Mallarmé and Mathematics: on Nothing and the Infinite

Mallarmé and Mathematics: On Nothing and the Infinite

Christian R. Gelder

The Centre for Modernism Studies in Australia The School of Arts and Media Faculty of Arts and Social Sciences

November 2016

A thesis in fulfilment of the requirements for the degree of Master of Arts by Research

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This thesis examines the poetry of Stéphane Mallarmé in relation to mathematics. I argue that this relationship has been a key feature of Mallarmé’s philosophical reception throughout the twentieth century. I show how Julia Kristeva, Jacques Derrida, Alain Badiou, Quentin Meillassoux and other figures position Mallarmé’s poetics against a single conception of mathematics: namely, the paradigm shifting work of the nineteenth-century mathematician Georg Cantor. For these philosophers, I claim, Mallarmé is “Cantor’s unconscious contemporary”. The rest of the thesis explores this relationship by reading Mallarmé’s and Cantor’s projects alongside one another. By looking at two fundamental concepts at work within both the poetry and the mathematics – the nothing and the infinite – I argue that Mallarmé’s poetry cannot solely be understood in terms of Cantor’s work. Instead, it recalls past and future conceptions of mathematics, making it a contemporary of various periods before and after his own time. I conclude by reflecting on the nature of contemporaneity, using this concept as a way of broadly understanding the points of similarity and dissimilarity that exist between poetry and mathematics.

4 Acknowledgments

First and foremost, my most sincere thanks go to my primary supervisor Sigi Jöttkandt, without whom there is no way this thesis ever would have reached completion. Her support, advice, occasionally cutting remarks and, dare I say it, friendship sustained my desire to write and to think, even when that desire came close to dissipating into pure nothingness. Thank you Sigi. Secondly, I must extend this gratitude to Sean Pryor, my co-supervisor. Nobody at any level of the University was able to tell me exactly what the role of a co- supervisor is, and yet I am sure that however one chooses to define it, Sean happily exceeded whatever was required of him.

Thirdly, thanks goes to my comrade Robert Boncardo, who discussed and read over every minor point of this thesis in great detail. Many late nights were spent scrutinising the most intricate details of all things Mallarmé, and his friendship is imprinted on this thesis. That same thanks goes also to Baylee Brits, Laura Lotti, Tanya Thaweeskulchai and Elizabeth King, who each were kind enough to read over sections of my work and lend me their extensive, but always overly generous, feedback.

Finally, I must thank my family: Mum, Dad and “the small one” Julian. Their proximity to my life shifted over the past two years, but their unconditional – some might even say naive! – support was always closest when I needed it most.

5 Table of Contents

Abstract

Acknowledgments

A Note on Translation

Introduction: Mallarmé the Mathematician?

Chapter 1: The Nothing that Is and the Nothing that is Not

Chapter 2: The Mystery in Numbers: On the Infinite, Chance and God

Conclusion: What is an Unconscious Contemporary?

Bibliography

6 A Note on Translation

All of Mallarmé’s poems appear in their original language; all his prose writing in English, with the original French quoted in the footnotes; all existing translations of academic and philosophical criticism are cited, with occasional modification; all other translations are my own.

Hear the voice of the Bard! Who Present, Past, & Future sees; Whose ears have heard The Holy Word That walk’d among the ancient trees

Introduction to Songs of Experience, William Blake

But poets should Exert a double vision; should have eyes To see near things as comprehensively As if afar they took their point of sight, And distant things as intimately deep As if they touched them.

Aurora Leigh, Emily Barrett Browning

8 Introduction: Mallarmé the Mathematician?

This thesis examines the poetry of Stéphane Mallarmé in relation to mathematics. Coupling these two topics together may seem strange. However, as I show in this chapter, there has been a consistent merging of the two throughout Mallarmé’s philosophical reception in the twentieth century. In fact, his poetry has often been read alongside or in relation to mathematics, and a very particular kind of mathematics at that. In this introduction, I examine the readings of Mallarmé produced by Julia Kristeva, Jacques Derrida, Alain Badiou and Quentin Meillassoux in order to demonstrate how he has been compared and contrasted to a temporally contemporaneous idea of mathematics known as set theory – with one single exception, discussed at the end of this chapter.

In the remaining two chapters, I attempt to take this relationship between Mallarmé and set theory seriously, scrutinising the lines of similarity and disjunction between the two. I read Mallarmé’s own remarks about mathematics, and also provide reinterpretations of his most famous poems, in order to see how it is possible to discuss and theorise his poetry in light of the mathematically oriented reception of his work. Even though, as I argue in this chapter, Mallarmé’s poetry and set theory both work with the same concepts, I ask whether Mallarmé’s poetry is best aligned to this contemporary idea of mathematics. I argue that when examining a potential homology between the two, the idea of mathematics must be at once ancient and modern, both from the past and from the future.

The central arguments of this thesis are thus threefold:

1) I demonstrate not only that the question of mathematics has been a key feature of Mallarmé’s philosophical reception, but also that there is a general consensus as to what kind of mathematics he is tied to. For the philosophers, I argue, Mallarmé is

9 “Cantor’s unconscious contemporary”, the latter being the founder of set theory.1

2) I suggest that this relationship of contemporaneity can be challenged through a reexamination of the poetry itself. I work through the two central concepts that Mallarmé and set theory share – the nothing and the infinite – to suggest that his poetry cannot solely be placed alongside any singular moment from the history of mathematics. Instead, I argue that he evokes and summons various temporal instances of mathematical history into his poetry, histories from his distant past and immediate future.

3) I then propose a different meaning of the word contemporaneity, specifically as it pertains to poetry. When comparing Mallarmé’s poetry to mathematics, I argue that the logic of contemporaneity at play implies similarity and difference, resemblance and dissimulation. I conclude by suggesting that this logic can be reflexively brought to bear upon the topics of poetry and mathematics themselves, providing a speculative point of separation between the two.

To begin, I will turn to the curious and fascinating history of Mallarmé’s historical reception in order to show how the question of mathematics has silently taken centre stage and comprises one of the central concerns orienting the poet’s place within twentieth- century philosophy.

The literary historian Thierry Roger notes that the first Belgian and French reviews of Mallarmé’s 1897 masterpiece Un Coup de dés were indeed mostly concerned with the quasi-mathematical nature of the poem’s images. Un Coup de dés itself stages a shipwreck, where a master who had once captained the sunken ship emerges from the wreckage clutching two dice in his hands. I will provide a longer reading of this poem in my third chapter, but for now it is important to note that it is a long meditation on the master’s

1 Alain Badiou, Briefings on Existence: A Short Treatise on Transitory Ontology, trans. Norman Madarasz (New York: SUNY, 2006), 124.

10 hesitation to throw the dice and on the unique number it could produce. Nothing takes place in the poem – it is not seen whether he actually does manage to throw them – but from the catastrophic circumstances of the shipwreck and the abyss of the sea there arises a constellation of stars – although Mallarmé qualifies that this constellations only perhaps appears. The poem, which continually references number, the infinite, chance and the throw of the dice, then closes with a maxim: “Toute Pensée émet Un Coup de Dés”.2 For these early critics, the most peculiar thing about Mallarmé’s peculiar poem was the way it invoked and played with some vaguely expressed idea of mathematics. Contrary to the later linguistic and political reception of his work, the first critics to view Un Coup de dés wondered why Mallarmé encoded this poem – one that was to pronounce upon the “destiny of a future poetry”3 – in a dice throw and a unique number. Roger writes, “given its title, the first considerations on the Coup de dés were of a philosophico-mathematical order, and not poetic”.4 These first considerations, then, were less concerned with Mallarmé’s use of natural language and more with some sort of homology between his poetry and the formal language of the so-called “mathematical order”.

The observation that Mallarmé’s work shared certain qualities with mathematics – superficial or otherwise – also did not escape Paul Valéry. Valéry was one of Mallarmé’s greatest disciples, yet he also developed a fascination with numbers when the engineer Pierre Féline introduced him to the latest developments in mathematics in 1889. Judith Robinson-Valéry notes how Valéry would sprawl copies of papers and formulations by the late nineteenth-century German mathematician Georg Cantor, founder of a radical vision of the foundations of mathematics called set theory, beside his bed, perhaps next to his copies of Mallarmé’s poetry.5 In an open letter written after Mallarmé’s death, Valéry took it upon himself to classify definitively a poet many thought unclassifiable, by situating him in the framework of an exact science. He writes:

2 Stéphane Mallarmé, Œuvres complètes I, Édition présentée, établie et annotée par Bertrand Marchal (Paris: Gallimard, 1998), 387. 3 Quentin Meillassoux, The Number and the Siren: A Decipherment of Mallarmé’s Un Coup de dés, trans. Robin Mackay (Falmouth/New York: Urbanomic/Sequence, 2012), 7. 4 “Les premières considérations sur le Coup de dés, au vu de la phrase-titre, seraient d’ordre philosophico- mathématique, et non pas poétique.” Thierry Roger, L’Archive du Coup de dés (Paris: Éditions Classiques Garnier, 2010), 53-54. 5 Judith Robinson-Valéry, “The Fascination of Science,” in Reading Paul Valéry: Universe in Mind, eds. Paul Gifford and Brian Stimpson (Cambridge: Cambridge University Press, 1998), 74.

One day I told him that he belonged to the family of the great scientists. I do not know whether the compliment was to his taste, for he did not have a conception of science that rendered it comparable to poetry […] but for my part, I could not fail to draw what seemed to me an inevitable parallel between the construction of an exact science and, on the other hand, Mallarmé’s evident design of reconstructing the whole system of poetry with the help of pure and distinct notions isolated by the delicacy and soundness of his judgement and disengaged from the confusion usually created, in minds that reason about literature, by the multiplicity of functions served by language.6

It is worth pausing to consider the way Valéry positions Mallarmé as a figure who ruptures with the confusion usually created by hermeneutic polysemy. Mallarmé does not engage in the kind of signifying excess his poetry is famed for here. Rather, Valéry suggests that Mallarmé’s words, his “pure and distinct notions”, seem closer in kind to the formal language of the exact sciences than natural language. This is an image of Mallarmé the scientist – or perhaps even Mallarmé the mathematician.

The philosophers soon engaged with exactly this topic. Jean Hyppolite, one of the foundational figures in mid twentieth-century French philosophical thought, first identified this by suggesting that readers of Un Coup de dés should look for points of comparison with modern mathematicians.7 Hyppolite’s essay does not focus on the type of mathematics we see emerge in later readings because he links Mallarmé instead to theories of entropy, cybernetics and information theory; yet his remark uncannily anticipates what came thereafter. In this chapter, I shall examine four philosophical meetings with Mallarmé: Julia Kristeva’s “Towards a Semiology of Paragrams”, Jacques Derrida’s “The Double Session”, Alain Badiou’s reading of the poet in Being and Event (2005) and Quentin Meillassoux’s

6 Paul Valéry, “Letter to Mallarmé,” in Paul Valéry, Leonardo, Poe, Mallarmé, trans. Malcolm Cowley and James R. Lawler (London: Routledge, 1972), 243, 251. 7 Jean Hyppolite, “Le Coup de dés de Stéphane Mallarmé et le message,” Les Études philosophiques, no.4 (1958): 466.

12 recent The Number and the Siren (2011).8 Each philosopher is preoccupied with a very different set of problems, and bringing them into conversation with one another may at first seem problematic. My purpose, however, is not to claim that they are all doing the same thing when they read Mallarmé. Rather, it is to show that they are united in how they figure his poetry against the same conception of mathematics. For these philosophers, I argue, Mallarmé must be understood as “Cantor’s unconscious contemporary”, a description I will open up for discussion throughout the rest of this chapter.9

The early stages of Julia Kristeva’s career were marked by a fascination with Mallarmé. From the early Sèméiôtiké (1969) to Revolution in Poetic Language (1984), she frequently drew on the poet to establish, exemplify and further her philosophical and political propositions. However, it is in “Towards a Semiology of Paragrams” – an essay that appeared in the journal Tel Quel in 1967 and was revised and republished as a chapter of Sèméiôtiké – that Kristeva first theorises a potential homology between Mallarmé’s poetry and mathematics. The essay is essentially a defense of “[l]iterary semiology” and structuralism from accusations of “staticism” and “non-historicism”. She writes that the point of semiological analysis is to discover “a formalism that corresponds isomorphically

8 One glaring exception from this list is Gilles Deleuze, who did occasionally, in a more roundabout manner than the above philosophers, link Mallarmé’s poetry to mathematics. I have decided to leave Deleuze out of this lineage because the idea of mathematics he invokes is different to the one that we shall see is at work throughout multiple other philosophical interpretations. In The Fold: Leibniz and the Baroque (1993), Deleuze reconstructs Leibniz’s differential metaphysics of the fold through, at least in part, his infinitesimal calculus and calculus of the infinite series, as well as other developments in mathematics contemporary to Leibniz. He also, very briefly I should add, links Leibniz’s fold to Mallarmé: “The fold is probably Mallarmé’s most important notion, and not only the notion, but, rather, the operation, the operative act that makes him a great Baroque poet”. I am not going to present the arguments for or judge the merits of Deleuze’s claim here, but I will mention that his articulation of the problem of Mallarmé and mathematics sits well outside the lineage I am attempting to trace. Moreover, where Mallarmé is indeed a central figure for Kristeva, Derrida, Badiou and Meillassoux, for Deleuze he is treated in a more occasional manner, perhaps because Deleuze was interested in what he thought was the superiority of Anglo-American literature over French poetry. Gilles Deleuze, The Fold: Leibniz and the Baroque, trans. Tom Conley (London/New York: Continuum, 2006), 34. For an excellent assessment of Deleuze’s way of reading mathematics and Mallarmé in The Fold, see Arkady Plotnitsky, “Algebras, Geometries and Topologies of the Fold: Deleuze, Derrida and Quasi-Mathematical Thinking (with Leibniz and Mallarmé),” in Between Deleuze and Derrida, eds. Paul Patton and John Protevi (London/New York: Continuum, 2003), 98-119. 8 This is Bennington’s argument in Geoffrey Bennington, “Derrida’s Mallarmé,” in Meetings with Mallarmé in Contemporary French Culture, ed. Michael Temple (Exeter: Exeter University Press, 1998), 126-142. 9 Badiou, Briefings on Existence, 124.

13 to literary productivity’s thinking itself”.10 In other words, Kristeva wants to think the deep structure of literary language, a task that “only two methodologies” can provide an adequate framework for: “mathematics and meta-mathematics”, and Noam Chomsky’s generative grammar.11 The latter is dismissed because its “philosophical foundation [is] derived from a scientific imperialism that grants generative grammar the right to establish rules for the construction of new linguistic [...] variations”.12 But the former is lauded since it is not subject to the same kind of scientific imperialism. Rather, mathematics and meta- mathematics are “artificial” languages, “more able to elude the constraints of a logic based on the Indo-European subject-predicate relation, and [...] as a consequence are better adapted to describing the poetic operation of language”.13

The ordinary use of natural language – the commercial use, to invoke Mallarmé’s phrase14 – is not what is at stake in Kristeva’s essay. She focuses instead on poetry because it is one of the few inherently subversive discourses: poetry, Kristeva argues, challenges and disrupts structures discerned in ordinary language. The inherently subversive capacity of the poem is often overlooked by linguistics, she thinks, and when poetry is spoken of, it is brushed aside as “ornamental”, “superfluous” or “anomalous”.15 Kristeva’s wager is to reject this model, to subvert what the linguist takes as normal, by placing poetry at the centre of her study. This leads her to formulate a “‘paragrammatic’ concept of poetic language”, one that “suggests three major theses”:

A Poetic language is the only infinity of code. B The literary text is double: writing-reading. C The literary text is a network of connections.16

10 Julia Kristeva, “Towards a Semiology of Paragrams,” in The Tel Quel Reader, eds. Patrick ffrench and Roland-François Lack, trans. Roland-François Lack (New York: Routledge, 1998), 25. 11 Kristeva, “Towards a Semiology of Paragrams,” 25. 12 Kristeva, “Towards a Semiology of Paragrams,” 25. 13 Kristeva, “Towards a Semiology of Paragrams,” 25. 14 Stéphane Mallarmé, Divagations, trans. Barbara Johnson (Cambridge/Massachusetts/London: The Belknap Press of Harvard University Press, 2007), 211. 15 Kristeva, “Towards a Semiology of Paragrams,” 26. 16 Kristeva, “Towards a Semiology of Paragrams,” 26.

14 I am going to outline briefly the arguments involved in the first two theses because at the end of section B Kristeva perfectly summarises the way in which she binds Mallarmé and mathematics together. For Kristeva, every use of language must be designated by the term “code” but there is a tendency, she thinks, to treat poetic language as a “sub-code of the total code”.17 However, to think of language hierarchically, even if poetry is placed at the top of that hierarchy, is to misunderstand the nature of language. She writes, “for us poetic language is not a code encompassing all others, but a class A which has the same power as the function ! (x1 … xn) of the infinity of the linguistic code”.18 Poetry thus designates one class of language. Yet it is a unique class: the words of a poem do not follow the strict relation of signifier to signified, nor are they submitted to the regular rules of grammar such as the subject-predicate relation. To read a poem is rather to experience “the exploration of the possibilities of language” and these possibilities are designated and formalised under an “ordered infinite code”.19 This is precisely what Kristeva means when she writes that “poetic language is the only infinity of code”.20 If the laws that are usually brought to bear upon other uses of language do not suffice when treating poetry in its singularity, then the only model of formalisation capable of capturing how the poem subverts the normal rules of language is mathematics – a discourse that not only escapes the “subject-predicate” relation, but also provides a concept of infinity capable of capturing poetry’s deep structure.

Kristeva opens her second section by invoking the language of set theory, a discipline of mathematics created by Georg Cantor in the late nineteenth century. I shall spend some time over the next two chapters explicating the central ideas of set theory, but for now it is worth mentioning that Cantor’s work sought to provide a universal backdrop for modern mathematics by abstracting all the particular properties of the discipline and transposing them into sets: for example, the numbers 1, 2 and 3 can be transposed into the set A {ø, 1, 2, 3}. In the set theoretical universe, everything is treated as a set, and thus subject to the same rules of manipulation. With this in mind, Kristeva writes that “the literary text inserts itself into the set of all texts”.21 Each literary text belongs to the completed infinity named,

17 Kristeva, “Towards a Semiology of Paragrams,” 27. 18 Kristeva, “Towards a Semiology of Paragrams,” 28. 19 Kristeva, “Towards a Semiology of Paragrams,” 28. 20 Kristeva, “Towards a Semiology of Paragrams,” 26. 21 Kristeva, “Towards a Semiology of Paragrams,” 29.

15 in the abstract, literature; it interacts with other members of that set while also occupying a uniquely singular place. Using the language of set theory metaphorically here, the second facet of Kristeva’s paragrammatic approach attempts to account for this “dialogue” between the original text and that of its history, between the text’s place in the set and the other members of that same set.22 What one finds in her essay, then, is a loosely modeled mathematical theory of intertextuality where a text occupies a structural position in a set determined by its relation with the other members. To further this point, she invokes a line from a tribute Mallarmé wrote for his friend and fellow writer, Villiers de l’Isle Adam:

Already Mallarmé knew that to write was “to claim in accordance with doubt – the drop of ink married to the sublime night – some duty to recreate it all, with reminiscence to prove that one is really there where one should be …” “To write” was for him “a summons to the world that it should equal its fear with rich figured postulates, as a law, on the pallid paper of such audacity…”23

Kristeva deliberately plays on the French word for “summons” – sommations – which means to summon something into the world, but also signifies the mathematical property of addition. She writes, “the book refers to other books and by the modes of summation (of application, in mathematical terms) gives those books a new way of being, elaborating thereby its own signification”.24 From this, she derives the fact that poetry does not follow a binary logic of 0-1, of the true and the false. Instead, it allows for a transgression into the realm of the 2: “without the 1 there would be no paragram based on the 2. The prohibition (the 1) constitutes meaning, but at the very moment of constitution meaning is transgressed in an oppositional dyad”.25 While she shifts in this passage from the language of set theory to that of logic, her point is that formalising literary language in logical or mathematical terms provides a way of thinking about how it escapes or transgresses the models of understanding propagated by traditional linguistics. As a model by which one can formalise the deep structure of poetic uses of language, “the paragrammatic sequence is a set of at

22 Kristeva, “Towards a Semiology of Paragrams,” 29. 23 Kristeva, “Towards a Semiology of Paragrams,” 30. 24 Kristeva, “Towards a Semiology of Paragrams,” 30. 25 Kristeva, “Towards a Semiology of Paragrams,” 31.

16 least two elements. The modes of conjunction of its sequences (Mallarmé’s summation) and the rules that govern the paragrammatic network can be expressed by set theory, by the operations and theorems that are derived from or relate to them”.26 Coupling Mallarmé’s poetry with the “operations and theorems” of set theory allows Kristeva to understand poetic language’s “total code”. Mallarmé and set theory are analogous here, but only insofar as together they describe a deep structure of language that otherwise eludes traditional linguistics.

Of all the readings I examine in this introduction, Kristeva’s was subject to the most criticism. At the time of its republication in Sèméiôtiké, the poet and mathematician Jacques Roubaud and his colleague Pierre Lusson published a twofold reply to the piece in the journal Action poétique – articles that have barely been discussed in either English or French. Their responses focus almost exclusively on Kristeva’s metaphorical use and misunderstanding of the mathematics she invokes, making ornamental-like mentions of Mallarmé where necessary. Their conclusions are devastating: “In the actual state of things, it would be without doubt preferable that paragrammatism abandon its reckless and unfounded references to the approaches and results of different scientific disciplines (linguistics, mathematics, and psychoanalysis [...]) and resolutely commit itself to its natural path: that of magical incantation”.27 Another equally critical appraisal of “Towards a Semiology of Paragrams” can be found in the physicists Alan Sokal and Jean Bricmont’s Fashionable Nonsense (1998). They dedicate an entire chapter to Kristeva’s essay, occasionally linking its claims with various remarks made elsewhere in her writings.28 One

26 Kristeva, “Towards a Semiology of Paragrams,” 31. 27 “Dans l'état actuel des choses, il serait sans doute préférable que le paragrammatisme abandonne ses références imprudentes et non fondées à la démarche et aux résultats des différentes disciplines scientifiques (linguistique, mathématique et psychanalyse [...]) et s'engage résolument dans la voie qui lui est naturelle: celle de l'incantation magique”. Jacques Roubaud and Pierre Lusson, “Sur la Sémiologie des paragrammes de Julia Kristeva,” Action Poétique 45 (1970): 36. 28 While their chapter on Kristeva is one of the few persuasive sections of the book, I do not want to give the reader the impression that I support the polemical backdrop underpinning Sokal and Bricmont’s project. In fact, I wholeheartedly agree with Arkady Plotnitsky’s pointed and necessarily cutting assessment of their work. Plotnitsky writes, “The criticism by these particular authors is disabled by (a) their lack of the minimal necessary familiarity with the specific subject matter, arguments, idiom, and context of many of the works they criticise; (b) their inattentiveness to the historical circumstances of using mathematical and scientific ideas in these works; (c) their lack of general philosophical acumen, which is necessary for understanding most of the works in question, whether one sees them positively or critically; and (d) their insufficient expertise in the history and philosophy of mathematics and science or even certain areas of mathematics and

17 of their objections concerns how Kristeva thinks poetic language cannot be subsumed into the binary logic of 0-1, the false and the true. Given that her main focus is on the link between poetic writing and set-theoretical formalisation, Sokal and Bricmont argue that she confuses the specific use of the 0-1 binary in logic with the set {0,1} and the interval [0-1]. They write:

Kristeva makes one correct assertion and two mistakes. The correct assertion is that poetic sentences cannot, in general, be evaluated as true or false. Now, in mathematical logic, the symbols 0 and 1 are used to denote “false” and “true”, respectively; it is in this sense that Boole’s logic uses the set {0,1}. Kristeva’s allusion to mathematical logic is thus correct, though it adds nothing to the initial observation. But [...] she seems to confuse the set {0,1}, which is composed of the two elements 0 and 1, with the interval [0,1], which contains all the real numbers between 0 and 1. The latter set, unlike the former, is an infinite set.29

As we have seen, Kristeva’s main concern lies in thinking a homology between poetry and set theory, using the latter to metaphorically describe the structure of literary writing as such. However, her argument does not strictly comply with the mathematics she invokes, leading her to mistake the set {0,1} – a facet of set theory that belongs to the paragrammatic network – with the interval [0-1] and the logical denotation of 0-1, the true and the false.30

Roubaud, Lusson, Sokal and Bricmont focus almost exclusively on Kristeva’s use of mathematics and each wonders why she forces a relationship between mathematics and poetry that may not necessarily be there in the first place. However, it should be noted that science themselves”. Arkady Plotnitsky, The Knowable and the Unknowable: Modern Science, Nonclassical Thought, and the “Two Cultures” (Ann Arbor: The University of Michigan Press, 2002), 112-113. 29 Alan Sokal and Jean Bricmont, Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science (New York: Picador, 1998), 40. 30 Kristeva did in fact reply to Sokal and Bricmont’s scathing assessment of her work. Interestingly, she does not defend her use of mathematics, suggesting instead that they misunderstand something about what it is to read literature: “I am not truly good at mathematics, this goes without saying. But when you told me that I have done a violence to literature, the verdict becomes fatal. It is not easy to do violence to literature (Je ne suis pas une vraie matheuse, cela va de soi. Mais quand on me dit que j’ai fait violence à la littérature, le verdict devient fatal. Ce n’est pas facile de faire violence à la littérature)”. Julia Kristeva, “Une désinformation,” Le Nouvel Observateur, no. 1716 (25th September to 1st October 1997): 122.

18 Kristeva also does not substantially examine the poetry she claims to be defending. When she reads Mallarmé’s use of the word “summons”, for example, she treats it in the same manner one would any other sentence: it is examined at the level of the signified, rather than as an instance of the “possibilities of language”. Aside from a few other ornamental references to Mallarmé and Lautréamont throughout the essay, Kristeva never really examines the poetry itself, relying instead on propositional theses that are neither derived from the poetry nor correctly discerned in the mathematics. Her analysis rests on the mere assertion of a structural analogy between the two. Even so, Kristeva’s essay does show one of the ways Mallarmé’s poetry has been philosophically aligned to mathematics, in this case set theory. She is, however, a long way from answering those early critics who queried why Mallarmé’s Un Coup de dés housed its message in a unique number.

One line of thinking that unites Kristeva with many of the other writers involved in Tel Quel is the belief that mathematics is inherently transgressive. In a 1969 interview with Jacques Derrida, Kristeva asks whether “logical-mathematical notation” might provide a challenge to the “expressive” and logocentric nature of ordinary signification.31 While the wording of Kristeva’s question implies an anticipation of what he will say – “would not”, she asks, “grammatology be a nonexpressive ‘semiology’ based on logical-mathematical notation rather than on linguistic notation” – Derrida instead urges us to be cautious.32 On the one hand, he thinks “the effective progress of mathematical notation [...] goes along with the deconstruction of metaphysics”; but on the other, he says, “we must also be wary of the ‘naive’ side of formalism and mathematism, one of whose secondary functions in metaphysics [...] has been to complete and confirm the logocentric theology which they otherwise could contest”.33 Anne Brubaker has recently argued that Derrida’s view of mathematics “lends a certain agentless agency to numbers”, as they are inscribed in a pre- linguistic nature that exists “before us”.34 But in a response to Brubaker, Karin Lesnik- Oberstein notes that Derrida’s treatment of mathematics is more ambiguous. She writes that

31 Jacques Derrida and Julia Kristeva, “Semiology and Grammatology: Interview with Julia Kristeva,” in Jacques Derrida, Positions, trans. Alan Bass (Chicago: The University of Chicago Press, 1981), 32-33. 32 Derrida and Kristeva, “Semiology and Grammatology,” 32-33. 33 Derrida and Kristeva, “Semiology and Grammatology,” 35. 34 Anne Brubaker, “Between Metaphysics and Method: Mathematics and the Two Canons of Theory,” New Literary History 39, no. 4 (2008): 878.

19 Derrida “does not establish it [mathematics] as such or adopt it as the solution to phonologism or logocentrism”.35 One way of clarifying how Derrida understands mathematics lies in his treatment of Mallarmé. Although I do not want to make any general claim about the relationship between deconstruction and mathematics, I do want to show how Derrida’s essay on Mallarmé’s prose-poem “Mimique”, “The Double Session”, reprises the homology between the poet and set theoretic mathematics. Derrida in fact extends properties that pertain to mathematical language to the general level of writing itself, and to Mallarmé in particular.

Before it was published in Tel Quel, “The Double Session” was given as a lecture in 1969 to the Group d’Etudes théoriques, a group Kristeva helped found one year earlier. Geoffrey Bennington writes that the aim of Derrida’s essay is to “interrogate the relation between literature and truth, and to show how literature escapes (or can escape) the hold of the ontological question in a way which can nonetheless provide something like a ‘formalisation’ of the question of truth”.36 The model of truth at stake for Derrida is taken from Plato’s work on mim!sis. “Mim!sis”, he instructs, “produces a thing’s double. If the double is faithful and perfectly alike, no qualitative difference separates it from the model”.37 This doubling is also inflected with a moral dimension: “obviously, according to ‘logic’ itself, according to a profound synonymy, what is imitated is more real, more essential, more true, etc., than what imitates”.38 In Derrida’s account of the Platonic schema, that which is imitated is true and that which imitates is not. Mim!sis thus entertains a profound symmetry with metaphysical ideas of truth per se and reflection upon mim!sis is done by way of truth itself: “It is in the name of truth, its only reference – reference itself – that mim!sis is judged, proscribed or prescribed according to a regular alternation”.39 Derrida does not pinpoint or identify a moment at which the logic of mim!sis ceases to function. Displacing mim!sis, he writes, “has not taken place once, as an event”. Nor does it

35 Karin Lesnik-Oberstein, “Reading Derrida on Mathematics,” Angelaki: Journal of the Theoretical Humanities 17, no. 1 (2012): 38. 36 Bennington, “Derrida’s Mallarmé,” 126. 37 Jacques Derrida, “The Double Session,” in Dissemination, trans. Barbara Johnson (Chicago: The University of Chicago Press, 1981), 187. 38 Derrida, “The Double Session,” 191. 39 Derrida, “The Double Session,” 193.

20 “take place in writing”. Rather, “this dis-location (is what) writes/is written”.40 To illustrate this, Derrida provides a formalisation of what he calls the “re-mark” through a stunning reading of Mallarmé’s “Mimique”, a piece that appears in his prose collection, Divagations (1897).

“Mimique” is a difficult work to classify because it is both a meta-reflection on the nature of literature and, more ordinarily, a review of a play about a stock-standard French literary character type, Pierrot the mime. In the stories and tales that preceded the play Mallarmé reviewed, Pierrot was generally portrayed as a comical figure. But in this particular restaging Pierrot is imbued with a sinister dimension, miming a performance where he tickles his wife Columbine to death. “Mimique”, as Theophile Gautier poetically remarks, is “the story of Pierrot who tickled his wife and thus made her laughingly give up her life”.41 Despite a series of coincidences that adds to Derrida’s reading – such as Mallarmé perhaps never seeing the original play and his review being first published in a book that appeared only in its second edition – he suggests that the narrative of the piece internally subverts the axioms of Plato’s mim!sis. We can see this clearly even in the brief outline of “Mimique” presented above. For example, what stable reality does the Pierrot mime copy if the event that takes place in the play – the murder of his wife – occurs during the mimed, imitated performance? The answer, for Derrida, is that Mallarmé’s piece semantically rehearses a kind of complicated undoing of Plato’s schema; it stages a reality that cannot be decided by the logic of mim!sis.

However, the semantic content of “Mimique” is not what is most important for Derrida. Instead, he is interested in the way it refers to the processes or operations of reference, particularly with respect to Mallarmé’s following sentence: “The stage illustrates but the idea, not any actual action, in a Hymen (of which flows Dream), tainted with vice yet sacred, between desire and fulfilment, perpetration and remembrance; here anticipating, there recalling, in a future, in a past, under the false appearance of a present”.42 The

40 Derrida, “The Double Session,” 193. 41 Quoted in Derrida, “The Double Session,” 203. 42 Mallarmé, Divagations, 140. “‘La scène n’illustre que l’idée, pas une action effective, dans un hymen (d’où procède le Rêve), vicieux mais sacré, entre le désir et l’accomplissement, la perpétration et son souvenir: ici devançant, là remémorant, au futur, au passé, sous une apparence fausse de présent”. Stéphane Mallarmé,

21 semantic content of the word “Hymen” already suggests what Derrida wants to theorise: “Hymen” refers to the undecidable point between the mime and the mimed, the inside and the outside, “between desire and fulfilment”. But the way the word is positioned within the very structure of “Mimique” means that it also serves as a point of indication, one that is not simply marked within the space of the poem, as a normal unit of language, but instantly remarked in order to refer to itself as a mark. For him, the semantic content of the “Hymen” does indeed hold multiple significations, but it also reflexively organises and distributes the semantic meaning. In other words, the “Hymen” points to the gap in meaning between the other signifiers – the between of the between, the gap of differentiated semantic units – which is the very condition of signification. In this sense, the referent of the “Hymen” is lifted since it does not serve any purpose other than to point to the undecidable space between the mime and the mimed; and yet, the “reference remains”.43 Derrida writes that “Mallarmé thus preserves the differential structure of mimicry or mim!sis but without its Platonic or metaphysical interpretation, which implies the being of something that is, is being imitated”.44 The “Hymen” therefore cannot be decided – that is, placed within – the Platonic logic of mim!sis. Derrida then goes on to connect this to the post-Cantorian mathematician Kurt Gödel’s concept of undecidability, which points to yet another connection between Mallarmé and mathematics.

First established in his 1931 paper, “On Formally Undecidable Propositions in Principia Mathematica and Related Systems”, Gödel formulates two logical proofs that concern the dominant and foundational systems of pure mathematics: Bertrand Russell and A.N. Whitehead’s Principia Mathematica (1910) and Ernst Zermelo and Abraham Fraenkel’s axiomatic reformulation of set theory – the latter being a series of minimal givens or “axioms” that underpin and define the basic operations of set theory. These two mathematical systems are chosen, Gödel argues, because they “are so far developed that you can formalise in them all proof methods that are currently in use in mathematics, i.e.

Œuvres complètes II, Édition présentée, établie et annotée par Bertrand Marchal (Paris: Gallimard, 1998), 178-179. 43 Derrida also suggests that this is the definition of fiction at play within this passage. Derrida, “The Double Session,” 211. 44 Derrida, “The Double Session,” 206.

22 you can reduce these proof methods to a few axioms and deduction rules”.45 For Gödel, the first proof relating to undecidability concerns any logico-mathematical system great enough to encompass arithmetic, and arithmetical operations. This system, he states, will produce one proposition or sentence within the system’s rules that paradoxically cannot be decided as true or false with respect to those rules themselves. Gödel’s second proof concerns the internal consistency of such a logico-mathematical system. For him, any system said to be internally consistent must also be incomplete, insofar as it will produce an undecidable proposition.

Derrida’s interest lies here: he is concerned with how Gödel proved that any axiomatic mathematical program can only be thought of as consistent if it is incomplete – and that, moreover, the system will produce one undecidable sentence that is “neither true nor false with respect to those axioms”.46 Derrida locates the word “Hymen” in Mallarmé’s “Mimique” as functioning in an analogous fashion to the undecidable sentence produced by the post-set theoretical mathematical systems of Russell and Whitehead and Zermelo and Fraenkel. This is because the “Hymen” refers both semantically to the inside and outside of the text (the internal workings of “Mimique” and the history of the Pierrot mime), as well as to the formal process that “composes and decomposes it”.47 In other words, if one takes the axioms of the Platonic system as the dominant model of truth – a model that clearly distinguishes between original and copy, truth and falsity, signifier and signified – then the “Hymen” is undecidable with respect to these axioms because it reflexively refers to the very model, the transcendental conditions, that it itself is a part of, while retaining its status as a piece of writing – that is, as a product of the very system it points to. The “Hymen” is thus both signifier and signified, model and copy, exposing the Platonic model that conditions its production, but it is also not signifier or signified, model or copy: it is undecidable with respect to this logic, and yet it is written inside it.

45 Kurt Gödel, “On Formally Undecidable Propositions of Principia Mathematica and Related Systems,” trans. B. Meltzer, 173. http://www.csee.wvu.edu/~xinl/library/papers/math/Godel.pdf. 46 Derrida, “The Double Session,” 219. 47 Derrida, “The Double Session,” 220.

23 Paul Livingston has rightly suggested that this analogy between mathematical and literary undecidability is predicated on three points. Firstly, Derrida takes natural language to “trade decisively on the capacity of a total system of signs, directed to the establishment of truth”.48 This means that one of Derrida’s suppositions concerns the fact that natural language as such is governed by the Platonic system; it is but one instance of the metaphysics of presence whereby language as a total system can be divided into signifier and signified, model and copy, truth and falsity. While there are no doubt other systems – for example, this model may not govern mathematics – undecidability will always be produced so long as there is a totalising system deciding upon its own rules and regulations. Secondly, and as a consequence of this, undecidability is a general principle: it applies to any mathematical system and, by Derrida’s philosophical extension, any natural language writing. Thirdly, the analogy applies insofar as both Derrida and Gödel attend to the “semantical effect of syntax”.49 Derrida certainly does not understand syntax in its usual sense, as a way of looking at the structure and order of a given language. Instead, he uses the term as a way of showing how the “Hymen” points to the gap between semantic units, to their possibility of distinction and the way they can be used in an ordered linguistic system. This space between semantic units is a “void” created by syntactic undecidability, but a void that “signifies”. The undecidable space remarked by the “Hymen” is thus “neither purely syntactic nor purely semantic”.50 This is to say that while Mallarmé’s “Hymen” cannot be wholly accounted for by the recognisable systems of meaning – its status is undecidable – it does reflexively refer to the other meaningful words themselves. Given this, the “Hymen” can then simultaneously take on semantic value, leaving the gap between syntax and semantics itself undecidable.

For Derrida, then, Mallarmé’s poetry is in a relationship with mathematics. But this relationship between mathematical and literary undecidability is not constructed in terms of an exact symmetry. Rather, it is one of “absolute extension”.51 Since both formal and natural languages are comprised of a total system of signs whose principle is generalisable,

48 Paul Livingston, “Derrida and Formal Logic: Formalising the Undecidable,” Derrida Today 3, no. 2 (2010): 225. 49 Livingston, “Derrida and Formal Logic,” 227. 50 Derrida, “The Double Session,” 222. 51 Derrida, “The Double Session,” 223.

24 Mallarmé’s poetry thus demonstrates how Gödel’s incompleteness theorem can be extended from formal language to the level of writing itself. Derrida concludes:

When this undecidability is marked and re-marked in writing, it has a greater power of formalisation, even if it is “literary” in appearance, or appears to be attributable to a natural language, than when it occurs as a proposition in logicomathematical form [...] If one supposes that the distinction, still a metaphysical one, between natural language and artificial language be rigorous [...] one can say that there are texts in so-called natural languages whose power of formalisation would be superior to that attributed to certain apparently formal notation.52

If Mallarmé is a mathematician, then it is only insofar as his poetry formalises mathematical propositions in a “superior” fashion to mathematics itself.53

Despite this forceful reading, Derrida does not provide a definitive answer to the question of Mallarmé’s relationship to mathematics. Instead, he points to the undecidability inherent to all language – be it natural or formal – and “Mimique” only serves as a “strategic” example used to illustrate a larger, more general phenomenon.54 Noting this, Jacques Rancière argues that Mallarmé’s work does not subvert the logics of mim!sis at all. Rather, it relies upon it:

Jacques Derrida not long ago lauded Mallarmé’s subversion of the Platonic system of the idea-model copy. But two things have to be distinguished. Mallarmé dismissed the art of representation of the idea-model, but maintained a mimetic

52 Derrida, “The Double Session,” 222. 53 Arkady Plotnitsky makes a related point in his excellent “Algebras, Geometries and Topologies of the Fold: Deleuze, Derrida and Quasi-Mathematical Thinking (with Leibniz and Mallarmé)”, noting that Derrida – and Deleuze – philosophically generalise various mathematical concepts. He writes, “[...] particular mathematical concepts can and have been converted into philosophical ones, just as certain philosophical ideas or arguments can be and have been converted into mathematics. The quasi-mathematical is defined by and defines this reciprocity, which thus also gives rise to both Deleuze's and Derrida's quasi-mathematics. The quasi-mathematical allows us to gain an understanding of a certain conceptuality that, while not mathematical, is irreducible in mathematics and perhaps makes it possible”. Plotnitsky goes on to read this quasi-mathematical stratum in both Deleuze and Derrida via Mallarmé, examining concepts of algebra, geometry and topology. Plotnitsky, “Algebras, Geometries and Topologies of the Fold: Deleuze, Derrida and Quasi-Mathematical Thinking (with Leibniz and Mallarmé),” 98. 54 This is Bennington’s argument in Bennington, “Derrida’s Mallarmé,” 126-142.

25 status for the poem: the poem imitates no model, but traces perceptibly the movement of the Idea, the idea as movement of its own breaking forth. That the idea is only fiction does not prevent there from being a first copy of its movement.55

In Rancière’s reading, Mallarmé draws upon an essential mim!sis that, while fictional, is still modeled on other forms of fictional, artistic practice, such as theatre and music. This, for Rancière, is the mimetic Mallarméan movement of the Idea. Instead of subverting mim!sis, he claims, “in Mallarmé, mimesis is at work on a much broader terrain than that which is circumscribed in the little text commented on by Derrida”.56 Derrida’s reading, then, once again ties Mallarmé’s poetry to mathematics, but also indicates that this relationship still requires further examination.

Alain Badiou is another philosopher who has written extensively on Mallarmé and, like Kristeva and Derrida, he also brings the poetry into conversation with set theory. In Badiou’s mature philosophical system, Mallarmé and mathematics are given a precise place. But what is unique about Badiou’s reading is that the relationship he establishes between the two is not one of analogy, extension or similarity. Rather, he frames the question in terms of disjunction and irreducible distance. In his grandiose philosophical treatise Being and Event, mathematics provides nothing less than the discourse on ontology. This claim has not been well understood, but it is perhaps Badiou’s most radical intervention into the history of philosophy.57 For Badiou, being is not mathematical or represented numerically as it is, for example, in Pythagorean thought. Instead, that mathematics is ontology “is not a thesis about the world but about discourse”.58 Badiou

55 Jacques Rancière, Mallarmé: The Politics of the Siren, trans. Steven Corcoran (London/New York: Continuum, 2011), 52. 56 Jacques Rancière, “A Singular Invention in Language and Thought,” in Mallarmé: Milner, Rancière, Badiou, eds. Robert Boncardo and Christian R. Gelder (London/New York: Rowman and Littlefield, forthcoming March 2017). 57 See, for example, the debate that occurred in Critical Inquiry between David and Ricardo Nirenberg and Badiou, A.J. Bartlett and Justin Clemens. The Nirenbergs suggest that Badiou’s articulation of mathematics as ontology belies a secret Pythagoreanism, whereby being is numerical. Badiou’s reply is blunt: “what a disappointment!” See, Ricardo L. Nirenberg and David Nirenberg, “Badiou’s Number: A Critique of Mathematics as Ontology,” Critical Inquiry 37, no. 4 (2011): 583-614. For Badiou’s reply and Bartlett and Clemens’ extended response, see Critical Inquiry 38, no. 2 – where the Nirenbergs also published another article criticising Bartlett and Clemens, taking exception to the tone directed towards them. 58 Alain Badiou, Being and Event, trans. Oliver Feltham (London/New York: Continuum, 2005), 8.

26 axiomatically decides that being is pure multiplicity – that being is not one – and then axiomatically posits again that mathematics is the discourse best equipped to speak about this multiplicity. Moreover, because mathematics is a discourse less constrained by the traps of interpretation that haunt natural language – because mathematics, as Justin Clemens puts it, is “the paradigm of deductive rationality”59 – it gives the philosopher a language to describe the being of being; a language subtracted from the excesses of signification. Perhaps unsurprisingly, his mathematics of choice is set theory, which he brings together under what Oliver Feltham calls the “Cantor event”.60 Each axiom and pivotal moment of progress that occurred throughout set theory’s development over the course of the nineteenth and twentieth centuries are transliterated by the philosopher into ontological statements. Set theory speaks of pure multiplicity – it only ever deals with multiples, and multiples of multiples – and thus conditions philosophy: it is a singular truth in its own right, but it also has intra-philosophical ramifications to which the philosopher must submit.

However, if it is to exist, philosophy cannot subject itself to just one condition. In fact, the history of philosophy for Badiou has been marked by a series of mis-recognitions whereby philosophers mistakenly give themselves over to one mode of thought at the cost of ignoring others. Perhaps Badiou’s most daring claim is that there are, and always have been, four conditions for philosophy – art, science (i.e. mathematics), love and politics – each of which is to be philosophically re-thought in its singularity. These conditions have the capacity to produce truths in their own right as they are unable to be reduced to any established or recognised forms of knowledge. As such, philosophy must rethink their unprecedented interventions. In terms of the scientific condition, mathematics alone thinks ontology. However, there comes a point where mathematics runs up against an internal limit it cannot pass beyond. When it is used to measure the quantitative difference between two transfinite cardinals – two completed infinite sets – the thinking of quantity and size in set theory breaks down. I shall return to this in my third chapter, but it is worth stating here that what Badiou admires about mathematics is that it integrally “‘knows’ absolutely what

59 Justin Clemens, “Had We But Worlds Enough, and Time, this Absolute, Philosopher…” Cosmos and History: The Journal of Natural and Social Philosophy 2, no. 1-2 (2006): 285. 60 Oliver Feltham, “Translator’s Preface,” in Badiou, Being and Event, xviii.

27 it is talking about” – and it does not talk about “what-is-not-being-qua-being”.61 Echoing the Platonic gesture where Plato “submit[ted] language to the power of poetic speech” once he had reached the limits of discursive thought, Badiou then replaces Plato’s Sun with Mallarmé’s Stars: so poetry returns at the moment mathematics reaches its limit.62

I need to elaborate on this point. In Badiou’s ontology, sets – groupings of numbers such as {Ø, 1, 2, 3} – comprise consistent multiplicities. Because set theory begins on the basis of the void – the subject of my next chapter – it proves that “the one is not”.63 And yet there is oneness and unity in the world. Oneness arises from the fact that mathematics gives an internally consistent account of presentation and for something to be presented it must be “count[ed]-as-one”; that is, unified as a set.64 Inconsistency is therefore prior to consistency, but consistency alone determines what is presented. Nevertheless, there exists a multiple, the “evental site”, that “falls under the count-as-one of presentation [...] but is not separately counted-as-one”.65 The evental site is thus “an entirely abnormal multiple; that is, a multiple such that none of its elements are presented in the situation”.66 The concept of the evental site has a kind of paradoxical relation to the normal set theoretical multiples that are presented. On the one hand, the evental site belongs to a normal multiple and can be presented therein. On the other hand, the elements of the site cannot themselves be presented. The evental site subsequently lies “on the edge of the void”: it is an “element presented by a situation, none of whose elements are in turn presented”.67

The evental site is one of the most interesting concepts in Badiou’s work since it is not synonymous with the event itself, the latter being his concept for the undecidable supplement to a situation (any consistent multiple) that creates the spark of a truth in the world. Rather, the evental site localises the point where an event can take place. If mathematics is a discourse predicated on consistency, then the event disrupts what is

61 Badiou, Being and Event, 8, 173. 62 Alain Badiou, Handbook of Inaesthetics, trans. Alberto Toscano (Stanford: Stanford University Press, 2005), 20. 63 Badiou, Being and Event, 52. 64 Badiou, Being and Event, 52. 65 Badiou, Being and Event, 174. 66 Badiou, Being and Event, 175. 67 Badiou, Being and Event, 175. A.J. Bartlett, “Site,” in The Badiou Dictionary, ed. Steven Corcoran (Edinburgh: Edinburgh University Press, 2015), 315.

28 considered consistent through a moment of sheer inconsistency. But the event is nonetheless “attached, in its very definition, to the place, to the point” of its site.68 Whether the event belongs to the situation is ultimately undecidable from the standpoint of the situation itself – and it is not an ontological category because it cannot be decided, established or constructed through set theoretical mathematics. In order to establish the event’s existence, a poetic naming is required. As Badiou writes, “to name a supplement, a chance, something incalculable, it is necessary to draw from the void of sense, in the absence of established significations, and to the perils of language. One must therefore poeticise, and the poetic name of the event is that which throws us outside of ourselves”.69 Badiou thus discerns a way of formalising what is involved in deciding on the undecidable through poetry, not mathematics.

In Being and Event, Mallarmé is given two roles. On the one hand, he is an example of a post-evental truth procedure, one that produced “the truth of French poetry after Hugo”.70 On the other hand, the truth procedure he was engaged in involved a poetic thinking of the event: “Un Coup de dés [is] the greatest theoretical text that exists on the conditions for thinking the event”.71 In “Meditation 19” of Being and Event, Badiou gives a “a very beautiful reading” of Un Coup de dés, “perhaps the best that has ever been made”, as Jean- François Lyotard unashamedly pronounces.72 Badiou focuses on the emergence of the constellation of stars that, “peut-être”, appear at the close of the poem. He reads this image as the model for the moment of decision that formalises whether or not an event has taken place and argues that “the constellation is subtractively equivalent, ‘on some vacant superior surface’, to any being which what-happens shows itself to be capable of, and fixes for us the task of interpreting it”.73 Since Mallarmé qualifies the emergence of the constellation with a perhaps – “RIEN [...] N’AURA EU LIEU [...] QUE LE LIEU [...] EXCEPTÉ [...] PEUT-ÊTRE [...] UNE CONSTELLATION” – Badiou suggests that the

68 Badiou, Being and Event, 179. 69 Alain Badiou, Conditions, trans. Steven Corcoran (London/New York: Continuum, 2008), 42. 70 Badiou, Being and Event, 404. 71 Alain Badiou, “Is it Exact That All Thought Emits A Throw of Dice?” trans. Robert Boncardo and Christian R. Gelder, Hyperion IX, no. 3, (2014): 74. 72 “C'est là une très belle lecture, peut-être la meilleure qu'on ait jamais faite du Coup de dés”. Philippe Lacoue-Labarthe, Jacques Rancière, Jean-François Lyotard, and Alain Badiou, “Liminaire sur l’ouvrage d’Alain Badiou ‘L'Être et l’événement,’” Le Cahier (Collège international de philosophie), no. 8 (1989): 234. 73 Badiou, Being and Event, 197.

29 undecidability of the constellation’s existence forces a decision to name the event poetically, something mathematics is unable to do.74

This would appear, then, to provide yet another account of the relationship between Mallarmé’s poetry and mathematics. Each condition treats a distinct topic, which means that their relationship to one another is always one of disjunction. Mathematics thinks ontology, whereas Mallarmé thinks the event; set theory is the paradigm of consistency, whereas Un Coup de dés accounts for the insistence of inconsistency. But the two are also made to interlock with one another, particularly around the concept of the evental site. Recall that, to use A.J. Bartlett’s words, whereas mathematics is a discourse concerned with consistency, the event is what “exposes its situation to the perils of inconsistency”.75 To find its support, the event is attached to the evental site, which lies “on the edge of the void”. The concept of the evental site is first discerned through set theoretical means but it also figures in the poetry. As Badiou writes,

In Un Coup de dés…, the metaphor of all evental-sites being on the edge of the void is edified on the basis of a deserted horizon and a stormy sea [...] The term with which Mallarmé always designates a multiple presented in the vicinity of unpresentation is the Abyss, which, in Un Coup de dés…, is “calm”, “blanched”, and refuses in advance any departure from itself, the “wing” of its very foam “fallen back from an incapacity to take flight”.76

The image of the Abyss that haunts the surroundings of the Master’s shipwreck metaphorically corresponds to the concept of the evental site since nothing can be differentiated within it. Like a completely abnormal multiple, nothing that is within the Abyss can be counted by the situation in which the Abyss is itself counted. This image is thus a poetic transposition of the evental site. So while set theory and Mallarmé are said to be irreducibly distinct in Badiou’s schema, they nonetheless interact with one another through this, and indeed other, concepts.

74 Mallarmé, Œuvres complètes I, 384-387. 75 A.J. Bartlett, Badiou and Plato: An Education by Truths (Edinburgh: Edinburgh University Press, 2011), 71. 76 Badiou, Being and Event, 192.

However, Badiou’s analysis may not completely answer the question of the relation between Mallarmé and mathematics. His doctrine of conditions does not imply that the philosopher must faithfully adhere to what is originally proposed in the mathematics or the poetry. Instead, philosophy’s job is to re-think the thought of the four truth conditions. As he states, “I think that literary events are indeed operative for philosophy, but when philosophy puts them as conditions for its own development, it nonetheless proceeds through operations of selection, change, or transformation. In my eyes, these operations are not exactly falsifications, but they are, after all, displacements”.77 In other words, the artistic condition, which in this case signifies “literary events”, does not imply that the philosopher must faithfully reconstruct the artistic work in its singularity. The relationship between the philosopher and the literary text operates through “selection, change, or transformation”, where the text is repurposed so that its potential philosophical ramifications can be revealed.

Quentin Meillassoux, who studied under Badiou, has in fact argued that there is nothing intrinsic to Mallarmé’s poem that suggests the undecidable must be decided upon. For Meillassoux, the final perhaps qualifying the emergence of the constellation in Un Coup de dés instead celebrates the undecidable in and for itself:

Badiou makes this PERHAPS the expression of a “promise”: nothing will have taken place except perhaps – in the future – a constellation […] But nothing of this kind [...] is expressly indicated in the Poem: the PERHAPS is neither realised nor invalidated – it is on the contrary hypostasised, celebrated for itself, erupting in the Heavens as an intrinsic property of the constellation.78

Badiou’s interest in Mallarmé’s poetry arises from its status as a truth – and so necessarily its intra-philosophical value is repurposed in order to arrive at his concept of the event.

77 Alain Badiou, Peter Hallward and Bruno Bosteels, “Can Change be Thought?” in Badiou and Politics (Durham/London: Duke University Press, 2011), 312-313. 78 Quentin Meillassoux, “Badiou and Mallarmé: The Event and the Perhaps,” trans. Alyosha Edlebi, Parrhesia, no. 16 (2013): 38.

31 However, as Meillassoux suggests, if one attends exclusively to the poetry itself, then a radically different answer may, perhaps, emerge.

Like Badiou, Meillassoux also negotiates a passage between Mallarmé’s poetry and set theory – although in the English speaking reception of his work, most commentators have focused almost solely on his relationship to science and mathematics.79 Meillassoux’s philosophical aim, his “one idea” as Badiou puts it, is to think the world without relying upon a subject to perceive it.80 This classical philosophical project sees him enact a kind of return to the real, to the “great outdoors”, in an attempt to cure contemporary philosophy from what he thinks is its current sickness: namely, correlationism.81 Correlationism refers to an unconscious axiom circulating at the heart of a number of diverse philosophies, from Heideggerian hermeneutics to analytic language philosophy. This axiom is predicated on an unbreakable bond between thinking and being, which Meillassoux claims has been operative since Kant. Correlationism suggests that a world independent from a thinking subject is impossible and that thought is therefore also impossible without a world to ground it. In other words, thought and being must be melded into one, such that an object only has purchase philosophically if it is entwined with a subject, and vice versa. Meillassoux’s philosophy systematically attempts to undo this correlationist supposition, hoping instead to prove that the world without us, the world in-itself, not only exists but also is rationally thinkable.

In his book, After Finitude (2008), mathematics plays two separate roles. The first concerns nothing less than the ability of mathematics to discern the properties of an object in-itself. Meillassoux begins by calling for a revival of the classical Lockean and Cartesian distinction between primary and secondary qualities. The secondary qualities of an object

79 Frank Ruda cautions that Meillassoux’s philosophy worryingly places us “in the age of scientists”. Others have made similar arguments to Ruda. Markus Gabriel, for example, accuses Meillassoux of instilling in science the “magical power of getting it right”. See, Frank Ruda, “The Speculative Family, or: Critique of the Critical Critique of Critique,” Filozofski vestnik XXXIII, no. 2 (2012): 69 and Markus Gabriel, “The Mythological Being of Reflection – An Essay on Hegel, Schelling, and the Contingency of Necessity,” in Markus Gabriel and Slavoj "i#ek, Mythology, Madness, and Laughter: Subjectivity in German Idealism (London/New York: Continuum, 2009), 86. 80 Alain Badiou, “Preface,” in Quentin Meillassoux, After Finitude: An Essay on the Necessity of Contingency, trans. Ray Brassier (London/New York: Continuum, 2008), vi. 81 Meillassoux, After Finitude, 7.

32 are those determined by human interaction. To use Meillassoux’s own example, the object “fire” can be said to possess the property of burning heat only when a subject can perceive the relation between the temperature of the burning fire and the way in which that temperature affects them.82 The secondary properties of fire are thus predicated on a bond between subject and object, conditioned by their reciprocal relation. The object’s primary qualities, by contrast, refer to the properties of the object as it exists in-itself; that is, without a subject. Meillassoux makes the striking claim that mathematics determines the primary qualities of an object. “We maintain”, he writes, “that the mathematisable properties of the object are exempt from the constraint of [correlationism], and that they are effectively in the object in the way in which I conceive them, whether I am in relation with this object or not”.83 This argument is not developed in After Finitude, but Meillassoux has noted that his entire philosophy rests on the ability of mathematics to describe, as it were, the real.84

The second role mathematics plays in After Finitude concerns Meillassoux’s radical articulation of contingency, something I shall explicate here. To challenge the dominance of correlationism, Meillassoux first draws on a series of scientific facts that speak about the world before the possibility of either thinking or being arose: for example, “the date of the origin of the universe”.85 These statements do not trouble scientific discourse, but the modern position – that is, the position “to be is to be a correlate” – does.86 In fact, he avers that correlationist philosophy cannot meaningfully understand this type of scientific discourse since it refers to an ancestral world prior to the correlation between thinking and being. To take up this challenge, to think the ancestral realm, “we must grasp how thought

82 Meillassoux, After Finitude, 2. 83 Meillassoux, After Finitude, 3. 84 Meillassoux has stated that he intends to “demonstrate – note here that this is still not done in After Finitude – that what is mathematisable is absolute”. Quentin Meillassoux, “Interview with Quentin Meillassoux,” in New Materialism: Interviews & Cartographies, eds. Rick Dolphijn and Iris van der Tuin, trans. Marie-Pier Boucher (Ann Arbor: Open Humanities Press, 2012), 79. The first attempt to establish this can be found in Quentin Meillassoux, “Iteration, Reiteration, Repetition: A Speculative Analysis of the Sign Devoid of Meaning,” in Genealogies of Speculation: Materialism and Subjectivity Since Structuralism, eds. Armen Avanessian and Suhail Malik, trans. Robin Mackay and Mortiz Gansen (London/New York: Bloomsbury, 2016), 117-198. In this essay, he begins to formulate a concept of the “sign devoid of meaning”. This sign, Meillassoux claims, is the condition for mathematical writing as such and does not participate in the meaning- laden language orienting our access to the world. 85 Meillassoux, After Finitude, 9. 86 Meillassoux, After Finitude, 28.

33 is able to access an absolute”.87 However, modern philosophy has entirely abandoned this quest, opting instead to absolutise the reality of the correlation or to view other claims to absoluteness, such as religious or poetic speculation, as foreign to the object of reason. But Meillassoux argues that reviving this pursuit is required in order to return to the real. “For if I cannot think anything that is absolute”, he states, “I cannot make sense of ancestrality”.88

To think the absolute is to think something necessary, a property of the world in-itself that must be. Meillassoux rejects the classical definitions of the absolute, such as Descartes’ God, and replaces them with a thoroughly materialist conception. Through a complicated philosophical argument, he posits that the only necessary facet of the world is precisely the opposite of necessity: namely, contingency. Contingency is “the eternal property of what is”, designating “the possibility whereby something can either persist or perish, without either option contravening the invariants that govern the world. Thus, contingency is an instance of knowledge; the knowledge I have of the actual perishability of a determinate thing”.89 Meillassoux’s idea of contingency does not signify the arbitrary nature of the world and his argument does not imply that the world will always be in a constant state of flux. Rather, contingency concerns the meta-laws governing our universe, which could, he thinks, be otherwise: they could “persist or perish”. Meillassoux furthers this by making a philosophical distinction between chance and contingency. Chance in its classical formulation – and not, as we shall see, its Mallarméan guise – signifies the random sequence of events that could or could not take place under the necessity of the correlation.90 In contrast, contingency is a far more radical concept because it names the fact that the correlation itself could be otherwise, that the laws upholding our universe need not necessarily be the way they are. Meta-laws, such as the laws of physics or the correlation, no doubt exist, but they are themselves subject to a higher law still: that of contingency.

87 Meillassoux, After Finitude, 28. 88 Meillassoux, After Finitude, 28. 89 Meillassoux, After Finitude, 52, 53-4. 90 Meillassoux, After Finitude, 96-101. The chance/contingency distinction was first established in Quentin Meillassoux, “Potentiality and Virtuality,” in The Speculative Turn: Continental Materialism and Realism, eds. Levi Bryant, Nick Srnicek and Graham Harman, trans. Robin Mackay (Melbourne: Re.Press, 2011), 224- 236. However, I would argue that this distinction, once crucial to Meillassoux’s work, has been all but abandoned since he began writing on Mallarmé.

Meillassoux continues to employ set theory to confirm the existence of the necessity of contingency. To do this, he cites and critiques the “frequentialist school” of probabilistic reasoning, which draws upon a defined totality of possible results to confirm the existence of “aleatory reasoning”.91 Meillassoux writes, “this probabilistic reasoning is only valid on condition that what is a priori possible be thinkable in terms of a numerical totality”.92 In this model, probability theory works with the assumption that the total number of results is knowable, even when that total is infinite. Yet taking set theory as his point of departure, Meillassoux, pace Badiou, claims that Cantor’s articulation of the infinite troubles and dismantles the frequentialist model. For Cantor, as I shall show in chapter three, set theory’s radical gesture lies in demonstrating the existence of infinite infinities, each one bigger than the last. Cantor takes infinite or what he calls transfinite sets, such as the set of all natural numbers, and finds a way to generate larger infinite sets out of them, a project that can be continued infinitely. There is, then, no largest infinite set, no knowable totality. For Meillassoux, this means that “the (quantifiable) totality of the thinkable is unthinkable”.93 The frequentialist model thus corresponds to Meillassoux’s idea of chance, since there is a necessary totality under which results can be predicted. But set theory demonstrates that there is no quantifiable totality, and this in turn corresponds to his idea of contingency. As Meillassoux puts it, “we can only move immediately from the stability of laws to their necessity so long as we do not question the notion that the possible is a priori totalisable”.94 But when the whole cannot be totalised, we “can no longer lay claim to any logical or mathematical necessity – which is to say, to any sort of a priori necessity” other than the necessity of the untotalisable itself.95

Meillassoux’s only other full-length book, The Number and the Siren (2011), takes Mallarmé’s Un Coup de dés as its subject.96 Here, he argues that, like set theory, the poem confirms the necessity of contingency – and, moreover, that it is built around an intrinsic

91 Meillassoux, After Finitude, 100. 92 Meillassoux, After Finitude, 101. 93 Meillassoux, After Finitude, 104. 94 Meillassoux, After Finitude, 105. 95 Meillassoux, After Finitude, 107. 96 Meillassoux, After Finitude, 41.

35 numerical structure. For Meillassoux, the unique number of Un Coup de dés is in fact a performative poetic device that Mallarmé uses to respond to the major socio-political events of his time: the death of God, which left the idea of the community split asunder, and the crisis of free verse against traditional form. Regarding the latter, Meillassoux wagers that Mallarmé secretly encoded his poem in order to give it form – but a unique form, one that cannot be another. To decipher this code, he performs a number of interpretative gestures on the poem. For example, he reads the imagery in Un Coup de dés to suggest that it plays out a drama between the seven stars of the constellation that perhaps emerge as the poem ends and the nothingness of the void. Yet his reading also extends beyond the poem’s semantic content when he counts, one by one, its number of words.97 All this leads him to believe that the poem has an intrinsic numericity embodied by the number 707, confirmed both at the symbolic level as well as through the formal operation of the word count. This unique number thus gives the seemingly formless poem form.

However, this code is by no means stable. For Meillassoux, it relies on an essential “quavering”, an undecidability, where it oscillates between – and ultimately dialectically synthesises – opposing numbers that might be deduced from counts working with other editions of the poem: 706, 708, etc.98 The number, which Mallarmé positions as the embodiment of the infinity of chance itself, is thus an explicitly Hegelian (rather than mathematical) infinite as it dialectically incorporates the number that it is not. However, as Meillassoux argues, this link between the infinite and chance suggests that Mallarmé also attempted to prove a similar thesis to his own: namely, that the only necessity is, in fact, contingency. Mallarmé’s number is an absolutely necessary entity, but its perennial undecidability produces nothing but contingency. Meillassoux writes, “as it fuses with Chance, the Number [...] escapes from the effects of Chance. It ceases to be hazardous and becomes necessary. Thus, in one sense, the Coup de dés does not abolish Chance, but in another, it does – for it abolishes Chance's capacity to produce nothing but contingent realities, through the exception of the unique Number”.99 Meillassoux appears to have abandoned his chance/contingency distinction here. But perhaps more importantly,

97 Mallarmé, Œuvres complètes I, 387. 98 Meillassoux, The Number and the Siren, 138. 99 Meillassoux, The Number and the Siren, 165.

36 Mallarmé’s poetry and set theory are seen to be doing the same thing: each provides a formalisation of the necessity of contingency. They are not strictly analogous as they are, for instance, in Kristeva. Thematically they both rely on a principle of numericity that produces an account of contingency, albeit in different ways. I shall return to Meillassoux’s reading in my third chapter, where I will argue that he misses one crucial aspect of the Mallarméan logic of chance.

The four readings I have examined thus far are certainly philosophically far apart from one another. What unites them, however, is how they read Mallarmé’s poetry in relation to the same conception of mathematics. In an essay that appeared in Briefings on Existence: A Short Treatise on Transitory Ontology (2006), Badiou perfectly summarises this coupling. For him, the radicality of the two discourses – that of Mallarmé’s poetry and Cantor’s set theory – lies in how they figure the infinite. Each makes the infinite into a number; a move he argues is as novel as it is revolutionary:

Yet can the infinite be a number? This is what Mallarmé, Cantor’s unconscious contemporary, contends in the poem. That the infinite is a number is what a set- theoretic ontology of the multiple finally made possible after centuries of denial and enclosure of the infinite within theology’s vocation.100

I want to use Badiou’s phrase, “Cantor’s unconscious contemporary”, to structure the arguments of this thesis. I am aware that Badiou does not completely conflate Mallarmé’s poetic infinite with its Cantorian, set theoretical counterpart; this is a distinction that is crucial to his philosophical system. But in this thesis I have argued that not only does a relationship between Mallarmé and mathematics exist, it is also one that has indeed continually been read in terms of the poet’s relationship, his contemporaneity, to Cantor. Kristeva, Derrida, Badiou and Meillassoux have all treated Mallarmé as “Cantor’s unconscious contemporary” at some stage in their work. So the question then becomes: are the two truly contemporaries of one another and, if so, what does it mean to be an

100 Badiou, Briefings on Existence, 124. My emphasis. I’ve chosen to translate “multiple” as “multiple”, rather than “manifest”. Norman Madarasz also chooses to translate Badiou’s use of “chiffre” as “integer,” whereas I have opted for “number” because it is both more faithful to the developments of set theory and more in line with Mallarmé’s own terminology.

37 unconscious contemporary? I shall address these concerns in the following two chapters. Nevertheless, the question of Mallarmé’s relation to mathematics is prescient, so much so that entire philosophical systems have been grounded upon it. For the philosophers, if Mallarmé is a mathematician, then he is Cantorian.

This thesis aims to test whether or not this is the case. By re-examining Mallarmé’s poetry, I want to see just how far the comparison with set theoretical mathematics can be taken. I want to examine the lines of analogy, harmony and discordance between the two by reading and restoring their projects alongside one another. Following Andrew Gibson’s work on Badiou and Beckett, I am not going to produce a theoretical critique of Mallarmé or Cantor, but rather look at “the problematic character of [the] relationship”.101 Before doing so, however, it is worth mentioning one figure who has already problematised this relation: the eclectic linguist Jean-Claude Milner, whose work remains relatively unknown in the Anglophone academy. Although Milner has acknowledged his general indifference to Cantorian mathematics, his career has been marked by an interest in understanding what conditions make mathematics and science possible in the modern world.102 In what follows, I shall show how Milner articulates a very different understanding of Mallarmé’s poetry and its relation to mathematics by looking at his work on post-Galilean science as well as a short piece he wrote on the poet.

Milner’s ideas about mathematics and science draw liberally on Alexandre Koyré, a French historian of science. In From the Closed World to the Infinite Universe (1957), Koyré posits that the fundamental makeup of epistemological knowledge underwent a massive reorientation under the unprecedented findings of Galileo in the seventeenth century. Traditional ideas of the finite Cosmos and the Ancient concept of phusis were rendered

101 Andrew Gibson, Beckett and Badiou: The Pathos of Intermittency (Oxford/New York: Oxford University Press, 2006), 4. 102 As Milner puts it, “mathematics is essential for mathematics itself, and for it alone”. Alain Badiou and Jean-Claude Milner, Controversies: A Dialogue on the Politics and Philosophy of our Time, trans. Susan Spitzer (Cambridge/Malden: Polity Press, 2014), 68.

38 inoperative when Galileo discovered that the universe was infinite, indefinite or limitless in nature:

This scientific and philosophical revolution [...] can be described roughly as bringing forth the destruction of the Cosmos, that is, the disappearance, from philosophically and scientifically valid concepts, of the conception of the world as a finite, closed, and hierarchically ordered whole (a whole in which the hierarchy of value determined the hierarchy and structure of being, rising from the dark, heavy and imperfect earth to the higher and higher perfection of the stars and heavenly spheres), and its replacement by an indefinite and even infinite universe which is bound together by the identity of its fundamental components and laws, and in which all these components are placed on the same level of being.103

Koyré continues to track “this scientific and philosophical revolution”, noting how Galileo’s fundamental achievement was to popularise this new vision of science. For Koyré, one of the key traits of Galilean science can be found in the specific role he accorded to mathematics. As Galileo himself famously wrote, “[p]hilosophy is written in this grand book – I mean the universe – which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the character in which it is written. It is written in the language of mathematisation”.104 Under the influence of Koyré, Milner thus seeks to establish what science designates when it is permanently bound to the language of mathematics, when the “grand book” of nature finds its script in formal language.

In Introduction à une science du langage (1989), Milner best illustrates the influence Koyré holds over his thinking:

103 Alexandre Koyré, From the Closed World to the Infinite Universe (Baltimore: The Johns Hopkins Press, 1957), 2. 104 Quoted in Michel Blay, Reasoning with the Infinite: From the Closed World to the Mathematical Universe, trans. M. B. DeBevoise (Chicago/London: The University of Chicago Press 1998), 1. In this book, which is in part a response and critique of Koyré’s, Blay argues that Galileo’s remark calls for a geometrisation of the universe, not strictly speaking a mathematisation.

By science, we understand here a discursive configuration that took form with Galileo and has not ceased to function since. Since A. Koyré, it has been characterised by the combination of two features: (I) the mathematisation of the empirical (mathematical physics would rather be called mathematised physics); (II) the constitution of a relation with techniques, such that techniques is defined as the practical application of science (whence the theme of applied science) and that science defines itself as the theory of techniques (whence the theme of fundamental science).105

While he eventually adds a third trait – the philosopher of science Karl Popper’s idea of falsification – it is the “mathematisation of the empirical” that proves most interesting to Milner, which provides the condition for “mathematised physics”. For the post-Galilean scientific subject, empirical reality is not defined by a sensible relation to the world or even by a situated agent operating in a spatio-temporal field. Instead, empirical reality is mapped and formalised by mathematical language.

For Milner, Galileo thus achieved nothing less than an entire redistribution of the sensible for the subject. In the infinite universe mathematics is bound to the way the subject sees the world, but in the pre-modern world, the universe was configured by what Milner calls the “gaze”.106 For stars to exist, they had to be perceived and then grouped and unified into unchangeable patterns or configurations. Inasmuch as the subject perceives what the limit of the universe is, constellations of stars in the sky are considered an object for science, relying first on a subject to perceive them if they are to take on a mathematical dimension: “Man looks at the starry sky and persuades himself that they are assembled into figures”.107

105 “Par science, on entendra ici une configuration discursive qui a pris forme avec Galilée et n’a pas cessé de fonctionner depuis. Depuis A. Koyré, on la caractérise par la combinaison de deux trais [sic]: (I) la mathématisation de l’empirique (la physique mathématique devant bien plutôt être dite physique mathématisée); (II) la constitution d’une relation avec technique, telle que la technique se déinisse comme l’application pratique de la science (d'où le thème de la science appliquée) et que la science se définisse comme la théorie de la technique (d'où le thème de la science fondamentale)”. Jean-Claude Milner, Introduction à une science du langage (Paris: Éditions du Seuil, 1989), 21. 106 Jean-Claude Milner, “The Tell-Tale Constellations,” trans. Christian R. Gelder, S: The Journal for the Circle of Lacanian Ideology Critique, forthcoming September 2017. 107 Milner, “The Tell-Tale Constellations,” forthcoming.

40 But with the advent of techniques (notably the telescope) and the axiom that the book of nature is written in mathematical language, stars that cannot be perceived by the gaze become more real than those that can. As Galileo himself writes: “It is assuredly important to add to the great number of fixed stars that up to now men have been able to see by their natural sight, and to set before the eyes innumerable others which have never been seen before and which surpass the old and previously known [stars] in number more than ten times”.108 Through the use of symbolic mapping, stars not accessible to the gaze could now be mathematically positioned, leaving the empirical realm – and indeed the “grand book” of nature – fundamentally determined by mathematics. But this comes at a price. The constellations that were previously considered objects of scientific knowledge no longer exist in the modern, post-Galilean universe. Rather, “[t]here only exists the stars that compose them. This is a lemma of modern science”.109

So why, Milner asks, did the poets of the nineteenth century resume writing about constellations? Among these poets, he “distinguishes” Mallarmé as the proper object of his study on mathematisation and poetry.110 This may not be all that surprising for readers of Milner’s work. From his Lacanian reading of Un Coup de dés in Les Noms indistincts (1983) to his political diatribe against the “strict Mallarméans” of the twentieth century in Mallarmé au tombeau (1999), Milner is yet another philosopher who has entertained a lifelong engagement with the poet.111 Where he differs from the others, however, is in how he articulates Mallarmé’s relationship to mathematics. For Milner, Mallarmé “says no” to the modern world: his poetry is not contemporary with the modern post-Galilean assignation of the role of mathematical writing, one that underwrites the birth of “mathematised physics”.112 The image of the constellation that, perhaps, appears at the close of Un Coup de dés and the sea upon which the shipwreck is occurring all point to Mallarmé’s refusal of modern science’s destruction of the Cosmos. “Constellations do not

108 Quoted in Koyré, From the Closed World to the Infinite Universe, 89. 109 Milner, “The Tell-Tale Constellations,” forthcoming. 110 Milner, “The Tell-Tale Constellations,” forthcoming. 111 “mallarméens stricts” Jean-Claude Milner, Mallarmé au tombeau (Paris: Éditions Verdier, 1999), 88. See also, Jean-Claude Milner, Les Noms indistincts (Paris: Éditions du Seuil, 1983), 48-9. 112 Milner, “The Tell-Tale Constellations,” forthcoming.

41 exist in the universe”, Milner writes, “but, nevertheless, they shine”.113 Far from being “Cantor’s unconscious contemporary”, Milner’s Mallarmé attempts to undo the post- Galilean configuration by empirically testifying to the unavoidable “brilliance” of star constellations.114

Even so, Mallarmé’s number – the unique number that cannot be another – still signifies for Milner the intimate bond poetry shares with mathematics. But this is a bond that is not modern. While Mallarmé says no to the modern concept of number in its post-Galilean configuration, he nonetheless recognises and plays with number’s “genealogy”, recalling older conceptions that exist outside the modern world. Nature, of course, exists for Mallarmé but, in reviving the constellations, poetry is positioned as both an exception and a limit to modern science: “let us understand by this that nothing will have taken place except that which takes place, namely Nature, as the place of science and technique – except the exception that constitutes a limit to it”.115 This exception is expressed through poetry’s unique relationship to mathematics and number, one that is not configured in the post- Galilean sense:

He [Mallarmé] makes mathematics his ally: “We must study our mathematicians”. The number as the limit of modern Nature and science is at once legitimate and possible on one condition: we must recall the genealogy of number. This genealogy brings us back to the constellations: “THE NUMBER / born of the stars”. Not, therefore, mathematised science, but mathematics. Mathematics in exception to science. Now, the number insofar as it is recalled, is verse.116

For Kristeva, Derrida, Badiou and Meillassoux, Mallarmé’s relationship to mathematics is absolutely modern and exists within the temporal milieu in which it was created. But for Milner precisely the opposite is the case. Mallarmé cannot be Cantor’s unconscious contemporary because he refuses the modern, preferring instead to “recall” the genealogy of number as such. The modern in its post-Galilean guise is, for Mallarmé, a limiting factor,

113 Milner, “The Tell-Tale Constellations,” forthcoming. 114 Milner, “The Tell-Tale Constellations,” forthcoming. 115 Milner, “The Tell-Tale Constellations,” forthcoming. 116 Milner, “The Tell-Tale Constellations,” forthcoming.

42 one that represses the unavoidable constellations. On the one hand, philosophical interpretations of Mallarmé have bound him to modern mathematics. But on the other, Milner’s argument suggests that there is a conscious rejection of this bind at play within his poetry.

In the following two chapters, I want to weigh up this alternative, to test it, to see if it is possible to fall on one side of the debate or on the other. Let me say something about the general method of this thesis. Although it is unlikely that either Mallarmé or Cantor knew of the other’s existence, there is a remarkable simultaneity in the overarching concerns of their work and the almost uncanny manner in which they were both taken up in the other’s country. French mathematicians at the turn of the century quickly consumed the German Cantor’s work. As Anne-Marie Décaillot has noted, Cantor would often write to the famous French mathematician Henri Poincaré. He also exchanged letters with two mathematicians from l’Association français pour l’avancement des sciences, Émile Lemoine and Charles- Ange Laisant, each one making it their mission to disseminate Cantor’s work throughout France.117 In the mirror image of this, the first to translate Mallarmé into German was a professor of mathematics, Kurt Reidemeister, the nephew of Richard Dedekind, one of Cantor’s key friends, colleagues and interlocutors.

Besides this kind of proximal contemporaneity, Mallarmé and Cantor also share remarkably similar thematic concerns. The two concepts that have plagued the history of mathematics – the nothing and the infinite – went through a paradigm-shifting revolution under Cantor’s watchful eye. As Charles Seife writes, “zero and infinity are two sides of the same coin – equal yet opposite, yin and yang, equally powerful adversaries at either end of the realm of numbers. The troublesome nature of zero lies with the strange powers of the infinite, and it is possible to understand the infinite by studying zero”.118 Tzuchien Tho has also stated that with Cantor’s “taming of the infinite, the void is thus also tamed [...] The conceptual power of the void is neutralised along with that of the infinite”.119 Mallarmé’s poetry enacts a very

117 Anne-Marie Décaillot, Cantor et la France: Correspondance du mathématicien allemand avec les français à la fin du XIXe siècle (Paris: Éditions Kimé, 2008), 9. 118 Charles Seife, Zero: The Biography of a Dangerous Idea (New York: Penguin, 2000), 131-132. 119 Tzuchien Tho, “The Void Just Ain’t (What it Used to Be): Void, Infinity, and the Indeterminate,” Filozofski vestnik XXXIV, no. 2 (2013): 28.

43 similar drama, one where the infinite and the nothing are continually made to interlock with one another. “We go, simply, to the edge of the ocean, where all that remains is a pale and indistinct line”, Mallarmé wrote in one of his brief, but important, pieces on fashion, “to gaze at what lies beyond our ordinary existence – that is to say, the infinite and nothing”.120 One might also think here of the image of the Abyss from Un Coup de dés, where the very thing that Mallarmé wished to discern – that is, concepts of the infinite and the nothing – are themselves indiscernible and “beyond our ordinary existence”. As I shall show in the following two chapters, the concepts of the nothing and the infinite are absolutely crucial to the overarching concerns of Mallarmé’s work and can be mapped, defined and clarified when placed alongside Cantor. It is to them that we now must turn.

120 “[…] nous allons, simplement, au bord de l’Océan, où ne persiste plus qu’une ligne ordinaire, c’est-à-dire l’infini et rien”. Mallarmé, Œuvres complètes II, 524.

44 Chapter 1: The Nothing that Is and the Nothing that is Not

“Modernity is defined by the fact that the One is not”121 – Alain Badiou

This chapter inquires into how the concept of the nothing figures in Mallarmé’s poetry and set theory. It is organised into three sections. In the first, I examine the manner in which the nothing operates throughout Mallarmé’s oeuvre. Drawing on his letters written during the infamous “spiritual crisis” of 1866, as well as the poems “Salut” and “À la nue…”, I argue that the concept of the nothing plays three interrelated roles in the poet’s career: firstly, it is an epistemological discovery; secondly, it is incorporated into the poem’s ontological structure; and lastly, this epistemological and ontological figuration is bestowed with a social dimension, orienting Mallarmé’s vision of poetry’s social destiny.122 In the second section of the chapter, I turn to set theory’s own concept of the nothing – the empty set – and examine how it has developed throughout the twentieth century. I conclude this section by looking at Jacques-Alain Miller and Alain Badiou’s work on mathematics and the nothing to suggest that both set theory and Mallarmé use the nothing as their starting point and that their conceptions of this are uncannily similar. In the final section, I provide a long reading of Mallarmé’s early poem, “Sonnet en -yx”, to see how far this analogy between the mathematical and literary nothing can be taken – all of which will bring me one step closer to determining whether Mallarmé truly was “Cantor’s unconscious contemporary”.123

After returning from a year teaching in London and recently married to his partner Marie in the summer of 1863, the trajectory of Mallarmé’s career took a sharp turn: what Rosemary

121 Alain Badiou, Number and Numbers, trans. Robin Mackay (Cambridge/Malden: Polity Press, 2008), 65. 122 For two of the most authoritative accounts of the spiritual crisis, see Pascal Durand, Mallarmé: Du sens des formes au sens des formalités (Paris: Éditions du Seuil, 2008), 35-53 and Bertrand Marchal, La Religion de Mallarmé (Paris: Librairie José Corti, 1988), 55-64. 123 Alain Badiou, Briefings on Existence, trans. Norman Madarasz (New York: State University of New York Press, 2006), 124.

45 Lloyd calls a “descent into the void”.124 During this summer, Mallarmé had finally broken off from his old master Baudelaire and resided in relative obscurity in Tournon, a small town in the south of France. Roger Pearson paints a particularly bleak picture of Mallarmé’s new living arrangements:

For the human being the first year in Tournon was difficult. Mallarmé loathed the town’s climate. The bitter cold in winter and the howling Mistral wind exacerbated his rheumatic condition, while the heat in summer made life unbearable until the school holidays began in mid-August. Scorpions were a scourge, requiring bed legs to stand in bowls of water. The town had no cultural life to speak of, and the townspeople seemed to him materialistic and dull. There were no poets.125

Mallarmé’s deep sadness in Tournon (as well as Avignon, where he moved in 1867) and his increasing disillusionment with Baudelairian poetics triggered what is now called his spiritual crisis – a crisis resulting in the “discovery of nothingness”.126 Over the course of this period, Mallarmé did not in fact publish any poetry. Most of the evidence for his descent can be found in the various letters he wrote to friends between 1866 and 1869. On the 3rd May 1868, he tells Eugène Lefébure: “either I’ll get worse or I’ll cure myself, I’ll disappear or I’ll remain”.127 This elocutionary disappearance was not the result of a purely existential affirmation of life’s meaninglessness. Rather, it took the form of an epistemological discovery. Through his descent into the void, Mallarmé believed he had “vanquished” God. In May 1867, the poet confided in his close confident Henri Cazalis, detailing the utter terror this discovery brought him:

I’ve just spent a terrifying year: my Thought has thought itself and reached a pure Concept. All that my being has suffered as a result during that long death cannot be

124 Stéphane Mallarmé, Selected Letters of Stéphane Mallarmé, ed and trans. Rosemary Lloyd (Chicago/London: The University of Chicago Press, 1988), 74. “ou j’irai plus mal ou je me guérirai, je disparaîtrai ou je resterai”. Stéphane Mallarmé, Correspondance: Lettres sur la poésie, ed. Bertrand Marchal (Paris: Gallimard, 1995), 384. 125 Roger Pearson, Stéphane Mallarmé (London: Reaktion Books, 2010), 42. 126 Stéphane Mallarmé, Œuvres complètes I, Édition présentée, établie et annotée par Bertrand Marchal (Paris: Gallimard, 1998), 1373. 127 Mallarmé, Selected Letters of Stéphane Mallarmé, 85.

46 told, but, fortunately, I am utterly dead, and the least pure region where my Spirit can venture is Eternity [...] But this was even more the case a few months ago, firstly in my terrible struggle with that old and evil plumage, which is now, happily, vanquished: God.128

Many have noted the distinctly Hegelian terms used throughout this passage and Mallarmé’s early intellectual development more generally.129 While that is not my interest here, there is nevertheless a tendency to force this moment of Mallarmé’s life into a comparison with a philosophical counterpart. In Mallarmé: poésie et philosophie (1994), Pierre Campion treats Mallarmé as a philosopher, although not one who deals with abstract ideas. Instead, he argues that the poet always reflexively philosophises about the nature of his craft, writing meta-poetic philosophical statements into his poetic production. For Campion, these letters give the first crucial indication of Mallarmé’s “philosophy”: they suggest that the poet is primarily a thinker of absence and that this absence arises in the first instance from his own Nietzschean declaration that “God is dead”.130 Eric Benoit has echoed this, writing that “the Mallarméan discovery of Nothingness is the discovery of the Nothingness of God”.131 Yet rather than reinstating a return of the One through a Nietzschean affirmation of Life, Mallarmé treats the nothing as an epistemological starting-

128 Mallarmé, Selected Letters of Stéphane Mallarmé, 74. “Je viens de passer une année effrayante: ma Pensée s’est pensée, et est arrivée à une Conception Pure. Tout ce que, par contre-coup, mon être a souffert, pendant cette longue agonie, est inénarrable, mais, heureusement, je suis parfaitement mort, et la région la plus impure où mon Esprit puisse s’aventurer est l’Éternité […] Mais combien plus je l’étais, il y a plusieurs mois, d’abord dans ma lutte terrible avec ce vieux et méchant plumage, terrassé, heureusement, Dieu”. Mallarmé, Correspondance, 342. 129 The question of Mallarmé’s supposed Hegelianism is the subject of an immense amount of literature: for some newer and older classics of the genre see, Jean-Pierre Richard, L’Univers imaginaire de Mallarmé (Paris: Éditions du Seuil, 1961); Lloyd James Austin, Essai sur Mallarmé, ed. Malcolm Bowie (Manchester/New York: Manchester University Press, 1995); Janine D. Langan, Hegel and Mallarmé (Maryland: University Press of America, 1986); and most recently, Barnaby Norman, Mallarmé’s Sunsets: Poetry at the End of Time (London: Leganda, 2014). Many of these readings acknowledge the largely speculative connection between the two – that is, that the poet began to use quasi-Hegelian language after visiting the “very Hegelian” Eugene Lefébure in the spring of 1866 – but it is also thought that Mallarmé could have read Edmond Scherer’s 1861 introduction to Hegel, Hegel et l’hégélianisme. “très hégélienne” Marchal, La Religion de Mallarmé, 58 and Quentin Meillassoux, “The Coup de dés, or the Materialist Divinisation of the Hypothesis,” Collapse VIII (2014): 837. 130 As Campion writes, “Nothingness is that which is absent, but in such a manner that this absence is pregnant and has powers of organisation in the scene of the poem [le Néant est ce qui est absent, mais de telle manière que cette absence soit prégnante et tension d'organisation au sein de poème]”. Pierre Campion, Mallarmé: poésie et philosophie (Paris: PUF, 1994), 16, 17. 131 “la découverte mallarméenne du Néant, c’est la découverte du Néant de Dieu” Eric Benoit, Néant sonore: Mallarmé ou la traversée des paradoxes (Genève: Droz, 2007), 41.

47 point for thinking.132 Where God is not, the nothing is – and the effects of this discovery ripple throughout his poetic career. “In the face of the Nothing which is truth”, he writes to Cazalis in April 1866, all of mankind’s “achievements”, all its “dreams” of God, of transcendence, of the One, and of the soul, are exposed to be dependent on a prior nothing.133 For Mallarmé, the nothing names the truth hiding behind everything that exists in the world.

Mallarmé continues to pursue this line of thinking throughout his time in Tournon, describing a series of never completed projects on the nothing that develop out of his discovery. In September 1867, he tells Villiers de l’Isle-Adam that he “arrived at the Idea of the Universe through sensation alone (and that, for instance, to keep an indelible notion of pure Nothingness, I was forced to impose on my brain the sensation of absolute emptiness)”.134 In this letter, the poet continues to outline two of these projects. The first concerns a poem about the relationship between Beauty and the world. This was inspired by a previous remark he made to Cazalis in 1866, one that highlights the aesthetic quality of the nothing: “having found Nothingness, I’ve found Beauty”.135 While Mallarmé’s work on Beauty is best expressed in his poem “Hérodiade” – first composed between 1865 and 1866 – the term diminishes in importance throughout his mature writings.136 The second reference concerns a never-completed book entitled “Sumptuous Allegories of Nothingness”.137 Although Mallarmé does not manage to complete either of these projects, the crowning achievement of this period in his life is the composition of “Sonnet allégorique de lui-même”, later revised under the name “Sonnet en -yx”. This poem is certainly one of the most sumptuous allegories of Nothingness in Mallarmé’s oeuvre and I shall turn to it in the final section of this chapter.

132 Badiou, Number and Numbers, 65. 133 Mallarmé, Selected Letters of Stéphane Mallarmé, 60. Modified trans. “[…] devant le Rien qui est la vérité”. Mallarmé, Correspondance, 298. 134 Mallarmé, Selected Letters of Stéphane Mallarmé, 81. “[…] je suis arrivé à l’Idée de l’Univers par la seule sensation (et que, par exemple, pour garder une notion ineffaçable du Néant pur, j’ai dû imposer à mon cerveau la sensation du vide absolu)”. Mallarmé, Correspondance, 366-367. 135 Mallarmé, Selected Letters of Stéphane Mallarmé, 65. Modified trans. “[…] avoir trouvé le Néant, j’ai trouvé le Beau”. Mallarmé, Correspondance, 310. 136 For a recent analysis of the relation between “Hérodiade” and Beauty see Norman, Mallarmé’s Sunsets: Poetry at the End of Time, 18-47. 137 Mallarmé, Selected Letters of Stéphane Mallarmé, 82. Modified trans. “Allégories somptueuses du Néant”. Mallarmé, Correspondance, 367.

For Mallarmé, then, the nothing is the precursor to all effects of oneness, the place from which the semblance of oneness originates. “Yes, I know, we are merely empty forms of matter”, he ceremoniously states, “but we are indeed sublime in having invented God and our soul”.138 While the death of God triggers a fundamental shift in Mallarmé’s work, the importance of the nothing passes beyond this epistemological level and begins to structure many of his larger ideas. As Benoit notes, “this discovery of Nothingness is [...] the properly poetic and linguistic deepening of verse, what Mallarmé sometimes called ‘condensation’ [...] words do not return to the things themselves, but to their absence, to their nothingness”.139 Here, he continues, one should think of Mallarmé’s famous passage from “Crisis of Verse”: “I say: a flower! And [...] there arises, musically, the very idea in its mellowness; in other words, what is absent from every bouquet”.140 In this example, Mallarmé alludes to the nothingness that hides behind the presentism of the word. The signifier “flower” does not guarantee the object it signifies, but is rather empty beyond its immediate musicality. One says flower and the idea emerges, while “the notion of object [...] is lacking”.141 Words thus conjure a sonorous presence, but this only highlights a more fundamental absence, that of the truth of the nothing. Mallarmé’s epistemological discovery of nothingness is first conceived in quasi-spiritual terms, implicating the idea of the self. But this idea is then transported into the matter of words themselves – the object that is necessarily absent from every word.

So far, what Mallarmé’s spiritual crisis reveals is his disillusionment with all the transcendent unities that form the basis of the world (such as God, the soul or even the

138 Mallarmé, Selected Letters of Stéphane Mallarmé, 60. “Oui, je le sais, nous ne sommes que de vaines formes de la matière, — mais bien sublimes pour avoir inventé Dieu et notre âme”. Mallarmé, Correspondance, 297. 139 “cette découverte du Néant est donc lié à l’approfondissement proprement poétique et linguistique du vers, ce que Mallarmé appelle plusieurs fois ‘la condensation’”. Ainsi, les mots ne renvoient pas aux choses mêmes, mais à leur absence, à leur néant”. Benoit, Néant sonore, 41. 140 Stéphane Mallarmé, Divagations, trans. Barbara Johnson (Cambridge/Massachusetts/London: The Belknap Press of Harvard University Press, 2007), 210. “Je dis: une fleur! et, hors de l’oubli où ma voix relègue aucun contour, en tant que quelque chose d’autre que les calices sus, musicalement se lève, idée même et suave, l’absente de tous bouquets”. Stéphane Mallarmé, Œuvres complètes II, Édition présentée, établie et annotée par Bertrand Marchal (Paris: Gallimard, 1998), 213. 141 Mallarmé, Divagations, 187. “la notion d’un objet […] qui fait défaut”. Mallarmé, Œuvres complètes II, 68.

49 One). But since Mallarmé is first and foremost a poet, the discoveries of the crisis bear directly upon language, which for him is inherently haunted by nothingness and absence. The epistemological discovery that a prior nothingness structures the world must therefore become part of the poem’s ontological structure. In his 1893 “Salut”, Mallarmé explores the nature of poetic writing in order to develop this point further. Jacques Rancière has noted that “there is a tendency to classify [‘Salut’] as one of the ‘Occasional verses’”.142 However, although it was indeed written for an occasion – a banquet held in the early days of 1893 for the journal La Plume – Mallarmé chose it to open his sole collection of poetry, Poésies (1897), thereby elevating its importance in the Mallarméan canon143:

Rien, cette écume, vierge vers Nothing, this foam, virgin verse A ne désigner que la coupe; Only to designate the cup: Telle loin se noie une troupe Thus, far off, drowns a Siren troop; De sirènes mainte à l'envers. Many, upended, are immersed.

Nous naviguons, ô mes divers We navigate, O my diverse Amis, moi déjà sur la poupe Friends, myself already on the poop, Vous l'avant fastueux qui coupe You the sumptuous prow to cut Le flot de foudres et d'hivers; Through winter wave and lightning burst;

Une ivresse belle m'engage A lovely drunkenness enlists Sans craindre même son tangage Me to raise, though the vessel lists, De porter debout ce salut This toast on high and without fear

Solitude, récif, étoile Solitude, rocky shoal, bright star A n'importe ce qui valut To whatsoever may be worth Le blanc souci de notre toile. Our sheet’s white care in setting forth.144

At first glance, the threefold descriptor that comprises the exclamatory opening to “Salut” is simply the description of a toast or salutation. The middle term, the foam, is that of the champagne overflowing from the cup, the “ivresse belle” Mallarmé offers his fellow poets at the banquet. Nothing, the foam and verse all appear upon first reading only to designate the cup through the supreme originary act of a toast. Bertrand Marchal has argued,

142 Jacques Rancière, Mallarmé: The Politics of the Siren, trans. Steven Corcoran (London/New York: Continuum, 2011), 1. 143 In his excellent translators’ commentary, Weinfield writes: “The poem offers a salutation, or greeting, to the poet’s audience, and taking the form of a drinking song, it raises a toast, or a wish for good health, to the reader. Indeed, the sonnet was originally entitled “Toast” and was read by Mallarmé at a banquet for the journal La Plume prior to its publication in the February 1893 edition of that journal”. Stéphane Mallarmé, Collected Poems: A Bilingual Edition, trans. Henry Weinfield (Berkeley/Los Angeles/London: University of California Press, 1996), 149. 144 Weinfield chooses to italicise the first line of the last tercet, something I have removed from my reproduction of the poem here. Stéphane Mallarmé, Collected Poems, 3.

50 however, that while the poem looks as though it is a superficial ode to friendship and drunkenness, its movement echoes a similar thematic trope present in neighbouring sonnets and compositions. “Salut” rehearses the very typical Mallarméan scene of a shipwreck, which serves as a kind of poetic metaphor for the socio-political dramas of Mallarmé’s time. First there is the toast to the “symbolic ‘Nothing’”, which lies underneath the foam of the water’s waves.145 Upon the foamy water is a ship, where Mallarmé himself is “déjà sur la poupe”. The poet, prone to punning on the English language, invites his fellow divers to join him on board as they sail atop the unseen nothingness lurking below. But the ship is negotiating a complex passage through “[l]e flot de foudres et d’hiver” that threatens the success of the poet’s journey. Rancière notes an immediate comparison with the penultimate sonnet of Poésies, “À la nue…”, in which an analogous scene is described. Taken together, these two poems illustrate the ontological and social dimension of the nothing in Mallarmé’s work:

À la nue accablante tu Hushed to the crushing cloud Basse de basalte et de laves Basalt and lava its form A même les échos esclaves Even to echoes subdued Par une trompe sans vertu By an ineffectual horn

Quel sépulcral naufrage (tu What shipwreck sepulchral has bowed Le sais, écume, mais y baves) (You know this, foam, but slobber on) Suprême une entre les épaves The mast supreme in a crowd Abolit le mât dévêtu Of flotsam and jetsam though torn

Ou cela que furibond faute Or will that which in fury defaulted De quelque perdition haute From some perdition exalted Tout l'abîme vain éployé The vain abyss outspread

Dans le si blanc cheveu qui traîne Have stingily drowned in the swirl Avarement aura noyé Of a white hair’s trailing thread Le flanc enfant d'une sirène. The flank of a young Siren girl.146

Rancière’s reading attempts to dispel the myth that Mallarmé authored impenetrable elitist poetry that could be dismissed in favour of clearly-articulated democratic writing. To do so, he reads the various images (the shipwreck, the foam, the abyss and the siren) at work within “Salut” and “À la nue…”, the latter being one of the most difficult poems in Mallarmé’s oeuvre. Rancière argues that the poem presents a series of oscillating

145 Bertrand Marchal, Lecture de Mallarmé (Paris: Librairie José Corti, 1985), 13. 146 Steven Corcoran’s translation in Rancière, Mallarmé: The Politics of the Siren, 1.

51 hypotheses that are affirmed or negated throughout its movement. He writes, “[r]eading [‘À la nue…’] reconstitutes not history, but the virtuality of history, or the choice between the hypotheses it proposes to us”.147 This is confirmed, he thinks, by the two unique commas that surround the word “écume”, which signify a kind of “or else” that “weighs the two terms of an alternative against one another”.148 The first choice Rancière proposes concerns that of the historical passage from the Romanticism of Victor Hugo and Alfred de Vigny to Mallarmé’s own time, which is metaphorised by the “sépulcral naufrage” against “Le flanc enfant d’une sirène”. The shipwreck is made to represent a “major drama [...] which swallowed up a ship to its last [...] bit of wreckage”.149 The alternative to the shipwreck is the “agitation” of the siren, signifying instead Mallarmé’s unique poetic program. Through the image of the foam, the poem thus presents the reader with a choice, one whose consequences implicate the historical destiny of poetry. As Marchal puts it, “is the foam the sign of a shipwreck, or does it betray a drowning siren?”150

“But”, Rancière writes, “this opposition between great drama and lightweight pantomime is doubled by another alternative”.151 This second choice concerns the “ineffectual horn” invoked in the first stanza, which suggests that the great drama “went unnoticed”, and that this drama was what “the surrounding world (vain chasm of billows) awaited but was denied”.152 Rancière therefore manages to read the poem in terms of a social drama that articulates not only the relationship between poetry and history, but also between history, poetic obscurity and social determination. In order to decipher what hypothesis the poem favours, Rancière turns to the image of the ship that opens “Salut”. As Robert Boncardo writes,

[the shipwreck’s] fortune, as well as its relation to the public of its time, can therefore be seen as being staged in the alternatives that the poem articulates: either the ship of the avant-garde ran aground and was lost in a social context

147 Rancière, Mallarmé: The Politics of the Siren, 2. 148 Rancière, Mallarmé: The Politics of the Siren, 2. 149 Rancière, Mallarmé: The Politics of the Siren, 2. 150 “L’écume est-elle le signe d’un naufrage, ou trahit-elle la noyade d’une sirène?” Mallarmé, Œuvres complètes I, 1205. 151 Rancière, Mallarmé: The Politics of the Siren, 2. 152 Rancière, Mallarmé: The Politics of the Siren, 2-3.

52 unresponsive to the song of these poètes maudits, or this poem constituted precisely what the “vain abyss” was seeking yet which it had been denied by the mediocre substitutes offered by “the social arrangement”.153

In other words, “À la nue…” and “Salut” are both commentaries on the complex relationship between the modern poem, in its opposition to Hugo and Vigny, and the social order: what the “surrounding world” desires from the poem and what the poem can in turn give to the surrounding world. But why is this choice necessary in the first place? Why does the audience to whom the poem is addressed desire poetry at all?

The important thing to note about Rancière’s reading is that the existence of the foam and the transcendence desired by the audience are predicated on the assumption that art, society and politics are now fundamentally determined by the nothing. What lies beneath the foam, as “Salut” recounts, is precisely “Rien”; what supports the poetic ship is again the nothing. Marchal, whose work is heavily present in Rancière’s book, develops this link between the nothing and the social. Citing a particular line from Mallarmé’s prose piece “Bucolic”, he also recognises the tension between the poet, the poet’s product, and the audience addressed by the poem. In “Bucolic”, Mallarmé writes “[i]t has to be said that the artist and man of letters, who goes by the unique name of poet, has nothing to do with a space reserved for the crowd or for chance; he’s a servant, in advance, of rhythms”.154 Since this “artist and man of letters” must attest to the fact that the nothing lies behind all forms of oneness – that this is in fact the “century of the death of God” – all social thought and programs pursued by poetry, according to Marchal, now “converge towards nothing”.155 The “servant of rhythms” is best equipped to address the social sphere because the modern poem recognises and diagnoses a society whose lack has now been made manifest by the inexistence of God. For Mallarmé, the social is thus founded upon a “central nothing” and the poet must rethink the “crowd”, his term for the generic makeup of the social sphere, on

153 Robert Boncardo, “Appropriations Politiques de l’Œuvre de Stéphane Mallarmé: Les Cas de Sartre, de Tel Quel, de Badiou et de Rancière,” (PhD thesis, Sydney University, 2014), 265. 154 Mallarmé, Divagations, 266. Modified trans. “Que l’artiste et lettré, qui se range sous l’unique vocable de poëte, n’a, lui, à faire dans un lieu adonné à la foule ou hasard; serviteur, par avance, de rythmes”. Mallarmé, Œuvres complètes II, 252. 155 Marchal, La religion de Mallarmé, 397.

53 the basis of this discovery.156 But the crowd itself, knowing that God is dead and the nothing is, desires a new form of transcendence, one that it briefly looks to poetry to provide.

To refer back to “Salut”, this poem exemplifies the two facets of the nothing I have been presenting here. On the one hand, as the first line suggests, Mallarmé begins the poem on the basis of the nothing. But on the other hand, the poem also enacts a socio-political drama, where Mallarmé rethinks the social in the century of the death of God. The poets are the ones who are able to sail atop the nothingness below the foam, navigating their way through the contemporary fog.157 However, “Salut” is also a commentary on the act of writing itself, on what the ontological status of the poem becomes after the discovery of nothingness. Marchal notes that the beautiful trinity of objects invoked at the beginning of the final tercet (“Solitude, récif, étoile”) allude to the concept of inspiration: aboard the ship, the poets are able to see the bright stars and rocky shoal that guide, and potentially misguide, their otherwise solitary existence.158 Each of these objects, these singular inspirations, press themselves onto the “blanc souci”. The “blanc souci” of the final line refers back to the poem’s opening, that of the “vierge vers”, where the whiteness of the page is corrupted by the black marks of the pen. The opening line of “Salut” thus describes the three-fold operation of poetic practice – the description of the ontological status of the poem – that begins firstly with the nothing, secondly with the foam (that is, the identifiable material of poetic inspiration), and thirdly with the virgin page – all of which refer back to this originary nothing from which poetry begins. “Salut” therefore embodies the structure of the nothing that finds its expression in Mallarmé’s work. First, there is the epistemological discovery that God is dead and that the nothing is. Second, this discovery is ontologically incorporated into the genetic makeup of the poem. And, third, the social destiny of the poem now becomes a matter of basing the assumptions and axioms of social life on a “central nothing”, a nothing that both is and is not.

156 Marchal, La religion de Mallarmé, 372. 157 Marchal, La religion de Mallarmé, 372. 158 Marchal notes, “[t]oast to the poets, toast to the readers, this poem is also a toast to all the sources of inspiration [Salut aux poètes, salut aux lecteurs, ce poème est aussi un salut à toutes les sources d’inspiration]”. Marchal, Lecture de Mallarmé, 15.

54 Mallarmé further clarifies the transposition of the nothing in the poem in his article “Crisis of Verse”, where he argues for the untotalisability, the incompleteness, of language. “Languages [are] imperfect insofar as they are many”, he writes, “the absolute one is lacking”.159 To ground this claim, Mallarmé cites the inherently oppositional nature of the words “jour” or “day” and “nuit” or “night”.160 For the poet, although the “sense” or signified content of the former word should in principle refer to the properties of day (in this case brightness or lightness), the “sound” of the word is in fact dark.161 Similarly, the sense of “nuit” is dark but its sound is light. The internal coherency of language is therefore skewed: words contain and exemplify their own internal contradictions as sound and sense never quite manage to align with one another.162 But Mallarmé argues that verse is capable of overcoming language’s inherent lack. As Jean-Claude Milner says of this passage, “[v]erse requires a supplementary component comprised of calculations, symmetries, plays of sonority and, running under it all, a design to create, by means of verse, this single word that language lacks, whose sound corresponds with its sense”.163 While I shall return to the question of sonority later, it is precisely language’s internal defect, its inherently flawed nature, which sparks what is perhaps the most famous attribute of Mallarmé’s linguistic theory: namely, the injunction to impersonality. Insofar as the task of verse is to supplement the natural lack of language, the poet must be completely removed from his product. Since, for Mallarmé, speech “has to do with the reality of things only commercially”, in poetic writing “one contents oneself with alluding to it or disturbing it slightly, so that it yields up the idea it incorporates”.164 The writing of poetry contains the possibility of accessing a higher order, that of the Idea, which is something the ordinary use of language cannot hope to grasp because it is oblivious to its own incompleteness. The

159 Mallarmé, Divagations, 205. “Les langues imparfaites en cela que plusieurs, manque la suprême”. Mallarmé, Œuvres complètes II, 208. 160 Mallarmé, Divagations, 205. 161 Mallarmé, Divagations, 211. “[…] le sens et la sonorité”. Mallarmé, Œuvres complètes II, 213. 162 Jean-Claude Milner has recently examined the difference between this view of language and that of Saussure’s; Mallarmé’s sound and sense conception is juxtaposed to the arbitrary relation of the signifier to the signified. While Milner does not express a particular preference for either idea of language, he nonetheless notices that they are distinct from one another. See Jean-Claude Milner, “Mallarmé Perchance,” trans. Liesl Yamaguchi, Hyperion IX, no. 3 (2014): 87-9. 163 Milner, “Mallarmé Perchance,” 94-5. 164 Mallarmé, Divagations, 208. “Parler n’a trait à la réalité des choses que commercialement: en littérature, cela se contente d’y faire une allusion ou de distraire leur qualité qu’incorporera quelque idée”. Mallarmé, Œuvres complètes II, 210.

55 “pure work” of poetry therefore “implies the disappearance of the poet speaking” – the elimination of the author – who, once removed from the work, leaves poetry alone and autonomous, existing by and for itself in order to supplement the inevitable defect of language.165

In his 1894 lecture delivered at Oxford and Cambridge, Mallarmé spoke about this unique property of poetry in relation to the nothing. “We know” and are “held captive”, he suggests, “by an absolute formula” that states “only what is, is”.166 But Mallarmé cannot ignore the attraction of a “lure” that something other than what is, in fact, is. He continues:

In light of a superior attraction like a void, we have the right to be lured on by nothingness: it is drawn out of us by the boredom of things if they are established as solid and preponderant – we frantically detach them and fill ourselves up with them, and also endow them with splendor, through vacant space, for as many solitary festivals as we wish. As for me, I ask no less of writing, and I’m going to prove it.167

This statement embodies all the aspects of the Mallarméan nothing I have been outlining. There is, for example, the “boredom of things” that necessitates the initial epistemological discovery of nothingness: the existence of the nothing relinquishes, indeed vanquishes, the boredom that arises when one is only conscious of “what is”, rather than what is not. There is also the image of the festival, the ceremony, which implicates the “central nothingness” on which the social bond is founded. In a short footnote to this passage, Mallarmé suggests that his choice of the phrase “in light of a superior attraction” was deliberately metaphysical but also “pyrotechnical”, like a “fireworks show”.168 These social rituals (the “solitary

165 Mallarmé, Divagations, 208. “L’œuvre pure implique la disparition élocutoire du poëte”. Mallarmé, Œuvres complètes II, 211. 166 Mallarmé, Divagations, 187. “Nous savons, captifs d’une formule absolue, que, certes, n’est que ce qui est”. Mallarmé, Œuvres complètes II, 67. 167 Mallarmé, Divagations, 187. “En vue qu’une attirance supérieure comme d’un vide, nous avons droit, le tirant de nous par de l’ennui à l’égard des choses si elles s’établissaient solides et prépondérantes — éperdument les détache jusqu’à s’en remplir et aussi les douer de resplendissement, à travers l’espace vacant, en des fêtes à volonté et solitaires. Quant à moi, je ne demande pas moins à l’écriture et vais prouver ce postulat”. Mallarmé, Œuvres complètes II, 67. 168 Mallarmé, Divagations, 197. “Pyrotechnique”, “un feu d’artifice”. Mallarmé, Œuvres complètes II, 76.

56 festivals” like the “fireworks show”) all imply the forms of social gathering the poet attempts to theorise. Think of the image of the fireworks, for example: in French, the term is written “feux d’artifice” and the poet plays on the word artifice to signify the anti- transcendence involved in its production. Fireworks are propelled into the sky yet do not pass beyond the world into the heavens – and, after appearing, quickly disintegrate into pure nothingness. This is just one example of the kinds of secular rituals Mallarmé thought could bind together a community split asunder by the inexistence of God. Finally, ontologically speaking, poetry, like everything else, has an origin but this origin is precisely nothing. As Mallarmé suggests in the above passage, it is the poet’s task to prove that the internal, self-contained system of poetry is itself founded on this very nothingness.

Thus far, I have attempted to show the threefold way Mallarmé figures the concept of the nothing. I suggested that it begins as an epistemological discovery, manifested and necessitated by the spiritual crisis of 1866. Given that this crisis “lures” the poet away from the “absolute formula that [...] only what is, is” and that behind all effects of unity and oneness there is only nothingness, the nothing then becomes the ontological starting point for the poem. As this epistemological discovery is transposed into the poem’s ontological structure, Mallarmé bases his social, political and a-theological thought around the nothing. However, the precise status of the nothing is still not completely determined. While I have shown the three bases on which Mallarmé thinks it, I am yet to show exactly how he does so – this will be the subject of the final section of this chapter. Before this, however, I want to address how mathematics deals with the nothing. I will begin by examining the aims and context of set theory in more detail and then point to a number of similarities that its concept of the nothing shares with Mallarmé’s own understanding of it.

According to Penelope Maddy, “the astounding achievement of the foundational studies of

Badiou notes the distinctly political implications of the image of the firework: “Mallarmé’s key image here is fireworks: commemorating, on July 14, the foundational riot, they project onto the sky a splendour of which the crowd is only the nocturnal ground”. Quoted in Boncardo, “Mallarmé in Alain Badiou’s Theory of the Subject,” Hyperion IX, no. 3 (2014): 5.

57 the late nineteenth and early twentieth century was the discovery that [...] fundamental [mathematical] assumptions could themselves be proved from a standpoint more fundamental still, that of the theory of sets”.169 Emerging out of the intellectual innovations of the late nineteenth century, set theory was founded by Georg Cantor and further developed by Gottlob Frege, both of whom attempted to provide a framework that could support modern mathematics. The philosophical importance of Cantor’s legacy arose from his unprecedented, paradigm-shifting work on the two central concepts of this thesis: his “taming”, to quote Tzuchien Tho, of the void and the infinite.170 As Maddy notes, if set theory is to provide a consistent and internally abstracted backdrop that could underpin modern mathematics, then all the previously established terms, givens, concepts, and customs employed by mathematicians up until then must be able to translate without fault into the set-theoretical universe. She writes, “[...] set theory provided a clear and compelling surrogate for what had heretofore been a troubling source of fundamental confusion and concern”.171 For Badiou, this means that Cantor’s approach “determines number as a particular case of the hierarchy of sets”.172 In other words, set theory further abstracts even the most basic mathematical concepts – such as number itself – so that number denotes a specific property of mathematics, whereas sets constitute its universal backdrop. By contrast, Frege sought to establish a logical basis for number. Working within the “paradise that Cantor [had] created”, as the German mathematician David Hilbert famously put it in 1925, Frege attempted to undo the Kantian tradition of the philosophy of mathematics, which saw arithmetical operations as synthetic in nature, that is, requiring a spatial or temporal reference outside itself in order to be understood.173 For him, mathematics was not a matter of synthetic judgements relating to the world but instead an analytic a priori truth. As Badiou observes, “Frege’s approach […] seeks to ‘extract’ number from a pure consideration of the laws of thought itself. Number, according to this point of view, is a universal trait of the concept, deducible from absolutely original

169 Penelope Maddy, Naturalism in Mathematics (Oxford: Clarendon Press, 1997), 1. 170 Tzuchien Tho, “The Void Just Ain’t (What it Used to Be): Void, Infinity, and the Indeterminate,” Filozofski vestnik XXXIV, no. 2 (2013): 27. 171 Maddy, Naturalism in Mathematics, 27. 172 Badiou, Number and Numbers, 8. 173 David Hilbert, “On the Infinite,” in Philosophy of Mathematics, eds. Paul Benacerraf and Hilary Putnam (Cambridge: Cambridge University Press, 1983), 191.

58 principles”.174 To establish this, Frege “devised the formal language of the Begriffsschrift”, which relied upon and could be transposed into Cantor’s set theoretical foundation.175 In order to discern exactly how mathematics dealt with the nothing at this particular moment in its history – a moment temporally contemporaneous with Mallarmé – I shall examine how both Cantorian and post-Cantorian set theory address the concept of the nothing, as well as the philosophical readings of these developments given by Jacques-Alain Miller and Alain Badiou in the 1960s.

Akihiro Kanamori writes, “[i]t is among the set theorists that the null class, qua empty set, emerged to the fore as an elementary concept and a basic building block. Georg Cantor himself did not dwell on the empty set”.176 Similarly, Joseph Warren Dauben suggests that “though it would certainly have been appropriate, [Cantor] said nothing about the empty set, nor did he attempt to distinguish between a zero and a set containing no elements”.177 Perhaps one reason for Cantor’s own reluctance to think a concept of the nothing could be that reflection on the empty set requires a prior degree of reflection on sets themselves, a task that sits uneasily with his early articulation of set theory. In his last great work, Beiträge zur Begründung der transfiniten Mengenlehre (Contributions in Support of Transfinite Set Theory) (1895), Cantor defined a set as follows: “By a ‘set’ we mean any collection M into a whole of definite, distinct objects m (which are called the ‘elements’ of M) of our perception [Anschauung] or of our thought”.178 The striking thing about this definition is that it is neither rationally grounded nor inherently mathematical. Rather, it relies upon a subject’s innate capacity to intuitively recognise sets at an empirical level. In

174 Badiou, Number and Numbers, 8. 175 Maddy, Naturalism in Mathematics, 4. 176 Akihiro Kanamori, “The Empty Set, The Singleton, and the Ordered Pair,” The Bulletin of Symbolic Logic 9, no. 3 (2003): 275. 177 Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Cambridge/ Massachusetts/London: Harvard University Press, 1979), 88. Here, I would like to briefly introduce the distinction between ordinal and cardinal numbers – something that will concern me more when I turn to the infinite in my next chapter. Ordinal numbers are used as a way to generalise natural numbers and refer to their sequential nature (namely, their order). In set theory, every finite set can be ordered in terms of their largest and smallest member and there is an uncountable amount of infinite sets that can be ordered too. Cardinals comprise another way to generalise the natural numbers, only this time it is in terms of their cardinality (or size). Cardinality is generally defined in terms of a set’s bijection, wherein two sets can be said to have the same cardinality if and only if they can be put into a one-to-one correspondence with one another. Crucially, Cantor proved that infinite sets can have differing cardinalities, meaning that there are transfinite numbers which are larger than others. I shall explore this further in my next chapter. 178 Quoted in Dauben, Georg Cantor, 170.

59 his previous 1883 paper, the Grundlagen, Cantor did stress the part-whole relation set theory makes manifest: “In general, by a ‘manifold’ or ‘set’ I understand every multiplicity which can be thought of as one, i.e. every aggregate of determinate elements which can be united into a whole by some law”.179 However, while this definition points towards what set theory does for pure or “free” mathematics – since sets bring together multiples under the category of the one – it still does not indicate what they are; it highlights instead what they do.180 With Cantor’s definition of a set never leaving the space of intuition, one can only imagine how difficult it was to define the empty set, the set that has no members. And yet the empty set plays a crucial role in set theory’s development.

Cantor’s reluctance to speak about the empty set was not without precedence. In his The Mathematical Analysis of Logic (1847) – a book that argues for the coexistence of logic and mathematics – George Boole introduced the 0 into his system “without explanation”, positioning it as a purely technical construction requiring no philosophical or reflexive analysis.181 Boole writes, “the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination”.182 However, as Kanamori notes, Boole’s later An Investigation into the Laws of Thought (1854) “assigned to ‘0’ the interpretation ‘Nothing’, the class consisting of no individual”.183 Boole’s work is thus a precursor to Cantor’s, setting precedence for an intrinsic concept of a prior nothing that can found logico-mathematical analysis. By contrast, Cantor’s friend and colleague Richard Dedekind, who sought to construct a structural basis for mathematics and number, categorically denied the existence of the nothing. He chose instead to base number on the 1, rather than the 0 (or the empty set). As Badiou writes, “Dedekind abhors the void and its mark, and says so quite explicitly: ‘[W]e

179 Georg Cantor, “Foundations of a General Theory of Manifolds: A Mathematico-Philosophical Investigation into the Theory of the Infinite (Cantor 1883d),” in From Kant to Hilbert: A Source Book in the Foundations of Mathematics Volume II, ed and trans. William Ewald (Oxford/New York: Oxford University Press, 1996), 916. 180 “[...] it [mathematics] deserves the name of free mathematics, a designation which, if I had the choice, would be given precedence over the now usual ‘pure’ mathematics”. Cantor, “Foundations of a General Theory of Manifolds,” 896. 181 Kanamori, “The Empty Set, The Singleton, and the Ordered Pair,” 274. 182 George Boole, The Mathematical Analysis of Logic: Being an Essay Towards A Calculus of Deductive Reasoning (New York: Cambridge University Press, 2009), 3. Kanamori also quotes this stunningly illustrative passage, found in the opening lines of Boole’s text. 183 Kanamori, “The Empty Set, The Singleton, and the Ordered Pair,” 274.

60 intend here for certain reasons wholly to exclude the empty system which contains no element at all’”.184 These “certain reasons” are not altogether clear, but what Boole and Dedekind’s work does illustrate is that to be accepted, the nothing must be declared. Comparing Frege to Dedekind, Kanamori again writes, “[w]hereas zero was crucial to Frege's logical development, the particular ‘base element’ for Dedekind was immaterial and he denoted it by the symbol ‘1’ before proceeding to define the numbers ‘by abstraction’”.185 To understand how set theory came to embrace the existence of the empty set, a brief look at its development throughout the late nineteenth and the early twentieth century is required.

Noting that set theory’s raison d’être was to further abstract mathematics in order to solve certain paradoxes, the late nineteenth-century French mathematician Henri Poincaré worried that set theory might give rise to a series of new, unforeseeable problems. “We have put a fence around the herd to protect it from the wolves”, he writes, “but we do not know whether some wolves were not already within the fence”.186 Poincaré’s warning eerily anticipated what was to come. Bertrand Russell, for example, famously shattered the second edition of Frege’s Foundations of Arithmetic (1884) with nothing more than a short, mostly admiring letter. Russell presented a paradox suggesting that set theory’s central inconsistency arose when one tried to establish the set of all sets. He phrased this in the following terms: “[l]et w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself? From each answer the opposite follows. Therefore we must conclude that w is not a predicate”.187 In posing this question, Russell examined the concept of a greatest cardinal; a set of all sets that would technically be the largest available or knowable by set theoretical means. However, if sets cannot be members of themselves (for example, the set {Ø, 1, 2} cannot be included in the set {Ø, 1, 2}), then the definition of the set of all sets does not exist without falling into Russell’s paradox.188

184 Badiou, Number and Numbers, 14. 185 Kanamori, “The Empty Set, The Singleton, and the Ordered Pair,” 275. 186 Quoted in Maddy, Naturalism in Mathematics, 29. 187 Bertrand Russell, “Letter to Frege,” in From Frege to Gödel: A Source Book in Mathematical Logic, 1879- 1931, ed. Jean van Heijenoort (Cambridge/Massachusetts/London: Harvard University Press, 1967), 125. 188 The terms “belonging” and “inclusion” denote two of the most basic operations in set theory. If the set A can be defined as {Ø, 1, 2, 3, 4}, it is correct to say that the element 1 belongs to the set A. Given a new set, set B defined as {Ø, 1, 2}, because all the elements of B are also in the set A, set B is included in set A. On

Contrary to the narrative Badiou presents in Being and Event, the subsequent axiomatisation of set theory by Ernst Zermelo in 1908 was not a direct reply to Russell’s paradox. In fact, Zermelo had privately written to Dedekind alerting him to its existence. The axiomatisation of set theory did, however, dilute its potentially shattering troubles by formalising the minimal amount of necessary givens needed to underpin its operations. Zermelo writes:

I intend to show how the entire theory created by Cantor and Dedekind can be reduced to a few definitions and seven principles, or axioms, which appear to be mutually independent [...] I have not yet been able to prove rigorously that my axioms are consistent, though this is certainly very essential; instead I have had to confine myself to pointing out now and then that the antinomies discovered so far vanish one and all if the principles here proposed are adopted as a basis.189

Zermelo’s axioms were a revelation for set theory, expelling Poincaré’s wolves from Cantor’s paradise. His second axiom concerned the empty set: it stated, very simply, that there exists a set with no members. In formal language, it is written as follows: "x #y ¬(y $ x).190 By giving the mathematical nothing a place within his axiomatic system, Zermelo hoped to

provide principles for generating new sets from old that will produce all the sets needed to recapture Cantorian/Dedekindian set theory, but won’t produce the paradoxical sets. For this to work, he must begin somewhere; the principles of

the question of consistency – and speaking of Saunders Maclane’s dissatisfaction with set theory and his subsequent development of category theory – Maddy writes, “the mathematical benefits provided by set theoretic foundations do not depend on set theory being provably consistent”. Maddy, Naturalism in Mathematics, 30. 189 Ernst Zermelo, “Investigations in the Foundations of Set Theory 1,” in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, ed. Jean van Heijenoort (Cambridge/Massachusetts/London: Harvard University Press, 1967), 200-1. 190 While this is its technical expression, the mark of the empty set is written as follows: ø. This was first used by the mathematical collective Nicolas Bourbaki, led by André Weil in 1939.

62 generation must have something on which to work. That starting-point is the empty set; the Empty Set Axiom asserts the existence of a set with no members.191

As one canonical textbook puts it, all the operations and axioms of set theory are established in order to “make new sets out of old ones”.192 The axiom of the empty set, then, has a very specific role: it formally posits the place from which subsequent sets can be generated. Every axiom has a generative function, but the axiom of the empty set provides the mathematician with a starting point. Zermelo and Russell both note that the empty set’s existence is “fictitious” and some have even argued that it requires extra-mathematical or philosophical justification.193 What I want to argue, however, is that just like Mallarmé these two late nineteenth-century projects posit the nothing and not the One as their foundational building block. Nothing is again the foundational precursor to oneness, the place from which oneness originates. Moreover, the empty set must be a subset of every subsequent set.194 In a gesture that echoes Mallarmé’s argument about the internal absence that belongs to every word, the nothing, as a subset belonging to all sets, comes to be a part of every single set generated thereafter: what is absent in every set.

Set theory thus begins its internal operations on the basis of the nothing, the empty set. In light of this, I want to examine another aspect of the relationship between post-Cantorian mathematics and the nothing: how is the empty set – the mathematical nothing – figured with respect to the constitution of formal language? To address this question, I will rehearse a debate that occurred between Jacques-Alain Miller and Alain Badiou in the Lacanian- Althusserian journal Cahiers pour l'analyse. The centrality of this debate cannot be

191 Maddy, Naturalism in Mathematics, 40. 192 P. R. Halmos, Naive Set Theory (New York: Springer, 1974), 4. 193 Maddy provides an excellent discussion of the relevant literature in Maddy, Naturalism in Mathematics, 39-43. 194 Halmos writes of the bizarre proof that the empty set belongs to every set: “[t]he empty set is a subset of every set, or, in other words, ø % A for every A. To establish this, we might argue as follows. It is to be proved that every element in ø belongs to A; since there are no elements in ø, the condition is automatically fulfiled. The reasoning is correct but perhaps unsatisfying. Since it is a typical example of a frequent phenomenon, a condition holding in the ‘vacuous’ sense, a word of advice to the inexperienced reader might be in order. To prove that something is true about the empty set, prove that it cannot be false. How, for instance, could it be false that ø % A? It could be false only if ø had an element that did not belong to A. Since ø has no elements at all, this is absurd. Conclusion: ø % A is not false, and therefore ø % A for every A”. Halmos, Naive Set Theory, 8.

63 underestimated. Recalling it over twenty years later in Number and Numbers (2008), Badiou, echoing a line from Rimbaud’s 1872 poem “Qu'est-ce pour nous, mon cœur”, states, “I am here! I am always here”.195 Crucial as it may be for Badiou, it also constitutes a seminal moment in psychoanalytic history since the article that prompted the debate, Miller’s “Suture (Elements of the Logic of the Signifier)”, was first given during Jacques Lacan’s Seminar XII and remained close to his teaching for some years. In rehearsing this debate, I want to show how each figure highlights the importance of the nothing in the constitution of formal language, in this case post-Cantorian mathematics.

In “Suture”, Miller seeks to establish the general structural logic of every logical system, a “minimal logic” that applies “to all fields of knowledge”.196 This “minimal logic” conditions, produces and describes every type of structure and “should be conceived of as the logic of the origin of logic”.197 Where each structure seemingly takes itself to be autonomous – be it political, social, economic, psychological, literary, or even mathematical – for Miller something of any structure’s inherent self-determination relies on a fundamental moment of misrecognition. In other words, an essential component of any structure’s structure is repressed from the outset. He calls this process “suture”: “suture names the relation of the subject to the chain of its discourse; we shall see that it figures there as the element which is lacking, in the form of a stand-in”.198 Although this might sound overly abstract, Miller’s fundamental interest lies in examining the “general relation of lack to structure”; a thematic he argues had been present, but not entirely manifest, in the work of Lacan up until that point.199

195 “J’y suis! j’y suis toujours”. Badiou, Number and Numbers, 24. Modified trans. 196 Jacques-Alain Miller, “Suture (Elements of the Logic of the Signifier),” in Concept and Form: Volume 1, Key Texts from the Cahiers pour l’Analyse, eds. Peter Hallward and Knox Peden, trans. Jacqueline Rose (London/New York: Verso, 2012), 92. Miller’s essay is one of the recurring motifs that structures Tom Eyers’ recent history of Continental (post)-rationalism and his presentation of it has greatly influenced my understand of the Cahiers pour l’Analyse. Tom Eyers, Post-Rationalism: Psychoanalysis, Epistemology, and Marxism in Post-War France (London/New York: Bloomsbury, 2013), 31. 197 Miller, “Suture,” 92. 198 Miller, “Suture,” 93. 199 Miller, “Suture,” 93.

64 The controversy surrounding Miller’s article comes from his claim that even the “logician’s logic” is guilty of disavowing its own lack.200 To support this, he performs a “heterodox” reading of Frege’s logical and meta-mathematical number theory.201 As Tho puts it, Frege’s question is “how to present a theory of arithmetic without already presupposing numerical concepts”.202 For Frege, the essence of number cannot be known in and of itself, relying instead on a fundamental principle of extension that charts a relation between concepts, objects and numbers. Objects only appear as empirically unified, and they are underpinned by their conceptual unity, their concept. If someone perceives a certain amount of individuated objects, then one implicitly has an idea of what concept supports the objects. Through this relation of the concept to the object, Frege defines numbers as an extension of this principle. Miller writes, “from this can be deduced the definition of the assignation of number: according to Frege ‘the number assigned to the concept F is the extension of the concept ‘identical to the concept F’”.203 This process of “redoubling”, as he calls it, creates a homology between truth and self-identity. The number representing the concept F is defined as that which is identical to the concept F (the concept F, empirically deduced from F amount of objects). As Miller puts it, “truth is. Each thing is identical with itself”.204

Miller’s critique of Frege examines the way he treats the nothing. “That a thing should not be identical with itself”, he writes, “subverts the field of truth, ruins it and abolishes it”.205 The number zero or the empty set is conceptually underpinned by nothing, which makes it the sole number that does not have an object to which it could be identical: “this concept, by virtue of being a concept, has an extension, subsumes an object. Which object? None”.206 Nevertheless, Frege persists in giving it a definition, famously arguing that the numerical nothing is that which is not identical to itself. For Miller, this definition “sutures logical discourse” and forces the number zero to be “rejected from the dimension of truth”. He writes, “[a]s for this place, marked out by subsumption, in which the object is lacking, there nothing can be written, and if a 0 must be traced, it is merely in order to figure a

200 Miller, “Suture,” 92. 201 Eyers, Post-Rationalism, 32. 202 Tho, “The Void Just Ain’t (What it Used to Be),” 35. 203 Miller, “Suture,” 95. 204 Miller, “Suture,” 95, 96. 205 Miller, “Suture,” 96. 206 Miller, “Suture,” 97.

65 blank, to render visible the lack”.207 The zero – the point of lack in Frege’s system of truth and identity – manifests itself according to Miller as the “first non-real thing in thought”.208

However, as soon as Frege exposes the necessary lack of his own structural discourse, Miller argues, he immediately represses it: Frege’s “lack happens to be lacking”.209 Miller suggests that although Frege gives 0 a place within his discourse, he paradoxically positions it as a stand-in for the one: “This system is thus so constituted with the 0 counting as 1. The counting of the 0 as 1 [...] is the general support of the series of numbers”.210 Miller deduces this by examining another aspect of Frege’s system, one that attempts to understand how number sequences are generated and how they can be defined. While Frege notes that 0 is the origin of number, he goes on to perform an operation that defines the number 1 as “n+1” and 2 as “n+1+1” and so on. Miller comments on this as follows:

The impossible object, which the discourse of logic summons as the not-identical with itself and then rejects as the pure negative, which it summons and rejects in order to constitute itself as that which it is, which it summons and rejects wanting to know nothing of it, we name this object, insofar as it functions as the excess which operates in the series of numbers, the subject.211

For Miller, as soon as the pure nothingness of the real is recognised – the nothingness that should be the base from which mathematics or logic departs – it is subsumed back into the 1: the n here being a mark that functions as a 1, and not as a 0. This, he concludes, leaves Frege’s theory guilty of a suture. In Miller’s analysis, the mathematical concept of the nothing renders the “logician’s discourse” subject to a higher logic, one that affects poetry just as much as it does mathematics: the logic of suture, where the subject comes to stand- in for the lacking lack.

207 Miller, “Suture,” 97. 208 Miller, “Suture,” 97. 209 Jacques Lacan, Anxiety: The Seminar of Jacques Lacan Book X, ed. Jacques-Alain Miller, trans. A. R. Price (Cambridge/Malden: Polity, 2014), 42. 210 Miller, “Suture,” 97. 211 Miller, “Suture,” 99.

66 Published in the following edition of the Cahiers, Badiou’s reply to this essay makes no attempt to conceal its target – and in fact, he is not alone in distancing himself from how Miller thinks formal language works.212 Badiou writes, “the thesis we are defending here aims only at delineating the impossibility of a logic of the Signifier [or Suture] that would envelop the scientific order and in which the erasure of the epistemological break would be articulated”.213 He aims to show that the property of non-identity Miller thinks is the moment of repression in Frege’s system does not produce the lack he claims it does. Insofar as Badiou takes Miller to be mistaken, he goes on to propose an alternative account of how formal language is composed.

In Miller’s reading of Frege, there must be a pre-numerical thing or object that can be empirically identified in order to extend the concept to the account for numbers. But for Badiou numerical marks “never [have] any recourse to any self-identical thing”. Rather, “the logico-mathematical signifier is sutured only to itself”, not to an external reference.214 Badiou thereby creates a clear distinction between natural language, whose lack is necessitated by the problems of interpretation and iteration, and formal language, which can only refer to exactly what it is. Where Badiou agrees that the logic of suture exists when an empirical thing is necessary to determine a structure’s internal operations, he thinks that the development of formal language happens through a different process. Badiou thus lists four mechanisms that are necessary to construct a formal language: “concatenation”,

212 Stunning summaries of this essay can be found in Eyers, Post-Rationalism, 84-94; Tho, “The Void Just Ain’t (What it Used to Be),” 38-43; Ray Brassier, “Science,” in Alain Badiou: Key Concepts, eds. A.J. Bartlett and Justin Clemens (Durham: Acumen, 2010), 61-72. To invoke Eyers’ assessment of Badiou’s reply: “Badiou contests that Frege’s use of the non-identical (x$x) does not in any sense produce a ‘lack’ and, in turn, there is nothing to suture”. Eyers, Post-Rationalism, 39. Jacques Bouveresse, a philosopher notable for introducing analytic philosophy into France, recalls that Miller’s reading of Frege was part of his motivation for dismissing the Cahiers’ cohort: “Given that at the time I was in the process of reading Frege and doing so seriously, I had the feeling that Miller had really understood nothing. He didn’t even really try”. Jacques Bouveresse and Knox Peden, “To Get Rid of the Signified: An Interview with Jacques Bouveresse,” in Concept and Form: Volume 2, Interviews and Essays on the Cahiers pour l’Analyse, eds. Peter Hallward and Knox Peden, trans. Tzuchien Tho (London/New York: Verso, 2012), 256. 213 Alain Badiou, “Mark and Lack,” in Hallward and Peden, Concept and Form: Volume 1, Key Texts from the Cahiers pour l’Analyse, 160 ft. 4. Eyers understands Badiou’s general position as follows: “while the question of the nonidentical may be relevant to the ways in which psychoanalysis understands the articulation of the subject and the signifier, it is fundamentally inadmissible to any discussion of mathematical logic”. Eyers, Post-Rationalism, 84. 214 Badiou, “Mark and Lack,” 165.

67 “formation”, “derivation” and the “systematic level”.215 Concatenation simply refers to the “raw material” of mathematical marks; that is, the tools for writing that exist in mathematics.216 Formation designates the syntactical rules that govern the manipulation of these marks. For example, one must agree on what a plus sign means so there can be a universal iteration of its use “without remainder” or interpretation.217 The process of derivation allows one to identify a “well-formed expression” – a specific formula in a formal language such as 1 + 1 = 2 – while the systematic level allows for the construction of a general model that regulates or underpins these operations (for example, the axioms of set theory).218 Badiou’s critique of Miller suggests that he only attends to this fourth level, ignoring these other stages of the development of formal language.

In terms of the nothing, then, Badiou is quite clear: “the zero is not the mark of lack in a system, but the sign that abbreviates the lack of a mark”.219 Insofar as the nothing is formulated in the first three orders of formal language and only then reintegrated on the systematic plane, the nothing “is sutured only to itself”: it has no lack and produces no lack.220 Moreover, because it is only what it is, the mathematical mark of the nothing, the empty set, is as close as one can get to signifying or giving presence to an actual nothing. As Burhanuddin Baki comments, the set theoretical “void is and is not nothing”.221 It is the nothing insofar as it does not take on any semantic meaning outside of itself. Yet it is not the nothing insofar as it still gives presence to that which has no presence by virtue of the fact that it is marked. Nevertheless, it is as close as one can get. In true Althusserian fashion, Badiou’s final point is that “there is no Subject of science” since formal language produces no lack to which the subject could be sutured.222 The autonomy and self-

215 Badiou, “Mark and Lack,” 160. This last term comes from Tho’s reading of Badiou, not from Badiou himself – although Tho does indeed argue that it is already implicit in Badiou’s discussion. Tho, “The Void Just Ain’t (What it Used to Be),” 40. 216 Badiou, “Mark and Lack,” 160. 217 Eyers, Post-Rationalism, 86. 218 Badiou, “Mark and Lack,” 161. 219 Badiou, “Mark and Lack,” 165. 220 Badiou, “Mark and Lack,” 165. 221 Burhanuddin Baki, Badiou’s Being and Event and the Mathematics of Set Theory (London/New York: Bloomsbury, 2015), 101. 222 The full quotation foreshadows Badiou’s later concerns, albeit without the idea of the subject we see emerge: “[w]e can claim in all rigour that science is the Subject of philosophy, and this is precisely because there is no Subject of science”. Badiou, “Mark and Lack,” 173.

68 sufficiency of mathematical writing – its status as an integrally rule-governed universe that has no subject to suture – leads Badiou to surmise that “Science is an Outside without a blind-spot”: it does not refer to anything outside itself, and the existence of the nothing, the point from which it begins, does not trouble its internal consistency.223

Accompanying this neat proposition is a footnote that invokes Mallarmé. Badiou writes:

If one wants to exhibit writing as such, and to excise its author; if one wants to follow Mallarmé in enjoining the written work to occur with neither subject nor Subject, there is a way of doing this that is radical, secular, and exclusive of every other: by entering into the writings of science, whose law consists precisely in this. But when literary writing, delectable no doubt but obviously freighted with the marks of everything it denies, presents itself to us as something standing on its own in the scriptural Outside, we know in advance [...] that it merely sports the ideology of difference, rather than exhibiting its real process.224

What Badiou remarkably notices here are the numerous similarities that exist between mathematics and the Mallarméan project, particularly when it comes to the nothing. Both imply the elocutionary disappearance of the poet and the mathematician; both aim to exist in themselves and for themselves; both rely on their total system for meaning (the words or marks that autonomously light each other up without reference to an outside); and both begin on the basis of nothing. Yet the young Badiou concludes that despite the seemingly Mallarméan virtues of mathematics, formal language is exempt from the iteration of the “ideological signifier”.225 That is, mathematical marks have a finite usage, whereas signifieds inhabit the signifier and thus mean different things in different contexts. There is, then, a real disjunction between how Mallarmé’s poetry and mathematics operate, even though both of them declare the nothing to be at the basis of everything. For Badiou, what Mallarmé desired, mathematics completes.

223 Badiou, “Mark and Lack,” 172. 224 Badiou, “Mark and Lack,” 172 ft. 24. 225 Eyers, Post-Rationalism, 87.

69 The last section of this chapter will examine just how far the comparison between the Mallarméan and the mathematical nothing can be taken, by offering a reading of the poem “Sonnet en -yx”. Before doing so, I shall make one final point about the nothing, particularly as it has morphed throughout Badiou’s philosophical system. In Being and Event, set theory provides the discourse on being qua being and allows for a presentation of what can be said of being as such. The set theoretical void, the empty set, thereby occupies a privileged position. As we saw in the introduction, sets comprise consistent multiplicities that are counted-as-one in the situation. However, that “the one is” is only an “effect of structure”.226 Because the mathematicians declare that the void exists through the axiom of the empty set, ontology is forced to admit that “the one is not, that the being of consistency is inconsistency”.227 “The nothing”, for Badiou, “names that undecidable of presentation which is its unpresentable”: one knows the nothing exists, but it cannot simply be presented after the structuration has occurred.228 The nothing is thus “subtracted” from presentation, but nonetheless insists within the space of set theoretic ontology.229 For the philosopher, the void therefore names the being of being; being’s “proper name”.230 What I want to highlight here is that the nothing, the empty set, marks the presentation of the unpresentable. In turning to “Sonnet en -yx”, I shall thus address the following question: to what extent is it possible to think of Mallarmé’s poetic nothing as functioning in an analogous fashion to the mark of the empty set? In other words, can a poem really present the unpresentable?231

226 As he continues, “[n]othing is presentable in a situation otherwise than under the effect of structure, that is, under the form of the one and its composition in consistent multiplicities”. Alain Badiou, Being and Event, trans. Oliver Feltham (London/New York: Continuum, 2005), 52. 227 Badiou, Being and Event, 53. 228 Badiou, Being and Event, 54. 229 Badiou, Being and Event, 55. 230 “This name, this sign, indexed to the void, is, in a sense that will always remain enigmatic, the proper name of being”. Badiou, Being and Event, 59. 231 Badiou, it must be said, in no way thinks the poem can achieve this. As he writes, “[n]aturally it would be pointless to set off in search of the nothing. Yet it must be said that this is exactly what poetry exhausts itself doing; this is what renders poetry, even at the most sovereign point of its clarity, even in its peremptory affirmation, complicit with death. [This] is because poetry propagates the idea of an intuition of the nothing”. In other words, for Badiou poetry can never demonstrate the existence of the nothing in the rigorous fashion that mathematics does. Rather, poetry’s nothing, for the philosopher, is purely based on intuition. Badiou, Being and Event, 54.

Thus far, I have shown that both Mallarmé’s poetry and post-Cantorian mathematics begin on the assumption that the nothing is prior to the existence of any unity or oneness. Through reading the work of Jacques-Alain Miller and Alain Badiou, I have also argued that the set theoretical nothing operates in a technical manner that reflexively illustrates how formal language is composed. The mathematical mark of the nothing refers only to itself and exists within a self-contained rule-governed universe. In the final section of this chapter, I shall investigate whether it is possible to view Mallarmé’s nothing in an analogous fashion. While at first glance both concepts of the nothing appear to share a number of similar qualities – such as impersonality and autonomy, as well as the place of origin – I shall test if this is in fact the case through a close reading of “Sonnet en -yx”. Here is the poem in its final version:

Ses purs ongles très haut dédiant leur onyx Her pure nails on high displaying their onyx L’Angoisse, ce minuit, soutient, lampadophore The lampbearer, Anguish, at midnight sustains Maint rêve vespéral brûle par le Phénix Those vesperal dreams that are burnt by the Phœnix Que ne recueille pas de cinéraire amphore And which no funeral amphora contains

Sur les crédences, au salon vide: nul ptyx On the credenzas in the empty room: no ptyx Aboli bibelot d’inanité sonore Abolished shell whose resonance remains (Car le Maître est allè puiser des pleurs au Styx (For the Master has gone to draw tears from the Styx Avec ce seul objet dont le Néant s’honore). With this sole object that Nothingness attains).

Mais proche la croisée au nord vacante, un or But in the vacant north, adjacent to the window panes, Agonise selon peut-être le décor A dying shaft of gold illumines as it wanes Des licornes ruant du feu contre une nixe. A nix sheathed in sparks that a unicorn kicks.

Elle, défunte nue en le miroir, encor Though she in the oblivion that the mirror frames Que, dans l’oubli fermé par le cadre, se fixe Lies nude and defunct, there rains De scintillations sitôt le septuor. The scintillations of the septuor.232

The way I propose to compare the mathematical nothing with the nothing in “Sonnet en - yx” is through the infamous “ptyx” that appears at the end of the first line of the second quatrain. The debate surrounding the “ptyx” centers on whether or not the word is truly meaningless. Here, we should also distinguish between the “Rien” that appeared in “Salut”

232 I have slightly modified the last line of Henry Weinfield’s otherwise excellent translation because, in order to force the rhyme, he writes “The scintillations of the one-and-six”. Given that this thesis is concerned with mathematics, it is potentially misleading to invoke the property of addition when citing an English translation of “Sonnet en -yx”. Otherwise, I have faithfully reproduced his version. Mallarmé, Collected Poems, 69.

71 and the “ptyx”. The word “Rien” follows a relation of signifier to signified, where it signifies that which is not. “Rien” thus has a decisive meaning, even though its meaning is nothing. In terms of the “ptyx”, however, I want to initially suggest that it has no signified to which it refers other than itself. Rather than meaning nothing, the “ptyx” is meaningless – and the way this is figured in the poem enables us to read it as a mark that is sutured only to itself. The “ptyx” therefore gives the poet a way to present the unpresentable in the poem in a manner outside of the signifier/signified relation. However, after working through this, I then turn to the question of sonority in “Sonnet en -yx”. I examine the importance of sound in the poem, as well as in other writings by Mallarmé, to show that as soon as one thinks of the “ptyx” as sutured only to itself, the inherent sonority of the word pulls it back into the sound/sense relation outlined in my first section. By looking at the function of sound in Mallarmé’s overall project, I argue that the “ptyx” must be posited as that which is sutured to the outside of itself, something I then link to an older conception of mathematics at work within the poem.

It might be useful to look firstly at what Mallarmé himself said about the “ptyx”. Although there have been numerous attempts to determine the etymological root of this word, he was quite comfortable with its obscurity.233 In May 1868 Mallarmé writes to Eugène Lefébure, “[...] I may write a sonnet and as I have only three rhymes in -ix, do your best to send me the true meaning of the word ptyx, for I’m told it doesn’t exist in any language, something I’d much prefer, for that would give me the joy of creating it through the magic of rhyme…”234 This phrase, “the magic of rhyme”, is a curious one: “magic” might obviously be juxtaposed to the careful and rigorous principles at work in mathematical reasoning or literary creation, but it could also imply the creation of something out of nothing. Yet given

233 Weinfield writes, “the [ptyx] does occur in Greek, where it seems to designate a fold (a favourite Mallarméan concept) and, by implication, a seashell”. Moreover, he continues to provide a list of other scholars who have fixed the word with an etymological meaning. Ellen Burt, for example, has noted that it could refer to “a seashell, a writing tablet, a fold, and a receptacle”. Mallarmé, Collected Poems, 217. See also Burt’s highly influential article: Ellen Burt, “Mallarmé’s ‘Sonnet en yx’: The Ambiguities of Speculation,” Yale French Studies, no. 54 (1977): 55-82. 234 Mallarmé, Selected Letters of Stéphane Mallarmé, 85. “[…] je fisse un sonnet, et que je n’ai que trois rimes en ix, concertez-vous pour m’envoyer le sens réel du mot ptyx, ou m’assurer qu’il n’existe dans aucune langue, ce que je préférais [sic] de beaucoup afin de me donner le charme de le créer par la magie de la rime”. Mallarmé, Correspondance, 386.

72 the poet’s comments about the relationship of the “ptyx” with the nothing, magic seems here to signify the opposite: the creation of a mark of nothing out of nothing.

Another remark Mallarmé made about the poem refers to its original title, “Sonnet allégorique de lui-même”, the allegory of itself. As I suggested in the first section of this chapter, the poet aimed at absolute impersonality with respect to the central nothingness of the poem. “Sonnet en -yx” is Mallarmé’s first attempt to achieve “an internal mirage created by the words themselves,” as he put it when sending the poem to Henri Cazalis.235 Moreover, the influence of Edgar Allan Poe’s The Philosophy of Composition (1846) on the poet’s general method and “Sonnet en -yx” in particular must be noted – an essay Mallarmé himself translated into French. Here, Poe writes that poetic composition should proceed “with the precision and rigid consequence of a mathematical problem”.236 This is certainly evident in “Sonnet en -yx”, where the words aim to take on all the superficial qualities of formal language mentioned above, gaining meaning only through the reciprocal relations they hold to one another in the so-called “scriptural Outside”. More than this, however, its narrative – or hypothesis – actually stages a drama between the nothing and the something, between various orders of nothingness and potential modes of presence. Given these features, I want to make the claim that “Sonnet en -yx” deserves a privileged status because, unlike the other instances of the nothing that appear in his work (such as the opening “Rien” of “Salut”), it is entirely centered on the problem the nothing presents to poetry.

Thematically speaking, “Sonnet en -yx” begins by giving presence to absence. The opening stanza is set in a “salon vide”, its temporality is that of “ce minuit” and there is a certain “Angoisse” fuelled “by the disappearance of light”.237 The reader is immediately thrust into a spatial and temporal situation defined in terms of absence. There is nothing orienting the reader other than that which is not there: what is not in the room, and the light that has disappeared. Gardner Davies has linked the opening setting of “Sonnet en -yx” to

235 Mallarmé, Selected Letters of Stéphane Mallarmé, 86-7. “[…] un mirage interne des mots mêmes”. Mallarmé, Correspondance, 392. 236 Edgar Allan Poe, “The Philosophy of Composition,” in Literary Theory and Criticism, eds. Leonardo Cassuto (Toronto: Dover, 1999), 102. 237 Alain Badiou, Conditions, trans. Steven Corcoran (London/New York: Continuum, 2008), 54.

73 Mallarmé’s work on the solar drama. Roger Pearson notes this as well, writing “[t]hat the solar drama was rarely far from Mallarmé’s mind” because of “his efforts to secure from Longmans the translation rights to A Manual of Mythology in the Form of Question and Answer published in 1867 by the Reverend George William Cox [...] His translation-cum- adaptation would appear eventually in 1880 under the title Les Dieux antiques, and subtitled ‘A New Mythology’”.238 Les Dieux antiques stages a Greek mythology of creation through the constant rising and setting of the sun. According to Rancière, this solar drama is aesthetically presented in the poem through “the pure act of appearing and disappearing, whose model is provided by the daily sunrise and sunset”.239 In terms of “Sonnet en -yx”, we might think about the solar drama as the first way Mallarmé thematises the tension between the nothing and the something. Yet the poem subverts its usual order. As Davies notes, “unlike the preceding ones, [“Sonnet en -yx”] does not offer us a direct evocation of the sunset. It is as if situated at the second stage of the drama [...] Anxiety maintains the memory of the vanished light and of all that it had inspired in the poet”.240 Davies links this to the “[m]aint rêve vespéral brûle par le Phénix”, the burnt light that cannot reappear, like the Phœnix itself, because no funeral amphora contains its ashes. The motif of the solar drama is in fact present in “Sonnet en -yx”, but the sonnet does not play out the traditional movement of appearance and disappearance that is found in his other poems. Rather, it begins in the drama’s second stage – that of absence – by thrusting the reader into a temporal and spatial situation continually defined in terms of that which is not.

The second thematic order of the nothing is that of death, which is metaphorised in the final two lines of the second quatrain. Here, the “[m]aître” or the poet “est allé puiser des pleurs au Styx” with an object that “Néant” subsumes. The room evoked in the first quatrain is empty because the poet who had previously occupied it has descended into the realm of death, the river Styx. This image echoes Mallarmé’s spiritual crisis; he continually evoked his own death during this period. For example, he writes to Theodore Aubanel that “[e]very

238 Pearson, Stéphane Mallarmé, 92-3. 239 Jacques Rancière, Aisthesis: Scenes from the Aesthetic Regime of Art, trans. Zakir Paul (London/New York: Verso, 2013), 98. 240 “[...] ce sonnet ne nous offre pas, comme les précédents, une évocation directe du couchant. Il commence à la deuxième étape du drame [...] l’Angoisse entretien le souvenir de la lumière évanouie et de tout ce qu’elle avait inspiré au poète”. Gardner Davies, Mallarmé et le Drame Solaire (Paris: Librairie José Corti, 1959), 108.

74 man has within him a Secret, many die without having found it and won't find it because, since they are dead, neither it nor them will have any further existence. I have died and been born again with the gem-encrusted key to my final spiritual Casket”.241 While the Master in “Sonnet en -yx” must return from the river Styx, death is first necessary to find the “gem-encrusted key”; the “seul objet” that is reborn alongside the poet. Yet like the order of absence described in the first stanza, “Sonnet en -yx” does not show the movement of life to death as it begins instead with the latter. The poet’s return from the river Styx is never shown; indeed, he appears already to have died. In other words, every presence in the opening of the poem is always already an absence, every something is always already a nothing, and every life is always already a death.

“Sonnet en -yx” appears immediately to privilege the nothing over the something, and yet the indeterminacy between the two still holds some degree of legitimacy. This is embodied in the second rhyme scheme: the “or”, translating as “gold” or “now” in English. But Mallarmé, who authored a book on the English language in 1878 entitled Les Mots anglais, was known to pun on English words throughout his work. For example, the poem’s first phrase echoes “Ses purs Anglais”, “its pure English”. The “or” thus points to the constant flickering between the something and the nothing that circulates throughout the poem’s images and themes: is there only absence and death or is there in fact something? This question is brought to bear upon the meaning or non-meaning of the “ptyx”. Despite Mallarmé’s own proclamations that the word is indeed meaningless – and despite the critical injunction to bestow the word with an etymological root – this word does not simply mean nothing, not least because Mallarmé often placed “Néant” or “Rien” into the poem to signify nothingness. Following a comma that separates the appearance of the “nord vacant”, the enjambment of this line gives the reader the same choice that is presented throughout the entirety of the poem: “un or”. With the deliberate plays on the English language in mind, this phrase materialises the subject matter of the poem: the one (the “un”) or the nothing, which is immediately passed over as the reader moves on to the next

241 Mallarmé, Selected Letters of Stéphane Mallarmé, 66. Modified trans. “Tout homme a un Secret en lui, beaucoup meurent sans l’avoir trouvé, et ne le trouveront pas parce que, morts, il n’existera plus, ni eux. Je suis mort, et ressuscité avec la clef de pierreries de ma dernière Cassette spirituelle”. Mallarmé, Œuvres Complètes I, 703.

75 line. Here, the enjambment causes a sensation akin to what Paul Valéry, following Mallarmé, deemed the poem’s ability to prolong a “hesitation between sound and sense”.242 As the reader enunciates this particular phrase, a flickering of the poem’s central concern emerges through a moment of sonic hesitation: is there the one, the “un”, or is there that which cannot be presented and must be passed over as one line ends and the next begins? In this sense, the type of nothingness at stake within the poem moves beyond the thematic level of representation. Mallarmé’s idea of the nothing in “Sonnet en -yx” is not simply descriptive, giving presence to absence through images of empty rooms and lights that have disappeared. Rather, through the “ptyx” Mallarmé transports the unpresentable into the very material of the poem itself, lifting the concept of the nothing from the level of description to that of material reality.

Before developing this argument, it is worth noting that what I have been calling the orders of nothingness in the poem are in fact catalogued by the “yx” rhyme. Starting with the “ptyx”, the following rhyme is “styx”, which as noted is an allusion to the kind of nothingness found in death.243 The next rhyme is “nixe”, a mythical water creature that appears and disappears in the mirror invoked in the third tercet, again echoing the movement of the solar drama. The etymological roots of the word “nix” lie in the eighteenth-century German word for nothing or nul – “nitch” or “nix” – which originated from the High Middle Germanic word “nihtes”.244 This particular order of the nothing thus signifies absence: the mythical nixe who is already “nix”. Lastly, Mallarmé uses the word “fixe” to fix the three orders of nothingness to one another. The unpresentable belongs to the “ptyx”, death to the river “styx”, and absence to the “nixe”.

242 Quoted in Giorgio Agamben, “The End of the Poem,” in Agamben, The End of the Poem: Studies in Poetics, trans. Daniel Heller-Roazen (Stanford: Stanford University Press, 1999), 109. 243 The river Styx was famously immortalised as a poetic theme by Hesiod in Theogony (700 B.C.), where it was tied to hatred, but also to a water nymph. We shall turn shortly to the presence of Ancient Greek themes within “Sonnet en -yx”. 244 Amidst a discussion of Johannes Kepler’s scientific treatise “On the Six-cornered Snowflake”, Samuel Frederick and Nilima Rabl write: “The gift of snow also provides the scientist with a cross-linguistic pun: the Latin word for snow (nix) is a homonym with the German word for nothing (nichts; commonly pronounced nix): ‘Ask a German what Nix means, and he will answer ‘nothing’...” See, Samuel Frederick and Nilima Rabl, “Dividing Zero: Beholding Nothing,” SubStance 35, no. 2, Issue 110 (2006): 74.

76 Moreover, the actual “x” sound produced by this rhyme orients the entirety of “Sonnet en - yx”. The poem’s volta occurs in the movement between the nothingness described in the first two quatrains and the emergence of the stars in last tercet, something I shall return to shortly. In the poem’s seventh line (the middle of the poem) the “avec-ce” works to create the sonic presence of the “x” sound, and links together each order of nothingness. As Marchal writes, it is in this sense that the “‘ptyx’ is thus par excellence the word that designates nothing”: its presence is everywhere in the poem and yet the “ptyx” has no presence beyond its immediate material inscription.245 It has no meaning, and yet its unique sonorous quality haunts the poem in a ghost-like fashion.

There may in fact be one definitive reason to suspect that the “ptyx” is an attempt to convey the presentation of the unpresentable – to place the mark of nothing – in the poem. In a fragmented essay entitled “Notes sur la langue”, which documents the sole remains of a linguistic project pursued but never completed by the poet between the years 1869-1870, Mallarmé makes one of his few explicit references to a philosopher.246 Here, he speaks of the “great and long period of Descartes,” an age that includes La Bruyère, Fénelon, “a hint of Baudelaire” and also, curiously, “mathematical language”.247 He concludes by placing himself at the closing point of this Cartesian age. As many have argued, this reference to Descartes shares the assumptions of Mallarmé’s broader ideas about language.248 Elizabeth Rechniewski writes, “influenced by Descartes’s Discours de la méthode [...] he [Mallarmé] explores the possibility of applying Science to Language, of ‘language reflecting on itself’”.249 Mallarmé’s highly heterodox reading of Descartes concerns what he calls “fiction”. “All method is a fiction”, the poet pronounces, and language is “the instrument of fiction”.250 For Mallarmé, the radical doubt of the cogito highlights the human’s innate

245 “‘Ptyx’ est donc par excellence le mot qui ne désigne rien”. Marchal, Lecture de Mallarmé, 179. 246 As Marchal comments, the title ‘Notes sur le langage’ is not Mallarmé’s original. Originally published in 1926 and 1929 by Mallarmé’s son in law, Dr. Benniot, its first title was “Diptyque” and “Diptyque II”. Mallarmé, Œuvres Complètes I, 1360. 247 “[...] la grande et longue périod de Descartes [...] Puis, en général: du La Bruyère et du Fénelon, avec un parfum de Baudelaire. Enfin, du moi, et du langage mathématique”. Mallarmé, Œuvres Complètes I, 872. 248 See, Durand, Mallarmé: du sens des formes au sens des formalités, 34-43; Marchal, La Religion de Mallarmé, 83-91; Elizabeth Rechniewski, “Ambiguity, the Artist, the Masses, and the ‘Double Nature’ of Language,” CLCWeb Comparative Literature and Culture 12, no. 4 (2010): 1-9. 249 Rechniewski, “Ambiguity, the Artist, the Masses, and the ‘Double Nature’ of Language,” 6-7. 250 “Toute méthod est une fiction [...] Le langage lui semble [...] l’instrument de la fiction”. Mallarmé, Œuvre Complètes I, 872.

77 capacity to invent and dream on the basis of a language haunted by nothingness and absence. Descartes, for him, is the first person to stumble across the common essence of the human: that singular animal capable of doubting and reconstructing reality on the basis of nothingness. Man, for Mallarmé, is thus a fictional animal. As Marchal puts it, “[m]aking language the instrument of fiction is to act as if words [...] say nothing other than absence and death, nothing other than nothingness; in other terms, it makes words the instrument of an absolute negativity, and the poem consumed by this negativity, equal to ‘Sonnet en -yx’, is the enclosure where language, by its own reflection, applies to become the place of nothingness”.251 What Mallarmé calls “fiction”, then, is simply another way of talking about the nothingness of “Sonnet en -yx”.

While “Notes” establishes the principles of a science, a linguistic science, capable of capturing the Cartesian concept of fiction, I want to be a bit more speculative in my reading, if only because Mallarmé continually references mathematics – something most commentators pass over. He continues his discussion of Descartes by writing a cryptic little note regarding the reception of the philosopher by international mathematicians, despite previously including “mathematical language” in his favourable description of the Cartesian lineage:

We have not understood Descartes; other countries have taken him over, but he did arouse French mathematicians. We have to take up his momentum, interrogate our mathematicians – and not use any foreign land, whether Germany or England, except as a counter-proof: helping us in that way with what they have taken from us. Besides, the hyperscientific momentum only comes from Germany, not England, which cannot adopt a pure science, because of God, whom Bacon, its legislator, respects.252

251 “Faire du langage l'instrument de la fiction, c'est faire comme si les mots [...] ne disaient rien d'autre que l'absence et la mort, rien d'autre que le néant; c’est, en d’autres termes, faire des mots l’instrument d’une négativité absolue, et du poème où se consomme cette négativité, à l'égal du sonnet en -ix, l'enclos où le langage, par sa réflexion même, s'applique à devenir le lieu du néant”. Marchal, La Religion de Mallarmé, 87. 252 “Nous n’avons pas compris Descartes, l’étranger s’est emparé de lui: mais il a suscité les mathématiciens français. Il faut reprendre son mouvement, étudier nos mathématiciens — et ne nous servir de la contre-

Poets, for Mallarmé, must study and interrogate the mathematicians, who are too hesitant to “adopt a pure science” because of their respect for God. Nevertheless, mathematical language belongs to the Cartesian lineage, a lineage Mallarmé believes to be founded on nothingness. He thus notes the paradoxical status of mathematics: on the one hand, mathematical language is founded on the nothing, yet on the other mathematicians themselves hold too much respect for God. If the poets are to “interrogate” the mathematicians, then it will be only insofar as they show how the existence of the nothing overturns, rather than confirms, God’s rule.

However, contrary to Mallarmé’s reading, Descartes’ own thought in fact categorically denies the existence of the nothing. In his The World or Treatise on Light (1633) – a text notable for its revised publication due to the Church’s condemnation of Galileo’s treatise on motion – Descartes refutes a physicalist notion of the void. He argues that the unperceivable matter existing between visible and tangible substances (such as a subject interacting with an object) cannot be thought in terms of nothingness or the void. Instead, he writes, “the spaces in which we perceive nothing by our senses are filled with the same matter as those occupied by the bodies that we do perceive”.253 From this, Descartes rejects the philosophical concept of the void tout court, unambiguously stating that “it is a contradiction to suppose there is such a thing as a vacuum, i.e. that in which there is nothing whatsoever”.254 As Alexander Koyré puts it, “the void, according to Descartes, is not only physically impossible, it is essentially impossible”.255 But following the introduction of zero into Western mathematical writing after the advent of Pythagorean mathematics and during Alexander the Great’s conquest of Babylon in 331, Descartes’

épreuve l’étranger, l’Allemagne , que comme d’une contre-épreuve: nous aidant ainsi de ce qu’ils nous ont pris. Du reste le mouvement hyper scientifique ne vient que d’Allemagne, l’Angleterre ne peut à cause de Dieu, que Bacon, son législateur, respecte, adopter la science pure”. Mallarmé, Œuvre Complètes I, 872-3. This is Mary Ann Caws’ translation. See Stéphane Mallarmé, “Descartes,” in Mallarmé in Prose, ed and trans. Mary Ann Caws (New York: New Directions, 2001), 76. 253 René Descartes, The Philosophical Writings of Descartes: Volume 1, trans. John Cottingham, Robert Stoothoff and Dugald Murdoch (Cambridge: Cambridge University Press, 1985), 86. 254 Descartes, The Philosophical Writings of Descartes: Volume I, 229. 255 Alexander Koyré, From the Closed World to the Infinite Universe (Baltimore: The Johns Hopkins Press, 1957), 101.

79 mathematical work necessarily draws on the nothing.256 Even so, his mathematical demonstrations are not enough to convince him of the nothing’s radical power. Descartes may indeed be the founder of rationalism, but his rationalism is routinely subjected to the whims of the Divine. As he writes in Principles of Philosophy (1644), “[o]ur doubt will also apply to other matters which we previously regarded as most certain – even the demonstrations of mathematics and even the principles which we hitherto considered to be self-evident”.257 For Descartes, God is the only necessary, eternal being and thus he alone holds the power to change any property of the world, even the strictest bastion of reason: mathematics.

It is interesting that the very thing Mallarmé reads into Descartes is what Descartes himself went to great lengths to deny. A more speculative point of relation between the two could perhaps lie in the materiality of the “yx” rhyme in “Sonnet en -yx”. Pearson comments that due to the rarity of the “yx” rhyme in the French language, “X marks the spot” and is therefore “a symbol of rhyme: like rhyme, it is a reflection, a mirrored V, the V… stands for ‘vers’”.258 For him, the letter “x” must be treated as a single signifier that stands in for the concept of rhyme. However, due to the proximity of “Notes” to the composition of the poem – as well as the importance of the nothing to this moment in Mallarmé’s work on Descartes and “Sonnet en -yx” – I am more inclined to understand this in mathematical terms. While Descartes did not technically formalise the Cartesian plane (this was done by rival mathematician Pierre de Fermat in the seventeenth century), by the nineteenth century it was generally attributed to him. Taken together, the “yx” rhyme points to the only admissible site of the nothing in the Cartesian doctrine: the y,x or x,y coordinates that refer to the middle point of a graph, a point that is also marked by a double 0, a pre-Cantorian mathematical nothing. In line with the close relationship the poem has with the English language, the “pt” sound that opens the “ptyx” could then be read as “point” – which is spelt the same way in French. This inscribes the “ptyx” with the following meaning: the

256 According to Robert Kaplan, it was Alexander the Great’s invasion of India in 326 BC and “the later routes of commerce from Alexandria” that gave the Greeks, and the Western world more generally, the “gift of zero”. Robert Kaplan, The Nothing that Is: A Natural History of Zero (Oxford/New York: Oxford University Press, 1999), 37. 257 Descartes, The Philosophical Writings of Descartes: Volume 1, 194. 258 Pearson, Unfolding Mallarmé: The Development of a Poetic Act, 145.

80 “pt” of “yx”, the point of the nothing on a Cartesian plane. Moreover, while the t is silent in the French “point”, it is enunciated in the “ptyx”. Because the next rhyme is “Styx”, in which the “t” is again enunciated, the “t” sound of the “ptyx” is retrospectively guaranteed, meaning that the t takes on a sonic presence it would not ordinarily have. Each letter of the “ptyx” is thus imbued with a decisive meaning, but a meaning that functions more like a formal language than a natural one. The “p.t.” refers only to the point, while the “y.x.” refers only to the point of the nothing on the Cartesian plane. Each letter then becomes sutured only to itself – and that which it is sutured to is the nothing. The “ptyx” could then be read as an attempt to present a kind of mathematical mark of nothing in the poem. It is not a signifier that signifies nothing, for the “Rien” of “Salut” serves this purpose accordingly. It is instead an attempt to present the unpresentable, to force a mark sutured only to itself into the poem through “the magic of rhyme”. Mallarmé thereby ends up looking just as much like Descartes’ conscious successor than he does Cantor’s unconscious contemporary. But because of Descartes’ denial of the void, Mallarmé is at once Cartesian and anti-Cartesian, Cantorian and pre-Cantorian, in the same breath. Perhaps this really is as close as one can get to placing a mathematical mark into a poem.

However, to argue that there is an immediate homology between the mathematical and literary nothing in “Sonnet en -yx” would miss one crucial aspect: the relationship this poem establishes between the nothing, music and number. While I have argued that the “ptyx” can be understood in a similar fashion to the mathematical mark of the nothing, I now want to suggest that the bond between the nothing and music in the poem pulls the “ptyx” back into the framework of natural language. The question of sound is first glossed in the opening phrase of “Sonnet en -yx”, “Ses purs ongles”, which contains a kind of homographic pun on the nature of poetry and sound: “Ses pur-son”, “its pure sound”. This pun is to be read on a graphic and not necessarily sonic level, but the graphic itself is not divorced from music. Mallarmé often compared the graphic scene of the poem to a musical score, and remarked that the common point of music and poetry could be found in writing. Furthermore, the line that follows the “ptyx” is “Aboli bibelot d'inanité sonore”. Henry Weinfield translates this as “Abolished shell whose resonance remains”, but it is worth giving the line a more literal translation, something like “abolished trinket of sonorous

81 inanity”. With this translation in mind, what one finds in the “salon vide” is the “abolished trinket” that has a contingent sonic quality.

The peculiar articulation of the above line, a-bo-li / bi-be-lot / d’i-na-ni-té / sonore, is also important to note, because each syllable is necessarily accentuated. Through the repetition of the “a”, “b”, “o” and “n” sounds, the line takes on the sonic quality of babble (babiller in French). The “ptyx” may indeed have no meaning beyond itself, but Mallarmé does highlight the sonic quality of the word. Here, as Emilie Noulet has suggested, the Greek etymology of the “ptyx” should be recalled. Although this etymology does not designate the word’s decisive meaning, the “ptyx” nevertheless finds one of its origins in the word for a conch, “one of those shells from which, when bringing it to our ears, we have the sensation of listening to the sound of the sea”.259 Mallarmé thereby frames the “ptyx” in terms of the sound/sense relation of natural language described above: the sense of the “ptyx” is meaningless, and yet its sound is full of rich associations and possible meanings. But this raises a larger question: why is the category of sound important for the poet? How does the inherent sonic property of the “ptyx” affect his conception of the nothing, especially when placed alongside the empty set? To answer this, I shall now examine Mallarmé’s broader writings on sound in Divagations.

In one of his character sketches, Mallarmé assesses the “challenge [...] inflicted on modern poets [...] by Richard Wagner”, a challenge based in part upon the success of Wagner’s annexation of music to theatre.260 Through a combination of the two, the “empty, abstract, and impersonal” stage is given life by the “flood dispensed by Music”.261 “Music’s presence and nothing more”, Mallarmé writes, “constitutes a triumph”.262 Wagner thus “reconciled a whole intact tradition, just about to fall into decadence, with the virginal,

259 Quoted in Octavio Paz, “Commentary on the ‘Sonnet in IX’ of Mallarmé,” in Mallarmé in the Twentieth Century, eds. Robert Greer Cohn and Gerald Gillespie (London/Ontario: Associated University Presses, 1998), 120. 260 Mallarmé, Divagations, 108. “Singulier défi qu’aux poëtes dont il usurpe le devoir avec la plus candide et splendide bravoure, inflige Richard Wagner”. Mallarmé, Œuvres complètes II, 154. 261 Mallarmé, Divagations, 109. Modified trans. “[…] que l’acte scénique maintenant, vide et abstrait en soi, impersonnel, a besoin, pour s’ébranler avec vraisemblance, de l’emploi du vivifiant effluve qu’épand la Musique”. Mallarmé, Œuvres complètes II, 155. 262 Mallarmé, Divagations, 109. “Sa presence, rien de plus! à la Musique, est un triomphe”. Mallarmé, Œuvres complètes II, 155.

82 occult energy surging up from his scores”.263 Yet Mallarmé also denounces Wagner, stating that he will not allow poetry to be outshone by new innovations in music. For Mallarmé, Wagner’s mistake was to “decorate the magnificence of the origin”, to use his new art to return to some original nationalistic mythos.264 Art, by the poet’s estimation, is fundamentally lost if it attempts to enact and revive an origin based on some predicate (like race, class, gender), simply because the origin of art itself cannot be tied to an original meaning: “I command that one say where [Music] comes from, what its first meaning was, and what its destiny is”.265 Rather than returning to an original myth, art’s purpose is to invent a new one. “If the French spirit, which is strictly imaginative and abstract, and thus poetic,” Mallarmé writes, “ever projects a glimmer of Truth, it won’t be like that: it rejects, and is thus in agreement with Art as a whole, which would rather invent, any existing Legend”.266 Art – in this case, the poem – invents a new cosmic “Fable” written on the Skies, which dissolves existing myths in order to reinvent them. Art is thus generative, creating a myth that is yet to come, and while often looking back at the past, it does not solely and uncritically adopt it.

Yet to take up Wagner’s challenge is not simply to reject his vision tout court. It is rather to match it, to laud what is admirable and dispel what is not. Mallarmé’s idea of poetry thereby brings music to letters in the same way Wagner brought music to the stage: “[...] everything moves towards some supreme bolt of light, from which awakens the Figure that No One is, whose rhythm, taken from the symphony, comes from the mimicking of each musical attitude, and liberates it”.267 This “Figure that No One is” to whom the poem addresses signifies Mallarmé’s distance from Wagner. The origin of nation is not to be recalled because the poem is oriented towards an abstract conception of the “Figure” as

263 Mallarmé, Divagations, 109. “[…] il concilia toute une tradition, intacte, dans la désuétude prochaine, avec ce que de vierge et d’occulte il devinait sourdre, en ses partitions”. Mallarmé, Œuvres complètes II, 155. 264 Mallarmé, Divagations, 111. “… [u]ne magnificence qui décore sa genèse”. Mallarmé, Œuvres complètes II, 157. 265 Mallarmé, Divagations, 110. “[…] je somme qu’on insinue d’où elle poind, son sens premier et sa fatalité”. Mallarmé, Œuvres complètes II, 156. 266 Mallarmé, Divagations, 111. “Si l’esprit français, strictement imaginatif et abstrait, donc poétique, jette un éclat, ce ne sera pas ainsi: il répugne, en cela d’accord avec l’Art dans son intégrité, qui est inventeur, à la Légende”. Mallarmé, Œuvres complètes II, 157. 267 Mallarmé, Divagations, 112. “alors y aboutissent, dans quelque éclair suprême, d’où s’éveille la Figure que Nul n’est, chaque attitude mimique prise par elle à un rythme inclus dans la symphonie, et le délivrant!” Mallarmé, Œuvres complètes II, 157-158.

83 such, a figure unable to be defined by any existing predicate. But to be oriented towards something, for the poet, is to already have “taken from the symphony”. Through the essential quality of rhythm, poetry mimics and thus repurposes the “musical attitude”.

Mallarmé presents a similar argument in “Music and Letters”, where he establishes that to ask whether letters exist is to ask the same question of music. Recall that for the poet, verse has the capacity to supplement the internal deficiencies of language. Languages are imperfect because there are many different ways of speaking and each has a separate, but still incommensurable, articulation of the arbitrary relation between sound and sense. Insofar as verse can modify or rearticulate this contingent property of language, it touches upon the “Idea”. In “Music and Letters”, this is phrased as follows: “that Music and Letters are two sides of the same coin; here extending into obscurity; there dazzling with clarity; alternative sides to the one and only phenomenon I have called the Idea”.268 The sound of poetry, its inherent sonic property, is intimately linked with music: a single phrase of poetry can, in the words of Alain Badiou, “become the equivalent of all that music provokes”.269

But Mallarmé, following Wagner, must orient the musicality of poetry out into the social sphere: music is one of the unifying factors that restores the missing social links brought about by the death of God. At the end of his sketch of Wagner, he writes, “[t]he City, which gave, for this experience of the sacred, a theatre, imprints on the earth its universal seal”.270 For Mallarmé, theatre is one of the names signifying the unbreakable combination of music and letter. But one should recall that the conception of the City that the theatre addresses is founded on pure nothingness. In order to orient poetry towards the “Figure that No One is”, poets must become the “Musicienne du silence”, giving the nothing in all its forms – absence, the unpresentable, death and even silence – a sonorous quality.271

268 Mallarmé, Divagations, 189. Modified trans. “[…] que la Musique et les Lettres sont la face alternative ici élargie vers l’obscur; scintillante là, avec certitude, d’un phénomène, le seul, je l’appelai l’Idée”. Mallarmé, Œuvres complètes II, 69. 269 Alain Badiou, “Is it Exact that All Thought Emits a Throw of Dice?” trans. Robert Boncardo and Christian R. Gelder, Hyperion IX, no. 3 (2014): 64. 270 Mallarmé, Divagations, 112. Modified trans. “La Cité, qui donna, pour l’expérience sacrée un théâtre, imprime à la terre le sceau universel”. Mallarmé, Œuvres complètes II, 158. 271 Mallarmé, Collected Poems, 43.

84 The musicality of the “ptyx” thereby takes on a new importance. Rather than a conception of the nothing that is sutured only to itself – that exists by and for itself in an autonomous universe – the relationship between the word’s musical quality and the nothing is immediately oriented towards that which is outside of itself: namely, the social sphere. This property of Mallarmé’s poetry is in fact radically distinct from the Cantorian mathematics of the nineteenth and twentieth centuries. Whereas set theory works with a formal language that exists by and for itself, music both grounds the specificity of poetry (poetry alone brings music to letters) and allows it to address this “Figure that No One is” – the generic instance of the social formation. While the signification of the mathematical nothing is entirely minimal and sutured only to itself, the literary nothing is maximal, sutured to that which is outside of itself. Yet the themes of “Sonnet en -yx” are not altogether removed from the history of mathematics. In the final tercet of the poem, the reader witnesses the emergence of the stars: “se fixe / De scintillations sitôt le septuor”. Until this final line, the poem had only presented nothingness, absence, death and the unpresentable. The “nixe” of the first tercet lies “défunte nue en le miroir”, as if conceding that only nothingness exists. But out of this pure night of nothingness, the final image in “Sonnet en -yx” is that of a constellation, superimposed over the darkness of the sky. Robert Greer Cohn notes that the “septuor” should be thought of as a “musical term” like Pythagoras’s “nombre”, which again recalls the relationship between music, nothing and number.272 Although I have attempted to show that Mallarmé’s vision of the nothing and its relationship to music distinguishes it from the set theoretical empty set, I want to end this chapter by looking at a potential link between “Sonnet en -yx” and Pythagorean mathematics, an era of mathematical history during which mathematicians and thinkers forged a similar bond between music, number and constellations.

The larger doctrine of Greek mathematics, in particular Pythagoreanism, cannot solely be attributed to single figures. So much of Pythagoras’s original writings are lost that it is virtually impossible to establish what he actually thought, leaving most commentators to

272 Robert Greer Cohn, Towards the Poems of Mallarmé (Berkeley/Los Angeles/London: University of California Press, 1965), 144.

85 rely on subsequent accounts found in Aristotle or Plato.273 Walter Burkert writes, “under the influence of the Platonic tradition all mathematical science in early Greece is called Pythagorean”.274 Many of the most important mathematical thinkers in Ancient Greece – from Pythagoras himself to Zeno and Eratosthenes – embody a kind of Pythagorean doctrine, where, despite the local disagreements and debates they may have had with one another, each figure internalised a number of unconscious axioms that defined the general idea of mathematics at the time. I want to examine two of these axioms, which are striking not least because they are radically removed from the modern, set theoretical, idea of mathematics – but also because they seem to be operative in Mallarmé’s poetry. The first concerns the relation between the stars, the movements of the planets, and number. Burkert notes that “the risings and settings of the stars were thought of as related to the geometry of the sphere, and the irregularities in the path of the planets were explained by the combination of mathematically perfect circular movements”.275 This “astronomical conception [...] became the basis of the worldview of Plato and Aristotle and indeed dominated people’s ideas of the world and their attitude to it until the time of Galileo”.276 The movement of the stars and the appearance of constellations in the sky directly influenced how Ancient Greek mathematicians thought about number’s relationship to the world. Here, we should immediately notice the similarities between this view and Mallarmé’s solar drama. In contrast to the modern view, Ancient mathematics aligns legends and myths about the origins of the constellations and the rising and setting of the sun to mathematically reasoned argumentation: the matheme and the mytheme are one in the Pythagorean doctrine, just as they are in Mallarmé.

Thomas Heath confirms that with the suturing of “the risings and settings of the stars” to mathematical innovation, “we find, if not the origin, a striking illustration of the

273 As Thomas Heath writes, “[i]t is improbable that Pythagoras himself was responsible for the astronomical system known as the Pythagorean, in which the earth was deposed from its place at rest in the centre of the universe, and became a ‘planet’, like the sun, the moon and the other planets”. Thomas Heath, A History of Greek Mathematics: Volume I From Thales to Euclid (Oxford: Oxford University Press, 1921), 163. 274 Walter Burkert, Lore and Science in Ancient Pythagoreanism, trans. Edwin L. Minar, Jr. (Cambridge/Massachusetts: Harvard University Press, 1972), 303. 275 Burkert, Lore and Science in Ancient Pythagoreanism, 299. 276 Burkert, Lore and Science in Ancient Pythagoreanism, 331.

86 Pythagorean doctrines”.277 The groupings of the stars are scientifically important because of their patterned unity and empirical variability. But more importantly, their movements and properties can be mapped: the geometrical qualities of the constellations are also an object for mathematical analysis. Yet the axiom that “astronomy is pure mathematics” implies another axiom, that of the relationship between music and number.278 Here one finds the conglomeration of all the ideas that converge around what is perhaps the most famous Pythagorean formula: the harmony of the spheres. Given that one of the astronomical phenomena the Ancients studied was the distance and velocity between planets, the Pythagoreans thought astronomical principles expressed the same ratios as those found in the mathematical structures of music. “Harmonic intervals correspond to harmonic relationships of distance and velocity”, Burkert writes, “and since a musical tone [...] implies a uniform motion, one can infer from all this a Pythagorean system of astronomy”.279 In other words, the stars sing. Their distance in relation to other stars emit musical notes and like the planets themselves, the Pythagoreans developed an axiomatic bond between music and number. This link, according to Heath, is the crowning achievement of Pythagorean doctrine: “The capital discovery by Pythagoras of the dependence of musical intervals on numerical proportions led, with his successors, to the doctrine of the ‘harmony of the spheres’”.280 Pythagorean mathematics, then, is far from an internal formal language that exists by and for itself. Instead, it is immediately oriented towards that which is outside of itself: its outside makes up its inside.

The Greek etymology of the “ptyx” certainly provides a first link between Mallarmé’s own time and that of Ancient Greece. Yet what one finds in “Sonnet en -yx” more specifically is a reprisal of these Pythagorean axioms. I have already shown just how crucial the category of music is for Mallarmé, how it gives language its texture and orients poetry towards the social sphere. But the only specific point at which the poet links music to number is found in the emergence of the stars at the end of “Sonnet en -yx”. The puns on sound, the orders of the nothing and the emergence of the stars all show how the “septuor”, the term that

277 Heath, A History of Greek Mathematics, 67. 278 Heath, A History of Greek Mathematics, 165. 279 Burkert, Lore and Science in Ancient Pythagoreanism, 352. 280 Heath, A History of Greek Mathematics, 165.

87 combines music and number, is far from sutured only to itself. Rather, it too is oriented towards that which is outside of itself, that which implicates all the thematics and relations between nothingness and the social. Mallarmé at this moment looks more like Pythagoras’s unconscious contemporary than Cantor’s. In set theory, the concept of the nothing has a highly specific role: it operates as a starting point that allows for the creation of new sets and this function does not trouble set theory’s constitution as a discourse. To recall Badiou again, “the logico-mathematical signifier is sutured only to itself”; it refers only to what it is, always within its own autonomous scriptural Outside.281 However, for Mallarmé, the inherent sonic quality of literature means that it continually wanders in the opposite direction. “Literary art”, he writes, is that which “attaches itself to everything”. It is the art that cannot help but be oriented towards that which is outside itself.282 For the poet, even the nothing in all its forms is subject to this law. In “Sonnet en -yx”, there is no pure nothingness, no nothingness that refers only to itself. Rather, the nothing is always already the son-thing.

But this analogy between Mallarmé and Pythagoras can again be complicated when we recall that the concept of the nothing was unthinkable in Ancient Greek mathematics. While Brian Rotman points out that “it was mathematics that saw the first attempt to incorporate [...] nothing into its range of signifiers”, by his account this occurred decidedly after the Pythagorean doctrine. Rotman continues to note in great detail the reasons why figures such as Aristotle, Zeno, and Socrates forbade the existence of the nothing.283 However, their resistance to it is not a resistance to origins per se, as Greek mathematics is predicated on the existence of an all-powerful, all-generating One. For Badiou, the One/nothing dichotomy is what separates the Ancient from the modern because, he claims, the entire edifice of Greek mathematics “is supported by the being of the One”. The result of this is that “it is impossible to introduce [...] that other principle [...] which is zero, or the void”.284 Instead of the nothing being the place from which the semblance of oneness occurs, for the Greeks the existence of the One trumps any conception of nothingness. Mallarmé’s

281 Badiou, “Mark and Lack,” 165. 282 Mallarmé, Divagations, 195. Modified trans. “ce sujet où tout se rattache, l’art littéraire”. Mallarmé, Œuvres complètes II, 74. 283 Brian Rotman, Signifying Nothing: The Semiotics of Zero (Stanford: Stanford University Press, 1987), 64. 284 Badiou, Number and Numbers, 7.

88 reinscription of certain Pythagorean motifs thus does not simply enact a turning away from the modern in favour of the Ancient. Instead, the detail and thoughtfulness provided in establishing the epistemological, ontological, and social existence of the nothing on the part of the poet show him positioning his poetry both inside and outside of his own time. Like the Pythagoreans, Mallarmé insists upon the “musicality of everything” and in “Sonnet en - yx” this insistence finds its form in the relationship between music, number and the stars.285 However, unlike the Pythagoreans, Mallarmé does this on the basis of the very thing that was considered impossible in Ancient Greek mathematics. “Modernity”, as Badiou writes, “is defined by the fact that the One is not”.286 And yet, paradoxically, Mallarmé’s status as a thinker of modernity entails a return to the other of the modern: namely, the Ancient. This return does not repeat the Ancient, however; to use Milner’s term, it simply “recalls” it.287

285 Mallarmé, Divagation, 208. “[…] la musicalité de tout”. Mallarmé, Œuvres complètes II, 210. 286 Badiou, Number and Numbers, 65. 287 Jean-Claude Milner, “The Tell-Tale Constellations,” trans. Christian R. Gelder, S: Journal of the Circle for Lacanian Ideology Critique, forthcoming September 2017.

89 Chapter 2: The Mystery in Numbers: On the Infinite, Chance and God

“‘Fixing the infinite’ is indeed the fundamental programme of Mallarméan poetics”288 – Quentin Meillassoux

“[...] I have been logically compelled [...] to the point of view which considers the infinitely great not merely in the form of something growing without limit [...] but also to fix it mathematically by numbers in the determinate form of the completed-infinite”289 – Georg Cantor

In this chapter, I revisit the original remark that prompted this thesis: Alain Badiou’s claim that Mallarmé and Cantor are unconscious contemporaries. Both thinkers, he argues, figure the infinite as a number, a radical and unique move that points to their similarity, their contemporaneity. But I ask whether the conceptual ideas of the infinite and number underpinning their usage of the terms are indeed the same. By examining three moments in Mallarmé’s career, I suggest that he forges an unshakable bond between chance and the infinite. On the other hand, I show how Cantor positions his mathematical concept of the infinite – what he in fact calls the transfinite – against a theological absolute infinite that serves to confirm God’s existence. I thus identify a number of crucial differences between the two figures in an attempt to complicate the view that regards them as contemporaries of one another. I conclude this chapter by surveying a larger trend within the late nineteenth and early twentieth century to ask whether Mallarmé’s idea of infinite chance could find another contemporary in the history of mathematics – this time from his immediate future.

Mallarmé’s 1869 Igitur is the first piece of writing he dedicates to chance and the infinite. Despite its high standing within the canon, Igitur is not strictly a poem or even a prose piece. Rather, it is a series of notes and impressions documenting the supposed narrative and philosophical basis of a poem Mallarmé never actually managed to complete. Un Coup

288 Quentin Meillassoux, The Number and the Siren: A Decipherment of Mallarmé’s Un Coup de dés, trans. Robin Mackay (Falmouth/New York: Urbanomic/Sequence, 2012), 140. 289 Georg Cantor, “Foundations of a General Theory of Manifolds: A Mathematico-Philosophical Investigation into the Theory of the Infinite (Cantor 1883d),” in From Kant to Hilbert: A Source Book in the Foundations of Mathematics Volume II, ed and trans. William Ewald (Oxford/New York: Oxford University Press, 1996), 890. Emphasis my own.

90 de dés, by contrast, was one of the last poems he wrote – and it too is concerned with the relationship between chance and the infinite. Even though Igitur and Un Coup de dés are separated by almost thirty years, it is surprising to see how much imagery and terminology they share: the infinite and chance, the dice throw and also mysterious references to number and mathematics. By examining a line of continuity between Igitur, Un Coup de dés and another piece of writing entitled the Book, I argue that Mallarmé has a consistent idea of the infinite that can be located in each of the above writings – and that, for him, the infinite always arises out of chance. Special focus will be given to the moments when Mallarmé discusses and evokes numbers, in order to provide an answer to the very first question put to his poetry: why did he encode his final poem – a poem that was to pronounce upon the “destiny of a future poetry”290 – in the quasi-mathematical images of a dice throw and a unique number?

Igitur opens with a character of the same name reciting his “devoir” to his ancestors, who have charged him with the task of annulling chance.291 Like the “Maître” that appears in Un Coup de dés, Igitur is a kind of poet-hero, and the ancestors he addresses are a set of “Caractères” that exist in a “grimoire”.292 A “grimoire” is a spell-book or a book of mystery, yet its etymology lies in the word grammaire or grammar. These mysterious grammarians thus represent Igitur’s poetic ancestors, figures from his past who ask him to carry out their legacy by overcoming chance. To do so, he plans to perform an absolute act that “existera en dehors — lune, au dessus du temps”.293 This act must produce an absolute and necessary occurrence, which through its necessity can negate the implications of a world where God has died and chance now reigns in his place. The significance of Igitur’s “devoir” lies in understanding whether or not he succeeds in performing this act: whether he fulfils his filial duty of annulling chance, or heretically rebels against the demands of his poetic ancestors.

290 Quentin Meillassoux, The Number and the Siren, 7. 291 Stéphane Mallarmé, Œuvres complètes I, Édition présentée, établie et annotée par Bertrand Marchal (Paris: Gallimard, 1998), 473. 292 Mallarmé, Œuvres complètes I, 473. 293 As Marchal writes, “Igitur, the ultimate inheritor of his race, finds himself the depository of an immemorial dream, that of abolishing chance by an absolute act. [Igitur, ultime héritier de sa race, se trouve le dépositaire d’un rêve immémorial, celui d’abolir le hasard par un acte absolu]”. Bertrand Marchal, Lecture de Mallarmé (Paris: Librairie José Corti, 1985), 261-2. Mallarmé, Œuvres complètes I, 473.

Igitur continues with a summary of “À peu près ce que suit”, a short paragraph outlining the argument of the poem.294 Here, Mallarmé first alludes to his highly specific concept of chance, which is, he writes, what guarantees the existence of the infinite – and which Igitur’s ancestors believe it is possible to overcome: “L’infini sort du hazard”.295 Importantly, Mallarmé spells hazard with a z, rather than with an s (le hasard being the proper word for chance in French). This alludes to the old English definition of “hazard” – modified from the old French, itself taken from Arabic – which does not just signify chance but, according to the OED, refers to “a game at dice in which the chances are complicated by a number of arbitrary rules”.296 The trope of the dice throw is central to Mallarmé, and in Igitur it provides the framework for the absolute act: “le Minuit où doivent être jetés les dés”.297 Igitur’s ancestors ask him to “finir en infini” with the dice throw, which is performed through a unique combination of “parole et geste”.298 When the bell of midnight sounds in its “vacante sonorité”, Igitur must throw a pair of dice to complete the act.299 Should the dice land on the number 12 – the numerical correspondent of midnight – Igitur will name the result. As Mary Lewis Shaw clarifies, “[s]hould [Igitur] succeed in his attempt to throw the perfect number 12 (a number that represents the union of temporal contraries, Midnight), he would paradoxically disprove the principle of chance, which by definition implies randomness, the opposite of causality”.300

To set the scene for this act, Mallarmé describes the spatial and temporal situation in which it can be performed. Igitur descends “les escaliers, de l’esprit humain” to enter a spiritual chamber blackened by the hour of midnight.301 His descent moves him away from the semblance of individual consciousness into a purely generic psychical realm: “the neutrality of a space of universal unconsciousness”, as Marchal puts it, “[the] only possible

294 Mallarmé, Œuvres complètes I, 474. 295 Mallarmé, Œuvres complètes I, 474. 296 “Hazard, n. and adj.” OED Online, Oxford University Press, June 2016. Web: 14 June 2016. 297 Mallarmé, Œuvres complètes I, 474. 298 Mallarmé, Œuvres complètes I, 474. 299 Mallarmé, Œuvres complètes I, 483. 300 Mary Lewis Shaw, Performance in the Texts of Mallarmé: The Passage from Art to Ritual (Pennsylvania: Pennsylvania University Press, 1993), 155. 301 Mallarmé, Œuvres complètes I, 474.

92 vector of the absolute Act”.302 This space allows him to speak for his entire poetic heritage, rather than to his own personal existential anguish, in a space where the “[le] présent absolu des choses” can shine.303 Beyond this spatial setting, Mallarmé also recounts the temporal lineage of Igitur’s life. He was “projeté hors du temps par sa race” – his poetic ancestors – and his place within this lineage “pèse sur lui en la sensation de fini”.304 Igitur was already anticipated to exist in a time beyond the finite lineage of his race, and although he admits to having attempted to be “présent dans la chambre”, his position as a figure out of time fills him with profound boredom.305 Yet he is still, according to his ancestors, the unique vessel that can perform this absolute act: “Il se sépare du temps indéfini et il est”.306 Igitur is thus positioned outside of normal time and space, existing only in “CIRCONSTANCES ÉTERNELLES”, as Un Coup de dés puts it.307

However, in a supreme gesture of authorial intrusion, Mallarmé pronounces that any absolute act – any possible result Igitur’s dice throw yields – would not negate chance. Instead, chance exists in a space beyond either affirmation or negation:

Bref dans un acte où le hazard est en jeu, c’est toujours le hazard qui accomplit sa propre Idée en s’affirmant ou se niant. Devant son existence la négation et l’affirmation viennent échouer. Il contient l’Absurde — l’implique, mais à l’état latent et l’empêche d’exister: ce qui permet à l’Infini d’exister d’être.308

This passage outlines the structural logic that defines Mallarmé’s idea of chance, which is not determined by any act of negation or even any gesture of affirmation. If Igitur were to roll a 12 on the dice, it would simply be a product of chance, not its negation. The same general principle holds when chance is affirmed: chance still insists, Mallarmé thinks, even when it is credited with producing the result of the dice throw. For Mallarmé, chance

302 “[...] la neutralité d’une espèce d’inconscient universel, seul vecteur possible de l’Acte absolu” Marchal, Lecture de Mallarmé, 263. 303 Mallarmé, Œuvres complètes I, 483. 304 Mallarmé, Œuvres complètes I, 498. 305 Mallarmé, Œuvres complètes I, 498. 306 Mallarmé, Œuvres complètes I, 499. 307 Mallarmé, Œuvres complètes I, 369. 308 Mallarmé, Œuvres complètes I, 476.

93 contains the “Absurde” within itself; it contains something other than the reality it produces. The contrary realities – the numbers of the dice throw that were not produced – are “latent”, remaining potential within what is actual. This logic establishes a constant, undecidable movement between what is (the result) and what is not (what the dice throw has not produced), which thus confirms the existence of the infinite. As Meillassoux writes of this passage, “[c]hance is credited with a power of contradiction (it ‘contains the Absurd’) that allows it to be what it is, as well as what it is not – and thus to be ‘infinite’ in the dialectical (rather than mathematical) sense”.309 The infinite is born from the innate undecidability central to chance’s structure: because one can never choose – one can never simply affirm or negate chance – the logic of chance passes beyond the finite and the concrete into an unresolvable dialectical oscillation between the actual and the potential.

Igitur’s dice throw therefore takes the form of a more fundamental question still: to throw or not to throw, to actualise or to remain potential. For Mallarmé, this Hamletian decision is of crucial importance. He elsewhere refers to Hamlet as “the latent lord”, latency of course being an inherent property of chance, and the question to be or not to be – to throw or not to throw – exemplifies the very structure of chance itself.310 As Eric Benoit puts it, “all the questions raised by the Mallarméan aesthetic are situated in a problematic of everything or nothing [...] the alternative always excludes the middle term”.311 But this excluded middle – the or – is central to Mallarmé’s idea of chance. The question is not whether Igitur throws the dice. Instead, it is a matter of the perennial possibility of either option becoming an actualised reality: the “or” of the to throw or not to throw is the very condition of chance, not its affirmation or negation. Upon this realisation, Igitur subsequently abandons his filial duty of performing the act, knowing instead that chance will insist regardless of what decision he makes. In a moment of pure defeat, he “secoue simplement les dés”.312

309 Meillassoux, The Number and the Siren, 30. 310 Stéphane Mallarmé, Divagations, trans. Barbara Johnson (Cambridge/Massachusetts/London: The Belknap Press of Harvard University Press, 2007), 125. “[…] le seigneur latent”. Mallarmé, Œuvres complètes II, 167. This argument is made by Marchal in Marchal, Lecture de Mallarmé, 263. 311 “Il est clair que toutes les questions soulevées par l’esthétique mallarméenne se situent dans une problématique du tout ou rien [...] l’alternative exclut tout moyen terme”. Eric Benoit, Mallarmé et le mystère du “Livre” (Paris: Honoré Champion, 1998), 55. Emphasis my own. 312 Mallarmé, Œuvres complètes I, 477.

94 Recognising that he cannot overcome the all-encompassing logic of chance, Igitur closes the grimoire and lays to rest on the tomb of his fallen ancestors.313

But all hope is not lost as Igitur replaces the idea of an absolute act with a new, capitalised Absolute: the Absolute is now nothing other than chance itself. As Mallarmé writes in his early notes to Igitur, “Le personnage qui, croyant [..] à l'existence du seul Absolu, s’imagine être partout dans un rêve [...] trouve l’acte inutile, car il y a un hazard a et n’y a pas de hazard — il réduit le hazard à l’infini — qui dit-il, doit exister quelque part”.314 At its most abstract level of existence, chance always incorporates its opposite, which in this case is necessity. There is thus both “un hazard” and “pas de hazard” within chance itself: the opposite of chance lies within its very structure, rendering it paradoxically the only necessary Absolute.315 Although Igitur remains in the tomb with his dead ancestors, he finds hope in the fact that the necessity of chance exists somewhere concrete – perhaps, he muses, for some figure from the future to stumble across.

Mathematics is also mentioned in Igitur. However, unlike the numbers that appear in the Book or in Un Coup de dés, Mallarmé invokes the mathematicians themselves. In the poem’s opening section, he writes: “Vous, mathématiciens expirâtes — moi projeté absolu”.316 It is difficult to know which mathematicians Mallarmé is referring to here. In Mallarmé’s Ideas in Language (2004), Heather Williams expertly tracks all the philosophical material he encountered during his study at university in order to examine

313 It is safe to say that the importance of this very specific logic of chance has gone largely unrecognised in the English-language reception of Mallarmé’s work. Mary Lewis Shaw, for example, thinks Mallarmé’s concept refers only to “randomness, the opposite of causality”. Mary Lewis Shaw, Performance in the Texts of Mallarmé, 155. Similarly, Malcolm Bowie, in his otherwise outstanding Mallarmé and the Art of Being Difficult (1978), writes that Un Coup de dés “is a complex picture of contingency and risk, and of those activities of will by which humans seek to discover pattern and purpose within their experience [...] Mallarmé’s ‘hasard’ is a condition of mind – the abidingly unstable medium of thought – and a condition also of the physical universe; the vulnerable and inventive self which is seen at work in the world [...] is the human itself”. Malcolm Bowie, Mallarmé and the Art of Being Difficult (London/New York/Melbourne: Cambridge University Press, 1978), 142. I am not sure where this idea is expressed in the poems, and Bowie himself gives no real indication. 314 Mallarmé, Œuvres complètes I, 841. 315 This thesis is extended to the level of narrative. Igitur does indeed end in the manner described above, yet in another section Mallarmé shows him throwing the dice, thus incorporating the opposite result into the narrative body. 316 Mallarmé, Œuvres complètes I, 474. It is worth noting that “expirâtes” means both to expire but also to die. This sentence could very well be interpreted as Mallarmé telling the mathematicians to die.

95 where the philosophically inflected terminology used for the first time in Igitur comes from. Words and phrases such as “idée”, “absolu” and, of course, “l’infini” appear out of the blue, and yet these ideas take on a fundamental importance throughout the rest of Mallarmé’s work. Williams’s conclusion is that the textbooks Mallarmé read at university hankered “after the good old days when philosophical language had been capable of profound expression, and the logical conclusion to this attitude was expressed by Antoine- Auguste Cournot, who argued that given the inescapable imperfection of philosophical language, mathematics was the pure science that every philosopher must aspire to”.317 One way of reading the poet’s dismissal of the mathematicians may indeed lie in his resistance to the model of education he encountered at university. For Mallarmé, it is poetry, not mathematics, which is the language every thinker must aspire to since it alone expresses the Absoluteness of chance. Nonetheless, for a poet who so consistently invokes number throughout his career, it is certainly important to note – as Paul Claudel has – that Igitur is the tale of “a wise man awaiting nothing else from science or from art (in short, from numbers)”.318 Whether it is a matter of the arbitrary sound and sense distinction explored in my previous chapter or of how letters come together to form a word, Mallarmé held to the idea that “through the miracle of infinity” words themselves are the very things that convey the logic of chance.319 But the question still remains, why does Mallarmé subsequently turn to number? Why do numbers, as opposed to words, take on such fundamental importance for the poet? To understand why, I shall now turn to the curious case of the Book, Mallarmé’s “Great Work”.

Much like Igitur, Mallarmé’s dream of forging a “Great Work” began in 1866, resulting “directly from the discovery of nothingness”.320 His letters from this period document a series of mysterious and often contradictory remarks about this project. Sometimes it is said to be a treatise on the nature of the World, while at others it is simply mentioned in passing with no defined content. Yet as the previous chapter suggested, the discovery of nothingness directly resulted from Mallarmé’s renouncement of God. The “Great Work”

317 Heather Williams, Mallarmé’s Ideas in Language (Bern: Peter Lang, 2004), 38. 318 Quoted in Paul Claudel and Robert Greer Cohn, “Two Notes on Igitur,” in Stéphane Mallarmé, Selected Poetry and Prose, ed and trans. Mary Ann Craws (New York: New Directions, 1982), 101. 319 Mallarmé, Divagations, 186. “[…] par le miracle de l’infinité”. Mallarmé, Œuvres complètes II, 66. 320 Mallarmé, Œuvres complètes I, 1372.

96 thus denotes the tool he uses to supplement and re-configure a community no longer beholden to traditional religious doctrines. In a letter addressed to Armand Renaud in the final days of 1866, Mallarmé alludes to his early plans for this project:

I've worked infinitely hard this summer, on myself first of all, creating, through the most beautiful synthesis, a world of which I am the God,%and on a Work which will result from that world, and will be, I hope, pure and magnificent [...] I’m allowing twenty years to complete it, and the remainder of my life will be devoted to an Aesthetics of Poetry. Everything is in draft form, I’ve only to determine the place of certain internal poems, a place which is predestined and mathematical.321

Setting the reference to mathematics aside for a moment, we should note that this project is a product of a world where man himself takes on the role traditionally occupied by God. The “Aesthetics of Poetry” similarly replaces the traditional teachings of Catholicism, so that the “Work” figures as a kind of Bible for this new world. It is, then, an attempt to conceive of a modern cult, where the determinism implied by God’s existence is replaced by an a-theological formation submitted to infinite chance.

Throughout the late 1860s, Mallarmé continues to write about this “Great Work”. But it is not until the 1880s that he composes a series of notes outlining the structural and thematic content of what he now renames the Book – notes collected, organised and published by Jacques Scherer in 1957. In 1885, Mallarmé writes to Paul Verlaine detailing his motivation for writing these notes, as well as describing some of its principles of creation:

[...] I’ve always dreamed and tried something different [...] to feed the furnace of the Great Work. What was it? That’s hard to say: a book, quite simply, in many volumes, a book which really would be a book, architectural and premeditated, and

321 Stéphane Mallarmé, Selected Letters of Stéphane Mallarmé, ed and trans. Rosemary Lloyd (Chicago/London: The University of Chicago Press, 1988), 72. “J’ai infiniment travaillé cet été, à moi d’abord, en créant, par la plus belle synthèse, un monde dont je suis le Dieu, — et à un Œuvre qui en résultera, pur et magnifique, je l’espère […] Je m’assigne vingt ans, pour l’achever, et le reste de ma vie sera voué à une Esthétique de la Poësie. Tout est ébauché, je n’ai plus que la place de certains poëmes intérieurs à trouver, ce qui est fatal et mathématique”. Mallarmé, Correspondance: Lettres sur la poésie, ed. Bertrand Marchal (Paris: Gallimard, 1995), 335.

97 not an anthology of chance inspirations, however marvelous… I’ll go further and say: the Book, persuaded that when all’s said and done there is only one, attempted unwittingly by whoever has written, even those of Genius. The orphic explanation of the Earth is the sole duty of the poet.322

The Book, now capitalised, exists to capture the totality of the world. “Everything in the world”, Mallarmé elsewhere remarks, “exists to end up as a book”.323 But this Book is immanently singular, a kind of meta-Book that captures and incorporates every other word ever written. By its very mandate, then, Mallarmé’s dream remains impossible, if only for the banal reason that the production of books and words ceaselessly continues. At the same time, though, Marchal argues that Mallarmé never stops writing the Book, as every written word intimately constitutes its makeup.324

The Book will thus always remain incomplete and “essentially ideal”, as Eric Benoit has put it.325 But this did not stop Mallarmé from providing numerous sketches and notes detailing its ceremonious rituals, the a-theological simulacrum of everything the poet admired about religion. One of the most curious aspects of these notes is just how prominent a role Mallarmé gives to numbers. Perhaps for the first time in his career, numbers are more important than words. For example, on page 3 of the Book we find only a series of seemingly arbitrary calculations: “480 = 96 X 5 [...] 480 480 = 960”.326 Sometimes these elementary equations refer to the most ordinary organisational details of the Book, such as its cost of distribution. At other times, however, the meaning is not so

322 Mallarmé, Selected Letters of Stéphane Mallarmé, 143. Modified trans. “[…] j’ai toujours rêvé et tenté autre chose […] pour alimenter le fourneau du Grand Œuvre. Quoi? c’est difficile à dire: un livre, tout bonnement, en maints tomes, un livre qui soit un livre, architectural et prémédité, et non un recueil des inspirations de hazard, fussent-elles merveilleuses… J’irai plus loin, je dirai: le Livre persuadé qu’au fond il n’y en a qu’un, tenté à son insu par quiconque a écrit, même les Génies. L’explication orphique de la Terre, qui est le seul devoir du poëte”. Mallarmé, Correspondance, 585-586. 323 Mallarmé, Divagations, 226. “[…] tout, au monde, existe pour aboutir à un livre”. Mallarmé, Œuvres complètes II, 224. 324 As Marchal writes in his editor’s notes, “[t]here are then two ways to conceive of the Book [...] In the first case, the Book is always to come; in the second, it is always already here, a virtual or ideal Bible, beyond all the real books [Il est alors deux façons de concevoir ce Livre [...] dans le premier cas, le Livre est toujours à venir, dans le deuxième il est toujours déjà là, bible virtuelle ou idéal, au-delà de tous les livres réels”. Mallarmé, Œuvres complètes I, 1375. 325 “… [e]ssentiellement idéal”. Benoit, Mallarmé et le mystère du “Livre”, 14. 326 Stéphane Mallarmé, Le “Livre” de Mallarmé, eds. Jacques Scherer (Paris: Gallimard, 1957), 3 bis (A).

98 clear. Nevertheless, Steven Cassedy has written that “[t]he place where we see Mallarmé’s mathematical imagination at its best is his writing about the Book”.327 In looking more closely at the Book, I want to examine why numbers play such a crucial role in its composition. I ask why, when in Igitur Mallarmé told the story of a “man awaiting nothing” from numbers, they seem to play such a foundational role here. That is, why are numbers invoked to structure the rituals of what is essentially a new religion dedicated to chance?

The notes for the Book outline a “ceremony of reading”.328 This ceremony or “séance” requires 24 participants, although in some of his calculations hundreds more are brought along to watch. Each séance is led by a 25th participant called the opérateur, who fulfils a kind of Priest-like duty, leading and instructing the poetic congregation. Mary Lewis Shaw argues that Mallarmé’s scattered notes can be divided into three separate categories. The first indicate “the number of the Book’s volumes and pages, its geometrical dimensions”.329 Here one might think of page 46, which simply contains the following description: “Un des 4 livres / ayant 5 motifs / différents % / distribués / 95 X 5 =”.330 In this note, we are given a very basic outline of the book’s structural determination: there are four different printed volumes of the Book, each with 5 different motifs, distributed 95 times. Mallarmé’s so- called “mathematical imagination” here is not strictly speaking mathematical at all; rather, these numbers refer to the distributional and organisational features of the Book. Second, there are “notes indicating the number of performances, the number and nature of spectators”.331 This second category is intimately linked to the third, the “notes revealing the correlations, or ‘identity’ between the various aspects of the text and its performance”.332 I am now going to briefly examine the second category of notes, since numerical marks feature most heavily in these. I argue that Mallarmé uses numbers to establish the Book’s form and that the geometrical and numerical structure of the

327 Steven Cassedy, Flight from Eden: The Origins of Modern Literary Criticism and Theory (Berkeley/Los Angeles/Oxford: University of California Press, 1990), 151. 328 This phrase is adapted from Meillassoux’s work. See Quentin Meillassoux, “Badiou and Mallarmé: The Event and the Perhaps,” trans. Alyosha Edlebi, Parrhesia, no. 16 (2013): 42. 329 Mary Lewis Shaw, Performance in the Texts of Mallarmé, 187. 330 Mallarmé, Le “Livre” de Mallarmé, 46 (B). 331 Mary Lewis Shaw, Performance in the Texts of Mallarmé, 187. 332 Mary Lewis Shaw, Performance in the Texts of Mallarmé, 187-8.

99 performances reprises the logic of chance first sketched in Igitur. All this, I claim, will set the scene for a discussion of the “unique Nombre” in Un Coup de dés.

The different performances detailed in the Book attempt to synthesise various art forms into the séance. In many senses, Mallarmé’s project relied upon extracting what was laudable in competing modes of artistic production – such as theatre or music, as we saw in the previous chapter – to further the advancement of poetry. Nowhere is this more evident than in the Book, which seeks to incorporate various structural features taken from theatre, ballet, poetry, drama, and so on. On one page, for instance, this structural indistinction is presented as follows:

The many varying structures of the ceremony are outlined in the above diagram.333 The opérateur is located at the lower end of the centre, where he occupies the place of the “Lect”, the reader. Ballet dancers and figures perform on the left hand side and another series of objects are placed on the right. On the top, where the stage is set, the “th.” (the theatre) is placed. These terms are continually found throughout the book, but in this particular diagram their structure is combined and each divergent art form is blended together under the rubric of the Mallarméan “séance”.

This is an important point because it pertains to the way numbers and geometric diagrams are utilised throughout the Book. Take, for instance, the diagram on page 95:

333 Diagram found in Mallarmé, Le “Livre” de Mallarmé, 102 (A).

100

In this diagram, the theatrical structure of the Book’s performance is linked to the number “12” – a number that continually reoccurs throughout the Book’s calculations.334 The “12 fois” corresponds to the four sections of the three lines, which is then redoubled underneath through the “...”. The 24 participants double this theatrical structure again and the “480+480” signifies how many audience members will watch the “séance”. While this is in some ways an arbitrary example picked out from a handful of similar pages, what I want to highlight is how these numerical and geometric equations refer not to mathematics as such, but rather to the inherent form and structure of the Book itself, which draws as much from a theatrical determination of space as it does from the rhythms of poetry. Moreover, the entirety of this diagram is oriented around the number 12, the same number that played a central role in Igitur. There are 12 lines and dots; the performance is to be held 12 times; the structure of the double – which is key to Mallarmé’s idea of chance – is exemplified by the 24 participants; there are 480+480 participants, a number that is a multiple of 12; and 4 plus 8 plus 0 equals 12.

Before examining the specific significance of this number, it is worth highlighting how Mallarmé consciously incorporates the Igiturian structure of chance into the Book’s form. Every ritualistic space follows a logic of opposition, where two alternatives are presented but neither is decided upon. (In the previous diagram, the 12 lines are mirrored and doubled

334 Diagram found in, Mallarmé, Le “Livre” de Mallarmé, 95 (A).

101 by the 12 dots.) Mary Lewis Shaw calls this the logic of “unity-in-duality”, but we might note that no unity is ever reached. Rather, the inherent oscillation of every ceremonial practice is celebrated for its own sake.335 Take, for instance, a little diagram that appears in the corner of page 41:

Here, the structure of the Book is presented precisely through the alternatives – what, echoing Benoit’s remark, is the “excluded middle” of the “ou”, the “or”.336 Neither option – walking horizontally or vertically first – is decided upon, but the “ou” separating the two signifies the structural logic of chance, wherein a religion that is dedicated to infinite chance contains the oscillatory logic of the infinite within itself: the latent is contained within the actual.337

How, then, is one to definitively understand the relation between numbers and the Book? In Mallarmé’s 1866 letter quoted above, he describes its organisation as “predestined and mathematical”, which at least partially attests to the way he aligns structure and form to number.338 However, the poet’s use of the term “mathematical” is very different to the way mathematicians understand mathematics. It simply refers to the form of the Book’s performances, the manner in which one “determine[s] the place of certain internal

335 Mary Lewis Shaw, Performance in the Texts of Mallarmé, 9. The concepts of unity-in-duality and identity- in-difference are the main threads of her book, and are not limited, she argues, to the Book. 336 Diagram found in Mallarmé, Le “Livre” de Mallarmé, 41 (A). 337 On page 91, Mallarmé in fact gives a 6 point outline of this structure, and we should note, of course, that 6 is half of 12: “(1) quatre livres (2) sont les deux moitiés (3) un, chaque (4) deux, la (5) avec la (6) l’autre en deux sens différents”. Diagram found in Mallarmé, Le “Livre” de Mallarmé, 91 (A). 338 Mallarmé, Selected Letters of Stéphane Mallarmé, 72.

102 poems”.339 Moreover, the numerical formalism of the Book is designed to be absolutely iterable without remainder. Each time the ceremony is enacted, the calculations are there to guarantee its repetition without difference, as difference is part of its very structural logic. Mallarmé thus draws on the capacity of numbers to provide a formal outline of the Book so that the differential structure, as well as the structuring ability, of chance can be repeated indefinitely.

Mallarmé therefore invokes number to describe poetry’s form, which must somehow contain the logic of infinite chance within itself. But what is the significance of the 12, and how does it relate to the “unique Nombre” of Un Coup de dés? At the dawn of the nineteenth century, ideas about poetry’s form in France were undergoing an “exquisite and fundamental crisis”.340 As Mallarmé pronounced to a cohort of Englishmen in 1894, “[v]erse has been tampered with”.341 This crisis was triggered by the death of Victor Hugo in 1885, the legislator of France’s “national cadence”, the alexandrine.342 While Hugo did not strictly adhere to the alexandrine in all his poems (as the Parnassians did, for example), he more or less stuck to variations on the 12 syllables of the alexandrine line, playing and modifying it where necessary. As Albert W. Halsall writes, “[f]ar from wishing to abandon the twelve-syllable alexandrine as an archaic throwback to French neo-classical tragedy, Hugo insisted [...] on the central function it was to serve in the Romantic drama”.343 In the opening of “Crisis of Verse”, Mallarmé writes the following of Hugo: “A French reader, his habits interrupted by the death of Victor Hugo, cannot fail to be disconcerted. Hugo, in his mysterious task, brought all prose%philosophy, eloquence, history%down to verse, and, since he was verse personified, he confiscated, from whoever tried to think, or discourse, or narrate, almost the right to speak”.344 But when Hugo died, the right of speech was no long confiscated; verse, instead, had nothing to lose but its shackles.

339 Mallarmé, Selected Letters of Stéphane Mallarmé, 72. 340 Mallarmé, Divagations, 201. “[…] une exquise crise, fondamentale”. Mallarmé, Œuvres complètes II, 204. 341 Mallarmé, Divagations, 183. “On a touché au vers”. Mallarmé, Œuvres complètes II, 64. 342 Mallarmé, Divagations, 204. “[…] la cadence nationale”. Mallarmé, Œuvres complètes II, 207. 343 Albert W. Halsall, Victor Hugo and the Romantic Drama (Toronto/Buffalo/London: University of Toronto Press, 1998), 69. 344 Mallarmé, Divagations, 202. “Un lecteur français, ses habitudes interrompues à la mort de Victor Hugo, ne peut que se déconcerter. Hugo, dans sa tâche mystérieuse, rabattit toute la prose, philosophie, éloquence, histoire au vers, et, comme il était le vers personnellement, il confisqua chez qui pense, discourt ou narre, presque le droit à s’énoncer”. Mallarmé, Œuvres complètes II, 205.

103

This supposed liberation, however, led French poetry into a period of poetic relativism, where the indistinct character of the word was favoured over the rigid determination of a pre-established meter. Clive Scott has examined the emergence of French free verse in great detail, writing that “[i]t is historically suspect to suppose that any single year in literary history enjoys the privilege of watershed, but 1886 might lay claim to such a privilege”.345 In 1886, the year after Hugo’s death, there was an explosion of free verse texts and journals, most of which came from Mallarmé’s close friends Gustave Kahn and Jules Laforgue. Mallarmé himself, however, was critical of the verslibristes. Responding to the question of what is “new” in free verse, the poet writes that he doesn’t “see%and this remains my intense opinion%an erasure of anything that was beautiful in the past”.346 For him, free verse secretly “benefits” from the alexandrine, despite its pretensions to have moved beyond metrical determination: “the recollection of strict verses haunts [their] approximations”.347 Mallarmé concludes his assessment of the rise of free verse with this devastating assessment: “but from this liberation, to hope for something else, or to believe, seriously, that every individual possesses a new prosody in his very breath [...] is a joke”.348 While one could no longer adhere only to the alexandrine after Hugo’s death, free verse certainly did not provide Mallarmé with the liberation it so confidently promised.

In terms of Un Coup de dés, how is one able to understand this relationship between number and the infinite with this crisis of verse in mind, a crisis that also concerns the number 12? It may help to view this crisis through the prism of Mallarmé’s logic of chance. For the poet, those who supported the re-inscription of the alexandrine after Hugo’s death believed, like Igitur’s ancestors, that poetry could overcome chance – that, in fact, poetry’s task consists precisely in this. Mallarmé writes that these figures believe they are “not in the infinite” whenever their “voice[s] encounter a rule” and that encountering and enforcing

345 See, Clive Scott, Vers Libre: The Emergence of Free Verse in France 1886-1914 (Oxford: Clarendon Press, 1990), 54, 63-8 for an exhaustive list of free verse publications in 1886 and the surrounding years. 346 Mallarmé, Divagations, 205. “je ne vois, et ce reste mon intense opinion, effacement de rien qui ait été beau dans la passé”. Mallarmé, Œuvres complètes II, 207. 347 Mallarmé, Divagations, 204. “Je dirai que la réminiscence du vers strict hante ces jeux à côté et leur confère un profit”. Mallarmé, Œuvres complètes II, 207. 348 Mallarmé, Divagations, 206. “mais, de cette libération à supputer davantage ou, pour de bon, que tout individu apporte une prosodie, neuve, participant de son souffle […] la plaisanterie rit haut”. Mallarmé, Œuvres complètes II, 209.

104 such a rule expels all the arbitrariness from language, all the latency that chance brings to bear upon the word.349 For Mallarmé’s ancestors, the blanks of the poem – those instances of chance that continually insist – can be negated precisely by the rigidity of metre. But by the same token Mallarmé thinks that those who advocate free verse affirm chance too quickly, without understanding its proper structure. He argues that despite free verse’s renunciation of the alexandrine, the verslibristes are more dependent on it than they believe. In fact, Mallarmé thinks that a discernable meter can always be identified regardless of whether it was consciously placed there or not. As Rosemary Lloyd comments, “[a]s a poet for whom the alexandrine was always the explicit or implicit starting point, [Mallarmé] considered the new form as inalterably tied to the twelve-syllable line, existing less in its own right than as a variation on that line”.350 However, for Mallarmé, free verse suffers from the opposite problem. Rather than believing poetry can overcome chance, free verse purports to be infinite in nature: any possible combination of words can be arbitrarily placed upon the page without the regulatory apparatus of meter intervening before the fact. And yet, as Lloyd continues, “he [also] believes the number of permutations to be strictly limited”. Despite free verse looking as though it is an affirmation of chance’s existence – and thus, by extension, the infinite – it remains strictly within the realm of the finite.351 Mallarmé himself confirms this in a letter addressed to Henri de Régnier, another advocate of free verse, when he states: “I don’t believe that the new combinations of the poetic line, since it was restructured, are infinite”.352 We must understand Mallarmé’s use of the word “infinite” in terms of the specific logic he accorded it. Free verse does not successfully “fix” the infinite because those who practice it believe they are able to move beyond the alexandrine. But, as Mallarmé insists, the potential will always remain latent within the actual, leaving free verse haunted by the very thing it so adamantly denies.

Infinite chance, as Mallarmé has understood it ever since he authored Igitur some 30 years prior, exists beyond the logics of negation or affirmation, beyond the split between the 12

349 Mallarmé, Divagations, 206. “[…] pas dans l’infinité des fleurettes, partout où sa voix rencontre une notation”. Mallarmé, Œuvres complètes II, 208. 350 Rosemary Lloyd, Mallarmé, The Poet and his Circle (New York: Cornell University Press, 1999), 188. 351 Lloyd, Mallarmé, The Poet and his Circle, 188. 352 Quoted in Lloyd, Mallarmé, The Poet and his Circle, 187.

105 of the alexandrine and the falsely infinite nature of free verse. If a poem is to be submitted to the structure and structuring effects of chance, then a new form must be invented, one capable of transcending the false opposition between these two poetic forms. Herein lies the key to solving the mystery of Un Coup de dés and its “unique Nombre”: how is it possible for a poem that looks like an object-lesson in free verse to embody both the logics of strict, determined meter and free, undetermined experimentation? In other words, how is it possible for a poem to contain its opposite within its very form and thus numerically fix the infinite?

It would not be possible to cover every aspect of Un Coup de dés in the short space I am giving it. Over the course of the twentieth century, no French poem has garnered as much commentary as Mallarmé’s masterwork, so much so that entire books are dedicated to the history of its reception.353 In what follows, however, I do want to clarify the relationship between the poem’s form, content, and the number, something that has been rarely examined. Un Coup de dés was first published in May 1897, one year before Mallarmé’s death, in the bilingual poetry journal Cosmopolis. Its initial design was modified to accord with the specifications of the journal, but the poem was republished in its originally conceived form in 1914, under direction from Mallarmé’s son-in-law Edmond Bonniot. Echoing the logic of the double that characterises the Book, Mallarmé’s original specifications demanded that the poem be printed on what Marchal calls “the double page”, two separate pages folded into one, totaling 50 x 33 cm in size (signified by a capitalised Page).354 The poem spreads over 11 double pages, not including its title page, and across these are a series of words that look as though they follow no particular organisational structure. The words seem to be thrown as if at random onto the concrete backdrop of the page. As Mallarmé writes, the Page “is taken as the basic unit, in the way that elsewhere the Verse or the perfect line is”.355 The poem, however, operates according to another internal logic. There are, by his own admission, several different organisational “motif[s]” and the “difference in the type faces” are used to signify the rising and falling of the voice when the

353 See Thierry Roger, L’Archive du Coup de Dés (Paris: Éditions Classiques Garnier, 2010). 354 “La double page”. Mallarmé, Œuvres complètes I, 1321. 355 Stéphane Mallarmé, Collected Poems: A Bilingual Edition, trans. Henry Weinfield (Berkeley/Los Angeles/London: University of California Press, 1996), 121. “[…] prise pour unité comme l’est autre part le Vers ou ligne parfaite”. Mallarmé, Œuvres complètes I, 391.

106 poem is read aloud.356 Moreover, some phrases are entirely capitalised, sprawled over the page in large font, while others are smaller, written in lowercase. And yet on first viewing, the poem does indeed appear to have no guiding form.

Mallarmé reluctantly authored a preface to Un Coup de dés, writing “I would rather that this note not be read, or, if glanced at, that it be forgotten”.357 This preface again references the crisis of verse, highlighting the poet’s contradictory feelings towards the habits of poetic tradition against the transgressive potential of free verse. He writes that Un Coup de dés expresses a “‘state’ that does not break with tradition at all”.358 Admittedly the tradition he is referring to concerns the process of publishing poetry in a journal, but he also writes that Un Coup de dés participates in a number of “particular pursuits that are dear to our time: free verse and the prose-poem”.359 As Larissa Drigo has noted, this affirmation does not mean that Un Coup de dés is simply an exercise in free verse or the prose poem, only that it participates in similar ideas. In fact, in an early draft of his preface, Mallarmé explicitly states that the poem is “executed according to habits in fact completely different from others which defy our tradition”.360 The form of the poem – its typological layout – is positioned as traditional and modern all at once, constrained by custom yet free in ambition.

Mallarmé continues to note that the poem utilises the backdrop of the blank white page, something he had previously linked to the nature of chance. He states:

The “blanks”, in effect, assume importance and are what is immediately most striking; versification always demanded them as a surrounding silence, so that a lyric poem, or one with a few feet, generally occupies about a third of the leaf on which it is centered: I don’t transgress against this order of things, I merely disperse

356 Mallarmé, Collected Poems, 122. “La différence des caractères”. Mallarmé, Œuvres complètes I, 391. 357 Mallarmé, Collected Poems, 121. “J’aimerais qu’on ne lût pas cette Note ou que parcourue, même on l’oubliât”. Mallarmé, Œuvres complètes I, 391. 358 Mallarmé, Collected Poems, 122. “[…] ‘état’ qui ne rompe pas de tout point avec la tradition”. Mallarmé, Œuvres complètes I, 392. 359 Mallarmé, Collected Poems, 122-123. Modified trans. “[…] poursuites particulières et chères à notre temps, le vers libre et le poëme en prose”. Mallarmé, Œuvres complètes I, 392. 360 Quoted in Larissa Drigo, “Folding and Unfolding the Infinite: Space-Time Relations in Mallarmé's Un Coup de dés,” S: Journal for the Circle of Lacanian Ideology Critique, forthcoming September 2016 (obtained via personal correspondence).

107 its elements. The paper intervenes each time an image, of its own accord, ceases or withdraws, accepting the succession of others; and, as it is not a question, as it usually is, of regular sound patterns or verses but rather of prismatic subdivisions of the Idea.361

Again, Mallarmé does not suggest that he is breaking with tradition. Rather, his aim is to disperse the function traditionally assigned to the blanks. Perhaps the most striking thing about this passage is that Mallarmé does not think the poem’s form can be detected through regular sound patterns or established identifications of verse.362 Un Coup de dés instead takes its form from the “prismatic subdivisions of the Idea”, the dimensional dispersions of the Idea scattered throughout the poem’s typographical iconography. As Marchal writes, “the form is here the mise-en-scene of the Idea” and the Idea, in this case, is that of infinite chance itself.363 What one finds in this preface are some clues that help foreground my argument: the poem neither breaks with tradition nor forecloses the modern penchant for free verse. Instead, both are at work within Un Coup de dés and these oppositional structures combine to comprise the Idea at the level of the poem’s form – what we can call, after the Book, its unique number.

Marchal notes that the logic of chance Mallarmé first described in Igitur is evoked at the beginning of Un Coup de dés. The opening Page simply reads, in exclamative capitals, “UN COUP DE DÉS” but then, on the second, we are launched into the same Igiturian tension between the generic psychical space – the ideal realm – and the material existence

361 Mallarmé, Collected Poems, 121. “Les ‘blancs’, en effet, assument l’importance, frappent d’abord; la versification en exigea, comme silence alentour, ordinairement, au point qu’un morceau, lyrique ou de peu de pieds, occupe, au milieu, le tiers environ du feuillet: je ne transgresse cette mesure, seulement la disperse. Le papier intervient chaque fois qu’une image, d’elle-même, cesse ou rentre, acceptant la succession d’autres et, comme il ne s’agit pas, ainsi que toujours, de traits sonores réguliers ou vers % plutôt, de subdivisions prismatiques de l’Idée”. Mallarmé, Œuvres complètes I, 391. 362 Paul Valéry does in fact recount that upon seeing the poem for the first time, it was as if he were directly experiencing the capacity of language to transcend the finite: “It seemed to me that I was looking at the form and pattern of a thought, placed for the first time in a finite space. Here space itself truly spoke, dreamed, and gave birth to temporal forms. Expectancy, doubt, concentration, all were visible things. With my own eye I could see silences that had assumed bodily shapes. Inappreciable instants became clearly visible: the fraction of a second during which an idea flashes into being and dies away; atoms of time that serve as the germs of infinite consequences lasting through psychological centuries – at last these appeared as beings, each surrounded with a palpable emptiness”. Paul Valéry, Leonardo, Poe, Mallarmé, trans. Malcolm Cowley and James R. Lawler (London: Routledge & Kegan Paul, 1972), 309. 363 “La forme est ici mise en scène de l’Idée”. Mallarmé, Œuvres complètes I, 1316.

108 of what is about to occur.364 A throw of dice, Mallarmé continues, “JAMAIS / QUAND BIEN MÊME LANCÉ DANS DES CIRCONSTANCES ÉTERNELLES / DU FOND D’UN NAUFRAGE”.365 The “CIRCONSTANCES ÉTERNELLES” of the dice throw serves two purposes. It first establishes the generic psychical realm in which the “drame”, as Gardner Davies calls it, of Un Coup de dés takes place.366 But secondly it also creates a hierarchy in the poem’s typography. The phrases that appear only in capitals are both within the “CIRCONSTANCES ÉTERNELLES” that the poem presents, but also outside them, pointing, as if from above, to the very circumstances they name. To make sense of this, it is worth quoting one more remark from Mallarmé’s preface: “Everything that happens is foreshortened and, as it were, in hypothesis; narrative is avoided”.367 Here, the poet rejects the traditional linearity of narrative, where one event guarantees the existence of the next. Un Coup de dés is rather constructed as a “hypothesis”, as a kind of immaterial psychical backdrop on which the events of the poem can be written. Of course, one might also note the quasi-mathematical connotations of the word “hypothesis”. As Marchal suggests, the term is quasi-mathematical in that it sets up an ideal construction, a realm that is not immediately provable or falsifiable.368 Instead, following the “CIRCONSTANCES ÉTERNELLES”, we find a “SOIT”, the only capitalised word on the 3rd Page – if we include the title Page in our count – which orients and thrusts the reader into this ideal, generic and psychical thought experiment: namely, the Mallarméan poem.369

The “hypothesis” occuring throughout the remainder of Un Coup de dés concerns a character much like Igitur, this time named “LE MAÎTRE”, who – in a reprisal of the themes and images from “Salut” and “À la nue…” – faces off against a raging “Abîme” that seems to have engulfed his ship.370 This is not explicitly stated in the poem; rather, it is suggested through a cascade of single lower-case words and phrases that are placed upon the page in descending order. The Master has escaped the shipwreck and floats on the sea’s

364 Mallarmé, Œuvres complètes I, 367. 365 Mallarmé, Œuvres complètes I, 369. 366 See, Gardner Davies, Mallarmé et le Drame Solaire (Paris: Librairie José Corti, 1959). 367 Mallarmé, Collected Poems, 122. Modified trans. “Tout se passe, par raccourci, en hypothèse; on évite le récit”. Mallarmé, Œuvres complètes I, 391. 368 Marchal, Lecture de Mallarmé, 263 369 Mallarmé, Œuvres complètes I, 370. 370 Mallarmé, Œuvres complètes I, 370.

109 foam, clutching two dice in his hands. There is no doubt that this scene echoes Igitur. Yet unlike Igitur, the Master does not hesitate to throw his dice. Instead, he hesitates not to throw.371 “Jadis”, Mallarmé tells us, the Master “empoignait la barre / de cette conflagration”, and hesitated before throwing to produce “l’unique Nombre qui ne peut pas être un autre”, completing his “destin”.372 But now the Master “hésite”, as if in reverse, because he is filled with radical doubt as to whether a simple throw of the dice can effectuate an absolute necessity. One should note that the line evoking “l’unique Nombre” is itself an alexandrine, echoing Igitur’s absolute act of rolling a 12 on the dice – something I shall return to shortly. The Master’s hesitation thereby delays “cette conjonction suprême avec la probabilité” that characterises the gesture of throwing. However, he nonetheless tempts “une chance oieuse”, that of the raging sea, and on the bottom of the 5th Page we read “N’ABOLIRA”, serving as an extended continuation of “UN COUP DE DÉS / JAMAIS / N’ABOLIRA”.373 On the 8th Page, a siren briefly appears. However, as soon as she is recognised, the siren immediately “évapor[e]” into the mist, although not before slapping a rock in the middle of the ocean with her tail, thus placing “une borne à l’infini”.374

Then, on the 8th and 9th Page, Mallarmé reveals his most iconic statement about the “unique Nombre”: “SI / C’ÉTAIT / LE NOMBRE / CE SERAIT / LE HASARD”.375 Instead of the Master overcoming his hesitation not to throw in order to produce a 12, idle chance has well and truly been tempted and exposes itself as the true number. This Page, the 9th, is written almost entirely in the conditional tense, as if to avoid the definitive logic of affirmation and negation outlined above. Moreover, Mallarmé writes of how this new number, that of chance itself, is “évidence de la somme pour peu qu’une”.376 Any act of counting the result serves only as evidence of chance’s logic insisting within the dice

371 As Marchal rightly points out, both hesitating to throw and hesitating not to throw are different and yet each produces the same outcome: hesitation. Mallarmé, Œuvres complètes I, 1316. 372 Mallarmé, Œuvres complètes I, 372-3. 373 Mallarmé, Œuvres complètes I, 374-5. 374 Mallarmé, Œuvres complètes I, 380-1. 375 Mallarmé, Œuvres complètes I, 382-3. 376 Mallarmé, Œuvres complètes I, 383.

110 throw. The “plume” with which the Master writes thus falls into a “rythmique suspens du sinistre”, as he realises chance never stops manifesting itself within verse.377

The poem concludes with a statement concerning the seemingly brute inaction that characterises a world dominated not by an infinite God, but instead by infinite chance. Chance exists and any divine action, such as rolling a 12 that would correspond with the “toque de minuit” the Master appears to be wearing, is rendered inert.378 “RIEN”, Mallarmé thus exclaims in bold capitals, “N’AURA EU LIEU / QUE LE LIEU”.379 The entire poem thereby appears to have simply been a study of the all-pervading inertia that defines a world submitted to chance. There is no narrative, only hypothesis; the Master can neither throw the dice nor keep them clenched in his fist: he can only hesitate. However, one must recall that the logic of chance always contains its opposite within itself. Nothing has taken place, Mallarmé continues – this time reprising the end of “Sonnet en -yx” – “EXCEPTÉ / PEUT- ÊTRE / UNE CONSTELLATION”.380 The Master gazes at the sky above him and sees, perhaps, a constellation shine through the blackness of the sky. In order to reverse the poetic gesture, which writes black letters upon the white page, the constellation supplements the black sky – the image of inaction itself – with the whiteness of the stars.381 Like the blanks of the poem that remind Mallarmé of chance’s existence, the blanks in the sky – the constellation that, perhaps, emerges – become a poetic figuration of the unique number and thus of chance itself.

How is it possible to understand the relationship between content and form here, to understand how the “unique nombre” of the “CIRCONSTANCES ÉTERNELLES” refers to the poem’s seemingly arbitrary distribution of words on the page? As I noted in my introduction, the philosopher Quentin Meillassoux has examined the relationship between the crisis of verse, the death of God, and the “unique Nombre” in his recent The Number and the Siren. Meillassoux follows a very similar line of argumentation to the one I have

377 Mallarmé, Œuvres complètes I, 370. 378 Mallarmé, Œuvres complètes I, 379. 379 Mallarmé, Œuvres complètes I, 384-5. 380 Mallarmé, Œuvres complètes I, 386-7. 381 Jacques Scherer first acknowledged this point in his excellent introduction to the notes for the Book. Jacques Scherer, “Physique du Livre,” in Mallarmé, Le “Livre” de Mallarmé, 50.

111 been pursuing here – although he puts almost no emphasis on the Book – and in so doing proposes that the poem is “coded”.382 This code, he claims, bestows Un Coup de dés with the semblance of form, determined by an operation of counting where each word of the poem is tallied together to produce an actual determination of the “unique Nombre”. He thus stumbles – as if, he thinks, by chance – on the number 707, which he claims holds both semantic and formal value in Mallarmé’s oeuvre.383 707 symbolises the seven stars of the constellation that, perhaps, appear at the end of the poem, as well as the “SI” of the “COMME SI” that opens and closes the 6th Page, which signifies both the 7th note of a musical scale and is also the French word for “if”. The 0, the central nothing of the poem, is represented by the “RIEN”, and confirmed through the multiple and varied writings Mallarmé dedicated to the concept. Yet Meillassoux recognises that the Mallarméan structure of chance does not allow for a fixed, necessary determination of the number. Instead, the 707 must contain its opposite; it must be instilled with a formal ambiguity in order to truly encapsulate the logic of infinite chance. Here, he draws upon certain ambiguities within Un Coup de dés: for example, that there are different editions of the poem, containing 706 or 708 words. Most importantly, however, Meillassoux also notices that the “peut-être” can be counted as either one or two words. He writes, “[...] this compound word – this unique word that succeeds in unsettling our count, from its margins, by exhibiting the undecidable nature of its procedure – the reader will doubtless have guessed, is none other than: PEUT-ÊTRE”.384 Meillassoux thus concludes that this code comprises the unique number since the constant oscillation between it and its others – the 707 and the alternative counts – exemplify the Mallarméan logic of chance. This reading aims to solidify Mallarmé’s unique position with respect to the crisis of verse. Un Coup de dés is not simply free verse since it is coded. The poem is on the one hand fixed by the 707, yet the internal ambiguity that pertains to the count renders it, on the other hand, without fixed form. Both options are in play while neither can be decided upon – which means, for Meillassoux, that the poem’s form is nothing less than that of infinite chance.

382 Meillassoux, The Number and the Siren, 3. 383 It is important to note that he does not count the poem’s final phrase: “Toute Pensée émet Un Coup de Dés”. Instead, he suggests that because it is comprised of 7 words, it functions outside the main logic of the count. Meillassoux, The Number and the Siren, 50. 384 Meillassoux, The Number and the Siren, 208.

112 Meillassoux’s code specifically relies upon a rejection of the number 12, which as we have seen is crucial to Mallarmé’s early work. Before turning to why he does this, it is worth noting that Mitsou Ronat – a French poet, linguistic and literary critic – proposed an earlier reading of the poem that also suggested it was coded. In stark contrast to Meillassoux, Ronat argues that the unique number is in fact 12. Her argument is essentially three-fold. First, while the poem runs across 11 double pages, Mallarmé’s title Page adds a 12th to its total. Second, the phrase “l’unique Nombre qui ne peut pas être un autre” is itself an alexandrine, comprised of 12 syllables.385 But it is Ronat’s third claim about the poem that has caused her argument to be discredited amongst scholars today. She suggests that each double page is made up of 36 lines – 36, of course, being a multiple of 12. However, Bertrand Marchal – and more recently Nikolaj d’Origny Lübecker – have noted that in early notes for the poem Mallarmé specifically commented that its pages were to be made up of 40 lines (40, of course, not being a multiple of 12).386 This is enough to convince Marchal and Lübecker that Ronat’s thesis is incorrect.

Meillassoux recognises the validity of Marchal and Lübecker’s criticisms, citing both their works. However, he rejects the 12 for two other reasons. The first concerns its role as the numerical correspondent of midnight in Igitur. Meillassoux correctly notes that Igitur gives up on throwing the dice, shaking them instead, which suggests to him that Mallarmé had given up on the 12 of the alexandrine. He writes, “[...] if, in the Coup de dés, Mallarmé had identified the Number with 12, he would simply have returned to the aporia of Igitur by arbitrarily choosing one of the options he had earlier rejected”.387 This is one of Meillassoux’s strongest arguments against the existence of the number 12, as Mallarmé does indeed link the 12 of the dice throw to Igitur’s fallen poetic forebears. But it is important to note that the composition of Igitur took place well before the “memorable” crisis of verse, so although it is true that Mallarmé does not simply fall on the side of the alexandrine (and thus the 12), it makes sense to suggest that a re-examination of its role

385 Although, to quote Meillassoux, this line is “a nonclassical alexandrine, since the caesura falls on the relative pronoun ‘qui’, but representative by this token of the audacious gestures Mallarmé makes in his treatment of verse”. Meillassoux, The Number and the Siren, 26 ft. 11. 386 Mallarmé, Œuvres complètes I, 1322 ft. 1; Nikolaj d’Origny Lübecker, Le Sacrifice de la sirène: “Un Coup de dés” et la poétique de Stéphane Mallarmé (Denmark: Museum Tusculanum Press, 2002), 24-5. 387 Meillassoux, The Number and the Siren, 33.

113 was necessitated by the crisis, rather than rejected out of hand. While Meillassoux does note that the 12 plays a central role in the calculations for the Book – writings that are synonymous with the crisis – he argues that it “appears to play no role in the content, properly speaking, of the Book”.388 What he means by “content” is not entirely clear and, given the already dubious status of Mallarmé’s scrawled notes, it does seem presumptuous to establish which sketches of the Book matter and which do not. However, as previously argued, numbers that have 12 as their root do appear to be crucial to this moment in Mallarmé’s career. While Meillassoux is aware of this, his argument requires him to more or less dismiss the number’s centrality.

Nonetheless, I do not think that Meillassoux’s code is mistaken. Rather, I want to suggest that if his unit of determination for the code is the word, the poem’s “unique Nombre” also contains an opposing unit, one determined by the nature of the crisis: namely, the syllable. The crisis of verse consists in an antagonism between the rhythmic determination of the line (in this case, the 12 of the alexandrine) and the free floating individuation of the word on the page. Meillassoux’s operation of the count therefore falls strictly on the side of free verse, since it takes the word as its metrical unit of individuation. What makes his argument so forceful is that the count also takes on semantic value. The 7, for example, corresponds to the number of stars in the constellation. Moreover, the word “peut-être” takes on a performative function: it expresses the very structure of indeterminacy that it materialises in the act of counting. Yet what if one was to produce a different count that corresponds more to the chance that insists within the alexandrine, rather than in the word?

In Le Coup de dés de Mallarmé: Un recommencement de la poésie (2005), Michel Murat argues that the alexandrine has “citational value” in Un Coup de dés.389 Although he queries whether the poem can truly be reduced to a commentary on free verse, he concedes that it best displays Mallarmé’s position on the crisis. Unlike Meillassoux, Murat does not

388 Meillassoux, The Number and the Siren, 26. My emphasis. In fairness to Meillassoux, he has revised his position on this point in a recent essay. In “The Coup de dés, Encoded Crypt”, he gives more weight to the Book, noting how the number 12 is important to its overall composition. See, Quentin Meillassoux, “The Coup de dés, Encoded Crypt,” trans. Robert Boncardo and Christian R. Gelder, Hyperion, forthcoming September 2016. 389 “L’alexandrin a donc une valeur citationnelle”. Michel Murat, Le Coup de dés de Mallarmé: Un recommencement de la poésie (Paris: Éditions Belin, 2005), 120.

114 take the individuated unit of the word as the key to unlocking the poem. Instead, his reading implies that it is an exercise in prose, but one where the alexandrine retains a certain unconscious presence that flickers in and out.390 Murat, like Ronat, notes that the phrase “l’unique Nombre qui ne peut pas être un autre” is an alexandrine.391 However, he goes further and proposes that a number of the capitalised meta-poetic lines appearing over the course of the poem are themselves alexandrines or variations and derivations thereupon. “SI / C’ÉTAIT / LE NOMBRE / CE SERAIT / LE HASARD”, for example, is a classic 6-6 alexandrine, whereas “UN COUP DE DÉS / JAMAIS / N’ABOLIRA / LE HASARD” is made up of a 4-2-4-3 structure, containing one extra drawn out syllable.392 These two capitalised phrases offer a certain oscillation not at the level of the word, but rather at the level of the syllable, where the alexandrine is both retained and denied, manifest and latent. Murat demonstrates that the descending cascade of single words and phrases on the poem’s 3rd Page can in fact be turned into “very characteristic” Mallarméan alexandrines, despite looking at first like free verse.393 Instead of formulating these lines in terms of their metrical currency, Mallarmé performs a “deconstruction” of traditional verse by presenting them only as individuated words not connected to their successor by the line. “Que l’Abîme blanchi étale furieux” therefore becomes “que / l’Abîme / blanchi / étale / furieux”; by extension, the word seems to take precedence over the syllable at the visual level.394

Sonically, however, the denied alexandrine haunts the poem: the latent is contained within the manifest because of Mallarmé’s deconstructive gesture. We hear the alexandrine when the poem is read aloud, even if only for a moment, while we see the word on the page. This opposition between the sonic presence of the alexandrine and the free-floating distribution of words on the page masks the fundamental choice that the poem gives to the reader: is the form of Un Coup de dés determined by sound or sight? As we know, for Mallarmé neither option can be definitively decided upon. Meillassoux favours sight, since he takes the scattered, individuated word as his unit of determination. But the logic of the “ou”, the

390 Murat, Le Coup de dés de Mallarmé, 114. 391 Whereas in the first edition of Un Coup de dés, Murat notes, that particular line is not an alexandrine. Murat, Le Coup de dés de Mallarmé, 113. 392 Murat, Le Coup de dés de Mallarmé, 114. 393 “… [t]rès caractéristique” Murat, Le Coup de dés de Mallarmé, 114. 394 Murat, Le Coup de dés de Mallarmé, 115.

115 “or”, is in fact sonically transmitted into the makeup of the poem. The repetition of the “ou” – its metonymic chain – is found in the poem’s key words. The 12 of the alexandrine in fact houses a hidden “ou”: the “ou” is part of the “douze”, the 12. Similarly, one can also hear the “ou” in the title of the poem: Un Coup de dés. The act of throwing, the “coup” of the “dés”, will never guarantee either option. The two dice that the Master clutches in his hand could produce a 1 and 6, resulting in a 7, but they could very well also produce a double 6, resulting instead in a 12. “Toute Pensée”, Mallarmé writes at the end of the poem, “émet Un Coup de Dés”. Every thought that attempts to fix the numerical form of the poem will itself be subject to the order of chance; that is, its opposite will be contained within itself, the denied number (the 7 or the 12) will remain potential within the actualised form of the poem. Sonically, then, the “ou” repeats and its presence is like that of the alexandrine: latent, yet manifest.

Murat goes on to show how other metres intervene at various points in the poem as well, but what is interesting is how the “citational value” of the alexandrine bears upon Meillassoux’s word count. There is no doubt a great deal of evidence to suggest that the poem is coded and, moreover, that this code is fundamentally determined by the number 7. But the alexandrine has a sonic presence that insists within the poem, flickering into being only in order to immediately disappear. The dilemma that Meillassoux poses between the 707 or the 12 – a dilemma that finds him on the side of the former – excludes what matters most to Mallarmé: the “ou”. Even if we accept Meillassoux’s count and the logic of infinite undecidability that accompanies it (the oscillation between the 707 and other counts), the denied alexandrine will always be latent within the manifest. The 12, which has its own logic of oscillation expressed through the lines that have 13 syllables, finds a way to penetrate into the actualised reality of the poem, to make its presence felt even though one cannot wholly recognise its existence. Through this oscillation between the alexandrine and free verse, between the syllable and the word, the poet fixes the infinite at the highest level of poetic abstraction available to him at the time. The “unique Nombre” is the poetic expression of the infinite and chance, but because what is actual always contains its opposite, the “Nombre” is by no means unique at all. Instead, every number – every possible number of words, sounds, syllables and even letters that could comprise the

116 poem’s form – is part of its structure. The necessity of the Number lies in its chance existence, one produced in the ideal realm, the quasi-mathematical construction, of the poem’s “hypothesis”.

Mallarmé does indeed, as Badiou rightly notes, figure the infinite as a number. But what we have seen is how specific and contextual his conceptions of the infinite, chance and number really are. The question I will now ask is whether his use of number is in any way comparable to Cantor’s paradigm-shifting work on set theory and the infinite. In other words, is Mallarmé’s unique number in any way contemporary to Cantor?

In the previous chapter, I noted that Cantor did not think about the philosophical or mathematical implications of the empty set. In this chapter, however, Cantor’s work on the infinite is now the focus. Throughout his career, Cantor maintained that his unique idea of the infinite, the “transfinite”, was the single most important developmental leap in the concept’s history. I shall examine three key moments in Cantor’s career. I begin by working through his 1883 publication, the Grundlagen, the first text explicitly devoted to the transfinite. Then I turn to his famous diagonal argument, which he employs to establish the existence of multiple transfinite numbers with differential sizes. Finally, I shall address the complex relationship between mathematics and religion in his work. In contrast to Mallarmé’s vision of the infinite, I show how Cantor’s specific figuration of the transfinite does not allow for the emergence of a doctrine of chance. For Cantor, the infinite is not born out of chance. Instead, the transfinite allows him to reinstall a notion of God. Furthermore, I argue that even when the question of God is removed from set theory, the infinite still cannot arise from chance as it does in Mallarmé’s poetry. The opposite is in fact the case: contingency is born from the transfinite.

“From now on I replace the symbol & [...] with ", because the symbol & is already variously used to designate indefinite infinites”.395 So writes Cantor in a footnote to the

395 Cantor, “Foundations of a General Theory of Manifolds,” 907 ft. 4.

117 Grundlagen, a work so profound in breadth that it represents nothing less than “a revolution in the history of mathematics”.396 While Cantor had began to develop a concept of the transfinite in previous papers on geometry, the Grundlagen is his first sustained attempt to formulate it under a branch of mathematics that will come to be known as set theory. Many of the definitions given in the Grundlagen are still accepted today and it would be no exaggeration to suggest that it represents the single most important work of nineteenth century mathematics – not least because it gives an account of a definite, well-defined, infinite that is neither limitless in nature nor absolute in idea.

Cantor begins by identifying two competing conceptions of the infinite circulating in nineteenth century mathematics. The first, which could be called the potential infinite, is simple enough to grasp. Take any sequence of numbers whatsoever: for example, the natural numbers. The basic idea behind this infinite is that the quantity of numbers never ends, since they continue to generate and expand indefinitely. As Cantor puts it, this infinite “has hitherto appeared principally in the role of a variable quantity, which either grows beyond all bounds or diminishes to any desired minuteness”.397 In other words, the potential infinite never encounters a limit. One can always add, “beyond all bounds”, another number to the previous. However, Cantor critiques this idea, stating that although the sequence of natural numbers may indeed continue indefinitely, this indefiniteness remains distinctly finite in nature. To call the indefinite quality of number generation infinite is to mistake the never-ending quantity of finite numbers for some kind of infinite temporal duration. Cantor concludes that this should be referred to as the “improper infinite” because it does not derive from mathematics, but rather from the inability of mathematics to ever define or grasp a concrete instance of infinity.398

The second concept of the infinite Cantor identifies does not share the same quality of indefiniteness as the improper infinite: “in modern times, in geometry and in the theory of

396 Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Cambridge/ Massachusetts/London: Harvard University Press, 1979), 118. 397 Cantor, “Foundations of a General Theory of Manifolds,” 882. 398 Cantor, “Foundations of a General Theory of Manifolds,” 882. Emphasis removed.

118 functions, another equally justified concept of the infinite has been developed”.399 This geometric infinite is known as the “point at infinity”, first theorised in the seventeenth century through the advent of differential calculus.400 This infinite can be defined as the ideal point where two lines on a plane – that is, a geometric representation of a grouping of numbers – meet.401 However, this is not a single point. As Cantor writes, “[...] while the point at infinity of the complex-number plane stands isolated from all the points lying in the finite region, here we obtain not merely a single infinite integer but an infinite sequence of them; they are clearly differentiated from each other, and stand in lawlike number-theoretic relations both to each other and to the finite integers”. Cantor admires this concept of the infinite as it is intrinsically mathematical, established by the operations of mathematics alone. When such an infinite appears, he writes, “I call it the proper infinite”, which should stand in distinction to its potential, improper counterpart.402 As much as he admires both of these, they are not his own. He writes that although these concepts have led “to great progress in geometry, in analysis, and in mathematical physics […] [they] must be carefully distinguished if we are to understand what follows”.403 To present Cantor’s unique intervention, some technical definitions are required.

The first definition is that of a well-defined set: that is, a set whose elements can be known and identified. The set of natural numbers from 1-5 (including 1 and 5) is well-defined because each element in the set is identifiable, in this case 1, 2, 3, 4 and 5. But the set of Mallarmé’s favourite numbers, for example, is not because one cannot reasonably know what his favourite numbers were without engaging in guesswork. For Cantor, “[...] every well-defined set has a determinate power”.404 The concept of power, or what is now called cardinality, refers to the size of a set. “Two sets have the same power,” he writes, “if they can be, element for element, correlated with one another reciprocally and one-to-one”.405 If two sets have the same power, then their elements can be placed into a one-to-one correspondence with one another, like so:

399 Cantor, “Foundations of a General Theory of Manifolds,” 882. 400 Cantor, “Foundations of a General Theory of Manifolds,” 883. 401 Cantor, “Foundations of a General Theory of Manifolds,” 883. 402 Cantor, “Foundations of a General Theory of Manifolds,” 882. 403 Cantor, “Foundations of a General Theory of Manifolds,” 882. 404 Cantor, “Foundations of a General Theory of Manifolds,” 883-4. 405 Cantor, “Foundations of a General Theory of Manifolds,” 884.

119

1 2 3 ' ' ' 4 5 6

As the above example shows, even though these two sets contain different elements – one the numbers 1, 2 and 3 and the other 4, 5 and 6 – they can be placed into a relationship of bijection with one another. Their elements can be matched one for one and their cardinality, their size, is thus the same.

To add a third definition, if sets are well-defined – and can be put into a one-to-one correspondence with other sets – then they can also be well-ordered. Cantor’s definition of a well-ordered set is as follows:

A well-ordered set is a well-defined set in which the elements are bound to one another by a determinate given succession such that (i) there is a first element of the set; (ii) every single element (provided it is not the last in the succession) is followed by another determinate element; and (iii) for any desired finite or infinite set of elements there exists a determinate element which is their immediate successor in the succession.406

This definition makes it possible to order sets in terms of their size. The largest and smallest elements of the set of natural numbers from 1 to 10 (including 1 and 10), for example, can be identified: in this case, 1 and 10. Every well-defined set can therefore be brought “into the form of a well-ordered set”, which in turn allows sets to be ranked according to their powers.407 This was a revolutionary move for Cantor to make but one that also introduced a series of complicated problems into modern mathematics. How, for example, could one speak about the infinite in terms of differential powers? If a set is

406 Cantor, “Foundations of a General Theory of Manifolds,” 884. 407 Cantor, “Foundations of a General Theory of Manifolds,” 886.

120 indeed infinite, then surely it would be just that. But Cantor argued that it is in fact possible to think about infinite sets in the same fashion as finite ones, as knowable and identifiable.

To establish the existence of an infinite set, Cantor relied upon what he called the “first principle of generation”.408 He began with the assumption that the sequence of natural whole numbers (1, 2, 3, 4, etc) is indeed limitless in nature. Obviously, he thought, it makes no sense to speak of a point at which the numbers cease to continue, because adding another to the previous is a process that can go on indefinitely. That is precisely the problem the infinite poses mathematics: numbers seem to never end. However, as Dauben writes, “Cantor believed there was nothing logically false about thinking of a new number which expressed the natural, regular order of the entire set”.409 To establish the existence of an infinite set Cantor posited that although the sequence of natural whole numbers is unlimited, they could nevertheless be succeeded by a larger number still, one that does not belong to the realm of finite numbers. He represented the sequence of natural whole numbers as !, and the new infinite number that followed it as ": the first non-finite number. As Cantor writes, “[t]he formation of the finite real integers thus rests upon the principle of adding a unity to an already formed and existing number; I call this principle [...] the first principle of generation”.410 In this sense, " is transfinite: it is the properly infinite successor of the finite natural numbers. Most importantly, it is also entirely definite and identifiable, just like finite sets.

Cantor did not stop there, however. By examining " itself, he was able to show how other transfinite numbers could be established through a second principle of generation. From this, he expanded the size of ", making it larger than before: for example, he added " and ! together, producing a different transfinite ordinal – a differently ordered transfinite number – written as 2". This “second principle of generation” applied to transfinite numbers instead of finite ones, and the principle could henceforth be expanded further.411 Paradoxically, this showed that ", the first transfinite number, is in fact only the smallest

408 Cantor, “Foundations of a General Theory of Manifolds,” 907. Emphasis removed. 409 Dauben, Georg Cantor, 97. 410 Cantor, “Foundations of a General Theory of Manifolds,” 907. 411 Cantor, “Foundations of a General Theory of Manifolds,” 908.

121 number that succeeds all finite numbers. Cantor states: “if any definite succession of defined integers is put forward of which no greatest exists, a new number is created by means of this second principle of generation, which is thought of as the limit of those numbers; that is, it is defined as the next number greater than them all”.412

Cantor continued to add a third principle, the “principle of limitation”, which “was designed to produce natural breaks in the sequence of transfinite numbers”.413 His worry was that the first transfinite number, ", would simply continue in the same limitless fashion as the finites and would thus be rendered indistinguishable from other infinite sets. The principle of limitation allowed him to establish the existence of a second number class equally as definite and identifiable as ", but which would operate as a means of safeguarding different infinite number classes. He writes, “[w]e accordingly define the second number-class (II) as the aggregate of all numbers # formable with the help of the two principles of generation”.414 In other words, the principle of limitation limits and orders the transfinite ordinals, disallowing any absolute or total mathematical infinite to emerge. Introducing a second number class – that is, the numbers larger than the first – guaranteed the existence of even greater number classes that could be definitely identified. This, in turn, produced multiple transfinite numbers, each of which corresponding to a different number class.

Establishing the existence of multiple transfinite numbers was both an enormous triumph and an earth-shattering problem for Cantor. The triumph, of course, was that the infinite, to use Cantor’s Mallarméan-like expression, had now finally been fixed: “[...] I have been logically compelled [...] to the point of view which considers the infinitely great not merely in the form of something growing without limit [...] but also to fix it mathematically by numbers in the determinate form of the completed-infinite”.415 Under his watch, for the first time in history the infinite could be said to be definite, identifiable, and multiple. Multiple transfinite numbers necessarily implied multiple infinities: there are in fact infinite

412 Cantor, “Foundations of a General Theory of Manifolds,” 907-8. 413 Dauben, Georg Cantor, 98. 414 Cantor, “Foundations of a General Theory of Manifolds,” 909. Emphasis removed. 415 Cantor, “Foundations of a General Theory of Manifolds,” 890. Emphasis my own.

122 infinites, all of which became accessible to mathematics for the first time. The infinite is by no means a unique number; instead, it is as common as any one of the finites. Yet the existence of different well-ordered infinites was a terrifying prospect. How, for example, could one speak about the different powers of different number classes? If there are, for example, two transfinite numbers, one larger than the other, how is it possible to measure their difference in size? To make this problem more precise, recall that the first transfinite number is the set of all the natural numbers, unifying their unbounded sequence. But what about the set of real numbers, real numbers being all the numbers along the number line (for example, (2 is a real number)? Intuitively, there are more real numbers than whole numbers, but if both sets are infinite, could they not be placed into one-to-one correspondence with one another?

Cantor’s diagonal argument is often invoked to prove that there is a difference in size between the infinite set of natural numbers and the infinite set of real numbers. It shows that there are countable infinite sets – sets that can be put into a one-to-one correspondence with the natural numbers – and uncountable (or non-denumerable) infinite sets that are definitively larger than ". While the argument can be framed to talk about the difference between real and natural numbers, it is in fact a more general one, “remarkable not only because of its great simplicity, but more importantly because the principle followed therein can be extended immediately to the general theorem that the powers of well-defined manifolds have no maximum”.416 To simplify Cantor’s original demonstration, take the set of real numbers greater than 0 and less than 1. This set is infinite, as the lines of decimal places can continue to be extended without a limit. But is it countable? Can it be placed into a one-to-one correspondence with the countable natural numbers? The first way to answer this question is to create a sequence of numbers greater than 0 and less than 1 using only the numbers m and w:

E1 = (m, m, m, m, m, … ), E2 = (w, w, w, w, w, … ),

416 Georg Cantor, “On an Elementary Question in the Theory of Manifolds (Cantor 1891),” in From Kant to Hilbert: A Source Book in the Foundations of Mathematics Volume II, ed and trans. William Ewald (Oxford/New York: Oxford University Press, 1996), 921.

123 E3 = (m, w, m, w, m, … ), E4 = (w, m, w, m, w, … ), E5 = (w, w, m, w, w, … ), … … … … … … …

This sequence can be extended indefinitely; but Cantor’s genius was to show that if one drew a diagonal line down the sequence and flipped each number that the diagonal crosses into its opposite (turning the m into a w and the w into an m), then a new number that cannot be identified in the original count emerges. Using the above example, this new number is:

E6 = (w, m, w, w, m, … )

Even if E6 is then placed into the original set (E1 to E5), the same operation can be performed again, constantly creating new numbers that render the sequence of real numbers greater than 0 and less than 1 uncountable. For Cantor, the remarkable simplicity of this argument proved that there are countable infinite sets, but also infinite sets that are larger and uncountable.417

Most of Cantor’s subsequent writing on mathematics examined the consequences of this development. As much as he recognised how revolutionary his work was, the creation of multiple, differential and determined infinite sets troubled Cantor, both mathematically and philosophically. For example, after, and even in the Grundlagen, he tried to ascertain the cardinality between the set of all natural numbers and the set of all real numbers. From this, he established what he called the continuum hypothesis, which is the assertion that there is no cardinal between these two sets. Proving this hypothesis was a constant problem for Cantor – and he often wrote of how he was on the verge of solving it, only to then discover that his solution contained some fundamental flaw. It was not until the twentieth-century mathematician Paul Cohen discovered that the continuum hypothesis was independent of set theory – that it could be neither proved nor disproved by the mechanisms and axioms of

417 See, Cantor, “On an Elementary Question in the Theory of Manifolds,” 920-22.

124 set theory alone. Nonetheless, Cantor considered it of the utmost importance and regarded his inability to solve it as a failure.

Beyond the kind of complicated mathematical problems transfinite sets posed, Cantor also pondered the philosophical implications of his thinking. The Grundlagen is not only a work of mathematics but something of a philosophical treatise as well, a facet regrettably expunged upon its translation into French.418 Dauben notes that, for Cantor, “[...] just as the finite whole numbers possessed an objective reality, so did the transfinite numbers. Their existence was naturally reflected in the matter and space of the physical world, and in the infinities of concrete objects”.419 This should already give some indication of Cantor’s philosophical position. By his own admission, he was both an idealist and a realist of the “Aristotelian-Platonic kind”, believing that transfinite numbers were obstinately understood by the human mind – but that this understanding reflected the structure of a prior objective reality.420 Transfinite numbers exist in-themselves, and yet it is precisely the human mind that is the condition for this knowledge.

Cantor was further troubled by the implications his work had for the traditional teachings of the Church. How could he, a Catholic, reconcile the transfinite with the supposed absolute infinity of God? To reconcile his faith with his work, Cantor distinguished between his own transfinite numbers and an absolute idea of infinity. For him, sets must be understood as “consistent multiplicities”.421 Barring the kind of connotations the word “consistent” has in logic and mathematics, the general idea is that sets provide a relatively stable way of unifying a series of divergent numbers, giving them the property of wholeness (or consistency). But Cantor already had a vague idea of Russell’s paradox, something I briefly

418 It was Gösta Mittag-Leffler, Cantor’s colleague, who advised him to remove the philosophical speculations of the Grundlagen for the French translation. This is a shame, as Cantor’s knowledge of philosophical history is indeed impressive. In the Grundlagen alone, he engages with Aristotle, Locke, Descartes, Spinoza, Leibniz and even Hegel (whom he implicitly critiques for calling the mathematical potential infinite a “bad infinite”). See Cantor, “Foundations of a General Theory of Manifolds,” 890-5. 419 Dauben, Georg Cantor, 98. 420 Cantor admitted this in a letter to Paul Tannery. Quoted in, Dionysis Mentzeniotis and Giannis Stamatellos, “The Notion of Infinity in Plotinus and Cantor,” in Platonism and Forms of Intelligence, eds. John Dillion and Marie-Élise Zovko (Berlin: Academie Verlag, 2008), 225. 421 Quoted in Ignacio Jané, “The Role of the Absolute Infinite in Cantor’s Conception of Set,” Erkenntnis 42, no. 3 (1995): 375.

125 discussed in my previous chapter. If there are indeed an infinite amount of transfinite sets, each different in size, then how is it possible to think the set of all transfinite sets? Cantor argued that there is such a thing as an absolute infinite or “inconsistent multiplicity”.422 Instead of claiming, as Russell does, that there is no set of all sets, for Cantor the absolute infinite is nothing other than the manifestation of God, an absolute figure that exists beyond the sequence of transfinite numbers. As he writes, “the true infinite or absolute, which is in God, admits no kind of determination”.423 This God is the opposite of the transfinite. Where the transfinite is definite and identifiable, God is indefinite and unidentifiable. Where transfinite numbers are consistent multiplicities, God holds within him an inconsistent multiplicity. God is unknowable, reflected neither in the objective world nor graspable by the powers of the human mind.

The distinction between the transfinite and the absolute structured much of Cantor’s later life. Having been denounced by many of colleagues and contemporaries for his work on the infinite, he found a home within the Church. He spent a great deal of time writing to Pope Leo XIII – a neo-Thomist who saw that the future of Catholicism lay in combining scripture with science – and he took solace in believing that his work served to confirm the existence of God.424 I mention these biographical details because they highlight a distinction between Mallarmé’s idea of infinite chance and Cantor’s transfinite. Both do indeed position the infinite as a number: Cantor shows how there are infinite transfinite numbers, just as Mallarmé argues that there are multiple numerical possibilities contained within the “unique Nombre”. Yet naming God as the absolute, as that which exists beyond the transfinite, dispels chance from the field of mathematics. Rather than viewing the transfinite as a challenge to the reign of God, Cantor claimed that his work only strengthened the transcendence of the divine. All the attributes of Mallarmé’s infinite – undecidability, chance, and contradiction – are negated by Cantor. Where Mallarmé’s work

422 Quoted in, Jané, “The Role of the Absolute Infinite in Cantor’s Conception of Set,” 375. 423 Quoted in, Jané, “The Role of the Absolute Infinite in Cantor’s Conception of Set,” 384. 424 For accounts of Cantor’s correspondence with Pope Leo XIII, see Anne-Marie Décaillot, Cantor et la France: Correspondance du mathématicien allemand avec les français à la fin du XIXe siècle (Paris: Éditions Kimé, 2008), 59-113 and Joseph Warren Dauben, “Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite,” Journal of the History of Ideas 38, no. 1 (1977): 85-108.

126 is predicated on the death of God, Cantor’s inaugurates God’s rebirth at the end of the transfinite sequence. Chance is annulled and determinism prevails once more.425

However, Meillassoux argues that set theory does indeed present a unique concept of contingency. When the questions of God and the absolute are removed from mathematics, Cantor’s work suggest a thesis more fundamental still, one explored in my introduction: “We will retain the following translation of Cantor’s transfinite: the (quantifiable) totality of the thinkable is unthinkable”.426 Without God existing at the end of the sequence, transfinite numbers cannot be totalised. They are in fact untotalisable: there is no set of all sets. With this in mind, I want to briefly revisit Meillassoux’s distinction between chance and contingency outlined earlier. Chance, for him, can be likened to probability theory, where something can be predicted if the totality of possible results are known. For example, if I roll a dice, then I can safely assume that the likelihood of rolling a 3 is 1 in 6, since it is known that there are 6 sides to a standard dice. But prediction becomes virtually impossible when there is no identifiable totality governing the probability of something appearing. For Meillassoux, the power of set theory lies in its ability to show “a fundamental uncertainty regarding the totalisable of the possible”.427 If the totalisable does not exist, he claims, then there is no reason for any necessity whatsoever, since prediction will always be subject to a higher rule: that of untotalisability. Meillassoux calls this contingency. Every conceivable possibility that could occur in the world and every model employed to predict such occurrences cannot escape the possibility that something outside the modelling may very well emerge, since what is possible cannot be counted. Meillassoux’s philosophical – and imminently a-theological – rethinking of set theory attempts to forge a relationship between the transfinite and contingency. However, he links them in the opposite manner to

425 This was reflected in Cantor’s personal life. He believed God had chosen him to expose the transfinite to the Church. Dauben puts it best: “Cantor did regard his work as a sacred mission. He believed that God had actively chosen to keep him in Halle, rather than allow him to have post in Göttingen or Berlin. By denying his hopes for a better university position, God had ensured that Cantor would better serve Him and the Roman Catholic Church. He had been given faith in the necessary truth of his mathematical discoveries by virtue of their having come from, and their being part of, God’s infinite wisdom. In return, Cantor hoped to aid the Church in correctly understanding the problem of infinity”. We are used to the slogan “the personal is political”. But here, it seems, the personal is mathematical! Dauben, Georg Cantor, 232. 426 Quentin Meillassoux, After Finitude: An Essay on the Necessity of Contingency, trans. Ray Brassier (London/New York: Continuum, 2008), 104. 427 Meillassoux, After Finitude, 105.

127 Mallarmé. For the poet, the death of God implies nothing less than the death of determinism. In its place, chance reigns. His concept of chance seeks to establish a logic where what is actual contains its potential within it: chance is Absolute since chance itself contains necessity. The constant movement between the two – between what is and what is not – guarantees the existence of the infinite: “L’infini sort du hazard”.428 But for Meillassoux, the infinite is prior – it is in fact axiomatically guaranteed before the fact. The infinite is not born out of chance here, as it is, for example, in Igitur. Instead, contingency is born out of the transfinite.

To conclude, it is worth reflecting on whether Mallarmé does indeed appear to be “Cantor’s unconscious contemporary”.429 As I have shown, both the poet and the mathematician fix the infinite as a number. However, now we can say that what they mean by “infinite” and “number” is very different. Mallarmé’s infinite is dialectical, whereas Cantor’s is sequential. Mallarmé’s Absolute is that of chance itself whereas, for Cantor, God alone is absolute. Mallarmé’s “unique Nombre” is both unique and it is not, because it contains its opposite within itself. Cantor’s transfinite numbers can themselves be infinitely produced – and are thus in no sense unique. While their terminology does suggest that Mallarmé and Cantor are philosophically contemporaneous with one another, the concepts that underpin these terms are radically distinct. They both fix the infinite, and yet how they do so is different. Is it then possible to suggest another “unconscious contemporary” for Mallarmé? One whose use of mathematics is founded on chance itself?

At the end of the nineteenth century and the beginning of the twentieth, chance became a privileged concept. Many diverse thinkers sought to rethink the fundamental ideas of their disciplines under its auspice. Mallarmé’s work is but one example of this: traditional ideas about poetry, for him, could no longer be sustained when faced with the prospect of a world whose ruler is no longer God, but rather chance. The French literary critic Pierre de Barneville opened his turn of the century study on rhythm in French poetry by stating: “For

428 Mallarmé, Œuvres complètes I, 474. 429 Badiou, Briefings on Existence, 124.

128 a long time, humanity had considered poetry as a supernatural gift and the poet as a sort of messenger of the gods”.430 But with Mallarmé, humanity's long held view must change. The poet, no longer God’s voice on earth, now serves chance instead. This idea extended well beyond poetry, however. In 1874, Émile Boutroux published his extremely influential dissertation, aptly titled The Contingency of the Laws of Nature.431 In this work, Boutroux develops a critique of scientistic positivism, examining relationships of causality as they appear in the natural world. For him, there is no necessity guiding any facet of cause and effect. Biological analysis cannot account for psychological experience – one does not explain the other – and as such there is no necessary link between the two. Necessity as a critical category must therefore be replaced with contingency, since no fundamental law binds together the various orders of the world. Boutroux’s work is a mix of spiritualism and philosophy, heavily relying on speculative argumentation. But chance also became a critical category for the hard sciences, particularly mathematical analysis. In 1900, the French proto-economist and mathematician Louis Bachelier submitted a doctoral thesis entitled Théorie de la spéculation, which in many ways is the foundational text of mathematical finance.432 Here, Bachelier mathematised the movements in price that were taking place in the French economy, leading him to develop a model of stochastic analysis. As Jon Roffe writes, Bachelier’s “most significant claim is that price movements are stochastic, which is to say, statistically random in character. For Bachelier, there is no fundamental law governing the fluctuations of price on the market”.433 Instead, where chance is again positioned as the dominant factor orienting the world, probability theory is asked to “tame” chance – to predict how things will take place in an essentially unpredictable environment.434

430 “De tous temps, l’humanité a considéré la poésie comme un don surnaturel et le poète comme une sorte de messager des dieux”. Pierre de Barneville, Le Rythme dans la poésie français (Paris: Librairie Académique Didier, 1898), 1. 431 Émile Boutroux, The Contingency of the Laws of Nature, trans. Fred Rothwell (Chicago/London: The Open Court Publishing Company, 1920). 432 Louis Bachelier, “Théorie de la Spéculation,” in Louis Bachelier's Theory of Speculation: The Origins of Modern Finance, trans. Mark Davis and Alison Etheridge (Princeton/Oxford: Princeton University Press, 2006), 15-79. 433 Jon Roffe, Abstract Market Theory (Hampshire/New York: Palgrave Macmillan, 2015), 11. 434 This phrase is drawn from Ian Hacking’s excellent book, The Taming of Chance (1990). Hacking provides a large survey-style summary of the various disciplines that embraced chance throughout the late nineteenth and early twentieth century. He chooses not to examine mathematics, but does reference Mallarmé in his

129

I mention this to illustrate a kind of general cultural climate that existed in late nineteenth- century and early twentieth-century France – a climate that Cantor is somewhat removed from. In fact, Mallarmé’s conception of the infinite has much more in common with disciplines that submit themselves to chance than with those that attempt to negate it. With respect to mathematics, if there is one text that resonates with Mallarmé’s work, then it is surely Émile Borel’s Le hasard (1914), written after Mallarmé’s death, in the poet’s future. Borel’s work extends over a vast variety of fields – from mathematics to the physical and biological sciences, and even to early sociological analysis. Yet, at times, it reads more like a manifesto dedicated to thinking a world where chance names the void left by God’s death. Borel himself was well versed in set theory and his work has had a lasting contribution in the discipline. In Le hasard, however, he is more interested in thinking a philosophico- mathematical idea of chance, which he views through the prism of mathematised probability theory.

Borel begins by separating chance and necessity at a global level. Since Plato, he thinks, scientists, metaphysicians and philosophers have more or less held to “the conviction that the world is not governed by the blind gods or by chance, but by rational laws”.435 However, Borel argues that while necessity sometimes looks as though it is an inevitable part of our world, chance always manages to find a way of insisting and persisting within a diverse range of mathematical, scientific, metaphysical and philosophical problems. With this in mind, he continues to make a further separation between statistical laws and natural laws. Natural laws signify the general laws of reason and are, by definition, necessary. But statistical laws make no such claim to necessity. They follow laws of probability and unlike the natural laws do not annul the very Mallarméan idea that a throw of dice will never abolish chance.

Probability and the laws of statistics must therefore account for chance. In its classical definition, probability refers to “the relationship of a number of favourable cases to the total conclusion on C.S. Peirce. Ian Hacking, The Taming of Chance (Cambridge: Cambridge University Press, 1990), 215. 435 “[...] la conviction que le monde n’est pas régi par les dieux aveugles ou par le hasard, mais par des lois rationnelles”. Émile Borel, Le hasard (Paris: Presses Universitaires de France, 1948), 3.

130 number of events”.436 But Borel notes that a critique of this definition is equally as classical. He writes that the glaring weakness of the above definition is that it presumes knowledge to be totalisable, that every possible event can be known, cataloged, listed – and tamed. Any form of randomness or radical disjunction the future may hold is lost, as if the present comprises the eternal, signifying everything that has and will come to be. The banal example Borel uses to illustrate how chance affects the classical definition of probability is the following: if 51 percent of children born this year are boys, then there is no reason, other than that drawn from pure empirical observation, why this should be the case the following year. Chance events could indeed occur – such as only boys being born one year – and probability theory must attempt to factor this into its disciplinary operations. In fact, for Borel, recognising chance leads to all sorts of fanciful ideas that can be mathematically confirmed. For example, in Le hasard he evokes his “infinite monkey theorem”, where he imagines a million monkeys writing on typewriters for 10 hours per day over the course of many years.437 Instead of producing endless amounts of meaningless writing, chance shows how it is possible, at least in theory, for these monkeys to compose works that belong to the “richest libraries in the world”.438 When the infinite nature of chance is taken into account, the probability of monkeys writing coherent, even beautiful, works becomes a possibility, albeit a miniscule one. In this sense, a probability theory submitted to chance looks increasingly inert, able only to show how the random and the arbitrary are possible. Charles Mauron, an early Mallarmé scholar who authored works on the relationship between the poet and psychoanalysis, has drawn a comparison between Un Coup de dés and Borel’s above example. Of the poem’s central thesis – that a throw of the dice will never abolish chance – he writes,

What would this mean to a mathematician? This, I presume: that, extraordinary as it may be, a fluctuation is still part of chance. In the language of probability, “miraculous” means “extremely rare”; but the rare is still possible. Everyone knows

436 “[...] la probabilité est le rapport du nombre des cas favorables au nombre total des événements”. Borel, Le hasard, 8. Emphasis removed. 437 This is its contemporary name, which Borel himself did not use. See Prakash Gorroochurn, Classic Problems of Probability (New Jersey: John Wiley & Sons, 2012), 208-214 for an overview of the problem’s initial framework, as well as its development throughout contemporary debates. 438 “Les plus riches bibliothèques du monde”. Borel, Le hasard, 123.

131 what Borel has called the “miracle of type-writing monkeys”. One may, if he desires, compare Mallarmé with Borel. Both meditated upon the strong improbability that there is such a thing as philosophic order, and both apparently class order, or what appears to be order, as a sort of quite improbable disorder.439

For Mauron, then, Mallarmé is Borel’s unconscious contemporary, despite Borel existing in Mallarmé’s future. Both figures – the poet and the mathematician – rethink, indeed submit, their respective disciplines to chance, to the random occurrences that, however statistically unlikely, are nevertheless possible. Chance reigns for Borel just as it does for Mallarmé, leaving mathematics and poetry more conscious than ever of the fact that the only possible order of the world is its very disorder.

But is it really the case that Borel submits mathematics to chance in the same manner as Mallarmé? In Le hasard, Borel seeks to ascertain what he calls “the unique law of chance”.440 This law is internal to probability theory, but we might think of it as Borel’s overall statement on the nature of chance itself. For Borel, “the aim of the theory of probabilities is to evaluate the probabilities of complex events by means of other, more simple events, whose probabilities we suppose are known”.441 Its goal is to gain certainty, to draw on what is known to determine the unknown. But chance implies the opposite of certainty, signifying instead uncertainty, randomness and disorder. With this opposition in mind, probability theory must not succumb to the temptation of believing that any chance event could take place at any moment. Instead, the discipline’s “purpose is to arrive at predicting with an almost absolute certitude – humanly absolute one could say – certain events whose probability is such that they merge with certitude”.442 Chance, of course, will always insist but, for Borel, deciphering chance’s unique law allows us to understand how it can be overcome. “This law”, he claims, “simply consists in the phenomena whose

439 Charles Mauron, Introduction to the Psychoanalysis of Mallarmé, trans. Archibald Henderson, Jr. and Will L. McLendon (Berkeley/Los Angeles: University of California Press, 1963), 128. 440 “[...] la loi unique du hasard”. Borel, Le hasard, 12. 441 “[...] la théorie des probabilités a pour objet d’évaluer les probabilités d’événements complexes au moyen des probabilités d’événements complexes au moyen des probabilités supposées connues d’autres événements plus simples”. Borel, Le hasard, 11. Emphasis removed. 442 “Son but, c’est d’arriver à prévoir avec une certitude presque absolue, humainement absolue peut-on dire, certains événements dont la probabilité est telle qu’elle se confond avec la certitude”. Borel, Le hasard, 11.

132 probability is sufficiently small to never occur; or, if one prefers, the phenomena that have not been and will never be observed by man”.443 In other words, the unique law of chance states that the possible can still be, in fact, impossible.

To determine this law, Borel develops his own number, one that draws a line between whether phenomena can be objects for probability theory or considered virtual impossibilities. This (unique) number is illustrated through some “simple calculations” about the visible universe – the universe “accessible to man” – and the dimensions of the atom, each of which shows how some numbers are so large or small that they can be excluded from the purview of probability.444 Borel cites Jean Perrin’s Les Atomes (1913) – which argues that the number of molecules by centimeters cubed cannot be more than 1025 – to show how cubed centimeters must be inferior to 1030. Similarly, he thinks, the furthest reach telescopes have is inferior to 10 million light-years, which, by his calculations, is 1028 centimeters. With this astrological example, Borel imagines the probability of two people picking out the same “corpuscle” – that is, the same minute particle – in our universe. He states, “if we admit that we can randomly select one of these corpuscles [...] the probability of choosing a given corpuscle will be 10-150; such will also be the probability of the two persons choosing the same corpuscle”.445 The statistical likelihood of this, for him, is “truthfully absurd”.446 But this absurdity can be furthered still, “if we imagine [...] a universe V2 which would be in relation to our universe V1 what the latter is in relation to a corpuscle, then a universe V3 which would be in relation to V2 as V2 is in relation to V1, and 447 so on until V1000”. This universe, V1000, would extend the previous statistical absurdity of two people choosing the same corpuscle even further, to that of 10-150000. While, under chance, this may be theoretically possible, Borel argues that it is in fact statistically

443 “Cette loi consiste simplement en ce que les phénomènes dont la probabilité est suffisamment petite ne se produisent jamais; ou, si l’on préfère, ces phénomènes n’ont été et ne seront jamais observés par aucun homme”. Borel, Le hasard, 12. Emphasis removed. 444 “[...] nous borner à quelques calculs fort simples sur les dimensions de l’univers accessible à l’homme comparées aux dimensions de l’atome, de manière à rendre plus concrète la signification de nombre très grands ou de nombres très petits”. Borel, Le hasard, 12. 445 “Si nous admettons que nous puissions choisir au hasard un de ces corpuscules [...] la probabilité de choisir un corpuscule déterminé sera 10-150; telle sera aussi la probabilité pour que deux personnes choisissent le même corpuscule”. Borel, Le hasard, 12-13. 446 [...] véritablement absurde”. Borel, Le hasard, 13. 447 “[...] si nous imaginions [...] un univers V2 qui serait par rapport à notre univers V1 ce qui celui-ci est par rapport à un corpuscule, puis un univers V3 qui serait par rapport à V2 ce que V2 est par rapport à V1, ainsi de suite jusqu’à V1000”. Borel, Le hasard, 13.

133 impossible. This number, for him, represents what cannot take place. “Such”, he closes, “is the meaning of the unique law of chance”.448

Borel thus seeks to overcome chance not by annulling or negating it, but by mathematising it: the unique number is not chance itself, but rather what renders chance impossible. He certainly attempts to rethink probability – and indeed, all of the hard sciences – on the basis of chance. But this is not done in order to render them inert or to expose their inadequacy in the face of what Mauron called disorder. Instead, Borel attempts to “tame” chance, to use Ian Hacking’s expression once again, to show how the problem chance poses can indeed be overcome. Like Mallarmé, Borel recognises its supreme power in a world where God no longer rules. And yet, rather than embracing chance, he tames it.

As I have argued, it is no doubt the case that both Mallarmé and Cantor figure the infinite as a number. But in itself, this is not enough to make them unconscious contemporaries. For Mallarmé, the infinite is born of chance, whose vehicle of expression is the words of poetry. Mallarmé does invoke the category of number to speak about poetry’s form, but he does so only to show how chance is contained within poetry’s structural, genetic makeup. The choice that Mallarmé presents between the 12 of the alexandrine and the indeterminate number of words in a poem illustrates a larger point: that the choice itself – the or – brings with it a new, a-theological infinite best suited to describe and reunify a world ruled by chance. In this respect, Mallarmé ends up looking more like Borel’s unconscious contemporary than Cantor’s. And yet where Borel sought to tame chance, Mallarmé’s singularity lies in his willingness to celebrate it, to hypothesise it, and to place himself as its guardian and guarantor. Chance is, for Mallarmé, and thus so too is the infinite.

448 “Tel est le sens de la loi unique du hasard”. Borel, Le hasard, 13.

134 Conclusion: What is an Unconscious Contemporary?

“For a long time it was thought that language had mastery over time [...] In fact, it is only a formless rumbling, a streaming; its power resides in dissimulation. That is why it is one with the erosion of time”.449 – Michel Foucault

“Thus under each Word in modern poetry there lies a sort of existential geology, in which is gathered the total content of the Name, instead of a chosen content as in classical prose and poetry. The Word is no longer guided in advance by a socialised discourse; the consumer of poetry, deprived of the guide of selective connections, encounters the Word frontally, and receives it as an absolute quantity, accompanied by all its possible associations. The Word, here, is encyclopaedic [..] and is reduced to a sort of zero degree, pregnant with all past and future specifications”.450 – Roland Barthes

In this thesis, I have shown that there are a number of real and foundational differences between Mallarmé’s poetry and Cantorian and post-Cantorian mathematics. In the first chapter, I inquired into how the concept of the nothing operated within both discourses, arguing that it constituted the place from which they begin. However, by looking at how the nothing was specifically figured in each, I suggested that the mathematical mark of the nothing is “sutured only to itself”, whereas the poetic signifier of the nothing is imminently sutured to that which is outside of itself.451 The mathematical nothing faces inward, whereas the poetic nothing is geared towards the outside, attaching itself to music and thus to the social sphere. In the second chapter, I examined how Cantor and Mallarmé understood the infinite at a conceptual level. Against Badiou’s claim that Mallarmé is “Cantor’s unconscious contemporary”, I argued that their concepts of the infinite – as well as related ideas of chance, God and number – are radically distinct from one another. Mallarmé’s infinite is predicated on a dialectical idea of chance in a world no longer determined by God. Cantor’s transfinite exists without chance or the dialectic, as for him transfinite numbers follow a sequence that ultimately ends with God. Despite working with the same two fundamental concepts – the nothing and the infinite – Mallarmé and Cantor

449 Michel Foucault, “Maurice Blanchot: The Thought of the Outside,” in Aesthetics, Method, and Epistemology: Essential Works of Foucault 1954-1984, ed. James D. Faubion, trans. Brian Massumi (New York: The New Press, 1998), 167. 450 Roland Barthes, Writing Degree Zero, trans. Annette Lavers and Colin Smith (New York: Hill and Wang, 2012), 48. 451 Alain Badiou, “Mark and Lack,” in Concept and Form: Volume 1, Key Texts from the Cahiers pour l’Analyse, eds. Peter Hallward and Knox Peden, trans. Zachary Luke Fraser with Ray Brassier (London/New York: Verso, 2012), 165.

135 have more that separates them than unites them. I have also been pursuing another line of inquiry. In taking Badiou’s remark seriously, I tried to show how Mallarmé could not only be viewed as Cantor’s “unconscious contemporary”, but also as a contemporary of Pythagoras, Descartes and Borel.452 In order to conclude this thesis, I want to spend a brief moment reflecting on what it means to be an unconscious contemporary – and how such a category might serve as a starting point to think the difference between mathematics and poetry at a general level.

In “What is the Contemporary?”, Giorgio Agamben makes a distinction between the contemporary and the present. Being in the present moment means existing in one’s own time, in taking the present world as the totality of all that exists. Contemporariness, Agamben suggests, is different. It is a “singular relationship with one’s own time, which adheres to it and, at the same time, keeps a distance from it”.453 To be a contemporary is thus to be in a relationship of both proximity and distance to oneself, to hold an uncanny mix of resemblance and disfiguration to one’s own present. Agamben draws on a vast number of materials from Nietzsche, the poet Osip Mandelstam and modern science to think contemporariness, but I want to focus here on particular image he invokes. “The Poet%the contemporary%must firmly hold his gaze on his own time”, Agamben writes, yet he must not be “blinded by the lights of the century”.454 For the poet, the present itself is figured as vast “beams of darkness”, fixing the gazes of all its inhabitants.455 And yet the brightness of the stars, of the constellations, still manage to find a way of cutting through this darkness:

In the firmament that we observe at night, the stars shine brightly, surrounded by a thick darkness [...] To perceive, in the darkness of the present, this light that strives to reach us but cannot % this is what it means to be contemporary. As such, contemporaries are rare. And for this reason, to be contemporary is, first and

452 Alain Badiou, Briefings on Existence: A Short Treatise on Transitory Ontology, trans. Norman Madarasz (New York: SUNY, 2006), 124. 453 Giorgio Agamben, “What is the Contemporary,” in What is an Apparatus? And Other Essays, trans. David Kishik and Stefan Pedatella (Stanford: Stanford University Press, 2009), 41. 454 Agamben, “What is the Contemporary,” 44, 45. 455 Agamben, “What is the Contemporary,” 45.

136 foremost, a question of courage, because it means being able not only to firmly fix your gaze on the darkness of the epoch, but also to perceive in this darkness a light that, while directed toward us, infinitely distances itself from us. In other words, it is like being on time for an appointment that one cannot but miss.456

How can we not see Mallarmé in this image – that singular poet who, perhaps, superimposes the luminous lights of the constellations onto the pure darkness of the sky? Not only does Agamben’s image capture something of Mallarmé’s work, it also helps clarify what it means to be an unconscious contemporary. For Badiou, the contemporaneity that Mallarmé and Cantor share signifies a relationship of proximity: both figures are said to be alike, close to one another, even though neither were aware of the other’s existence. But Agamben suggests that to be a contemporary is to be both similar to something or someone and yet different, to see both likenesses and dissimilarity unconsciously reflected in the other. Mallarmé is indeed Cantor’s unconscious contemporary, but only if we understand that the word “contemporary” does not denote unconditional unity. Mallarmé and Cantor figure the infinite as a number, how they do so, however, is radically different. The same holds for Mallarmé’s relationship with other mathematicians from his past and future. Mallarmé and Pythagoras link music to number, but Mallarmé does so on the basis of the nothing, a category the Pythagoreans deny. Mallarmé and Borel both rethink their disciplines under the auspice of chance, but for Borel chance is to be “tamed”, whereas for Mallarmé it is to be celebrated.

“There’s no such thing as a Present”, Mallarmé writes, “no%a present doesn’t exist [...] Uninformed is he who would proclaim himself his own contemporary, deserting or usurping with equal imprudence, when the past seems to cease and the future to stall, in view of masking the gap”.457 Here, Mallarmé rejects the logic of contemporaneity implicit in Badiou’s remark. The poem, instead, exists to rupture with the present. As Mallarmé

456 Agamben, “What is the Contemporary,” 46. 457 Stéphane Mallarmé, Divagations, trans. Barbara Johnson (Cambridge/Massachusetts/London: The Belknap Press of Harvard University Press), 218. “[…] il n’est pas de Présent, non — un présent n’existe pas […] Mal informé celui qui se crierait son propre contemporain, désertant, usurpant, avec impudence égale, quand du passé cessa et que tarde un futur ou que les deux se remmêlent perplexement en vue de masquer l’écart”. Stéphane Mallarmé, Œuvres complètes II, Édition présentée, établie et annotée par Bertrand Marchal (Paris: Gallimard, 1998), 217.

137 elsewhere states, “I consider the present period as a form of interregnum for the poet, who has no business to get involved with it”.458 In rupturing with the present, the poet both “recalls” the past and reaches out to the future in the poem itself.459 For Mallarmé, this is the poem’s essential rhythm, its unique ability to gather different temporal rhythms – that of the past and the future – and place them into the disrupted present sphere, into the very makeup of the poetic signifier. Like the rising and the setting of the sun that characterises Mallarmé’s solar drama, the poem allows different epochs – from Ancient Pythagoreanism to Borel’s future manifesto dedicated to chance – both to appear and disappear in its very lines. Mathematics, on the other hand, follows a different trajectory, one where its marks are sutured only to themselves, only to their present. Its development is that of continuity, expansion and paradigm shifts. It is contemporary only to itself, and unable to suture its language to the ghosts of the outside, the ghosts of the past and future. For Mallarmé, then, the poem thus stretches over the plane of time to erode the present by recalling the past and following the future. This, for him, is the poem’s essential rhythm – and it is poetry’s rhythm alone.

458 Stéphane Mallarmé, Selected Letters of Stéphane Mallarmé, ed and trans. Rosemary Lloyd (Chicago/London: The University of Chicago Press, 1988), 144. “Au fond je considère l’époque contemporaine comme un interrègne pour le poëte, qui n’a point à s’y mêler”. Stéphane Mallarmé, Correspondance: Lettres sur la poésie, ed. Bertrand Marchal (Paris: Gallimard, 1995), 587. 459 Jean-Claude Milner, “The Tell-Tale Constellations,” trans. Christian R. Gelder, S: The Journal for the Circle of Lacanian Ideology Critique, forthcoming September 2017.

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