected with the infinite. In this sense, David is both closer to any sense of the infinite than anyone else, including and especially Eugenio with his analytic philosophy, and further away. He cannot speak of what characterizes or creates him, and he is above all unable to articulate infinity, because his way of relating to the world is nominalist and he is fixated on singularity (David would be no more capable of comprehending a transfinite number than Eugenio). This is not to say that Eugenio is without a conception of numerical identity or has a naïve concept of number. Quite the contrary. Eugenio’s statement belies the fact that he does have a concept of number, indeed, Eugenio’s statement reveals a singular number. Eugenio’s unique number is the brick, the algorithm from which all other numbers and the structure of the universe extends, which is not only an extrapolation from his choice of metaphor but also resonates with his notions of labour and purpose. David’s number, on the other hand, is the wand, which he becomes obsessed with towards the end of the novel, wearing a cloak and carrying a wand and purporting to be a magician. And what is the wand? The number one that isn’t a number one: the properly singular number without a name. This preoccupation with embodying the infinite is continued in the 2016 sequel to The Childhood of Jesus. The Schooldays of Jesus is more overtly concerned with the Gnostic gospels and other apocryphal texts, including the Acts of John. 65

Eugenio’s position, here, is of course a Hegelian one, and this explains his opposition to – and perhaps even incomprehension of – David’s nominalist predicament. In Hegel’s discussion of the infinite in The Science of Logic, a ‘bad’ infinity is one that is never ending, whereas a ‘good’ infinity is a complete totality. Eugenio’s example of endless recursion echoes the eternal or ongoing being of a bad infinity, whereas the bricks, which enclose and seal a space, could be seen as a totality and hence a ‘good’ infinity.

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In this book, Simón, Inés and David have reached Estrella and settle on a fruit farm, where David runs amok with a band of children and the two adults earn their keep picking fruit. Soon enough, the issue of David’s schooling comes up, and, after a failed attempt to have David individually tutored, he is sent to the unorthodox Academy of Dance where he is taught by Juan Sebastián Arroyo and his wife Ana Magdalena, whose names seem to reference J.S. Bach and his second wife. Here numbers again become central to the novel, as the children at the Academy are taught dances that are associated with numbers. In the Acts of John, Jesus initiates a circular dance at the Last Supper, the night before the crucifixion. This dance is accompanied, in this apocryphal text, by a hymn: Grace danceth. I would pipe; dance ye all. Amen. I would mourn: lament ye all. Amen. The number Eight (lit. one ogdoad) singeth praise with us. Amen. The number Twelve danceth on high. Amen. The Whole on high hath part in our dancing. Amen. Whoso danceth not, knoweth not what cometh to pass. Amen.66

In The Schooldays of Jesus, Coetzee again addresses the foundations of numbers, and the relation between numbers and language. At the Academy for Dance, numbers have become numerological: they are divine and, as in the hymn from the Acts of John, are invoked only in a vatic register. In this most recent novel, Coetzee departs from the mathematical foundations of numeracy completely, to explore the limits of the orphic or oracular dimensions of the numeral as a presentational mark. As in Coetzee’s earlier work, these allegories of the divine child demonstrate an extended and complex engagement with sequence, with what constitutes a unit and how it is naturalized in some continuing count. During the composition of Dusklands, Coetzee made notes from Henri Poincaré’s Mathematical Creativity. The first quote that Coetzee transcribed in his notes relates to the syllogism. ‘Imagine a long series of syllogisms,’ Poincaré writes, and that the conclusions of the first serve as premises of the following. We shall be able to catch each of these syllogisms, and it is not in passing from

66

Katherine John, The Gnostic Gospels of Thomas, Mary and John (Start Publishing, 2012),

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premises to conclusions that we are in danger of deceiving ourselves. But between the moment in which we first meet a proposition as conclusion of one syllogism, and that in which we reencounter it as premise of another syllogism occasionally some time will elapse, several links of the chain will have unrolled; so it may happen that we have forgotten it, or worse, that we have forgotten its meaning.67

According to Poincaré, the true danger of a syllogism is in the forgetting of the origin in the chain of logic. The syllogistic peril indeed relates to the chain that language forms, but what Poincaré emphasizes, here, is the fact that the falsity of a sequence may come not from the structure in itself but the acceptance of some original statement as true in itself. Such an original statement may be the assertion of an allegory, as we saw in the naming of the cloud as an angel with a flaming sword, a conviction that this is the only world, or a concept of number or the infinite. Poincaré made a famous distinction between intuitionist mathematicians (who he also aligned with geometry) and logicist mathematicians (who he aligned with analysis).68 The latter do not think spatially, but the former, who jump to conclusions faster, at times with an absence of reasoning, do think spatially. Intuitionist mathematicians do not arrive through due process at their conclusions, but instead see their conclusions. We might call David an intuitionist, rather than logicist. In David’s world the logic of probability (perhaps the most important normative logic) is absent and all depends on intuition, a fact that is equally applicable to Eugenio for whom numerical truth extends not from logic but a prerational apprehension of truth. Each opposing world view and epistemology contains its own ‘unique number’ that founds not only all other numbers but a whole metaphysics of individuation. A syllogism occurs not simply when one extracts a conclusion from a piece of information in an illogical or erroneous process, Poincaré reminds us, but only where the original piece of information

67

68

This quote by Henri Poincaré from Mathematical Creation is found in: J.M. Coetzee, Reading notes including materials for Dusklands. 1960s, Subseries A: Long Works 1960s-2012, Folder 99.3. Harry Ransom Centre, University of Austin, Texas, 34. The second Poincaré quote from Coetzee’s notes runs as follows: ‘When a sudden illumination seizes upon the mind of a mathematician, it usually happens that it does not deceive him; … [if false] we almost always notice that this false idea, had it been true, would have gratified our natural feeling for mathematical elegance.’ Ibid., 40.

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or the original ‘jump’ is forgotten. The overall effect of this is not to necessarily validate intuitionism over analysis but instead to illustrate the means by which various numerical regimes emerge based on conceptions of the nature of an initial ‘jump’, the forgetting that facilitates a ‘naturalized’ world. The variant intuitions of numerical identity – and the truth of number – held by David and Eugenio suggest that the origin of the numerical structure of societies rests in a kind of ‘forgetting’ of the origins of number, just as the residents of Novilla too have arrived with only the shadiest notions of their prior lives. This of course resonates with the story of the biblical Jesus that the title refers to: the man who is both finite and infinite, human and divine, mortal and immortal. This is, although it has never been called this, a transfinite subject position. Jesus is also, in quotidian terms, the origin of our chronological count. The birth of Jesus sets the measure of years in play: it is his birth that enables the jump from zero (and perhaps away from some, now forgotten count) to one. Here, numerical identity makes a transcendental and unavowed jump that finds its echoes in allegory but never presents itself, or, in other words, never reveals its own name.

Counting as One, Rather than Counting to One In the Heart of the Country and The Childhood of Jesus, and its sequel The Schooldays of Jesus, were written in vastly different circumstances, more than three decades apart, at either end of Coetzee’s career. Yet in each novel Coetzee is preoccupied with what is countable as ‘one’, a mathematical enterprise that pertains to the roots of the linguistic, the numerical and the subjective. This is not so much a question of learning to count to one as an issue of finding the originary, self-contained unit from which one can institute the very concept of a unit, and from which one can hope to count upwards. The concept of the ‘count-as-one’ is used by the French philosopher Alain Badiou to describe the relation between a unit and a broader structure, and the means by which such an operation comes to define a situation. Badiou does not imply a nominalist or universalist perspective on what counts as one. Rather, Badiou is referring to a very specific action in set theory. The ‘one’ is not a stable unit for Badiou,

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but is, rather, an operation.69 Each ‘situation’ ‘admits its own particular operator of the count-as-one. This is the most general definition of a structure; it is what prescribes, for a presented multiple, the regime of its count-as-one’.70 Lorenzo Chiesa has pointed out that Badiou’s ‘count-as-one’ is bound up with ‘the relationship between structure and metastructure, presentation and representation’.71 The count-as-one is an operator that inscribes the structure of a situation. It is this count that Coetzee is preoccupied with in both of these very different novels: the significance of a ‘one’ in traversing presentation and representation, as an operator that emerges from a certain situation. In In the Heart of the Country, we see the incommensurability between the pure ‘Esperanto’ of numbers and the capacities for language to form equations and to count. The answer to the question ‘who counts?’ is connected to the question of how to communicate, and, subsequently, Magda’s monologue is repeatedly confounded by the untranslatability of numbers into situations of singularity, or, in other words, the absence of a true conversion between the pronoun and the numeral. In The Childhood of Jesus, we are again confronted with a problem of translation between a singularity and a generic figure: David exists in a dangerous border zone, where he can recognize numbers, but only as actually existent and hence as singularities rather than as generic placeholders in a stable schema. David’s unique brand of nominalism has significant repercussions in terms of his ability to respond to rules and conventions. This link between numeracy and social structure pertains to one of the most important sociocultural issues in the twentieth century: the idea of 69

Sigi Jottkandt provides an excellent definition of Badiou’s concept of the ‘count-as-one’ which bears quoting at length: Briefly, the count-as-one is founded upon the structuring that, in presenting the void through a nominal decision, originarily specifies which elements of a ‘set’ are in a relation of belonging. The name founds the law of the situation, although Badiou is always quick to point out that, even as it purports to name the void, every eventual naming is inevitably an illegal misnaming. In emphasizing that what passes as a complete representation or identity is only ever a semblance, an ‘as if,’ Badiou reminds us that every identity, in so far as it is tied inextricably to that original evental misnaming, can only be counted ‘as’ one – rather than actually being a one.

70

71

See: Sigi Jottkandt, ‘Love’, in Alain Badiou: Key Concepts, ed. A.J. Bartlett and Justin Clemens (Oxford and New York: Routledge, 2010), 75. Alain Badiou, Being and Event, trans. Oliver Feltham (London and New York: Continuum, 2007), 24. Lorenzo Chiesa, ‘Count-As-One, Forming-Into-One, Unary Trait, S1’, Cosmos and History: The Journal of Natural and Social Philosophy 2, no. 1–2 (2006), 69.

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an originary language – the language that Magda, perversely, lusts after. ‘Like the North African Augustine, who resolved to write in Latin, in order to spread his thoughts throughout an unholy empire, the South African Coetzee became a student and master of English,’ Stephen Kellman writes, ‘the greatest imperial language of them all’.72 There are a series of questions that the split in the nature of the ‘land of the poet’ – a linguistic versus the cultural homeland, a form of ‘originary’ speech – presents for the postcolonial artist, educated in the classics of European and North American literature. How does an artistic language travel into contexts where there is no recognition of the predicates of that artwork? And what becomes of the artist emerging from ‘the Provinces’, the further reaches of Empire for whom the natural language that literature experiments with is absent? In these questions about the postcolonial form and responses to European modernism, Coetzee will be preoccupied with what constitutes literary inheritance and the disruption of this inheritance. Paul Sheehan situates Coetzee’s writing in terms of ‘geomodernism’, a term which describes ‘a way of putting classical modernism into dialogue with postcolonialism, through the politics of place’.73 Sheehan’s reading of Coetzee as ‘geomodernist’ involves a reconfiguration of the notion of provincialism and the relation of the provinces to art. This revisioning of modernism’s relation between the ‘margin and the metropole’74 allows for a recognition of a modernism that stems precisely from the position of the colonial subject, who is ‘drawn to the heart of Empire yet at the same time repelled by it’.75 The ‘geoliterary spirit’ of Coetzee’s South African origins is, then, the ‘fulfillment’ of modernism rather than its ‘betrayal’.76 Much like the young Borges, Coetzee’s preoccupation with the space or split between centre and province, and the intermingling between these two spheres of artistic possibility, structurally mirrors a search for the transmissibility of language between A and B, communication as a kind of conduction but also particular pattern that may be interpreted (or misinterpreted, or not even considered for

72

73

74 75 76

Steven G. Kellman, ‘J.M. Coetzee and Samuel Beckett: The Translingual Link’, Comparative Literature Studies 33, no. 2 (1996), 162. Paul Sheehan, ‘The Disasters of “Youth”: Coetzee and Geomodernism’, Twentieth Century Literature 57, no. 1 (2011), 26. Ibid. Ibid., 29. Ibid.

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interpretation) by the interlocutor in a whole host of different ways, dependent upon unique linguistic make up. Of course, the notion of communication from A to B, and the conduction that facilitates this, is also the question of what computers can do to poetry. This as much an issue of how computers might mimic humans and the human art of poetry, as it is of the extent that a human is like a computer. To what extent do humans possess a capacity to recognize and share universal art? And how, then, would we recognize such universality, when our operative concept of the classic is imbricated in a construction of centre and periphery that originates from conquest? South African fiction becomes a retrospective hermeneutic endeavour and the issue of what it means to write, and to communicate, becomes the necessary subject. In Watt, Beckett consciously works through such a task, unfolding the surface through permutative monologues that make the ‘dimensionless and immaterial point[s]’ of language appear. In Coetzee’s In the Heart of the Country and The Childhood of Jesus, number is, again, at the heart of a formal allegory, governing the writing of writing. It is that mediator between structure and metastructure, presentation and representation. What constitutes the ‘One,’ or, the failure to constitute a ‘One’, renders the form of the narrative as well as the implications for the characters’ existence within a community. In this, we perhaps see not only a geomodernism but a mathematical modernism, facilitated by transfinite allegory that traverses the boundaries between the individual, finite capacity to count and the production of generic subjectivity.

Conclusion – X: Literary Infinities after Zeno and Cantor

The works of Jorge Luis Borges, Samuel Beckett and J.M. Coetzee deliver some of the clearest examples of a certain type of literary enquiry, an aesthetic endeavour that I have, in this book, called a ‘generic literature’. This literature can be described as generic partly by virtue of its stakes in formal selfreferentiality, whereby the text enquires into its own processes of composition, even as that composition – its plot, dialogue, imagery and structure – takes shape. This self-referentiality is, most often, a version of formal allegory as it has been described by Paul de Man. In these instances of generic literature, the text stages, interrogates and intervenes in the very formulae that underpin the extensity and continuity of description and the consistency of relation between the world, the fictional world and language. In the work of Mallarmé, Borges, Beckett and Coetzee, this is achieved by the appearance of number in the text and, conversely, by the use of number to explore the numerical apparatuses that undergird fiction. But above all, this is achieved through a subversion of the finite regimes of number that undergird prose in favour of the modern, transfinite world of number. Transfinite allegory, and more generally the recourse to a mathematics that is inflected with the actual infinite, offers prose a different mode in which to progress, to offer an account and to suture the abstract and the specific. It is this claim to a generic fiction, and for the ontic significance of numeracy to representational regimes, which brings this preoccupation with number into the wider modernist and late modernist project of the twentieth century. Prose fiction can think with and through its own numbers. Literary Studies now needs to develop a relation to mathematics that can comprehend these literary transfinites too. The occlusion of number in literary criticism is only

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abetted by the paradoxes and impasses that have historically plagued the identity of the numeral. Numbers are the foundation of the system of ‘arithmetic’ (a term derived from the Greek portmanteau of arithmos – number – and techne – art) and yet the very definition of numerical identity has remained controversial throughout modern and ancient mathematics: How can a number be defined within the system of arithmetic? As marks of presentation numbers exist only as their notations, not as objects that can be subjected to empirical investigation. Numbers never function the way a standard sign in natural language does, as a signifier with a relation to that which is signified. Rather, a number presents nothing other than itself. In other words, a number does not possess a referent or signified, but rather is itself the signified. This is the tautologous havoc that presentation wrecks upon representation. In this regard, mathematical signs are still regarded as properly outside of ‘natural language’ and not subject to the laws of natural language; their problematic status regarding standard representation in natural language belies their traditionally vexed status as objects of study in mathematics. Literary Studies now needs to grapple with these numerical vexations if it is to retain a critical awareness and sophisticated grasp of writing’s other: the mathematical mark. There is an irony in the fact that it is only through ‘doubling’ the ‘sense’ of numbers that Cantor can prove the uncountability of the reals and the countability of natural numbers. The proof of countability of numbers, which is the first step in the development of transfinite numbers, relies on a subversion of the purity of the count: What happens when counting numbers are used to count numbers? An actual infinity is produced by virtue of a system of measure being applied to itself. In a formulation that Borges, Beckett and Coetzee would each have enjoyed, this is an instance of a measure measuring the measure. The exact nature of the infinitude of a system is extracted from a truth that resides within the system’s presentation and not elsewhere, not at the end of a number line, or at any end whatsoever. Instead, it is the measure of measure through one-to-one correspondence that allows for the development of a transfinite number: a number that presents the size of an infinity without being inaccessible to finite mathematics. I have argued here that this doubling is echoed in literature through a kind of internal allegory. This morphological link is the first step en route to an idea called ‘transfinite allegory’, which

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describes a form through which literature presents an actual infinity, rather than a transcendental or exclusively metaphorical infinity. One can also use a morphological metaphor to model the relation between literary transfinites and mathematical transfinites: the lemniscate. The lemniscate models separation and connection between mathematics and literature through the form of a reciprocity as well as a reflection. Of course, a morphology pertains to shape, to a mark, to the materiality of a sign. In this sense, a morphology necessarily sits outside the bounds of tolerance of both literature and mathematics. It is the absent third category of textual existence: the mark. To a certain extent, suturing metaphor to morphology echoes the techniques of a generic literature itself by locating significance in the arbitrary materialization of the mark. This metaphor mediating literature and mathematics produces its own visual allegory of separation and reciprocity. The formal conduit of what is metaphorically depicted in the lemniscate is ‘transfinite allegory’. What is a transfinite allegory? In simple terms it is an allegory for which the a-logos – the other ‘order’ or text contained within a fiction – is not some other story or moral truth but instead a representational, descriptive or imaginative limit or a conditioning factor for language. Importantly, this is not some romantic ‘never-attainable’ infinitude, but one that is materially available within the text: the a logos is manifest in the stories or novels, not alluded to or felt. The most obvious example, perhaps, is the ‘Golden Book’ in ‘The Library of Babel’ or the memory of Ireneo Funes in ‘Funes, His Memory.’ In the second chapter, I argued that Funes’s attempt to create a number system, and indeed his very unusual paralysis itself, contribute to a story that revolves around the possibility of presentation, without necessarily purporting to incorporate that presentation within the bounds of the story. This story incorporates the limits of its own medium, revolving around a world of infinite detail and infinite recall and the consequences of such capacities. Funes embodies both the possibility that gives rise to fictional worlds (imagined realities, access to a tangible world beyond the one we exist in, potentially infinite worlds accessed through the mind) as well as the numerical limitations that underlie the possibilities available to fiction: a simultaneously genetic and generative kernel. The story is brought to life by two forms of consciousness: one finite and one

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infinite, one belonging to the narrator and the other belonging to a paralysed, extraordinary Uruguayan youth. I have argued that this constitutes one mode of transfinite allegory: the construction of a fiction that points outside of its own borders, to an autonomous existence, which is embedded but not contained by the boundaries of the stories. This story achieves generic form because it evokes measure without definition on two different levels: Funes embodies an uncountable consciousness, one which is presented in a finite narrative that actualizes an infinite fictional world without totalizing that world, or commuting it to a form of ‘endlessness’. In other words, it indexes the numerical potentials undergirding the possibilities of representation and yields an image of infinite imagination in the smallest possible form, through a recursive short story. Number in Beckett’s work also achieves a generic status for the fiction. Mathematics in Molloy provides the textual material in which Beckett traces the non-semiotic jump between reason and material (partial, recurrent negation), mind and extension (proprioception), and language and referent. As in Borges’s stories, Beckett’s novels index the numerical stakes in description through a kind of radicalization of naturalism. In Watt, this happens through cycles not of description or narration but permutation. In Beckett’s later short prose works we find a transition from measure to magnitude, and a transition from a concern with the content of phenomena to conditions for phenomena. This enables Beckett to radicalize the tenets of naturalism and approach a fiction that allegorizes the emergence of images and fantasies of the essential literary domain: the imagination. Number is the enigma of prose, because it is its allegorical ‘other’. It has the capacity to index a novel by virtue of the fact that it is exempt from the text and representational economy that composes that novel. Equally, number is the enigma of prose because it is the constitutive force behind the development of narrative insofar as it cleaves to the genre of the novel from the eighteenth century onwards. The fact that literary criticism sees number as ‘obviously’ related to poetry but not prose is a blindness that emerges from a failure to grasp both the broad formal and thematic significance of number to narrative. Prose narrative depends on progress and on enumeration and resolution, and the role of syntax as well as novelistic length rely on

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continuity and accumulation. As Franco Moretti has noted, the word ‘prose’ comes from ‘pro-vorsa’, which means ‘forward motion’.1 This literary ‘provorsa’ is at its heart a numerical movement. The act of counting – the process of moving along the number line – underlies the pulse of this progression of narrative, verisimilitude and action. If the fiction of the twentieth century was to disrupt the linearity inherent in prose, it needed new radical forms of counting in order to do so. Moretti’s formulation demarcating poetry to symmetrical form and prose to forward motion illustrates the different stakes in number for poetry and for prose: issues surrounding the number line are inherent to prose form, even where it does not have the same metrical investment of poetry. In the representational currencies of narrative words constrain number, which bursts the seams of referentiality, a very different effect from the expressive currencies of poetry, where number as metric can constrain the word. This is seen in Coetzee’s In the Heart of the Country, where the numbers index the paragraphs of the story, one of several modes of counting that bring order or disorder to Magda’s narrative. Indeed, the numbers that accompany the paragraphs index Magda’s own linguistic and representational predicament: numbers are never properly incorporated into her own attempts to communicate and relate through language. This novel questions the impact of zero in language: zero as the placeholder for ‘no-one’ or one that cannot be ‘counted’. A similar problem of the constitution of the ‘one’ is found in The Childhood of Jesus, where we witness the subjective and social consequences of a child who, like Magda, quite literally embodies a singularity by virtue of existing prior to ‘one’, prior to the social, biopolitical and pedagogical regime of countable numbers. As the allegorical child Jesus, David initiates a chronological count that extends not from some objective foundation for numbers but from his own being, in a generic account of the foundations of thought and being in concepts of number. *

1

Franco Moretti, Distant Reading (New York: Verso, 2013), 162.

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Transfinite allegory comes good on the claim by Friedrich Nietzsche in Writings From The Late Notebooks: Since Copernicus, man has been rolling from the centre towards X.2

The fiction of ‘transfinite allegory’ or a ‘generic fiction’ is a literature of what Nietzsche refers to as X. These literary transfinites are the textual form appropriate to the age that Nietzsche describes. They express the form of narrative and the form of imaginative and artistic endeavour that no longer has a proper end but is not – crucially – endless, or does not involve the experience of endlessness. X, in Nietzsche’s formulation, is the algebraic substitute for the new displacement of the human after Copernicus and, indeed, after so many of the modern scientific revolutions, Cantor’s development of the transfinites preeminent among these. It is fascinating that the displacement of the human, into a vast universe wherein there is no clear sacral, and no narrative of transcendental order, is represented as an algebraic mark in this aphorism. The absence of a clear end or trajectory is registered by a mark that stands in for a potentially delimitable variable. X is not a future deity, a different ideal, or a nihilist future. Instead, it is a placeholder for the generic. This pursuit of ‘X’, or the journey outwards to ‘X’, echoes the fundamental task of ‘reaching’ the infinite, which involves formalizing a relation to something that one cannot ‘arrive at’. To paraphrase the words of David, in The Childhood of Jesus, the infinite must be actualized without us ever having ‘been there’. The journey to ‘X’ evokes both Cantorian method and the paradoxes that would beset Cantorian set theory at the beginning of the twentieth century. In 1901, a pernicious flaw was found in Cantorian set theory. The British philosopher and logician Bertrand Russell discovered a paradox in Cantor’s set theory, a paradox that was discovered around the same time by Ernst Zermelo, who did not publicize his discovery. Russell would elaborate on this paradox in the first volume of Principia Mathematica, which he co-authored with Alfred North Whitehead and published in 1903. Russell and Whitehead would develop ‘Type Theory’ in response to the paradox, whilst Zermelo would – with Abraham Fraenkel – go on to develop the ultimately much more successful Zermelo-Fraenkel Axiomatic Set Theory. 2

Friedrich Nietzsche, Writings from the Late Notebooks, ed. Rüdiger Bittner, trans. Kate Sturge (Cambridge and New York: Cambridge University Press, 2003), 84.

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Russell’s paradox follows the same lines as earlier logical and mathematical paradoxes, most famously ‘Richard’s paradox’, the ‘Burali-Forti paradoxes’ and the ‘liar’s paradox’. In his ‘Introduction to Mathematical Philosophy’, Russell explains his paradox as such: Normally a class is not a member of itself. Mankind, for example, is not a man. Form now the assemblage of all classes which are not members of themselves. This is a class: is it a member of itself or not? If it is, it is one of those classes that are not members of themselves, i.e. it is not a member of itself. If it is not, it is not one of those classes that are not members of themselves, i.e. it is member of itself.3

The paradox arises when one asks whether this set – the set that contains all sets that are not members of themselves – is a member of itself or not. In the same sense as Richard’s paradox, this set appears to be necessarily both a member of itself, and not a member of itself: hence the contradiction. The paradoxes appear as soon as one tries to apply mathematical systems to themselves, or, in other words, to internalize the principles of set theory (or another system like arithmetic) within set theory itself.4 One of the most 3 4

Bertrand Russell, Introduction to Mathematical Philosophy (London: Spokesman Books, 2008), 136. Similar to Richard’s paradox is the ‘Burali Forti’ paradoxes. The ‘Burali-Forti’ paradoxes were discovered in 1897, and bear the name of the mathematician who uncovered them: Cesare BuraliForti. Burali-Forti noticed that any demarcation of a set of all ordinal numbers is another ordinal number itself. Ordinal numbers classify sets according to their order type, and thus a set of ordinals necessarily produces another ordinal, constantly producing ‘one extra’ object outside the set. In a similar fashion to Richard’s paradox, what happens here is that the set can never be contained, because the very action of creating the set produces another, extra ordinal, one that contains – and is thus larger – than all of the ordinals in the set. Moreover, the simple arithmetical intervention of adding one to that new ordinal number creates an even bigger ordinal number, and so the set is never stable anyway. Richard’s paradox, or the Richardian paradox, was detailed in a paper in 1905 by the French mathematician Jules Richard. The Richard paradox is particularly relevant, given that it concerns the distinction between mathematical entities and the language that is used to describe them, as well as mathematics and metamathematics. Russell’s description of Richard’s paradox is as follows: Consider all decimals that can be defined by means of a finite number of words; let E be the class of such decimals. Then E has ℵ0 terms; hence its members can be ordered as the 1st, 2nd, 3rd, …. Let N be a number defined as follows: If the nth figure in the nth decimal is p, let the nth figure in N be p + 1 (or 0, if p = 9. Then N is different from all the members of E, since, whatever finite value n may have, the nth figure in N is different from the nth figure in the nth of the decimals composing E, and therefore N is different from the nth decimal. Nevertheless we have defined N in a finite number of words, and therefore N ought to be a member of E. Thus N both is and is not a member of E. in: Bertrand Russell, ‘The Theory of Types’, in From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1932, ed. Jean Van Heijenoort (Cambridge: Harvard University Press, 1967), 153–154.

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important paradoxes that predates Russell’s paradox is the liar’s paradox, of Ancient Greek origin, and simpler perhaps by virtue of its semantic nature. The paradox is contained in the sentence: ‘This sentence is false.’ The paradox is here found in the fact that if the sentence is indeed false, then what it states is true. However, if the content relating to the sentence is true (indeed, the sentence is false), then this would contradict the assertion that the sentence makes. What is notable about these paradoxes is that they are related to a discrepancy between a principle and its provability, which is a problem relating to the impossibility of consistency between entity and identity that arises from a certain type of categorization: the divisive order necessary to set theory, that is generic but not of genera, not pertaining to the constitution of differences. Russell’s paradoxes are versions of the classic mathematical paradoxes developed by the pre-Socratic philosopher, Zeno of Elea. Zeno of Elea devised the paradoxes to prove that the movement and progression that we perceive with our senses is actually an illusion. These paradoxes remain famous for their ability to separate sensory perception from the rational senses, splitting perception, which grasps movement, from reason, which understands its impossibility. Zeno’s paradoxes take the form of short narrative conundrums, the most famous of which is ‘Achilles and the Tortoise’. In this story, the tortoise challenges Achilles to a race. Given their different abilities, Achilles allows the tortoise a nominal head start of, say for instance, 10 metres. The race starts, and as Achilles advances 1 m, the tortoise, going much slower, advances only 10 cm. As Achilles advances another 10 cm, the tortoise thereby runs 1 cm. This process continues, with Achilles seeming to get closer to the tortoise, but every time he reaches the last point that the tortoise was at, the tortoise has nonetheless moved a miniscule amount. Zeno’s paradoxes would be restaged in novelistic form by the Triestine writer Italo Svevo, the pen name of Ettore Schmitz, in 1923. Svevo’s Zeno’s Richard’s paradox is a ‘semantic antinomy’, because it relies not on a logical paradox but rather thwarts definitions of truth and falsehood: Richard opens up a certain class, and then proves that class impossible on its own terms. Richard’s paradox involves the possibility of clearly and completely defining a set of numbers, and, like the other paradoxes, proposes that one imagine the constitution of such a set. Once one has listed all the numbers definable by a certain principle, then one can claim a set x of all the countable numbers and demarcate y as the number which is not countable by the rule of definition. What has happened here is that y has inadvertently been defined and as such must be included in x, although this is inevitably a paradox. As such, one can never complete the set.

X: Conclusion: Literary Infinities after Zeno and Cantor

189

Conscience is offered as a paradox to its readers. Here, a man whose namesake is Zeno of Elea presents us with a literary paradox that reflects the contradictions of the soul. Zeno’s Conscience is written in the voice of Triestine businessman Zeno Cosini, initially at the suggestion of his psychoanalyst, though the volume has grown to become an act of defiance against that analyst and what Zeno takes to be his spurious diagnoses. It is Zeno’s characteristic combination of blindness and insight that leads him to believe that he can psychoanalyse himself, and the novel sets out from this paradoxical premise, with Zeno occupying the position of both analyst and analysand. The novel broadly performs a kind of illegal import of the logical paradoxes of Zeno of Elea into the realm of desire, intention and the self, applying procedures that are meant to be strictly metaphysical to the vagaries of the individual (and deeply flawed) psyche. Zeno’s confessions revolve around the split between sense and logic, movement and stasis that fascinated Zeno of Elea. Zeno finds himself perpetually suspended, both entrapped and enchanted by the pleasures and affirmations that he seeks. Zeno thus experiences the simultaneous oppression of immobility (he is controlled by desire) and the delight in the fulfilment of desire. Zeno is our repeatedly captivated subject, whether by his love for cigarettes or his pursuit of women. He conceives of freedom as a paradox: ‘Complete freedom consists of being able to do what you like, provided you also do something you like less. True slavery is being condemned to abstinence: Tantalus, not Hercules.’5 There is an important irony, here. Tantalus, whose name is the root word of ‘tantalise,’ is indeed condemned to abstinence for angering the gods: he would eternally stand in a pool of water, but every time he bent to drink from the pool the water would recede. Over him hung a tree with low-hanging fruit, but every time he reached for the fruit the branches would raise themselves, keeping the fruit just out of his reach. Zeno, in spite of himself, is condemned to exactly the same thing: he is more akin to Tantalus than to Hercules. Although Zeno has his pleasures perfectly well in reach, pleasure itself remains out of reach. Zeno can drink the proverbial water or eat the proverbial fruit, but he finds no consummation there. Indeed, his only genuine pleasure is from expectation, 5

Italo Svevo, Zeno’s Conscience (New York: Vintage International, 2003), 104.

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anticipation and the cleansing power of the fantasy of independence from desire. Cigarettes are for Zeno singularly pleasurable at the moment at which he gives them up. This is a strange inversion of Tantalus’s situation: Zeno accesses his pleasures, but satiation will only come from their denial and the innocence wrought of that denial. Hercules was repeatedly purified of his sins through hard labour or service. He can do what he wants but he must do something he ‘likes less’ in order to have this privilege. But Zeno Cosini is subject to stasis or paralysis despite the fulfilment or attainment of that which he is seeking. Speaking of his lack of talent with the violin, Zeno tells us: Even the lowest creature, when he knows what thirds are, or sixths, knows how to move from to the other with rhythmic precision, just as his eye knows how to move from one color to the other. With me, on the contrary, when I have played one of those phrases, it sticks to me and I can no longer rid myself of is, and so it intrudes into the next phrase and distorts it.6

Although Zeno attributes his bad playing of the violin to an unspecified ‘sickness’, this inability seems to be a general absence of expression or creative spontaneity. The only way that Zeno can master a piece of music is to tap his feet in the rhythm; once he plays one phrase, he cannot move to the next without some ‘distortion’ of the following phrase by the former. This is a reflection of Zeno’s difficulties on a minor scale: he is perhaps one of the more famous characters who suffer from a difficulty of propulsion. He is thus the consummately modern man in the tradition of Joyce’s Leopold Bloom, not least because Zeno is subject to an irresolvable contradiction between expectation and actuality, the ideals he aspires to and the inconsistencies and fragmented desires he is subject to. He is also the consummate Eleatic subject because desire is only realized through its perpetual denial or perpetual extinguishing. Just as in Zeno of Elea’s paradoxes, where motion proves the impossibility of progress, in this novel desire proves its own impossibility. Borges, Beckett and Coetzee have each looked to Zeno’s paradoxes in their fiction in order to explore literary propulsion: the capacity of prose to move

6

Ibid., 115.

X: Conclusion: Literary Infinities after Zeno and Cantor

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forward, which Svevo allegorizes so masterfully in Zeno’s Conscience. Zeno’s paradoxes have a central role in Borges’s essay ‘Kafka and His Precursors’. In this essay, Borges explains the paradoxes as the model for the form of Kafka’s The Castle: A moving body at point A (Aristotle states) will not be able to reach point B, because it must first cover half of the distance between the two, and before that, half of the half, and before that, half of the half of the half, and so on to infinity; the form of this famous problem is precisely that of The Castle, and the moving body of the arrow and Achilles are the first Kafkaesque characters in literature.7

As in Svevo’s novel, in this essay Borges mixes categories. The philosophical paradoxes are recognized, properly, as narratives with characters, and Kafka’s novel is associated with logical paradoxes. Zeno of Elea is Kafka’s precursor, here, which is an apt scheme with which to circumvent direct influence but acknowledge formal correlation. Borges offers us this tantalizing comparison without elaborating on it, moving instead to find other precursors for Kafka’s work, including in Browning’s poetry, Kierkegaard’s writing and an apologue of Han Yu. In Svevo’s novel, Zeno needs a force greater than himself to make decisions; he needs destiny to confront him, baulking at his own susceptibility to desire and circumstance. ‘I suddenly started shivering, and with a certain satisfaction I thought I had a fever,’ Zeno tells us, ‘It was not death I desired, but sickness, a sickness that would serve me as a pretext to do what I wanted, or that would prevent me from doing it.’8 This is related to Zeno’s paradoxes in the same way that – in Borges’s mind – Zeno is a precursor to Kafka. Zeno’s desire for hindrance, in order to attain some kind of fulfilment, mirrors the impossibility of progress seen in both Zeno of Elea’s narrative paradoxes as well as Kafka’s The Castle. Beckett’s Endgame also has recourse to Zeno’s paradoxes to express the impossibility of progress despite apparent onward movement. Beckett uses 7

8

Jorge Luis Borges, ‘Kafka and His Precursors’, in Selected Non-Fictions, trans., Esther Allen, Suzanne Jill Levine, and Eliot Weinberger (New York: Viking, 1999), 363. Svevo, Zeno’s Conscience, 206.

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Zeno’s paradox of the millet seeds to refer to the contradiction between the constancy of time and the impossibility of accumulation. Hamm, despairingly, takes note of the following: Moment upon moment, pattering down, like the millet grains of… (he hesitates) …that old Greek, and all life long you wait for that to mount up to a life.9

In the paradox of the millet grains, Zeno presents another challenge to the reliability of sense perception. A bushel of millet will make a sound when it falls on the ground. Yet, if a single grain of millet falls on the ground, we will not notice the sound. For Zeno, this reveals a problem with accumulation: If one grain makes no sound, how, then, does a bushel make noise? Ignoring the objection that makes recourse to thresholds of sense perception, this paradox offers an illustration of Hamm’s sense of futility – despite the expectation warranted by the logic of accumulation, adding the silence of one grain to the silence of the next grain never clearly ‘mounts up’ to a life, never offers the promise of substance or plenitude. Coetzee, too, directly engages with Zeno’s paradoxes, in his 2007 novel Diary of a Bad Year. In this novel, the narrator is a writer, who must produce a series of ‘Strong Opinions’ for an essay collection. As the novel draws to a conclusion, he replaces his initial set of opinions on politics and society with a set of far more unorthodox musings, one of which contains a reflection on Zeno’s paradox of ‘Achilles and the Tortoise’ and Isaac Newton’s failure to resolve the paradox: This is the dark possibility at the heart of the paradoxes of Zeno. […] By inventing a way of summing up the infinite number of infinitesimal steps on the way to the target and reaching a finite total, Isaac Newton believed he had overcome Zeno’s paradox. But there are depths to the paradox that go beyond Newton. What if, in the interval between the newly attained Nth step and the never yet attained – never attained in the history of the universe – (N + 1)th, the arrow were to lose its way, fall into a hole, vanish?10

9 10

Samuel Beckett, ‘Endgame’, in Endgame and Acts without Words (New York: Grove Press, 1958), 70. J.M. Coetzee, Diary of a Bad Year (Melbourne: Text Publishing, 2007), 76–77.

X: Conclusion: Literary Infinities after Zeno and Cantor

193

What follows this passage on ‘Achilles and the Tortoise’ is a reflection on Borges’s ‘Funes, The Memorious’. Borges’s story is used, here, to illustrate the fact that ‘the order we see in the universe may not reside in the universe at all, but in the paradigms of thought we bring to it’.11 Here, in a radical sense, Coetzee is dismantling the expectations of his novel to challenge the fundamental conventions of narrative: the authority of the narrative voice, the correlation between the world and our perception of it, the necessity of perspective in developing opinion. What if opinions became marked by contingency, rather than certainty? And what if logical clarity became marked by possibility, rather than probability? The overturning of the security of sense perception explored by Coetzee is fundamental to Zeno’s paradoxes, as it is fundamental to the possibility of a generic literature. A fiction that replaces predicates, locations and progressive narrative enumeration with allegory of its own composition, an allegory that measures the measure of fiction, is a generic literature, with Zeno as it’s ‘precursor’ – to use Borges’s term. The work of Borges, Beckett and Coetzee fulfils this generic literature through a composition attentive to its own mechanisms of numeration, and this allegory that ‘measures measure’ is a transfinite allegory. The ‘measuring of measure’, here, does not produce new mechanisms of narrative progress, so much as reveal the contingency of movement, and the conditions of possibility involved in forward movement. These Zenoian reflections by Svevo, Borges, Beckett and Coetzee show literature moving from the centre to X, achieving the modern journey instigated by Copernicus and perhaps secured in the mathematical domain by Cantor, who allows ‘X’, insofar as X also stands in for the lemniscate, to become an actual destination rather than a transcendental spirit.

11

Ibid., 77.

Works Cited Ackerley, C.J. Introduction to Watt, by Samuel Beckett, vii–xix. Edited by C.J. Ackerley. London: Faber and Faber, 2009. Ackerley, C.J., and S.E. Gontarski. The Grove Companion to Samuel Beckett. New York: Grove Press, 2004. Agamben, Giorgio. Homo Sacer: Sovereign Power and Bare Life. Translated by Daniel Heller-Roazen. Stanford, CA: Stanford University Press, 1998. Alazraki, Jamie. Borges and the Kabbalah: And Other Essays on His Fiction and Poetry. Cambridge: Cambridge University Press, 1998. Alexandrescu, Vlad. ‘Y a-t-il Un Critère d’Individuation Des Corps Physiques Chez Descartes’. ARCHES: Revue Internationale Des Sciences Humaines 5 (2003): 67–80. Arp, Hans, Samuel Beckett, Carl Einstein, Eugene Jolas, Thomas McGreevy, Georges Perlorson, Theo Rutra, James J. Sweeney, and Ronald Symond. ‘Poetry Is Vertical’. In Manifesto: A Century of Isms, edited by Mary Ann Caws, 529–532. Lincoln and London: University of Nebraska Press, 2001. Attwell, David. ‘Editor’s Introduction’. In Doubling the Point: Essays and Interviews, edited by J.M. Coetzee, 1–15. Cambridge: Harvard University Press, 1992. Badiou, Alain. Being and Event. Translated by Oliver Feltham. London and New York: Continuum, 2007. Badiou, Alain. ‘Infinity and Set Theory: How to Begin with the Void’. Paper presented for The European Graduate School Lectures, Saas-Fe, Switzerland, 2011. http:// www.egs.edu/faculty/alain-badiou/articles/infinity-and-set-theory/. Badiou, Alain. Number and Numbers. Translated by Robin Mackay. Cambridge: Polity Press, 2008. Badiou, Alain. On Beckett. Translated by Nina Power and Alberto Toscano. Manchester: Clinamen Press, 2003. Badiou, Alain. The Century. Translated by Alberto Toscano. Cambridge: Polity Press, 2007. Barthes, Roland. Writing Degree Zero. Translated by Annette Lavers and Colin Smith. New York: Hill and Wang, 2012. Baudrillard, Jean. Art and Artefact. Edited by Nicholas Zurburgg. London: Sage, 1997. Beckett, Samuel. ‘All Strange Away’. In Samuel Beckett The Complete Short Prose 1929–1989, edited by S.E. Gontarski, 169–182. New York: Grove Press, 1995.

Works Cited

195

Beckett, Samuel. Company, Ill Seen Ill Said, Worstward Ho, Stirrings Still. Edited by Dirk Van Hulle. London: Faber and Faber, 2009. Beckett, Samuel. ‘Endgame’. In Endgame and Acts Without Words. New York: Grove Press, 1958. Beckett, Samuel. ‘Imagination Dead Imagine’. In Samuel Beckett The Complete Short Prose 1929–1989, edited by S.E. Gontarski, 182–185. New York: Grove Press, 1995. Beckett, Samuel. Molloy. Edited by Shane Weller. London: Faber and Faber, 2009. Beckett, Samuel. Proust and Three Dialogues with Georges Duthuit. London: Calder and Boyars, 1987. Beckett, Samuel. ‘Quad’. In Collected Shorter Plays. London: Faber and Faber, 1984. Beckett, Samuel. ‘Recent Irish Poetry’. In Disjecta: Miscellaneous Writings and a Dramatic Fragment, edited by Ruby Cohn, 70–77. London: John Calder, 1983. Beckett, Samuel. The Letters of Samuel Beckett, Volume 1: 1929–1940. Edited by George Craig and Dan Gunn. Cambridge and New York: Cambridge University Press, 2009. Beckett, Samuel. ‘The Way’. Carlton Lake Collection, Harry Ransom Humanities Researrch Center, The University of Texas at Austin’ Box 17, Folder 3, 1981. Beckett, Samuel. Watt. Edited by C.J. Ackerley. London: Faber and Faber, 2009. Benacerraf, Paul. ‘What Numbers Could Not Be’. The Philosophical Review 74.1 (1965): 47–73. Benjamin, Walter. ‘The Work of Art in the Age of Its Technological Reproducability’. In The Work of Art in the Age of Its Technological Reproducibility and Other Writings on Media, edited by Michal W. Jennings, Brigid Doherty, and Thomas Y. Levin, 19–56. Cambridge and London: The Belknap Press of Harvard University Press, 2008. Berensmeyer, Ingo. ‘“Twofold Vibration”: Samuel Beckett’s Laws of Form’. Poetics Today 25.3 (2004): 465–495. Bloch, William Goldbloom. The Unimaginable Mathematics of Borges’ Library of Babel. Oxford: Oxford University Press, 2008. Bolin, John. Beckett and the Modern Novel. Cambridge and New York: Cambridge University Press, 2013. Borges, Jorge Luis. ‘A Defense of Bouvard and Pécuchet’. In Selected Non-Fictions, edited by Eliot Weinberger, 386–390. Translated by Esther Allen. New York: Penguin, 1999. Borges, Jorge Luis. ‘From Allegories to Novels’. In Selected Non-Fictions, edited by Eliot Weinberger, 337–341. Translated by Esther Allen. New York: Penguin Books, 1999. Borges, Jorge Luis. ‘Funes, His Memory’. In Collected Fictions, 131–138. Translated by Andrew Hurley. New York: Penguin Books, 1998.

196

Works Cited

Borges, Jorge Luis. ‘Joyce’s “Ulysses”’. In Selected Non-Fictions, edited by Eliot Weinberger, 12–16. Translated by Suzanne Jill Levine. New York: Penguin, 1999. Borges, Jorge Luis. ‘Kafka and His Precursors’. In Selected Non-Fictions, edited by Eliot Weinberger, 12–16. Translated by Suzanne Jill Levine. New York: Penguin, 1999. Borges, Jorge Luis. ‘The Book of Sand’. In Collected Fictions, 480–484. Translated by Andrew Hurley. New York: Penguin, 1998. Borges, Jorge Luis. ‘The Doctrine of the Cycles’. In Selected Non-Fictions, edited by Eliot Weinberger, 115–123. Translated by Esther Allen. New York: Penguin, 1999. Borges, Jorge Luis. ‘The Library of Babel’. In Collected Fictions, 112–119. Translated by Andrew Hurley. New York: Penguin Books, 1998. Borges, Jorge Luis. ‘The Lottery in Babylon’. In Collected Fictions, 101–107. Translated by Andrew Hurley. New York: Penguin, 1998. Bourbaki, Nicolas. Elements of Mathematics: Theory of Sets. Berlin and Heidelberg: Springer-Verlag, 1968. Boyer, Carl B. History of Analytic Geometry. Mineola, NY: Dover Publications, 2004. Brassier, Ray. ‘Behold the Non-Rabbit: Kant, Quine, Laurelle’. Pli 12 (2001): 50–81. Brassier, Ray. Nihil Unbound: Enlightenment and Extinction. Hampshire and New York: Palgrave MacMillan, 2007. Broberg, Gunnar. ‘The Broken Circle’. In The Quantifying Spirit in the Eighteenth Century, edited by J.L. Heilbron, Tore Frängsmyr, and Robin E. Rider, 45–73. Berkeley and Los Angeles: University of California Press, 1990. Broome, Peter, and Graham Chesters. An Anthology of Modern French Poetry 1850– 1950. Cambridge and New York: Cambridge University Press, 1976. Brulotte, Gaëtan. ‘Sade and Erotic Discourse’. Paragraph 23.1 (2000): 51–62. Burton, Robert. The Anatomy of Melancholy. New York: The New York Review of Books, 2001. Cantor, Georg. Contributions to the Founding of the Theory of Transfinite Numbers. Mineola, NY: Dover, 1955. Cascardi, Anthony. ‘Mimesis and Modernism: The Case of Jorge Luis Borges’. In Literary Philosophers: Borges, Calvino, Eco, edited by Jorge J.E. Garcia, Carolyn Korsmeyer, and Rodolphe Gasché, 109–127. New York and London: Routledge, 2002. Chaitin, Gregory. ‘Epistemology as Information Theory: From Leibniz to Ω’. The Alan Turing Lecture on Computing and Philosophy presented at E-CAP’05, European Computing and Philosophy Conference. Malardalen University, Vasteras, Sweden, 2005. Chatelet, Gilles. To Live and Think Like Pigs. Translated by Robin Mackay. New York: Sequence Press, 2014.

Works Cited

197

Chiesa, Lorenzo. ‘Count-As-One, Forming-Into-One, Unary Trait, S1’. Cosmos and History: The Journal of Natural and Social Philosophy 2.1–2 (2006): 68–93. Chomsky, Noam. Aspects of the Theory of Syntax. Boston: The Massachusetts Institute of Technology, 1965. Code, Murray. Process, Reality and the Power of Symbols: Thinking with A.N. Whitehead. Hampshire and New York: Palgrave MacMillan, 2007. Coetzee, J.M. ‘A Note on Writing’. In Doubling the Point: Essays and Interviews, edited by David Attwell, 94–96. Cambridge: Harvard University Press, 1992. Coetzee, J.M. ‘Farm Novel and “Plaasroman” in South Africa’. English in Africa 13.2 (1986): 1–19. Coetzee, J.M. In The Heart of the Country. London: Vintage, 1999. Coetzee, J.M. The Childhood of Jesus. London: Harvill Secker, 2013. Coetzee, J.M. ‘The Comedy of Point of View in “Murphy”’. In Doubling the Point: Essays and Interviews, edited by David Attwell, 31–39. Cambridge: Harvard University Press, 1992. Coetzee, J.M. ‘The English Fiction of Samuel Beckett: An Essay in Stylistic Analysis’. PhD diss, University of Texas at Austin, 1968. Coetzee, J.M. Youth. London: Vintage, 2003. Cohn, Ruby. A Beckett Canon. Ann Arbor: University of Michigan Press, 2001. Connor, Steven. ‘Hilarious Arithmetic: Annual Churchill Lective at the University of Bristol’. Steven Connor, n.d. http://stevenconnor.com/hilarious.html (accessed 30 November 2014). Connor, Steven. ‘“What’s One and One and One and One and One and One and One and One and One and One?” Literature, Number and Death’. Steven Connor, n.d. http://stevenconnor.com/oneandone.html. Copeland, Rita, and Peter Struck. ‘Introduction’. In The Cambridge Companion to Allegory, edited by Rita Copeland and Peter Struck, 1–15. Cambridge and New York: Cambridge University Press, 2010. Cordingley, Anthony. ‘Samuel Beckett’s Debt to Aristotle: Cosmology, Syllogism, Space, Time’. Samuel Beckett Today/Aujourd’hui 22 (2010): 181–195. Courant, Richard. ‘Foreword’. In Mathematics in Western Culture, edited by Morris Kline. Oxford and New York: Oxford University Press, 1953. Culik, Hugh. ‘Mathematics as Metaphor: Samuel Beckett and the Esthetics of Incompleteness’. Papers on Language and Literature 29.2 (1993): 131–151. Dauben, Joseph Warren. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 1990. De Larquier, Jeanne-Sarah. ‘Beckett’s Molloy: Inscribing Molloy in a Metalanguage Story’. French Forum 29.4 (2004): 43–55.

198

Works Cited

De Man, Paul. Allegories of Reading: Figural Language in Rousseau, Nietzsche, Rilke and Proust. New Haven and London: Yale University Press, 1979. Dennet, Daniel. Darwin’s Dangerous Idea: Evolution and the Meanings of Life. New York: Touchstone, 1995. Eliot, T.S. ‘Gerontion’. In Collected Poems 1909–1962, 29–32. Orlando: Harcourt, Brace and Company, 1991. Euclid. The Thirteen Books of the Elements. Edited by Sir Thomas L. Heath. 2nd ed. New York: Dover Publications, 1956. Fisher, Mark. ‘The “Hype” in Hyperstition’. Abstract Dynamics, 2004. http:// hyperstition.abstractdynamics.org/archives/003428.html (accessed 30 June 2014). Flint, F.S. ‘Imagisme’. Poetry: A Magazine of Verse 1.6 (1913): 198–200. Foucault, Michel. The Birth of Biopolitics: Lectures at the Collége de France 1978–1979. Edited by Michel Senellart. Translated by Graham Burchell. Hampshire and New York: Palgrave MacMillan, 2008. Foucault, Michel. The Order of Things: An Archeology of the Human Sciences. London and New York: Routledge Classics, 2002. Fraser, Graham. ‘The Pornographic Imagination in All Strange Away’. Modern Fiction Studies 41.3 (1995): 515–530. Fraser, Z.L. ‘Badiou Dictionary: Generic’. Form and Formalism, 2011. http:// formandformalism.blogspot.com.au/2011/03/generic-entry.html (accessed 30 June 2014). Gaifman, Haim. Chapter 0: The Easy Way to Gödel’s Proof and Related Topics, 2007. http://www.columbia.edu/~hg17/Inc07-chap0.pdf (accessed 10 January 2017). Gallagher, Catherine. ‘The Rise of Fictionality’. In The Novel, Volume 1, edited by Franco Moretti, 336–363. New Haven and London: Princeton University Press, 2006. Gibson, Andrew. Beckett and Badiou: The Pathos of Intermittency. Oxford and New York: Oxford University Press, 2006. Gontarski, S.E. ‘Introduction, From Unabandoned Works: Beckett’s Short Prose’. In The Complete Short Prose 1929–1989, edited by Samuel Beckett, xi–xxxi. New York: Grove Press, 1995. Goodman, Nelson, and Willard Van Orman Quine. ‘Steps Toward a Constructive Nominalism’. The Journal of Symbolic Logic 12.4 (1947): 105–122. Gordimer, Nadine. ‘The Idea of Gardening’. In The New York Review of Books. (2 February 1984), n.p. http://www.nybooks.com/articles/1984/02/02/the-idea-of -gardening/ (accessed 2 December 2016). Gray, Jeremy. Plato’s Ghost: The Modernist Transformation of Mathematics. Princeton, NJ and Woodstock: Princeton University Press, 2008.

Works Cited

199

Hallward, Peter. ‘Generic Sovereignty: The Philosophy of Alain Badiou’. Angelaki: Journal of the Theoretical Humanities 3.3 (2008): 87–111. Harris, Roy. ‘Introduction to the Bloomsbury Revelations Edition’. In Course in General Linguistics by Ferdinand de Saussure, xiv–xxiii. London and New York: Bloomsbury, 2013. Hassan, Ihab. ‘Joyce-Beckett: A Scenario in Eight Scenes and a Voice’. In Paracriticisms: Seven Speculations of the Times, 63–77. Champaign: Illini Books, 1984. Havil, Julian. The Irrationals: A Story of the Numbers You Can’t Count On. Princeton, NJ and Woodstock: Princeton University Press, 2012. Hayles, N. Katherine. Electronic Literature: New Horizons for the Literary. Notre Dame, IN: University of Notre Dame Press, 2008. Hayles, N. Katherine. How We Became Posthuman: Virtual Bodies in Cybernetics, Literature, and Informatics. Chicago and London: University of Chicago Press, 1999. Hayles, N. Katherine. ‘Subversion: Infinite Series and Transfinite Numbers in Borges Fiction’. In The Cosmic Web: Scientific Field Models and Literary Strategies in the 20th Century. Ithaca: Cornell University Press, 1984. Haynes, John, and James Knowlson. Images of Beckett. Cambridge: Cambridge University Press, 2003. Heilbron, J.L. ‘Introductory Essay’. In The Quantifying Spirit in the Eighteenth Century, edited by Tore Frängsmyr, J.L. Heilbron, and Robin E. Rider. Berkeley and Los Angeles: University of California Press, 1990. Herren, Graley. ‘Samuel Beckett’s Quad: Pacing to Byzantium’. Journal of Dramatic Theory and Criticism 15.1 (2000): 43–60. Hofstadter, Douglas R. Gödel, Escher, Bach: An Eternal Golden Braid. 20th Anniv. London and New York: Penguin Books, 2000. Jameson, Frederic. ‘Third World Literature in the Era of Multinational Capitalism’. Social Text, no. 15 (1986): 65–88. Johnston, Peter. ‘“Presences of the Infinite”: J.M. Coetzee and Mathematics’. PhD diss, Royal Holloway, University of London, 2013. Jöttkandt, Sigi. First Love: A Phenomenology of the One. Melbourne: Re.Press, 2010. Jottkandt, Sigi. ‘Love’. In Alain Badiou: Key Concepts, edited by A.J. Bartlett and Justin Clemens, 73–82. Oxford and New York: Routledge, 2010. Kanamori, Akihiro. ‘Cohen and Set Theory’. The Bulletin of Symbolic Logic 12.3 (2008): 351–378. Kannemeyer, J.C. J.M. Coetzee: A Life in Writing. Translated by Michiel Heyns. Melbourne and London: Scribe, 2012.

200

Works Cited

Kant, Immanuel. Critique of Pure Reason. Translated by Marcus Weigelt. London: Penguin Classics, 2007. Kellman, Steven G. ‘J.M. Coetzee and Samuel Beckett: The Translingual Link’. Comparative Literature Studies 33.2 (1996): 161–172. Kenner, Hugh. The Pound Era. Berkeley and Los Angeles: University of California Press, 1971. Kirkpatrick, Gwen. The Dissonant Legacy of Modernismo. Berkeley and Los Angeles: The University of California Press, 1998. Kline, Morris. Mathematical Thought from Ancient to Modern Times. Vol. 1. Oxford and New York: Oxford University Press, 1990. Knowlson, James. Damned to Fame: The Life of Samuel Beckett. Damned to Fame: The Life of Samuel Beckett. London: Bloomsbury, 1996. Krauthausen, Karin. ‘Paul Valéry and Geometry: Instrument, Writing Model, Practice’. Configurations 18 (2010): 231–249. Kristeva, Julia. Desire in Language: A Semiotic Approach to Literature. New York: Columbia University Press, 1980. Lacan, Jacques. The Seminar of Jacques Lacan: Book XI The Four Fundamental Concepts of Psychanalysis. Edited by Jacques-Alain Miller. Translated by Alan Sheridan. New York and London: W.W. Norton and Company, 1998. Lanier, Jaron. ‘Two Philosophies of Mathematical Weirdness’. American Scientist (May–June 2006): 269–271. Lautman, Albert. Mathematics, Ideas and the Physical Real. Translated by Simon B. Duffy. London and New York: Continuum, 2011. Lukács, Georg. ‘Narrate or Describe?’ In Writer and Critic and Other Essays, edited and translated by Arthur Kahn, 189–226. London: The Merlin Press, 1978. Lukács, Georg. ‘Preface’. In Writer and Critic and Other Essays, edited and translated by Arthur Kahn, 7–24. London: The Merlin Press, 1978. Macaskill, Brian. ‘The Logic of Coprophilia: Mathematics and Beckett’s “Molloy”’. SubStance 17.3 (1988): 13–21. Mackay, Robin. ‘Editorial Introduction’. In Collapse, Vol. 1, edited by Robin Mackay, 3–10. Oxford: Urbanomic, 2007. Mallarmé, Stéphane. ‘A Throw of the Dice Will Never Abolish Chance’. In The Number and the Siren: A Decipherment of Mallarmé’s Coup de Dés, edited by Quentin Meillassoux, 225–237. Translated by Robin Mackay. Falmouth: Urbanomic, 2012. Mallarmé, Stéphane. ‘Crise de Vers’. In Mallarmé: The Poet and His Circle, edited and translated by Rosemary Lloyd, 227–235. Ithaca and London: Cornell University Press, 1999.

Works Cited

201

Martinez, Guillermo. Borges and Mathematics. Translated by Andrea Labinger. West Lafayette, IN: Purdue University Press, 2012. Meillassoux, Quentin. After Finitude: An Essay on the Necessity of Contingency. Translated by Ray Brassier. London and New York: Continuum, 2008. Meillassoux, Quentin. ‘Contingence et Absolutisation de l’Un’. Paper presented at Conférence Donnée À La Sorbonne, Lors D’un Colloque Organisé Par Paris-I Sur ‘Métaphysique, Ontologie, Hénologie’, Paris, 2008. Meillassoux, Quentin. The Number and the Siren: A Decipherment of Mallarmé’s ‘Coup de Dés’. Translated by Robin Mackay. Falmouth: Urbanomic, 2012. Mellamphy, Dan. ‘Alchemical Endgame: “Checkmate” in Beckett and Eliot’. In Alchemical Traditions from Antiquity to the Avant-Garde, edited by Aaron Cheak, 548–638. Melbourne: Numen Books, 2013. Merrell, Floyd. Unthinking Thinking: Jorge Luis Borges, Mathematics and the New Physics. West Lafayette, IN: Purdue University Press, 1991. Miller, J. Hillis. Zero Plus One. Valénzia: Biblioteca Javier Coy d’estudis nordamericans, 2003. Miller, Tyrus. Late Modernism: Politics, Fiction, and the Arts Between the World Wars. Berkeley and Los Angeles: University of California Press, 1999. Miller, Tyrus, and Aleš Erjavec. ‘Editorial’. Filozofski Vestnik 35.2 (2014): 9–11. Monegal, Emir Rodiguez. Jorge Luis Borges: A Literary Biography. New York: Paragon House Publishers, 1988. Mood, John J. ‘“The Personal System” – Samuel Beckett’s Watt’. PMLA 86.2 (1971): 255–265. Moreas, Jean. ‘The Symbolist Manfesto’. In Manifesto: A Century of Isms, edited by Mary Ann Caws, 50–52. University of Nebraska Press, 2000. Moretti, Franco. ‘Conjectures on World Literature’. New Left Review 1 (2000): 54–68. Moretti, Franco. Distant Reading. New York: Verso, 2013. Murphet, Julian. ‘Beckett’s Televisual Modernism’. Critical Quarterly 51.2 (2009): 60–78. Murphet, Julian. ‘Introduction: On the Market and Uneven Development’. Affirmations: Of the Modern 1.1 (2013): n.p. Murphet, Julian. ‘The Mortification of Novelistic Discourse in Beckett’s “Trilogy”’. Paper presented at Beyond Historicism: Resituating Samuel Beckett, Sydney, Australia, 7–8 December 2012. Murphet, Julian. ‘Voice, Image, Television: Beckett’s Divided Screens’. Scan: Journal of Media Arts Culture 3.3 (2006): n.p. Nasar, Sylvia. A Beautiful Mind. New York: Simon and Schuster, 1998.

202

Works Cited

Okamuro, Minako. ‘Alchemical Dances in Beckett and Yeats’. Samuel Beckett Today/ Aujourd’hui 14 (2004): 87–104. Otis, Laura. Introduction to Literature and Science in the 19th Century, edited by Laura Otis, xvii–xxix. Oxford and New York: Oxford University Press, 2002. Perloff, Marjorie. ‘The Pursuit of Number: Yeats, Khlebnikov, and the Mathematics of Modernism’. In Poetic License, 71–99. Chicago: Northwestern University Press, 1990. Plotnitsky, Arkady. Complementarity: Anti-Epistemology After Bohr and Derrida. Durham: Duke University Press, 1994. Plotnitsky, Arkady. The Knowable and the Unknowable: Modern Science, Nonclassical Thought, and the ‘Two Cultures’. Ann Arbor: University of Michigan Press, 2002. Pound, Ezra. The Spirit of Romance. London: J.M. Dent and Sons, 1910. Quine, Willard van Ormen. ‘Universal Library’. In Quiddities: An Intermittently Philosophical Dictionary, 223–225. Cambridge: Harvard University Press, 1987. Rabaté, Jean-Michel. 1913: The Cradle of Modernism. Malden and Oxford: WileyBlackwell, 2007. Rabaté, Jean-Michel. ‘Joyce, Husserl, Derrida: Calculating the Literary Infinite’. Journal of Romance Studies 7.3 (2007): 27–45. Ramey, James. ‘Synecdoche and Literary Parasitism in Borges and Joyce’. Comparative Literature 61.2 (2009): 142–159. Rider, Robin E. ‘Measure of Ideas, Rule of Language: Mathematics and Language in the 18th Century’. In The Quantifying Spirit in the Eighteenth Century, edited by Tore Frängsmyer, J.L. Heilbron, and Robin E. Rider, 113–140. Berkeley and Los Angeles: University of California Press, 1990. Riera, Gabriel. ‘“The One Does Not Exist”: Borges and Modernity’s Predicament’. Romance Studies 24.1 (2006): 55–66. Rota, Gian-Carlo. ‘Mathematics, Philosophy and Artificial Intelligence: A Dialogue with Gian-Carlo Rota and David Sharp’. Los Alamos Science 12 (1985): 94–104. Roudiez, Leon. Introduction. In Desire in Language: A Semiotic Approach to Literature, edited by Julia Kristeva, 1–23. New York: Columbia University Press, 1980. Russell, Bertrand. Introduction to Mathematical Philosophy. London: Spokesman Books, 2008. Russell, Bertrand. ‘The Theory of Types’. In From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1932, edited by Jean Van Heijenoort, 150–183. Cambridge: Harvard University Press, 1967. Sade, Marquis de. Juliette. Translated by Austryn Wainhouse. New York: Grove Press, 1968. Salisbury, Laura. ‘What Is the Word: Beckett’s Aphasic Modernism’. Journal of Beckett Studies 17 (2008): 80–128.

Works Cited

203

Sarlo, Beatriz. Jorge Luis Borges: A Writer on the Edge. Edited by John King. 2nd ed. London and New York: Verso, 2006. Sawada, Joe, and Denis Wong. ‘A Fast Algorithm to Generate Beckett-Gray Codes’. Electronic Notes in Discrete Mathematics 29 (2007): 571–577. Scott, Clive. French Verse-Art: A Study. Cambridge and New York: Cambridge University Press, 1980. Senelart, Michell. ‘Course Context’. In Security, Territory, Population, edited by Michel Foucault, 369–403. Translated by Graham Burchill. New York: Picador, 2007. Sheehan, Paul. ‘The Disasters of “Youth”: Coetzee and Geomodernism’. Twentieth Century Literature 57.1 (2011): 20–33. Sigler, L.E. Fibonacci’s Liber Abaci/Leonardo Pisano’s Book of Calculation. New York: Springer-Verlag, 2003. Stiegler, Bernard. Technics and Time, 1: The Fault of Epimetheus. Translated by Richard Beardsworth and George Collins. Stanford, CA: Stanford University Press, 1998. Svevo, Italo. Zeno’s Conscience. New York: Vintage International, 2003. Symons, Arthur. The Symbolist Movement in Literature. New York: E.P. Dutton and Company, 1958. Szeman, Imre. ‘Who’s Afraid of National Allegory? Jameson, Literary Criticism, Globalization’. The South Atlantic Quarterly 100.3 (2001): 803–827. Tiles, Mary. The Philosophy of Set Theory: An Historical Introduction to Cantor’s Paradise. Mineola, NY: Dover, 2004. Valéry, Paul. ‘Mallarme’s “Coup de Dés”’. In Selected Writings of Paul Valéry, 217–220. Translated by Malcolm Cowley, C. Day Lewis, and Jackson Mathews. New York: New Directions Publishing Corporation, 1950. Valéry, Paul. ‘The Graveyard by the Sea’. In Selected Writings of Paul Valéry, 41–51. Translated by Malcolm Cowley, C. Day Lewis, and Jackson Mathews. New York: New Directions Publishing, 1950. Valéry, Paul. ‘Verlaine and Poincaré’. In Mathematical Lives: Protagonists of the Twentieth Century, edited by Claudio Bartocci, Renato Betti, Angelo Guerraggio, Roberto Lucchetti, Kim Williams, 25–27. Berlin and Heidelberg: Springer-Verlap, 2011. Wallace, David Foster. Everything and More: A Compact History of Infinity. 2nd ed. New York: W. W. Norton and Company, 2010. Warner, Julian. ‘Semiotics, Information Science, Documents and Computers’. Journal of Documentation 46.1 (1990): 16–32. Weil, Simone. Lectures on Philosophy. Translated by Hugh Price. Cambridge and New York: Cambridge University Press, 1978.

204

Works Cited

Weissert, Thomas P. ‘Representation and Bifurcation: Borges’ Garden of Chaos Dynamics’. In Chaos and Order, edited by N. Katherine Hayles, 223–244. Chicago: Chicago University Press, 1991. Weyl, Hermann. ‘The Mathematical Way of Thinking’. In The World of Mathematics: Volume Three, edited by James R. Newman, 1832–1856. Mineola, NY: Dover Publications, 1956. Whitehead, Alfred North. Science and the Modern World. New York: The Free Press, 1925. Wickman, Matthew. ‘Robert Burns and Big Data; Or, Pests of Quantity and Visualisation’. Modern Language Quarterly 75.1 (2014): 1–28. Williams, Heather. ‘Mallarmé and the Language of Ideas’. Nineteenth Century French Studies 29.3 (2001): 302–317. Worringer, Wilhelm. Abstraction and Empathy. Chicago: Ivan R. Dee, 1997. Wright, Edmond. ‘Jorge Luis Borges’s “Funes: The Memorious”: A Philosophical Narrative’. Partial Answers: Journal of Literature and the History of Ideas 5.1 (2007): 33–49.

Index abstraction 6, 12–13, 16, 18, 26, 49, 54, 62–3, 83–4, 127, 137, 156, 170–1 accumulation 108, 110, 113, 131, 143, 185, 192 Achilles and the Tortoise 188, 192–3 Ackerley, Chris 92, 101, 130 Actes sans Paroles I and II (Beckett) 122 actual infinities different sizes 3 Platonism 33 quantification 11–12 self-reference 36 A Defense of Bouvard and Pécuchet (Borges’s essay) 50, 84 Agamben, Giorgio 165, 173 aleph (‫ )א‬letters, 3, 31 aleph-null (‫א‬0), 169 Aleph, The (Borges’s collection of stories) 51. See also Borges, Jorge Luis Alexandrescu, Vlad 102 alexandrine (dodecasyllable) 37–9 algebraic geometry 102 All Strange Away and Imagination Dead Imagine (Beckett) 93, 120, 122, 125, 136–8 grammar and style 120, 133, 138 pornography 127, 129, 134 topological marks 131–2, 134–7 Allegories of Reading: Figural Language in Rousseau, Nietzsche, Rilke and Proust (de Man) 85 allegory 44–8, 77–8, 85,87,179,181 alphabet 70–4, 79, 97, 128, 131, 136 ambition 5, 7, 15–16, 22 analogy 12, 60, 104, 112 Anatomy of Melancholy (Burton) 71 Aristotle 4, 9, 97, 191 Arp, Hans 99 Artifices 58 Attwell, David 145

Badiou, Alain 11, 17–18, 22, 26, 34, 36, 41, 46, 141, 176–7 Baudrillard, Jean 129 Beckett and Badiou: The Pathos of Intermittency (Gibson) 18 Beckett, Samuel 1, 3–6, 11, 16, 19, 21–2, 48, 89–97, 99–117, 119–30, 135–8 Benacerraf, Paul 157–9 Benjamin, Walter 63 Berensmeyer, Ingo 89–90 Bianchi, Alfredo 57 binary function 59 Biographical History of Philosophy (Lewes) 51 Blindness and Insight (De Man) 85 Borges, Jorge Luis 2, 4–6, 19, 21, 48–55, 57–8, 62–3, 65–6, 69, 71–5, 81–7 Borges and Mathematics (Martínez) 54 Brassier, Ray 9, 170 Broome, Peter 41 Burton, Robert 71–2 calculation 62–4, 75, 103 Calvino, Italo 151 Cantor, Georg development of ‘actual infinities 4, 11–12, 21, 26–36 doubling, forms of 34–6, 45, 47–8 Gray’s association 7–8 on transfinite numbers 2–4, 11–12, 20–1, 24, 26, 29, 31–5 Cartesian rationality 101–3, 127, 136 Cascardi, Anthony 2, 53–4, 58 Castle, The (Kafka) 191 Cauchy, Augustin-Louis (1789–1857) 28 Cernuda, Luis 154 chance and determination 45, 48 Chatelet, Gilles 116 Chesters, Graham 41 Chiesa, Lorenzo 177

206 Childhood of Jesus, The (Coetzee) biblical infancy gospels 160 mathematical nominalism 160, 167, 169–71, 177 numeracy 167–8, 174, 177 Chomsky, Noam 143 circularity/circulation 96, 99–104, 108, 122 Coetzee, J.M. 2, 4–6, 19, 22, 48, 89–91, 106, 139–147, 151, 159–62, 174–9, 181–2, 185, 190 collected essays 155 education 140 Nobel Prize 165 universal pronouns 156 Zeno’s paradoxes 192 Cohn, Ruby 122–3 Connor, Steven 13, 16–17 Contributions to the Founding of the Theory of Transfinite Numbers (Cantor) 31 Copeland, Rita 46–7 count-as-one 141, 176–7 counting 1, 17, 28, 32, 38, 48, 52, 60–1, 64, 85, 90, 93, 104, 107, 110, 112, 114–16, 119, 122, 126, 130, 141, 147, 157, 167, 176, 182, 185 Critique of Pure Reason (Kant) 130–1 Croce, Benedetto 83, 85 Culik, Hugh 92 Cybernetics and Ghosts (Calvino) 151 Darío, Rubén (1867–1916) 55 Dauben, Joseph Warren 3–4 de Man, Paul 44–7, 85–7, 181 deductive method 8, 47 del Casal, Julian 55 Descartes, René (1596–1650) 101–2 Diary of a Bad Year (Coetzee) 192 Dictionary of Philosophy (Mauthner) 51 Doctrine of Cycles, The (Borges) 51 Don Quixote 167–8 Dusklands (Coetzee) 174 En Attendant Godot (Beckett) 122 Endgame (Beckett) 89, 95, 123, 191 Enlightenment 9–12, 64, 77, 125, 134–5

Index enumeration 48, 52, 94, 105, 107, 110–11, 113, 115, 145, 147, 184, 193 epistemology 3, 47, 76–8, 85, 87, 140, 157, 169, 175 equations 24, 150–2, 156, 177 eternal recurrence 51–2, 72 Eternal Return (Nietzsche) 51 European mathematicians 2 European modernism 21, 55, 139–40, 178 Fin de Partie (Beckett) 122 finitude 12, 48, 52–3, 62–3, 65–6, 68, 85, 93, 124, 152, 168 First World War 49 Fraenkel, Abraham 186 French metrical tradition 39 French Verse Art: A Study (Scott) 39 From Allegories to Novels (Borges’s article) 83 Funes, His Memory (Borges) 58–64, 66, 68, 84–5 Gaifman, Haim 30 Gallagher, Catherine 95–6 Gauss, Carl Friedrich 33 generic 42, 84, 92 geometry 26, 92, 99–102, 127, 129–30, 135, 148, 175 geomodernism 178 Gibson, Andrew 18–20, 34 God absolute infinite 31, 53 and destinies 83 ‘gateway to 78–9 will and intervention 52 Gödel, Escher, Bach (Hofstadter) 29 Gontarski, S.E 121 Good Soldier, The (Ford) 140 Gray, Jeremy 7–8, 34 Grundlagen der Geometrie (Hilbert) 7 He and His Man (Coetzee) 162 Heilbron, J.L. 10 Hilbert, David 7, 23, 27, 33 History of the War of 1914–1918 (Hart) 51 Hofstadter, Douglas 29–30 Imagisme (Pound’s essays) 57

Index Imagists 49, 57 Incompleteness Theorems (Gödel) In the Heart of the Country (Coetzee) 141, 145–60 infinite in Ancient Greek thought 24 auratic conception 19 irrational number 31 in literary modernism 8, 18, 45 logos alogos 35 Mallarmé on 36, 42 Meillassoux on 36, 43 in Romanticism 20 sizes 32 symbolic attribution 13 whole numbers 28 Jameson, Frederic 43, 47 Jolas, Eugene 99 Joyce’s Ulysses (Borges) 1, 50 Kafka and His Precursors (Borges’s essay) 191 Kant, Immanuel 93, 120, 130–2, 134–5 Kaun, Axel 5, 91, 144 Kellman, Steven G. 178 Kenner, Hugh 41 Khlebnikov, Velimir (1885–1922) 18 Kline, Morris 31 La Nacion 58 labyrinths 69–70, 74, 76, 78, 80–1 Lacan, Jacques 154 late modernism 4 Lautman, Albert 118, 135 Le Symbolisme (Moréas) 25 Letter to Axel Kaun (Beckett) 91 letters M C V 74 Library of Babel, The (Borges) 69–70, 73–4, 76–9, 82, 84, 87, 183 Golden Book 78, 84, 87, 183 Life of Samuel Johnson (Boswell) 51 literary ‘transfinite 36, 45–7, 181, 183, 186 literary modernism 4–5, 8, 57 literary theory 13, 48, 143 literary transfinites 1–2, 4–6,16,18–21, 23–4, 44, 46–8,54,160, 181–3, 186 Locke, John 12

207

Lukács, Georg 111–13, 116–17, 120–1, 124, 126, 132 Macaskill, Brian 92 Mackay, Robin 101 Mallarmé, Stéphane actual infinity 21, 26, 34 allegorical form of infinity 43, 45–6 coded poetry 24, 36, 39–42 Malone Dies (Beckett) 89, 94 Marti, Jose 55 Martinez, Guillermo 54 Marx, Karl 10 mathematical modernism 7–8, 54, 179 mathematical paradoxes 187 Mathematics and the Imagination (Kasner and Newman) 51 Meillassoux, Quentin 21, 30–1, 36–40, 42–6 Mellamphy, Dan 22, 95 Merrell, Floyd 54 metaphor 2, 5, 8, 12, 14–15, 19, 23, 40, 43, 49 Miller, J. Hillis 13–15 Mind of Man, The (Spiller) 51 Mirón, Salvador Díaz 55 modernism Anglophone 19 classical 178 European 19, 55, 57, 139–40, 178 Geo- 179 literary 4–5 mathematical 7–9, 16, 21, 33–4, 47, 54, 179 music 8 Modernismo 4, 55–6 Molloy (Beckett) 1, 22, 89–90, 92, 94–104, 113, 125, 136–7, 155, 184 Monegal, Emir Rodiguez 69 Mood, John J. 106, 108 Moréas, Jean 25 Moretti, Franco 13, 15–16 morphological metaphor 93, 182–3 multiple totalities 54 Nájera, Manuel Gutiérrez 55 Narrate or Describe (Lukács’s essay) 111

208

Index

narratives 9–10, 21, 44, 46–7 natural language 5, 12, 87, 104, 128, 178, 182 naturalism 21, 49, 77, 96, 105, 111–13, 116–17, 119, 121–2, 126, 138, 184 Nietzsche, Friedrich 51–2, 154, 186 Nine Billion Names of God (Asimov) 74 Nosotros (literary magazine) 57 notation system 45, 74 numbers 1–4, 10–11, 13, 16–17, 28–33, 37, 39–40, 45 and pattern 1 cardinal 3, 31–2 Connor’s work 16–17 irrationals 6, 28–9, 31, 35, 65–7, 69, 173 irreconcilable concepts 13 Moretti’s work 15–16 natural 3, 29–30, 32–3, 48 paradoxes 6, 27, 36, 182, 186–93 philosophical extension 13 quotidian 6, 13, 17, 55, 150, 166, 176 real 3, 27–30, 35, 48 numerical identity 13, 17, 28, 33, 102, 152, 156–7, 159, 171, 173, 176, 182 objectivity 6, 121, 135, 137–8 On Computable Numbers (Turing) 30 ontology 76–7, 81, 87, 145 Parnassians 55–6 Passage de Verlaine (Valéry) 24, 26 Perloff, Marjorie 18–19 permutation 22, 53, 70, 73, 75, 78, 105, 107, 110–11, 113–14, 116–17, 119, 125, 184 plaasroman genre 145–6 Plato’s Ghost: The Modernist Transformation of Mathematics (Gray) 7 Platonism 33–4 Poetry is Vertical (Arp and McGreevy) 99 Poincaré, Henri (1854–1912) 21, 24–6, 174–5 Pound, Ezra 57 primordial metaphor 5, 49, 56 Principia Mathematica (Russel) 51

Prisma: Revista Mural (journal) 56 Proust, Marcel 127–8 quantification 10–11, 16, 22, 48, 65, 141, 171 Quantifying Spirit in the Eighteenth Century, The (Heilbron) 10 Quine, Willard 169 Ramey, James 50 realism 50–1, 111–13, 117, 126 representation 1–2, 12, 15, 20–2, 25, 40, 47 retentional finitude (Bernard Stiegler) 62–3, 65 Riera, Gabriel 64 romanticism 16, 18–20, 33, 57–8, 183 Rousseau, Jean- Jacques 154 Russell, Bertrand 51, 186–8 Sagado, Cesar Augusto 50 Sarlo, Beatriz 57 Schooldays of Jesus, The (Coetzee) 141, 173–4, 176 Scott, Clive 39 semi-colonial modernism 4 set theory 2, 20, 27, 30, 32, 99, 118–19, 127, 143, 157, 176, 186–8 Sheehan, Paul 178 Silva, José Asunción 55 Stein, Gertrude 1, 151 Stiegler, Bernard 62, 64–5 Strange Attractors (Coetzee) 143 Struck, Paul 46–7 stylostatistics 144–5 Svevo, Italo (pen name of Schmitz, Ettore) 188, 191, 193 syllogisms 174–5 symbolism 21, 25, 39, 41, 44, 47, 49 Symons, Arthur 41 Szeman, Imre 47 Tiles, Mary 30 transfinite allegory 4, 20–2, 45, 48, 51–3, 83–4, 179, 181–4, 186, 193 transfinite numbers 2–4, 11–12, 20–1, 24, 26, 29, 31–5, 51, 53, 65–6, 182 Ultraism 49, 55–8

Index Un Coup de Dés Jamais N’Abolira le Hasard (Mallarmé) 21, 31, 36–45 uncountability 28–30, 36, 66, 68–9, 184 Unimaginable Mathematics of the Library of Babel, The (Bloch) 53 unique number 38–40, 42–6, 82, 173, 175 Unnamable, The (Beckett) 89, 94 Valéry, Paul (1871–1945) 4, 21, 24–6, 36, 159 vers libre 37 Vitruvian Man (Da Vinci) 125, 134 Waiting for Godot (Beckett) 89 Wallace, David Foster 33 Watt (Beckett), 105–20 Way, The (Beckett) 104 Weierstrass, Karl (1815–1897) 28 Weyl, Hermann 137 What Numbers Could Not Be (Benaceraff ) 157 Whitehead, Alfred North 18, 51, 76–8, 186

209

Work of Art in the Age of its Technological Reproducibility, The (Benjamin) 63 Wright, Edmond 68 Writer and Critic (Lukács) 120 Writings From The Late Notebooks (Nietzsche) 186 Yeats, W.B. (1865–1939) 18 Youth: Scenes from a Provincial Life II (Coetzee) 142–3 Zarathustra (Nietzsche) 51 Zeno’s Conscience (Svevo) 189, 191 Zeno of Elea 188–91 Zeno’s paradoxes 28, 188–93 Zermelo-Fraenkel Axiomatic Set Theory 186 zero 6, 13–14, 29, 91, 94, 148–52, 156, 159, 169, 176, 185 Zero Plus One (Miller) 13–14