The creation of stories is a fundamental human activity. By placing events in relation to one another we structure our experience and generate meaning. The narratives we believe convince us by their proximity to what we perceive as real, whether this perception is grounded in our specific knowledge of the world or conjured in our heads as the story is told. Concerns with meaning and structure are particularly intense in mathematics and here, while the desire for precise understanding is the main propulsive force, it is the combination of abstraction and rigour that gives this desire wings, allowing new connections to be seen from the vantage points offered. The volume under review contains fifteen essays exploring some of the contact-points between the areas of mathematics and narrative; my primary focus will be on narratives related to literary fiction.
The notion of mathematics may induce a vague (or possibly quite precise) sense of unease in some readers; however, I think the consideration of narrative through the lens this science offers can be enlightening, not least because we’re dealing with two of the oldest ways our species has created to make sense of the world. As art develops via comparisons with other forms of expression, exploring how its individual notions intensify or fade in changing contexts, so the dialogue between mathematics and narrative offers potentially new spins to questions concerning the notions of truth and structure in fiction. In addition, this book contains many different stories from the history of mathematics, ‘opening the box’ as it were, to show that acts of creation in this science can be as intense and disruptive as in great art, a notion far removed from the idea of mathematics as one dry formula remorselessly following the next.
Whether it’s the ‘given’ truths (axioms) at the core of particular theories, or the thousand-and-one-page proofs that combine formal techniques with ’ghost-like leaps of creativity’ (Gowers) to demonstrate how a particular fact relates to a host of others, the notion of truth is a defining characteristic of mathematics, this ‘knowledge of what always is’ in the words of Glaucon from Plato’s Republic. On one level this appears to rule out any connection with something so time-critical, so steeped in temporality as the idea of a narrative. G.E.R. Lloyd’s essay addresses this point by drawing attention to the fact that both proofs and narratives share a common basis in the idea of an ordered (not necessarily chronological) sequence, whether one traced by the tightly-knit steps of a logical argument or seen in a story’s flow of words and events. As Doxiadis mentions in his introduction, this particular essay is fundamental as it not only provides a common ground for comparison between proofs and narratives but also demonstrates that we are not at risk of some sort of category mistake by wanting to draw comparisons between them. The sources for Lloyd’s analysis are a number of ancient Greek texts concerned with geometry; apart from the enjoyable intricacy of geometric diagrams, one is also struck by the fact that both mathematics and narrative make use of non-textual elements. As examples of the latter, there’s the use of photographs in the novels of W.G. Sebald and the increasing sophistication of the graphic novel.
Lloyd’s focus on order opens up a number of questions on the use of this concept in fiction, one that operates on a number of levels, from the large-scale structures of paragraphs and chapters to the mechanics of individual lines as exemplified by Joyce’s statement: ‘I have the words already. What I am seeking is the perfect order of the words in the sentences that I have’. It is also intriguing to consider ways in which certain authors have manipulated this sense of order to weave deeper resonances into their fictional worlds. For example, Vladimir Nabokov’s Pale Fire follows the references in the poem’s commentary to yield a non-linear but still precisely ordered path into the novel, while the loose, unnumbered chapters of B.S. Johnson’s The Unfortunates (the novel is supplied in a box) offers each reader several equivalent entry-points into the text.
In literary fiction we’re exposed to a particular stream of words and, though our responses are partially related to our individual linguistic wiring, what these combinations of symbols and spaces imply are all we have: the written story is its telling. The details supplied, the uses of metaphor (‘He sat on the balcony and watched the afternoon die’ from Marquez’ In Evil Hour), the interplay of syntax and sense: all of these touch us, in part, because in stepping into the mirror-world the story offers, we have partially surrendered to the Coleridgean ‘suspension of disbelief for the moment’ (Biographia Literaria) about how language operates and yielded to the words streaming through us. The situation in mathematics is different: since ideas are of prime importance in this arena, the formulae representing them ideally minimize the gap between meaning and expression so that the underlying thoughts are presented as clearly as possible. While there is a difference in what is being transmitted, there are several commonalities between mathematical expositions and narratives about how something is presented, one part of a larger question about what constitutes a particular style.
The notion of clarity (mathematical or otherwise) lies at the heart of Timothy Gower’s discussion of the notion of ‘vividness’ where he considers a comparison between the show (‘He sheeted every mirror in the house’) and tell (‘He hated himself’) aspects of literary representation, making the point that it is the combination and calibration of the two that yields the greatest effect. Using the opening chapter of DeLilo’s White Noise, he notes how the build-up of concrete details acts as the preparation (the ‘canvas’ as it were) for the abstract summary we’re presented with (‘told’) as conclusion. This particular trope can also be seen in mathematics where a number of examples are presented (the ‘show’ part) before the abstract structures underlying them are defined (‘told’). Two other points that are particularly stimulating are his discussion of the relative absence of metaphor in mathematics and the possibility of constructing imaginary histories ‘not true and not intended to be believed’ of intellectual developments.
There is a sub-genre of literature that’s deeply entwined with the notion of dreaming, whether it’s the Middle English vision of Langland’s Piers Plowman (‘Then began I to dream’) or the multi-volume classical Chinese work The Dream of the Red Chamber. The shadow this particular word casts in the contexts of fiction and mathematics is one of the topics considered by Barry Mazur in his Visions, Dreams and Mathematics. Focussing on the work of the 19th century mathematician Leopold Kronecker, his Jungendtraum (dream of youth) in particular, Mazur investigates a particular narrative within mathematics best understood by viewing the main protagonist as an idea, one that finds its true shape through the efforts of a number of individuals. The situation where ‘the disembodied dream takes over … and the idea continues its journey’ is related to the quality (the ‘depth’ perhaps) of this waking-dream and its function in connecting otherwise disparate areas of mathematics. Taking his cue from the title of a Delmore Schwartz short-story, Mazur also considers what is implied by thinking and dreaming ‘in a new way’ and, in the light of his discussion, it is useful to consider how particular writers have shaped our conceptions of what is possible in fiction. As James Woods states in How Fiction Works: ‘Novelists should thank Flaubert the way poets thank spring: it all begins with him’.
Mazur’s essay offers a brief survey of the types of stories that can be found in mathematics (‘origin’ or ‘purpose,’ for example). In Margolin’s Mathematics and Narrative: A Narratological Perspective the role that mathematics plays in the creation of narratives is considered. Margolin discusses six categories that range from the artistic portrayal of mathematicians to the use of mathematical concepts in technical narrative theory. This essay is one of the few in this book to mention Oulipo, the French school set in motion by Raymond Queneau (among others) that considered writing a ‘ludic and combinatory process’ and utilised fixed rules, mathematical or otherwise, to derive both the overall structure of a narrative along with its individual details. Georges Perec, for example, did not use any words containing the letter ‘e’ in his novel A Void while in his shorter Exeter Text only words containing ‘e’ were permitted – an approach generalised to every English vowel (along with several other constraints) in the individual chapters of Christian Bök’s Euonia. The fifth group of Margolin’s classification, one he considers in detail, is concerned with the commonalities between mathematics and narrative such as the use of sequences, the relative freedom of invention in both areas and, a concept painfully absent from our discussion so far, the notion of beauty.
The poet John Keats famously combined the latter with the notion of truth (‘Beauty is truth, truth beauty’) and this correspondence is often raised when the aesthetic impact of mathematics is considered; the British mathematician G.H. Hardy in A Mathematician’s Apology (perhaps one of the saddest books about the joy of mathematics) writes that ‘the mathematician’s patterns, like the painter’s or the poet’s, must be beautiful’. The writer’s view of truth, on the other hand, is provisional, as in Dickinson’s ‘Tell all the truth but tell it slant’. In his essay Gowers remarks that there is little use of metaphor in mathematics and it is this particular trope in language, this ‘slant’ approach connecting two disparate things (one of them mathematical), that can be found in the work of the Argentinean Jorge Luis Borges, a writer whose short-stories, while remarkable for their imaginative use of mathematical concepts, are equally fascinating for their precise and hallucinatory style (‘There was a time when I could visualize the obverse, and then the reverse. Now I see them simultaneously … it is as though my eyesight were spherical’ – The Zahir).
Everything we experience is finite; part of its being is its end. Perhaps as a consequence of this, the notion of something without limit appears in a number of contexts such as the religious (the idea of ‘infinite love’) or the psychological (‘I could be bounded in a nutshell and count myself a king of infinite space’). One of the most intriguing areas of mathematics (once described as a ‘paradise’) is a technical theory of the infinite developed by the mathematician (and part-time Shakespeare scholar) George Cantor. In this theory, he proved the existence of not a single infinity but a precise hierarchy of ‘transfinite numbers’, each one demonstrably larger than the last. Many of Borges’ stories have at their heart a central metaphor that bridges these notions and our finite world. A library composed of hexagonal rooms fills the universe, a book with an infinite number of pages, a straight line considered the circumference of an infinite circle: these are some of the remarkable structures that animate his fiction. It is also worth noting that David Foster-Wallace wrote a small book about the history of infinity (Everything and More).
Sadly, there is not space in this review to discuss all of the essays in this book in detail; however, I would like to briefly mention Peter Gallison’s Structure of Crystal, Bucket of Dust where he discusses Haussmann’s rebuilding of Paris in the nineteenth century and considers the influence of ‘place’ on how particular mathematical narratives are constructed – a useful addition to the field of psychogeographical literature.
It is intriguing to see the visionary work of the mathematician Alexander Grothendieck quoted in a number of essays (Harris, Mazur, Corfield, Plotnitsky) illuminating different arguments like glints thrown from a stream. Grothendieck is justly famous not only for his intense devotion to mathematics but for the ground-breaking perspectives he opened-up utilizing a very general, highly abstract approach. Even from a non-technical perspective there is a curious beauty to many of the terms he introduced such as ‘Yoga of Weights’ and ‘Nuclear Space’. As Allyn Jackson writes, ‘he saw the act of naming mathematical objects as an integral part of their discovery’ (Comme Appelé du Néant – As If Summoned from the Void:The Life of Alexandre Grothendieck) – a viewpoint having some connection to Flaubert’s idea of le mot juste.
Circles Disturbed offers a number of perspectives into a new region that’s just beginning to be explored, one formed at the intersection of mathematics and narrative. For the mathematician, I’d suggest this book stresses the importance of an historical background and a renewed emphasis on the stories behind concepts and their relation to meaning. More generally, the essays in Circles Disturbed suggest new ways to consider the importance of form (at whatever level) in writing; not just as an organisational tool, but also as a living component in those tales that hold us and won’t let go.