Secrets of Creation - Robin Mackay

Published in 2020 by: Urbanomic Media Ltd, The Old Lemonade Factory, Windsor Quarry, Falmouth TR11 3EX United Kingdom The Secrets of Creation project was funded by Arts Council England. © Urbanomic Media Ltd. All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or any other information storage or retrieval system, without prior permission in writing from the publisher. ISBN (Print Edition): 978-0-9575295-1-9 www.urbanomic.com d_r0

Introduction: The Madness of Dramatization But I think for the communication of what happened and what we can draw from it, that will have to wait until we put it into a publication. Because it’s going to take us a long time to work out exactly what happened and put it in order. — RM, Tape #19, 28:58 Given any concept, we can always discover its drama. — Gilles Deleuze, ‘The Method of Dramatization’ An experiment in (or without) method, Secrets of Creation traverses the most exalted terrains of higher mathematics and makes the most unreasonable demands on art, only to terminate inconclusively with the construction of an assemblage whose rudimentary gesture scarcely reaches the level of elementary arithmetic and whose aesthetic indifference disbars it from claiming any artistic merit. A project designed as an antidote to the PR slickness and spectacular vaunting of science typical of the ‘sci-art’ that was a feature of the cultural landscape in the early 2000s,1 its focus on the insistence of the problem—‘the urgency of comprehension before any attempt at a solution’2—meant that, rather than art serving the cultural promotion of science under cover of a celebratory invocation of vague notions of their common ‘creativity’, art was knotted together with science and philosophy in a thoroughly perplexing fashion.

The artist Conrad Shawcross and the mathematician Matthew Watkins spent a week in discussion, with no expectation that anything would be ‘produced’, the intention being to present to the public the process of collaborative research itself, in the form of whatever it left behind. The question posed to the two residents in effect invited them to extend their existing practice: In Watkins’s case, presenting his speculative research on prime numbers to nonmathematicians, and in Shawcross’s, creating artworks from meticulously engineered systems that employ space and motion to explore scientific and philosophical questions. It also opened onto the broader social question of the widening gulf between those with scientific training and a majority subjected to a technological regime with little comprehension of the underlying concepts that make it possible. We hoped to attack the question of whether the work of artists can somehow help to cross this gulf—as the sci-art credo assumes—through an approach at once philosophical and oriented toward pragmatic experimentation. Looking back, the philosophical interest of the project documented here lies not so much in this misguided question with which it was announced—How can artists and artworks stage a ‘dramatization’ of abstract concepts that allows them to be grasped by non-experts— nor in the comedy of its eventual unfolding into a fraught attempt to demonstrate, by means of miniature wooden chariots and CCTV cameras, ‘how 2 x 5 is a novel approach to reality’, but in the way that the project engaged with the concept of dramatization itself: thematically, by retracing the most important open question in number theory back to an elementary concept of arithmetic and its anthropological origins, and immanently, in the collaborators’ attempt to present these fundamental concepts in a way that would engage the sensory-motor system. A Platonism Of Problems In an introduction to his work, Albert Lautman, one of the few thinkers to have linked the mathematics of his own time with contemporary developments in philosophy, insists that although

[i]t may seem strange to those who are used to separating the ‘human’ sciences from the ‘exact’ sciences, to see brought together in the same work, reflections on Plato and Heidegger, and remarks on the law of quadratic reciprocity or the distribution of prime numbers […] this rapprochement of metaphysics and mathematics is not contingent but necessary.3 A mathematical thinker of his time—irrevocably marked by Gödel’s consummation of the failure of foundationalism, inspired by the Bourbakian flowering of structuralism in mathematics, an enemy of the constructivism and conventionalism that would deprive mathematical objects of any claim to independent reality—Lautman shared with his friend and colleague Jean Cavaillès4 a conviction in ‘the solidarity that unites the nature of the mathematical object with the singular experiment of its elaboration through time.’5 That is, although an outlier of the early twentieth-century French epistemology emblematised by Gaston Bachelard’s theorisation of science as a dynamic transcendental structure,6 Lautman, like many of that current, interrogated the historicity of mathematics and the reality of its objects, regarding mathematics as an experimental discipline7 dealing in genuine discovery rather than arbitrary invention. To reconcile history and eternity in a stance that excludes classic Platonist idealism (the ‘conception of an immutable universe of ideal mathematical beings’)8 and a collapse into constructivism, Lautman rather surprisingly calls upon the philosophy of Martin Heidegger. Mathematical concepts are historical manifestations of a dialectic of Ideas which, although it cannot be expressed without them, surpasses them as Being does beings. The questions posed to this dialectic, the framing of its Ideas according to a particular epoch and circumstance, bring forth mathematical entities whose difference from the Ideas they ‘incarnate’ must be honoured. It is the problematic Being of the dialectic, rather than mathematical beings, that is eternal. In an articulation that fuses a Platonism, and perhaps Hegelianism, with a Heideggerian gesture, Lautman claims that [t]here is therefore a sense in which one can speak of the ‘participation’ of distinct mathematical theories in a common dialectic that governs them.

The Ideas of this dialectic must be conceived as Ideas of possible relations between abstract notions […].9 Even though the indeterminacy of these dialectical dynamisms means that the Ideas that emerge from them continue to transcend their actualisations, they find a favoured material in the mathematical concept: It seems to me that [the] meaning [of mathematical concepts] lies in their attachment to a metaphysics (or dialectic) of which they are the necessary extension. In short, they constitute the matter the closest to Ideas. It doesn’t seem to me at all that this would belittle mathematics, on the contrary it confers upon them an exemplary role.10 The exemplary role of mathematics, then, relates paradoxically to its being a concrete testing ground. Philosophy may well contemplate and muse upon what Lautman terms the possible liaisons between the notions of an Ideal dialectic—continuous and discontinuous, local and global, whole and part, etc.—and may even produce ‘metaphysical sketches’ of them.11 But it is mathematics that offers the concrete means to construct precise determinations of these liaisons, and generate fields of possible solutions to the problems they describe: [T]hought necessarily becomes involved in the elaboration of a mathematical theory as soon as it claims to resolve in a precise way a problem that could be raised in a purely dialectical way […].12 As well as providing an account of the reality of mathematical objects, then, Lautman’s approach affords the philosopher willing to attend to actual mathematics a unique opportunity to contemplate the process of the genesis of the real from an Ideal register characterised as ‘problematic’. In this Platonism of the problem,13 the presence of fundamental unresolved themes perpetually organises the objects of contemporary mathematics and constitutes the ground of the possibility of mathematical objects as such. A Perplicated Drama

Although the theory of Ideas first presented by Gilles Deleuze in ‘The Method of Dramatization’ [1967]14 is certainly endebted to Lautman, it renounces mathematics’ privilege as primary field of actualisation, ‘the matter the closest to Ideas’.15 The materials in which Ideas find their actualisation are now more various, and seem to play more of a part in determining the course of actualisation. There is not merely emanation but a mutual interplay between Ideal differentiations and their expression in actuality on an embryological model (‘the world is an egg’). In Deleuze’s account of the passage to actualisation, the Idea is incarnated via a movement of ‘dramatization’ which develops its virtual differentiations into the qualitative and organisational characteristics of actual phenomena. The ‘spatio-temporal dynamisms’ that express Ideas are to be found ‘in a physical experiment, no less than psychic experiments of the Proustian variety’, along with biological and even seismic phenomena,16 the suggestion being that every objective phenomenon and every concept attests to a distributive field of Ideal events, and so every domain in which there is a construction of objects involves the dramatization of problems-Ideas. The prerogative of Ideal intimacy with which Lautman endowed the precision of mathematics gives way to a ‘perplication’ that ‘usher[s] the Idea into culture’ at large.17 Which Way? Which Way? Deleuze speaks of dynamisms being ‘reprised’ in different systems, across different domains—‘all sorts of resonances […] between physical, biological, and psychic dynamisms’18—where Lautman spoke of ‘schemata which, in order to be drawn out, must be embodied’ in mathematical theories.19 In the course of our project somehow it became conventional to speak of Ideas being ‘drawn down’, as if through a sort of witchery, into experience. The orientation of the vector of emanation became ever more difficult to determine. Kinaesthetic gestures and concepts belonging to everyday experience (alongside, in front of, between) were mobilised to explore the fundamental Ideas that lie behind elaborate

mathematical developments. At the same time we addressed historical genesis: Does the domain of number arise from our manipulations of physical reality? What are the bridges between the anthropological and the mathematical? What encounters enable us to operate the counteractualisation that takes us from everyday experience to abstract motions with no precedent in lived reality? Our attention was drawn to multiplication, variously described as an ‘innovation’, a ‘machine’, or a ‘technology’ that is foundational for the discovery of the terrain of the prime numbers and their irregular distribution—that ultimate residuum of irregularity, a ‘mess’ of ‘holes’ which are also atomic factors, and whose analysis, as documented by Watkins, ultimately brings to light some startling unexplained connections between number theory and physics. Inspired by the physical presence of Shawcross’s systems, the modelling of this mathematical gesture of multiplication in such a way as to make its remarkable properties available to the sensory-motor system became an engineering mission. The attempt to attain a balance between visual expression, effective motion, fidelity to the mathematical gesture, and technical (not to mention time and budget) constraints, became a kind of pragmatic dramatization in itself, under pressure on both sides by Watkins’s concern for mathematical elegance and Shawcross’s preference for the impossible and the productively malfunctional. At the same time other processes were underway. Every salient point of the Idea that made an incursion into the actual, as if imbued with a magnetic force, attracted inessential determinations that fastened themselves onto the model, inflecting its development and introducing new twists into its actualisation. Imaginative leaps, metaphors, and metonymies flourished and were winnowed, wheat of elucidation separated from chaff of deviation. Having suspended all a priori judgments, however, we became increasingly uncertain of the existence of such things as faulty metaphors, irrelevant associations, or ‘ethnological red herrings’. An unexpected surreality grenade lobbed by Shawcross brought a tinge of pure delirium to the final days of the project, injecting an animal spirit into the proceedings that proved surprisingly fecund. In fact, at every point

where the process really took off, it swiftly entered the realm of nonsense. The full transcript boasts a Carrollian gallimaufry of props ranging from peanuts to boars to quantum chaotic oscillators, from cosmic trombones to floating orbs, from threevens to Jupiterian meters. Perhaps the madness is not surprising, given that we were constantly driven to see the most apparently simple and self-evident things as cryptic dramatizations of the most profound and disturbing problems: Does one exist? Can there ever be two of anything? What if we were starfish? Does a pistachio without a shell count as a pistachio? Where did that vibrating thing come from? The Unpredictability Of Their Coming This process reprised or extended the Ideal dynamism that connects higher mathematics and physics with simple arithmetical operations by ‘drawing down’ those operations into a kinaesthetic and mechanical lexicon. This relay was in fact anticipated in the title Secrets of Creation, which referred both to Watkins’s intuitions in regard to the profound mysteries of the prime distribution, and to the general question of how any creative process, whether artistic or scientific, arrives at unforeseen outcomes—how novel syntheses occur on the way to actualisation. This question, at least, can be said to have been answered, performatively. As to the mysteries of the prime distribution, success came in the form of the generation of further questions. Lautman himself addresses the distribution of primes as an important mathematical fact: The existence of primes in the series of whole numbers has always seemed to present a type of mathematical fact that is as objective, as independent of any prior intellectual construction, as the most manifest physical facts. The passage from 15 to 16 and that from 16 to 17, for example, are achieved by one and the same act—the addition of one to the preceding number—and yet the second operation gives a very different result from the first, since 17 is prime and 16 is not. So that what confers upon the primes their objective character is the unpredictability of their coming.20

For Lautman, the importance of Riemann’s formula for the distribution of the primes is ultimately that it sheds light upon the structural affinities between analysis and arithmetic, and thus upon the dialectic of the continuous and the discontinuous.21 Watkins’s claim is different, but arguably of a ‘Lautmannian’ order: it is the dialectic between addition and multiplication that makes possible the concept of primality; consequently, within the apparently simple operation of multiplication there must be some fundamental problematic at play which, in the process of its extravagant dramatization, unfolds the infinitely uneven landscape of the primes and, in turn, reveals the presence of the oscillating ‘machine’ of which they are the music. Again, Lautman too pays some attention to the multiplication operation, using it to demonstrate the importance in mathematical thought of a gesture of ‘dissociation’: ‘the Idea of multiplication contains both the formation of arithmetic products and the action of operators on a domain of elements distinct from these operators’,22 and thus indicates ‘[t]he distinction […] between the intrinsic properties of an entity or notion and its possibilities of action’. This distinction is effectively invisible in ordinary arithmetic but becomes visible in the case of, e.g., multiplying a vector by a whole number. This is raised in the context of an enumeration of ‘methods of division’, where Lautman suggests that generalisation in mathematics, as in experimental physics, often proceeds not from a subsumption of the particular under the general, but from a ‘dissociation’ that reveals as complex what was previously seen as simple. According to Lautman this type of dissociation demonstrates ‘the close connection of critical reflection and effective creation’23 when ‘concrete experiences present themselves to the intelligence as resulting from the exceptional encounter of certain notions whose separation can be carried out abstractly’.24 Much of the dialogue in the transcript can be understood as an attempt to discover a cognitive lever that will enable the notion of multiplication to be prised open like this. Early on, through a discussion of one of Shawcross’s early works, the notion of feedback emerges as a suitable metaphor to identify the ‘possibilities of action’

unleashed by the introduction of multiplication. Feedback, however, leads inevitably to the question of time. When Is Dramatization? For Lautman, unlike Cavaillès, the chronological timeline of historical epistemology is secondary to the genetic time of emanation.25 Yet this genesis of actual theories via the formation of Ideas from dialectical liaisons, the time of the incarnation of structural schemas, or, we might say, the time of mathematical anamnesis, remains somewhat enigmatic. There is an axis that crosses chronological time, punctures it, but whose movement remains distinct from and anterior to it. For Deleuze this would be the axis of differenciation, the time in which the differentiated virtual body of the Idea is dramatized into (or generates) a field of solutions. For Lautman, [t]he extension of the dialectic into mathematics corresponds […] to what Heidegger calls the genesis of ontic reality from the ontological analysis of the Idea. One thus introduces, at the level of Ideas, an order of before and after which is not that of time, but rather an eternal model of time, the schema of a genesis constantly in the making, the necessary order of creation.26 This question of another temporal axis, in a related sense, repeatedly makes its appearance in the discussions below, naturally following from the notion of oscillations or ‘spiral waves’ that seem to be ‘generating’ the prime distribution: ‘We’re not talking about historical time, we’re not talking about clock time, we’re talking about something outside that […] But again it comes back to this question of time […] what is the time parameter? […] it’s a different kind of time […] you can’t escape from the time thing…’. Perhaps ultimately the ‘surprising connections between mathematics and physics’27 could profitably be addressed from Lautman’s philosophical perspective, by seeing in mathematics and physics two declinations of the same ultimate dialectical schemas that shape reality.28 Equally, the emergence of themes and concepts during the week-long residency seemed to obey a temporality of its own, and certainly not a linear one.

For if the questions of number theory presented by Watkins lend themselves to being understood in terms of the movement of dramatization, so does the collaborative process. In its experimental attempt to incarnate mathematical ideas into experiences available to the non-specialist, Secrets of Creation seemed to recapitulate (or even reverse) the unfolding of fundamental dialectical Ideas into mathematical theories. A Week Is A Long Time In Number Theory This volume contains, firstly, the participants’ re-presentation of the research at the symposium held at the end of the residency period. But in an experiment where methodology itself is in question and the presentation of results matters less than the process of research, there is every reason to preserve the loose talk, divagations, and digressions that actually took place, and so there follows an edited transcript of the week’s discussions, which covers a far larger range of topics than is addressed above. In particular, Shawcross’s account of his own experience of the artistic process and of the genesis of his works serves as a corrective to any notion of a straightforward relationship between scientific model and artwork. What emerges instead is a general admission of the heterogeneity of science and art and the dangers of forcing them into too close a proximity—along with the suggestion that ‘a new discipline, a new type of activity’ might serve the purposes of ‘dramatizing abstract concepts for non-experts’. The need for such a discipline is repeatedly brought into focus throughout the discussion by Watkins’s remarks on the cultural ramifications of the image of number, ‘math fear’, and the problems of mathematics education. If the result of the inability to culturally ‘integrate’ scientific thinking is endemic illiteracy in the fundamental cognitive gestures that built the machines upon which the modern world runs, further experiments in developing this new discipline of dramatization may contribute toward a corrective.29

We were extremely fortunate in our choice of collaborators for this project—an artist whose systems, built as a means to think through metaphysical and philosophical questions, function at an oblique angle to the self-important discourses of contemporary art, and a mathematician more likely to be found presenting workshops at festivals and psychedelic fayres than in a lecture hall. Playing the role of ‘dark precursor’ between these two for the period of the residency was quite an experience, and one it has been fascinating to subject to ‘redaction at a distance’. As well as thanking Conrad and Matthew for their participation in this experiment and their commitment to the process, I would like to thank the team who joined me in facilitating the residency and contributing to the discussion: Paul Chaney, whose extraordinary ability to design, find materials for, and construct just about anything, no matter how absurd, at ridiculously short notice, was crucial; Kenna Hernly, whose work behind the scenes ensured that the whole week went smoothly and was properly documented; Andrea Poças, who generously volunteered her time to support the project; and photographer Catherine Frowd, whose presence with her camera during the latter half of the week not only ensured that we had visual documentation of the process, but in the moment inspired all those present to become more animated and demonstrative. Thanks also to Don Mackay who photographed the wall drawings and diagrams, and Elaine Tam for her work on the images. Robin Mackay, Plymouth, September 2020 1. For a review of the trend and its historical relationship to ‘science PR’ see C. Sleigh and S. Craske, ‘Art and Science in the UK: A Brief History and Critical Reflection’, Interdisciplinary Science Reviews 42:4 (2017), 313–30. In their programmatic conclusion the authors call for ‘neutral spaces—neither art nor science galleries—for transdisciplinary display and criticism, such that the rules of politesse that govern the home turf of either discipline do not pertain’ (327).

2. J.-M. Salanskis, ‘Idea and Destination’, in P. Patton (ed.), Deleuze: A Critical Reader (Oxford: Blackwell, 1996), 57–80: 67. 3. A. Lautman, Mathematics, Ideas and the Physical Real, tr. S.B. Duffy (London and New York: Continuum, 2011), 197. 4. See J. Cavaillès, On Logic and the Theory of Science, tr. K. Peden and R. Mackay (Falmouth and New York: Urbanomic and Sequence Press, forthcoming 2021), and J. Cavaillès and A. Lautman, ‘Mathematical Thought’ [1939], tr. R. Mackay, <https://www.urbanomic.com/document/mathematical-thought/>. 5. Cavaillès and Lautman, ‘Mathematical Thought’. For a detailed philosophical introduction to Lautman’s thought see also J. Petitot, ‘To Remake the “Timaeus”’ [1987], tr. R. Mackay, <https://www.urbanomic.com/document/jean-petitot-remaking-thetimaeus/>. 6. E. Barot, ‘L’objectivité mathématique selon A. Lautman: Entre Idées dialectiques et realité physique’, Cahiers François Viète 6 (2003), 3–28: 6. 7 . Ibid., 4. 8. Cavaillès and Lautman, ‘Mathematical Thought’. 9. Ibid. 10. Ibid., emphasis mine. 11. ‘[A] non-technical, non-mathematical, so to speak metaphysical sketch’, Salanskis, ‘Idea and Destination’, 67. 12. Lautman, Mathematics, Ideas and the Physical Real, 218. As I have described elsewhere, ‘[t]he successive posing of questions and generation of concepts operates an “enframing” of the Being-IdeaProblem, historicising it by constraining it to bring forth “cases of solution” [beings] to which it remains irreducible but without which it would remain the object of a sterile and mute contemplation’. R. Mackay, ‘Editorial Introduction’, Collapse 3: Unknown Deleuze (Falmouth: Urbanomic, 2012). 13. Or, according to Barot, more of a ‘differential’ reappropriation of esoteric Platonism, a Plotinian doctrine of ‘emanation’. Barot, ‘L’objectivité mathématique’, 7. 14. G. Deleuze, ‘The Method of Dramatization’, tr. M. Taormina, in D. Lapoujade (ed.), Desert Islands and Other Texts 1953–1974 (Los

Angeles: Semiotext(e), 2004), 94–116. 15. ‘The conceptual link between Deleuze and Lautman is that the former theorized ideal actualization in a general way, while the latter studied its philosophical logic and technical examples within the framework of mathematics alone. Deleuze privileges mathematical language, making use of mathematical terms to describe the properties of all Ideas, but he thinks the Idea in a general. transcultural way: indeed, one aspect of his conception is that the Idea is essentially pluridisciplinary’. Salanskis, ‘Idea and Destination’, 58. 16. Deleuze, ‘The Method of Dramatization’, 98. 17. Salanksis, ‘Idea and Destination’, 77. 18. Deleuze, ‘The Method of Dramatization’, 115. 19. Cavaillès and Lautman, ‘Mathematical Thought’. 20. Lautman, Mathematics, Ideas and the Physical Real, 213 [translation modified]. 21. Ibid., 218. Lautman claims that the pursuit of the prime distribution function lends weight to his thesis that ‘mathematical reality does not lie in the greater or lesser degree of curiosity that isolated mathematical facts may present, but only in the dependence of a mathematical theory with respect to a dialectical structure that it incarnates’. He points to how, historically, the research moves from a specific investigation into more abstract procedures that bring ‘a more hidden structure into play’ (Ibid., 213–14 [translation modified]). 22. Ibid., 40. 23. Ibid., 32–34. 24. Ibid., 33. 25. Barot, ‘L’objectivité mathématique’, 19. 26. Cavaillès and Lautman, ‘Mathematical Thought’. 27. See <http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/surprising.htm >. 28. See Barot, ‘L’objectivité mathématique’, 21. 29. The ‘hands-on’ kinaesthetic tendency that pervades Secrets of Creation should give pause for thought for the fact that, at the sharp end of this cultural deficit, while research consistently indicates that

spatial and kinaesthetic awareness and physical manipulation of shape in early childhood is a key contributor to later mathematical cognitive ability, we are increasingly educating and entertaining our children through flat screen images. See S. Gifford, ‘The Importance of Shape and Space in the Early Years’, NRICH, < https://nrich.maths.org/14544>, and the papers listed in the bibliography of this article. For a wider perspective on this question see M. Fisher, ‘Touchscreen Capture’, Noon: An Annual Journal of Visual Culture and Contemporary Art 6 (2016), available at <https://egressac.wordpress.com/>.

Exploring the Boundary, Seeing It Fail, Breaking Things Open Public Symposium, Urbanomic Studio, 13 March 2010—edited transcript robin mackay: The Secrets of Creation residency project emerged out of work on Urbanomic’s ‘Journal of Philosophical Research and Development’ Collapse, the aim of which is to bring together artists, scientists, people working in various disciplines and practices, together with philosophers, the conviction being that philosophy has to be nourished from outside, and that philosophical thought occurs in all disciplines: that, whatever you’re doing, at certain points you encounter philosophical problems and need to employ or invent philosophical concepts. Since its inception one of the features of Collapse has been extended interviews, and one of the first of these, in Volume I, published in 2006, was with Matthew Watkins, where he spoke about his work collecting, collating, and synthesizing research on what he himself called the ‘mysterious links’ between prime number theory and physics. It was a fascinating discussion, and I felt that Matthew was precisely the kind of broad-minded thinker appropriate to this notion of Collapse as a trans- or indisciplinary journal. Then, much later on, in 2012 I worked with Conrad Shawcross on a piece for Collapse 5. What interested me about Conrad’s work, and what fed into this project, was the fact that he was working with scientific and philosophical concepts. He was, in effect, materialising these concepts in his works, which themselves are also pieces of precise engineering. And I was interested in the way that this made something otherwise abstract available to the senses, available as an experience, sometimes with unexpected results. Because, obviously, one of the problems of looking at modern and contemporary science, and mathematics in particular, is that it’s

complex and requires a certain amount of time to be initiated into, so that most of us can’t really take on its concepts fully. What I was very interested in was how one could ‘draw down’ these concepts into some kind of work which would make them available to non-experts, so that you could actually in some sense experience a mathematical concept, say—and what role artists might play in that process. So, at a certain point, it became an idea to bring these two together to address, or rather to enact, this question of how artists and artworks can ‘dramatize’ abstract concepts and allow them to be grasped by those who are not formally trained. This is what we’ve been discussing and working on this week, in a process that’s been intense, exciting, sometimes frustrating, sometimes enlightening, and sometimes quite funny. I think the main conclusion we drew during the earlier part of the week was that this was a very flawed question—but the process of working out what was wrong with the question was itself productive,

suggested new questions, and opened up a process in between Conrad and Matthew in which we were able to examine in more depth what artists and scientists have or don’t have in common in terms of their way of looking at the world, and the tensions that emerge when they try to collaborate. So I’m not sure whether we answered the question, but we certainly ended up revealing a lot more problems which were fertile and significant in various ways. When we got to the middle of the week, we discovered we weren’t happy with the question ‘How do artists and artworks dramatize abstract concepts?’ because no one was sure whether they did—or should. So it became more of an experimental hypothesis: If artists were to do such a thing, then how would it work? And from that emerged an interesting creative process. We’re going to explain what all that was about, explore a bit further the themes that have emerged, and talk about this interesting interplay between our two residents and our tentative groping towards a process that would involve both parties—with, I think it has to be said, a great deal of goodwill on both sides. But first I’d like to turn to the residents to say a little about their respective work. Firstly, Matthew, since I spoke to you for Collapse I, you’ve been working on a trilogy of which the first volume is about to appear, and from which our project takes its title: Secrets of Creation. The first volume is called The Mystery of the Prime Numbers.1 What is impressive to me about the book is that not only is it a genuine attempt to try and explain some very complex mathematical concepts to a general audience without requiring any mathematical background, it also contains some very speculative and provocative hypotheses about what lies behind those concepts. Could you explain to us, as well as you can in the limited time we have, what that project is all about? matthew watkins: The first thing I should make clear is that the book may appear to be part of a current trend—there’s been a proliferation of popular science literature and other media, and there a lot of books in circulation that are trying to effectively promote or

generate excitement or enthusiasm about mathematical or scientific discoveries. I’d like to separate myself from that, because this isn’t simply an attempt to promote mathematical ideas as something everybody should be interested in, or that are somehow very important or exciting. The project is motivated by the philosophical implications of what I was finding. I didn’t specialise in number theory in my academic work. I left academia years ago, then found myself reentering the world of mathematical research some time later, mainly because there seemed to be some strange philosophical implications to what was being discovered about the distribution of prime numbers. After years of creating and curating and expanding an online web archive, which deals with this from a fairly high-level academic point of view, various people had asked me to create a more accessible version based on my apparent abilities to explain some quite difficult ideas to ordinary people. This seems to have come about because most of my friends are not involved in mathematical sciences and, just based on wanting to explain to my friends what I’m doing, I’ve developed a repertoire of techniques and visualisations and metaphors which seem to work, to explain some fairly difficult ideas that are normally considered inaccessible to non-mathematicians. With a bit of imagination and effort I’ve found you can actually get a lot of these things across. So basically, driven by the desire to bring some very important philosophical questions into public discussion, I’ve set out to selfpublish this trilogy. rm: The distribution of prime numbers is at the core of it. Could you explain this distribution from which these as yet unsolved mysteries emanate? mw: Most people will have learned what prime numbers are at school, and some of you will have forgotten or may just about remember, but probably won’t remember why it was important or can’t see why it would be important.

Prime numbers are those counting numbers that can’t be divided. They’re indivisible. The simplest way I can find to explain it is in terms of a pile of beans or a pile of pebbles. Some piles of pebbles can be divided into smaller piles of equal size. Twenty pebbles can be divided into four piles of five or ten piles of two. Twenty-one can be divided into three piles of seven. Twenty-two can be divided into two piles of eleven. But twenty-three pebbles is just twenty-three pebbles. You can’t do anything like that with it. And what’s interesting about this is that it completely transcends anybody’s opinion, anybody’s culture. It doesn’t matter what symbols you use to write the number twenty-three. You can write it in Roman numerals, binary, Mayan hieroglyphs, or anything you like. It’s one of these inarguable truths that some numbers, which we can represent as a pile of similar objects, can be subdivided, and others can’t. It seems a fairly straightforward matter then, for example, to write the numbers along a line, in a way that you may remember from mathematics lessons. Just create a number line and mark out the numbers that can be subdivided and the ones that can’t. And if you start doing this, one of the first things you notice is that there’s no obvious pattern. There’s no regularity. It looks completely arbitrary. And yet it’s just there, it’s just coded into the nature of reality. Unlike almost anything else that you encounter in the world, as I said, it just transcends opinion or culture or time or space. You could almost imagine replacing the entire contents of the universe with something else and it would still be true. There’s not much you could say that about. You could even imagine the laws of physics being different. So we’re looking at something really quite fundamental, in the truest sense of that word—bearing in mind that number is the interface whereby we comprehend the world and break it down into categories of similar things which we can name and talk about and manipulate. This irregularity in the sequence of prime numbers creates a sort of unease, in that people sense there’s something here. They can sense, if you display this sequence visually, that it’s not regular, and yet they feel it should be, because we’re so used to a mathematics that is, to most people’s minds, associated with order and regularity, with an ability to pin things down in that sort of way. And yet the

prime number distribution defies that. And that’s the starting point, really. Mathematicians have been struggling, since Ancient Greece, to try and make some sense of this, and there’s been a body of theory accumulated which is still being added to. There were some watersheds in the mid-nineteenth century where certain things were discovered, which I discuss in the book, but effectively we’re still a huge distance from really understanding what we’re looking at here. audience member: Are you saying, then, that if you project the sequence of prime numbers however far you want, there are no predictable or recurring sequences of intervals? mw: That’s right, exactly. It’s an absence of regularity…or, in fact, almost the inverse, negative image of regularity, because you’ve stripped out all the regularity. You remove all the even numbers. You remove all the multiples of three. You remove all the multiples of four, five, six…. Anything that you associate with rhythm or regularity has been removed, and you’re just left with all these holes. And the holes make this jumble, this messy-looking pattern, and this has been troubling the Western psyche, which wants to nail everything down conceptually, and generating, as I said, this huge proliferation of very difficult mathematics. At one level, you’re just dealing with counting numbers. You’re dealing with a concept that can be explained to a seven-year-old really, just playing with pebbles—and yet the world’s best mathematicians have been banging their heads against the deeper problems at the heart of this matter for centuries—millennia!—and really just getting more confused, in some ways. If I was to show you what some of what the mathematical research looks like, in terms of how it’s presented, it’s horrific! The whole reason for my book is to try and gently bring people into this discussion, because if you open up a book about the theory of the Riemann zeta function, it’s aesthetically very displeasing and it looks like some sort of alien language, and you have to be initiated

into it. You have to dedicate years of your life to be able to understand even some of it. rm: The Riemann zeta function is the most important mathematical tool for understanding the distribution of the prime numbers, and yet proof of Riemann’s hypothesis, since it was stated in 1859, has proved elusive. Essentially, there’s something very fundamental here about reality as understood through a number system, which, we all accept, is one of the most important and basic ways in which we understand reality. An unknown which has only been deepened by all of the research. And it’s this that you’re trying to get across to a general reader, but you also have your own hypotheses about it; not only its mathematical implications but also its wider social implications, because the way in which we understand number has an impact upon the way in which we understand the world. mw: That’s exactly it. rm: Matthew was talking there about the aesthetic dimension of mathematics and the fact that, if you open up a mathematics book, you’re unlikely to be seduced visually. And also about his trial-anderror development of a repertoire of techniques and ways of visualising and metaphorizing what he wants to get across to friends or to readers. Conrad’s work, I think, also often involves this element of visualising the nonvisual, indeed that might best describe its relation to scientific and philosophical concepts. Would you agree with that, Conrad? cs: Yes, mine is very much a different approach, but we share some similar motives, Matthew and I, in the sense that we’re both trying to visualise these invisible aspects of reality, and we’re reliant on a certain amount of metaphor and visual devices to communicate ideas. A lot of my work would come under the category of ‘systems’. I make systems, or I make models of things. These systems don’t

necessarily work, and are not necessarily accurate, in the sense that they’re not a correct model of something. But I have to say that, over the years since I started doing this, I’ve usually been under the impression when I started a journey with a piece that it was quite a good model of something. It’s only in hindsight that I’ve realised that the model is quite flawed. So it’s as much about my own personal journey of discovery as it is about the viewer’s experience. I think one of the problems of putting the viewer first is that you have to make an assumption as to who that viewer is. And depending on your own experience, you either underestimate or overestimate who that person is. There is always a shifting idea of who the viewer would be in terms of your own prejudices and your own experience. Generally, when I was younger, I underestimated my viewers, my audience, because I assumed that they had a level of knowledge similar to my own experience. So you just have to make things for yourself, really, otherwise it becomes overcomplicated. Anyway, I have gone on certain journeys. For example, I’ve made a lot of rope machines. When I was a teenager and when I was a young student, I became very preoccupied by rope machines, and I wanted to make a system like that. I didn’t know why I wanted to make that system, but I wanted to make a machine that would make a rope, and I was really obsessed with that process. And through actually making a rope machine, I realised this was very much about my own perception of time as either a line or a cycle. It was only later that I realised that this wasn’t really a mode of time at all; it was more of a metaphor for time and the way you perceive it. This is one of the things that has preoccupied me in my work, trying to visualise time. One of the things that’s been quite interesting this week is that I’ve been trying to push Matthew to help me try and find a way of doing that that is less metaphorical. But I think one of the conclusions that I’ve had to come to this week is that, unless you really understand mathematics and write in the language of mathematics, you can’t really understand time. You can only experience it through metaphor and through your own consciousness. In a way this is quite discouraging, but also quite

encouraging, because it just means most of us are in the same boat, in that sense. rm: Perhaps an apt metaphor for what’s been happening during the week, given your rope machines, is that it’s been a kind of tug of war. It’s been very interesting to see Matthew pulling you, sometimes willingly and sometimes unwillingly, toward his very precise way of understanding these concepts, and then you pulling back to regain the freedom you need in order to make something that is more than simply a model. Because, as you say, your works are never simply models to illustrate a mechanism or demonstrate a point. cs: Yeah. But actually, just listening to Matthew just now, I realised there’s another common link, in the sense that Matthew’s book isn’t just an educational treatise, as a lot of these books are. It’s not a way of bringing maths to the masses. It asks a lot of questions and brings up a lot of problems, rather than giving answers, which I hope is similar to my practice. It’s not just a pure model of how scientific knowledge can encompass everything: it doesn’t say that it won’t, but it’s hinting at what might be implied by the idea that we haven’t ever been able to find this way of solving this scatter of the primes. That there are bigger things at play than mathematics, possibly. The failure of my systems, the failure of my models, is where the real interest lies. If I had actually made a true model of time, if I had stumbled upon that, it wouldn’t be an artwork. They’re not interesting as true models, for me, in terms of their status as art. mw: This is quite a major point. Where is the boundary? When does a model that illustrates a scientific point become an artwork? We were exploring that boundary for a good part of the week, really, and that hasn’t been entirely resolved. rm: Although the actual project and its outcome ended up staging precisely that problem, didn’t it? At a certain point, I think it would be fair to say that Conrad challenged everyone else on the team to make this model of what we were discussing. Conrad challenged us

to make the model which he then, later on, responded to in a very unexpected way. And so we did get a kind of image of a process in which one would begin with a mathematical concept and try, in a very straightforward way, to create a model of it. But then the fact of making a physical model of it would introduce all these other factors, which would then feed back onto the way in which you were thinking about the subject and end up somewhat derailing the modelling process. For me that was fascinating to see. Perhaps we should try to explain the specific problem that we ended up homing in on. It concerned the fact that one of the ways to approach this problem of the prime number distribution is through the contrast between addition and multiplication. Perhaps, Matthew, you could explain a little about why something that might seem rather elementary opens onto problematic domains. mw: Well, addition and multiplication have several things in common. Both of them take two numbers and somehow combine them to create another number. Both of them are something that you learned fairly early on in your life. Even if you never really got the hang of doing them very well, you still understood what they were supposed to do. Both of them are represented by a simple symbol: ×, +. There are buttons on your calculator, there’s the plus button and there’s the times button…. So you’re easily lulled into thinking that these operations, as a mathematician would call them, are of the same category, that they’re the same type of thing. They’re arithmetical ways of combining a pair of numbers to create a new number. But something that becomes apparent when you think about this enough is that there is something categorically different about addition and multiplication. And it’s the tension between them that creates the philosophical problem associated with the prime numbers. The irregular scatter of the prime numbers that I described before is only present when you lay them out along a number line. Now, a number line in itself encodes the notion of addition. You’re taking space and dividing it up into equal intervals. And the act of addition can be dramatized—that’s the word we’ve been using this week—by simply walking along the number line. If I want to add three to seven,

I can just walk three steps and then walk seven steps. I’ve walked ten steps. That’s fairly straightforward. Or I can do it the other way round: I can walk seven steps and then three steps. But in each case, the three and seven are playing the same role: they’re a number of steps. I can count three pebbles and seven pebbles. I have ten pebbles. This is uncontroversial. But if I want to dramatize three times seven, something else has to occur. I have to walk three steps, and then three steps, and then three steps, three steps, three steps, three steps, three steps. I’ve had to keep track of how many times I’ve walked three steps. I’ve walked three steps seven times. The three and the seven are doing completely different things. The three is the number of steps, and the seven is the amount of times you’ve repeated the act of taking the three steps. That’s what we call an iteration: you’re taking the idea of adding and you’re turning it on itself. You’re applying the number system to itself. I’ve thought about this a lot, because prime numbers only really come into being when you have multiplication. If you just had addition—and, if you think about it, no one really knows how numbers arose in human culture; there’s a lot of speculation and theory and it doesn’t really help to dwell on that—it would appear that counting came first so you could keep track of things, and then adding things together would have been the next innovation. Now, you could just have counting and adding for a very long time until it finally occurred to you to multiply. If you just had counting and adding, there would be no prime numbers. The notion wouldn’t be there, It just wouldn’t be present. Because to have prime numbers you have to have the notion of divisibility. Dividing a pile of twentyfour stones into three piles of eight—well, that’s just multiplication backwards! Three times eight is twenty-four. So you’re looking at the answer—twenty-four—and thinking: What times what is that? So if I’m looking at twenty-three pebbles, what times what is twenty-three? Twenty-three times one? Well, that doesn’t count because anything times one is itself. So it is the fact that no two numbers can be multiplied to make twenty-three, in a nontrivial way, that makes twenty-three prime.

So the act of introducing multiplication brings with it this scatter of prime numbers. Suddenly, as soon as you have this idea of counting numbers rather than things, this crucial innovation, then things go a bit weird. We’ve evolved what I call a cultural operating system—a way of sorting out the world, breaking it down into recognisable objects. For example, how many people are sitting here in front of me? I have this concept, immediately, of a person, and I can scan my visual field, and I can separate out the various textures and colours, and see there’s a person, there’s a person, there’s a person, and I can start counting them. That’s fine. That works really well if you’re counting your children or your cattle or pebbles on a beach or whatever you want to count. But at some point, somebody had the idea of counting numbers: of actually taking this thing that’s scanning the world and processing it, and saying, what if we turn this on itself? So, how many eights make twenty-four? Well, what’s an eight? You can count rabbits, you can count pebbles, you can count people, but counting eights? There’s only one eight, surely? But then where is it? It’s another level of abstraction. You’re counting numbers. And in terms of our project this week, this suggested the partially successful metaphor of feedback. rm: Because multiplication introduces a kind of self-reflection into number. And I think this was the point, during the week, Conrad, when you picked up on this and operated a displacement on it by linking it back to a work that you had once made. cs: There were two interesting conversations around this, one was about Narcissus, and the other one was around this piece I once made, which was just the thing you’ve probably all done with a microphone and a speaker, where you put the microphone near the speaker and you get that feedback system, where the output is fed back into the input and you get a high-pitched squeal. If you do the same with a video camera and a monitor, mounting one in front of the other, then depending on the angle of the camera in relation to the monitor, you can create a kind of spiral effect. If you zoom in, at a

certain point you’ll get to the event horizon, if you like, where it becomes quite shaky and it will go into this Doctor Who-like gloopy, liquid place, which is basically an infinite progression of screens within screens within screens. mw: The output has been fed back into the input, leading to an unpredictable nonlinear dynamical system. cs: And this actually seems kind of extraordinary when you see it at first, but one thing that I think was a really interesting comparison, that’s just to do with our everyday experience, is that every time you look into the mirror and you see your own eyes and perceive yourself, as you’re standing in front of yourself, even for the first time as a four- or five-year-old, when you have that first conscious moment, it’s that same thing that is happening inside of you. You’re processing yourself inside your own head, and you basically have a Russian doll, an infinite regress of yourself, inside yourself. It has that same kind of fractal repetition to a notional infinite point. You can do it with animals, too, with cats, and you hope that they’ll have this epiphany, but it just doesn’t happen! audience member: Some animals do! cs: They do, but they think it’s someone else. audience member: No, no! cs: Okay, well, you’ve got a very smart cat. audience member: No, there’s lots of research on that…. cs: But with children, anyway, it’s one of those first moments, and you can link it with a lot of theories of the origins of consciousness and the myth of Narcissus….

rm: We arrived at the notion that multiplication was the narcissism of number. cs: Yes, that was one catchphrase. rm: At this point I felt there was a little bit of magic that happened where there was this sudden connection, and a displacement of these ideas we’d been talking about. From this there developed the idea that the way in which number operates upon itself could be figured in terms of video feedback, and we began to slowly develop this model you can see illustrated on the wall here, in which you would see two and three side by side, addition, and then you would see three operating on two, multiplying it.

mw: This was me trying to find some point of contact, so I wasn’t pretending this was going to be a work of art. This is me suggesting a physical model that we could start from, as a point of departure. So, we have some sort of object. We were eating a lot of peanuts this week, and there were lots of peanuts lying around on the table, so peanuts became this sort of archetypal unit for the week. But later on, Conrad pointed out that, actually, the peanut contains two kernels….

cs: It was quite nice, because we were using it as an example of singularity but then, of course, the peanut is two entities inside a shell. mw: From a pure mathematical point of view, you would just dismiss that. You would say, that’s irrelevant. Let’s just use a pebble then. But actually, I’m quite interested in this kind of unexpected subversion of these ideas, because I’m not a traditional mathematician…. The idea that I was trying to put forward was that, if we put a camera in front of a peanut and split the signal into two monitors, then we’d see two peanuts. This is the act of doubling something. This represents twoness. Anything you put in front of this, you get two of them. And if we then put a second camera and, say, we split that into three, that would show three peanuts…. We could have used any two numbers. We could use 29 and 168, but it would have been far more expensive to build! So, based on the budget limitations, we limited ourselves to the first two different numbers bigger than one, so we had 2 and 3. You have a set of two monitors and a set of three monitors. It’s a pretty straightforward model. You’ve got a peanut before the cameras, here you see two peanuts, and here three peanuts. Two peanuts plus three peanuts equals five peanuts. Yeah, wow! But the innovation is, what if we move this three-peanut camera in front of the two peanut monitors? Let’s just take this whole thing, the threepeanut set-up, as a single complex, and let’s just shift it over here in front of the other monitors. Now, this camera isn’t looking at the peanut anymore. It’s looking at the output of the pair of peanuts as a single thing. So this peanut represents unity, or one, and these complexes represent the counting of a certain number of something. At the moment, this is counting three peanuts. If we move it here it is counting three pairs of peanuts. But the peanut is just representing unity so, effectively, it is counting three twos. So what we end up with on the screen is two peanuts here, two peanuts here, and two peanuts here, on each of the three monitors. And, depending on how you set the cameras up, you can even get it so that you can see the

peanuts are framed. So that, even though there are two peanuts here, you see that they’re sort of a dual. The twoness of them is preserved within the screen. So the first configuration of the assemblage represents the fact that two plus three equals five, and the second configuration represents that two times three equals six. Now, of course, these facts are of no importance whatsoever. But the point is that this is trying to dramatize the categorical difference between addition, where the numbers are playing the same role, and multiplication, where one is playing a very different role from the other. In one, a number is pointing at an actual thing, in the other, a number is pointing at a number. So this was just a simple attempt to dramatize the categorical difference between addition and multiplication. Then we spent time discussing how you’d most effectively do this, just in terms of creating a model, as well as the more interesting issue: Could this ever be a piece of art, or is it always going to be the sort of thing you’d find in a science museum?

rm: At this point, I think it’s fair to say that some of us got quite excited and Conrad went a bit quiet, so I think it would be interesting to hear, Conrad, what were you thinking when we began to make this thing? cs: It was quite interesting for me, because I’m quite used to being this egotistical artist, working on my own and just pushing forwards on my own. So initially I was being led by this idea, which I wasn’t used to, in that sense. But I was interested by it, although at first, I was thinking: Are we going to make this artwork that is pretty boring? Is it going to be a model? It’s not going to create problems; it’s just going to be a demonstration of this thing. And I didn’t think that that in itself was going to be that interesting. I also thought it might be a bit clunky, because we were going to make a thing that went back and forth between these two positions. I was just worried about the way it was going to realised. But then, there was this enthusiasm, and I did like the idea of this branching effect with this system; it had this branching system which had a time element to it that I liked. I thought, this is actually good. We’ll make this model and then I’ll just respond to it or I’ll twist it into something that’s maybe more ambiguous. For me, the success of a conceptual piece of art is in its potential breadth of interpretation, and there’s a certain death-of-the-author that I like that is a measure of the success of a piece for me, in that I really relinquish control of it and how it’s interpreted once I finish it. But, for Matthew, it’s the conveying of the idea is paramount, and that ambiguity is the enemy of his intent. rm: This was one of the major differences we had to work through, or with. cs: Yeah, this was a real obstacle that we were getting to. Basically, it came out of a dinner we had on Wednesday after a talk I gave in Penzance. I’ve always been preoccupied with numbers, on a basic level—in a philosophical way but not coming at it with as an expert in

maths, just an epistemological or phenomenological concern. I’ve made pieces that are about specific ratios of numbers, looking at them in an aesthetic way, visualising musical intervals. So there’s been a lot of that kind of philosophical or metaphysical looking at what a nine is to an eight, or what a seven is to a five. And I’ve read a lot of books on the history of zero, like Robert Kaplan’s The Nothing That Is, or Sadie Plant’s Zeros + Ones, and those sorts of preoccupation with the history of number are really interesting to me.2 One of the things that happened—again, out of this fruitful exchange—was that I saw that one of the things we were really dealing with was that, in a way, we just needed to be more objective and distanced about what we were looking at. Because we’re so familiar with numbers, we’re so conditioned to using them in our everyday life, that we can’t really think about what they are anymore; they’re such an inbuilt mechanism. It took us years, when we were children, to learn numbers, because actually they’re very complicated things. You may have a more innate gift that enables you to perceive what they really are, but most children just learn them by rote. But the more you think about them, the more the simple idea of them collapses. When you’re looking at linguistics, writing in Chinese about English is easier than writing about English in English. I thought that what we needed to do was to create an objective platform for looking at numbers and, instead of using some other abstract system of numbers, I started thinking about primeval man and what that the first occasion of thinking about a number would be like as an experience, and the collective nouns of animals came up in discussion. In the fifteenth century, the nun Dame Juliana Barnes wrote a whole list of quite fanciful and poetic names for animals, published in the Book of St Albans; there are all these fanciful terms, many of which are quite humorous. But in actual fact there are a lot of these collective terms that go back a long way. And I was interested in the fact that some of these collective nouns represented specific numbers of animals. We did a bit of research and found that most of them refer to a group of animals but not a specific number.

But I was interested in this idea: if you were primeval man and you just had a tiny community of people in some remote wilderness, and you were confronted by a couple of rabbits that came by every evening at dusk. They ran past your field and you always saw this same amount of animals going past. Or three hawks would fly over your head every dawn. You wouldn’t have an idea of number, because you had no need for them. You would have no consciousness of what a number was. So you would call that family of animals that existed together a name. For example, a brace of rabbits is two rabbits. It refers to a specific number of rabbits. A lease of hawks is three hawks. So you’d have these. Another interesting one is a singular of boar—which is still a collective noun; there aren’t many words for animals that exist on their own.

So we created this sort of strange animal system for numbers that was trying to move completely away from the number line one, two, three, four, five…. You’d have a squadron of birds that was seven, but they would have nothing to do with the sequence of numbers, but just to do with some sort of basic patchwork of reality that you would build up as an early human. rm: It brought into relief even more strongly the idea that you could apply numbers to themselves. When you are referring to these named groups that are very concretely anchored in reality, it makes it even more patently bizarre to apply them to one another. As you can see by the multiplication table, when you multiply braces and leases, you’re dealing with non-fungible animal units, and you get absurd clusters of words. cs: If you’re going to multiply something by two or three, you’re twoing something or you’re threeing something. But we were then using the term ‘bracing’, and likewise ‘threeing’ became ‘leasing’. We created the whole times table, which Matthew filled in. mw: Yes, we were getting quite whimsical. It had been a long week! rm: But I think it’s worth dwelling on that just for a moment. This is the next episode when, in this tug of war, the rope got pulled the other way. Conrad was getting really excited and Matthew was saying, well, I’m not quite sure this has any relevance. But I think, gradually, we all began to warm to the menagerie. mw: I’ve read a little about this in ethnomathematics, where in ethnographers’ studies or early encounters with indigenous peoples from various parts of the world, various words for numbers or various peoples’ dealings with numbers have been recorded. And, in some cultures, if you’re talking about three people, you use one word for three, while if you’re talking about three cattle, you’d use a different

word, or if you were talking about three sticks, you’d use a different three word again. That’s really quite common. For us, it’s obvious that you’d use the same word, but if you’ve just got used to looking at this configuration of people or animals or sticks and seeing it in a certain way, you’d give that unit of that assemblage of things a single name. Why would you relate them to one another? You’d only relate them if you had a reason to abstract out the threeness, to strip out the uniqueness of the thing that’s being counted. This is what the Western mathematical project, rooted in Ancient Greek thinking, has done: taught us to abstract out anything particular about what is being counted, and just deal with the numbers as pure tokens which can be manipulated as things in themselves. rm: So when you’ve got specific number words for objects that belong to lived reality, it’s an obstacle to that. mw: The turning of the numbers on themselves is really the next stage in that process, where you’re able to count threes or eights or

something. And the absurdity of this act and/or its innovative nature, is something I’m trying to bring to the fore. Conrad was sensing this, I think, and as we discussed these animal terms, I gradually began to see what he was doing. Because you can have a brace of rabbits —I’m still not sure if they can be alive: I think a brace of rabbits are usually dead rabbits, but that doesn’t matter, I know that different cultures did have similar types of words—so, you can have a brace of rabbits but you can’t have a brace of anything else, because the word is specific to rabbits. audience member: Brace of pheasants? mw: You can have a brace of pheasants, that’s true. Okay. Brace of pheasants, brace of rabbits…. But suppose—oh, look, there’s a lease of hawks, and there’s another one! It’s still not a brace of leases. Brace applies to specific animals, not to concepts. You can’t have a brace of concepts, any more than you can have a brace of braces. A lease is a concept, as is a brace. rm: It’s as if Conrad came from the opposite direction to the same point that you were arriving at with your schematic model. cs: And in this table, you’ve got these impossible multiplications…. You’ve got a leasebracehawk or a braceleasehawk, which represents six in the traditional number system. But a braceleasehawk basically breaks down into a two and a three and a one. They are the primary numbers. So if you take any sort of larger number, you can break it down into those primaries. mw: It very quickly becomes completely unwieldy. The point is that no one ever did this. It’s showing that if you did have a more earthy or rooted approach to number, and you then thought, let’s start combining these, let’s start counting the numbers themselves, you’d end up with either these long, ridiculous-sounding words, or you’d end up with these strange sort of…at some point Conrad was imagining these genetic mutant combined rabbit-hawk-pig things.

cs: I had in mind the example of how zero wasn’t accepted by the Roman Catholic church, because it represented something unholy. The idea of nothing was an idea of hell, the idea of Satan. They only accepted it because of money, because the capitalist system couldn’t operate in Roman numerals nearly as efficiently as it could in a decimal system. Similarly, with the idea of multiplication, if your system of numbers was based on these groups of animals, you couldn’t put a rabbit and a hawk together because you’d have this unholy animal that would be the stuff of nightmares. audience member: To that point, Roman numbers are a bit like braces of leases too, right? When you think about it. Arabic numbers kind of condense everything…. mw: Yeah. audience member: Multiplication in Roman numbers must be difficult. mw: Try and do fractions in Roman numerals! That’s formidable. Because fives have their own thing going on. They suddenly become a V, and then you’ve got ten of them and there’s an X. And there’s nothing in the numbers themselves that does that. audience member: Could I just explain how it was told to me that these numbers were written down? That they came from hands. That you drew hands, so you have one, two, three four fingers, and there’s your five [V made by thumb and forefinger]. Then you’ve got six, seven, eight nine. Then you’ve got another V. Crossing the thumbs then gives you the X for ten. That’s how it was explained to me. Whether that’s true.... mw: Okay. Yeah. That starts to get complicated when you get to fifty or a hundred….

audience member: Which is L. mw: But, even so, it’s probably something like that. Often, counting does come back to something quite physiological, or something to do with encounters with the world at a very physical level. Pure mathematics has gotten as far away from that as it possibly can, and that’s the world I’m coming from, but I’m still interested in where it originated. rm: At this point in the process, you were confronted with something. As you said, you’re not an academic mathematician and you’re extremely open-minded in the way you think about numbers, and you pose a lot of questions in your book that a lot of academic mathematicians would not consider. But at this point, you were being faced with these cultural aspects of number and these associations which, traditionally, a mathematician would simply say are irrelevant to mathematics. I’m interested in hearing about your experience of this process, and the unexpected consequences that came from taking these abstract concepts and bringing them down to this very concrete level and making them into physical objects, and how that affected the way that you thought about these problems. There were various points at which we reaped unexpected rewards in terms of imaginative associations that sprung from the way we were handling this problem, and then there were also unexpected consequences which came from the sheer fact that we made a thing, this clunky assemblage that doesn’t work particularly well and doesn’t look like a work of art, but it demonstrates the concept in a fairly idiosyncratic way. mw: There’s the small matter of the peanut, in the sense that we grabbed something that looked fairly unitary and solid to represent oneness, and then, after a while, it was realised that the thing we’d chosen is a perfect representation of twoness and of the fact that there is no unity. Anything, ultimately, that we want to name can be broken down into parts and has subcomponents. So, when you try

and physicalise the idea, there are always these things spinning out that you wouldn’t have thought about. rm: A part of what the artistic process is about is that you have to choose something, some concrete thing or things, as material to use to think with, but then that material feeds back on what you want to do. mw: Whereas the mathematical approach is just to bracket all of that stuff out, because it’s just distracting. You just want to break everything down to its pure, structural skeleton, which is the same regardless of what you’re counting. The whole point is that it doesn’t matter what you’re counting, whether you’re dealing with numbers or variables, Xs and Ys that represent those numbers, or even more abstract things which represent those. Ultimately, you may be able to trace that back to somebody counting their cattle, seemingly. But you’re not supposed to be thinking about that. The cattle are supposed to have gone a long time ago, and you’re just dealing with the structure itself. And because we were in a studio full of peanuts and cocktail sticks and zip ties and things, it kept getting brought back to these materials, which Conrad would take and go off in a different direction with and, if I was prepared to follow that, it would lead to some interesting insights sometimes. But when we actually tried to physicalise this assemblage of cameras and screens, it did lead to some unusual, unexpected questions about the nature of the relationship between addition and multiplication, which may not lead anywhere, but there was one question in particular that I would never have thought of had we not tried to build a model of this. So that’s an unintended consequence of the dramatization, whether it’s art or not. But to explain what I mean by that, we’d have to have the thing in front of us. rm: Before we do that, I’d like to reverse the question and ask Conrad—not that you necessarily had any solid expectations of what was going to happen, I don’t think any of us did, but what were the

unexpected consequences of the way in which we set up this joint residency? cs: The end result has been totally unexpected. As you know, before we came, I had these preconceptions of building something immediately and using things I was already familiar with, and maybe just getting some number-crunching figures from Matthew, helping me make some algorithm and ending up creating some big shape, turning the room into an XYZ graph, or something. But actually, what this week has been about is challenging all my preconceptions. It’s actually been nice to feel on the back foot with something like this, to be a bit out of my depth in terms of the maths and the logic behind it, and processing it in my own way. Because I actually think this system of animal multiplication that I’ve got, it fails so dramatically, and it’s so unwieldy and so complicated and also quite humorous, but I think that, in that, it’s actually quite a useful model. It really highlights some of the things about numbers that are forgotten. These systems, these ubiquitous systems that underpin everything, that we rely on, they’re just so intrinsically part of us that you can’t really sense them anymore. Something like this is, hopefully, highlighting that to some degree. And it’s allowed me to articulate some things that I’ve been preoccupied with but haven’t had the time or the means to tap into. Being here and having this springboard and the luxury of this time to just debate and thrash things out has actually unearthed those things and brought them more into my consciousness. rm: This friendly tug of war, again, can be described in terms of Matthew’s quest to create successful visualisations and successful metaphors that would get his ideas across, with your role being to inject this element of the risk of ‘failure’, which injects a kind of frailty but also an uncontrolled plenitude of meaning that comes from all the cultural associations, all the different types of mechanisms that operate in culture and are rejected by mathematical structures. We’re going to have a Q and A, and we’ve brought out the…I don’t know what to call it…‘The Thing’, we haven’t thought of a name yet.

cs: Where’s the peanut? rm: The peanut? Where is the peanut? audience member: Here! [Laughter] [Someone brings over a peanut] audience member 2: There should be music, and a spotlight… audience member 3: Yes, we need Jean-Michel Jarre. rm: I think I should just point out again that the idea is not that this is the sole end result of the whole process. It is one among several exhibits. cs: This is a realised version of the theory that Matthew came up with, through various sorts of coincidental discussions…just from being in the room together for this week. It’s such a hybrid of bits that feed back to different elements of those discussions…. How we arrived at this is very difficult to trace. It’s quite fascinating to see this as the outcome. But this essentially is, as much as possible—apart from the rabbit—as precise a demonstration as Matthew and Paul could make of this multiplication and addition problem. rm: And again, I think the physical constraints ended up introducing unexpected twists to the idea. mw: Yeah, I’ll talk a bit more about that actually. We’ll just demonstrate what the thing does, though. It doesn’t just show five rabbits. Could you just kill the lights, somebody, so we can see it a bit better? audience member: You need some extra light on the rabbit.

mw: Oh, that looks really good. You can see the rabbit. That’s fine. Okay, so we have this wheeled complex which represents three. At the moment, this wheeled complex has a camera placed on it which is pointing at an image of a rabbit. And then there is another fixed complex which similarly has a camera on it pointing at the rabbit. But if we just happen to wheel this over here, this camera is now pointing at two rabbits, and we see six rabbits. So, in the first position, we have addition represented, where these numbers two and three play the same role. They’re both just replicating something—they’re counting rabbits. In the second position, two and three are playing different roles: the two is counting rabbits and the three is counting twos. In fact, you can even see the edges of the screens, and you can even see Conrad’s head, actually…[Laughter] audience member: Two Conrads? That’s two too many. mw: It’s messing up the mathematics… [Laughter] And similarly, you can imagine, if we built a more elaborate version, that you could slide the two in front of the three, and demonstrate that two threes give you the same result as three twos. But these two cameras would then show three screens, each with a rabbit on. And the other important point is [swaps out rabbit for a hawk, to general laughter] it doesn’t matter what it is, a rabbit or a hawk or a peanut, the mathematics are the same. Now, we were originally going to do this where the two- and threecamera complexes were pointed directly at the thing. And Robin made the point: Shouldn’t we have one in here somewhere? So we introduced the single-screen camera. Rather than the two cameras pointing at a picture of a hawk, they’re pointing at the picture of a hawk on the screen. So this complex here illustrates another layer of philosophical subtlety to this distinction. cs: It’s a representation. It’s called oneing.

rm: My point was that the legume or animal in question always has to be initially brought into the number system in order to be used by the other operators. mw: This nicely illustrates a point I’ve been trying to make for years to people, which isn’t relevant to mathematics but is relevant to the way we apply numbers to the world: that there really are no two of anything. The world is full of biologically generated forms which seem similar—two oak trees, two human beings, two rabbits—and mass-produced objects—two cigarette lighters or two skyscrapers. And we look at them and we see the similarity between this thing and that thing and we say, ah, there’s another one, and therefore there are two. But in fact, if you look at them closely enough, each one has its own features, each one has its own history, and the twoness isn’t there in the world, it belongs to our choosing to relate them to one another. So the fact there are no two of anything means

you can’t really have two rabbits or hawks; but if you introduce this element of the first, oneing camera, you see how the hawk has been reduced to a sort of flattened image of a hawk. It’s been brought into the number system. It’s been reduced from its physical actuality, with its distinct characteristics to a mere token. Something that you can then count. So you can count with hawks, you can count with pebbles, or with any other tokens. So, the one brings the thing into the number system, and only then can there be two of it. And we could have had seventeen screens here and thirty-nine screens there, if we’d had more money…. rm: And more time, I think. [Laughter] mw: …and more wire and more electricity…and we could be illustrating that seventeen plus thirty-nine equals…what is it? Sixtyeight or something. No, probably not. Fifty-six! [Laughter] And then we could roll it in front of that and then show you what seventeen times thirty-nine equals…. rm: Another aspect of what this apparatus introduced into the problem was that the actual image is degraded, so you do get this notion of abstraction with the repeated application of number, which we didn’t think about in advance. mw: No, no. rm: That was an added bonus. cs: We did a bit, in the sense that we knew the feedback system could only hold on to a certain amount of information. mw: But to actually see it—you’ve got an actual three-dimensional thing here, then a reasonably good representation of it—and then you roll this second complex out and you just end up with a vague blur. You know what you’re looking at, but only because you’re referring it back to the original secondary image.

This reminded me of how using numbers to count something, which is the most innocuous use of number; using numbers to add things together; and then using numbers to multiply things, are different stages in the evolution of how we apply number; and in doing so we remove ourselves further and further from the world we’re dealing with. The best way I could think of to explain this is that, if you could imagine a vet operating on a sheep that was in need of medical attention, the actual details of that particular sheep would be evident to that vet. They’d be dealing with a unique creature that’s metabolising and sentient and had its own unique history and characteristics. How that sheep differed from other sheep would be very, very apparent to the vet. If you’ve got a shepherd counting their flock to make sure that all the sheep are there, they’d have to register each sheep individually. To count you have to at least register that it’s there, otherwise you can’t count it. So the individuality of the sheep is further away—that’s that one, that’s different from the one I just counted over there—the individuality of the sheep is receding, but it’s still present. Now, if you were a large scale sheep farmer who had ninety different flocks of sheep in different fields, and each had twenty sheep, and you were thinking how many sheep you had, you’d multiply ninety by twenty, eighteen hundred sheep, but the individuality of the sheep is long gone. You’re not dealing with them anymore. It’s that degrading of the image. This was an unintended consequence. If we’d been using highquality gear and designing it in a more effective way, we wouldn’t have thought of that. rm: Perhaps that’s the ‘failure’ of the model that Conrad has been talking about. mw: Which leads to an interesting new insight. cs: And I think that this tracking motion here [moving the ‘wheeled complex’ back and forth], I was expecting this to be quite clunky, but actually it creates quite a dramatic effect. There’s this kind of liminal stage, a transition between these two states.

mw: I went out for a walk this afternoon and I was thinking about this. I don’t know if this is pretty wacky and is even a valid mathematical question, but I’d never even have thought to ask it had we not built this. We see what’s going on there: two plus three is five. [Moves wheeled complex out as far as it will go.] We see what’s going on here: two times three is six. But what’s going on in between? What’s going on between the two positions is clearly something you can explain in terms of optics and physics. In terms of metaphor, how do you get from + to ×? Is there anything in between? I’ve been talking about them all week as these categorically different things. There’s addition. There’s multiplication. There’s this huge leap. But in that leaping, do you pass through anything that you can talk about? Is there some in-between thing whereby addition can be morphed into multiplication? I can’t think of any mathematically coherent way of expressing it…. rm: This suggested another poetic extension of the project which you were talking about earlier… mw: Oh, yes. As a result of thinking about this, one feature of this we never got round to because it was all a bit rushed, was to introduce some sort of mechanism where a user, a viewer of this device could pull a handle or turn a crank or something to make these things move back and forth so they didn’t actually have to physically grab hold of it and potentially break it. So we were talking about levers and springs and then I think Conrad suggested you could just have a disc which you rotated with a rod, and it would move it between these two points. But, for some reason, this afternoon, thinking about that, and about this issue of moving between addition and multiplication, it occurred to me—and this is completely arbitrary and has nothing to do with the nature of the mathematics itself—but it occurred to me that the symbol for addition is + and the symbol for multiplication is ×. So I had this idea that you could have a cross with a handle on it. It would be a plus

sign in this state, and then you’d turn it a quarter turn and there’d be a four-to-one gear ratio that would then move it to there. So then it would be a cross. You see multiplication. You turn it to a plus. And, again, it’s only the fact that we built this that you’d even think about doing that. rm: I think that’s delightful evidence that there has been some crosscontamination here. Matthew, perhaps to his great horror, has begun to think like an artist. [Laughter] audience member: I’m a very weird sort of historian, so I know the answer to some of your questions, as far as history can tell us. Firstly, the relationship between the plus sign and the multiplication sign is quite deliberate. That was, I think, done by one of the great Arabic mathematicians—I can’t remember which—one of whom was al-Jabr, where we get the word ‘algebra’ from. The other thing was that the transition from adding to multiplication happened with farming, because, like you said, you would count: How many sheep have I got? I’ve got ten. I have ten ewes, ten female sheep. What if they have twins? How many sheep am I going to get? There’s your transition. Because suddenly, what were ten have produced far more! That is exactly multiplication, in that sense. cs: We didn’t really go very far with it, but we did talk about this. And on this graph, there’s the idea of reproduction, hybridisation…. A lot of my pieces have this extrusion of time, where time is occupying one of the three dimensions and you extrude it through space. You give it a representation. And one idea that came up was the idea of a timeline for a pistachio nut—another nut we were eating, we were in this room for a week and the nuts seemed to form the basis of lots of conversations—so you have your pistachio nut here which then grows into a tree, and then your tree is here, and from that tree comes another nut… audience member: Many of them!

rm: It’s fascinating to think that this kind of poetic association forged between number and the spectacle of these processes of biological multiplication, in fact, might have been humans’ way into thinking multiplication. cs: And I suppose, one of my things in terms of creating this kind of failure—this graph that was getting into all these incredibly complicated realms—was to create a parallel world where each of these singulars or these braces or leases, these names, these symbols of numbers, actually had genders as well. And then it was combinations of both male and female braces and leases, so that you’d have this sort of sub-table…. To a certain extent, this is what the boar-brace-lease system does in, obviously, quite a humorous, flawed way. It’s attempting to create an objective language to highlight the fluency of our own languages.

audience member: Do you think it’s significant that all the maths and multiplication and addition is going on this side of the screen? So, could we say that the maths is not really a part of reality? rm: We could say that numbers, the number system, is itself a kind of machine so that, once the real object is taken into this machine, it enters into the reality of number instead. mw: The act of counting, the act of adding, and the act of multiplying, as seen in the assemblage, are stages of moving away from the real object. But in each case, you’re compounding a particualr sort of conceptualisation of what is actually out there. You never really deal with what’s out there. You process your sense-data and then you reduce it to categories of similar objects that can be counted. But, yeah, you think, oh, look, there’s five rabbits—there’s not five rabbits, there’s this unique, miraculous self-organising chunk of matter that is sentient and metabolising—oh, and look, there’s another one, and another one, and another one. And it’s not even another one! Each one is its own unique miracle. The fact they look a bit like each other is because there’s this process of reproduction. They can reproduce themselves! That makes it even more miraculous! And yet we look at them as just five rabbits. What’s the problem? [Laughter] So we are continually breaking the world down into chunks that we can relate to one other, but those relations are not in the world. We’re projecting them. So it’s all in here [Gestures to his head]. But it’s a good point actually. I hadn’t thought about that. You’re just looking at the front end, at the screens. If you go behind it, there’s a rabbit and there’s a load of electronics. I don’t quite know what that means… [Laughter] It only works that way [looking at the assemblage front on]. You’re trying to get behind the circuit board and see what the process is. cs: And on the drawing we’ve got this critical strip of oneing that happens at a certain point inside the camera. mw: Something happens!

cs: At some point during the circuitry, something in the quantum action of things going on inside the motherboard, it gets oned and then it gets twoed and then it gets threed. [Conrad knocks hawk off its pedestal with his elbow as he pushes the wheeled complex into multiplication position.] mw: That’s zero! [Laughter] Yeah, that’s an interesting point. If there’s nothing on the plinth, then there’s zero on the screen, and zero plus anything is zero, and if you multiply anything by zero it’s still zero…. There are metaphors here that we didn’t intend. audience member: Matthew, do you have any fear or phobia of any particular number? Or do you know of any mathematician that would have? And is there a disorder? [Laughter] rm: We did have a conversation about superstitions, didn’t we?

mw: I don’t. I don’t really have…. I did have a thing about twentythrees for a while, but it’s just a phase that everyone goes through. [Laughter] cs: I haven’t been through that phase yet. mw: It was back in the early nineties, I think. audience member: What about infinity? mw: Infinity is not a number. This is a mistake that has been propagated for far too long. [Laughter] Children will often ask their parents, What’s the biggest number, in the same way they’ll ask, What’s the biggest animal, What’s the tallest mountain? And some parents will say, there isn’t one! Which is the correct answer. And others will say, oh, yeah, it’s called infinity. And that has led to endless confusion. [Laughter] I’m sure I do, at some level, have a subliminal set of feelings and relationships with each of the integers. [Laughter] I remember being a child and doing simple arithmetic and sitting next to my friend, working on our sums, and we’d actually be chatting about which numbers we liked and which ones we didn’t and why. It was as if we were talking about our friends’ different personalities. And then that gradually gets ironed out of you as you get older—if you’re going to be successful as a mathematician. [Laughter] rm: You spoke about your feeling that the fact that certain people diagnosed as autistic can have a very particular relationship to numbers, and particularly with prime numbers, might somehow be an interesting piece of evidence regarding the significance of prime numbers. mw: Oh yes. To complete my answer to your question: I don’t know any mathematicians who would admit to having strong feelings about any particular number. They might secretly. But Robin just mentioned something that we were discussing on the first day. A lot of this stuff

is covered in the first chapter of my book, where I discuss the general issue of how humans relate to numbers, before I get into the more detailed matters. There have been some very interesting studies of people with various conditions which would fall on what I think is called the ‘autistic spectrum’ these days—the language is a bit unclear: there’s Asperger’s, there’s autism, and no one is quite sure where these boundaries are—but there are people with various conditions whose perceptions of reality seem radically different from the rest of ours, and one thing which seems to show up a lot is an affinity with number, a sort of tendency to gravitate towards numbers as something familiar and stable or a source of some sort of comfort in the world. And prime numbers seem to show up quite regularly as having some sort of different texture or quality. It’s very difficult to translate this into something we can talk about, if you’re not in that state. There’s one very, very interesting writer called Daniel Tammet, who wrote a book called Born on a Blue Day.3 He has a very severe sort of Asperger’s-type condition, but he also has this savant condition, he’s one of these supergeniuses who can learn languages in a few hours and perform ridiculously complicated calculations. There are about three people in the world who have both and he’s one of them. So he’s a kind of living Rosetta Stone: he experiences the world in this radically different way, but he can also invent these incredibly effective ways of describing it to us. He’s featured in the media a bit over the last few years and he’s written this book where he talks about his feelings towards numbers. He talks about prime numbers being sort of signposts. When he’s drifting off to sleep, he’s in this landscape of number and if he feels a little bit lost or uncomfortable and sees a prime number it will be a sort of pointer somewhere. He also says they’re smooth like pebbles whilst other numbers have different textures. There’s been various work by neuroscientists on children and people with various abnormal conditions—people who have had brain damage, people who have had severed lobes and things—and how they relate to number. And all of it seems to indicate, to me— and this is a huge leap as a mathematician—that some of these

people are actually experiencing something about the world, about the realm of number, that the rest of us aren’t. It’s not just a disorder. It’s not just something short-circuiting. Synesthesia you could understand, perhaps—smelling colours and tasting sounds and things—as just crossed wires, but this seems to be something else. It’s almost as if they’re reaching into the ‘true’ world of number— whatever that would be—while the rest of us are just looking at reflections of it. But they can’t generally communicate much about it to us. So I think we just have to maintain humility and accept we don’t really know what we’re dealing with, which is my stance towards the number system. rm: You talked about the Rosetta Stone, and this facility to experience the abstract world in a more intuitive way and be able to express it is the ultimate prize in terms of deciphering all the questions we’ve been discussing this week. If we could develop some kind of apparatus or artwork, some kind of pedagogical process that would allow people to really inhabit the world of number, what difference would that make to the way in which we understand number and, through that, the way we understand the world? That’s really the speculative horizon of your work, and of this question about the dramatizing of abstract concepts. mw: That seems to be the promise that this approach holds. It wasn’t what we set out to do and it wasn’t what anyone expected. Making this assemblage, there was some talk about ‘What is this?’ It isn’t art. It’s something else. It’s sort of an illustration. It’s informed by the arts, in a sense, or immersed in this discussion of artistic expression. It’s informed by science. Conrad is someone coming from the arts who’s drawing on elements of science. I’m somebody coming from the sciences and sort of dabbling in art a little bit, and so we’re in this strange in-between world. It’s not really clear what it is. rm: Several times during the week you talked about this process and brought up the possibility of there being some other discipline.

mw: Yeah, some new thing which is neither art nor science, and which hasn’t even got a name yet. This process of integration is an important thing that came up during the week. Robin and I particularly were talking a lot about this. There’s a gap opening up. There was a time when the most advanced scientific ideas known to humanity were more or less accessible to everyone because they were fairly simple, and then they got more and more specialised. Leibniz, in the seventeenth century, was supposedly the last human being to actually have a complete picture of everything that was known in the sciences, after that there was just too much. Today, even in one area, no one mathematician has a grasp of the whole of mathematics. No one biologist has a grasp of the whole of biology. You can only see a tiny little part of it. And most ordinary people are separated by this vast chasm from this incredibly specialised scientific work that’s going on because, in order to grasp the results, you need to study for years just so as to be able to understand the language and its symbols. Something’s wrong here. I suppose that’s an ethical statement, but it doesn’t seem right that the world is being driven by the mathematical sciences to such an extent that everyone’s being left behind. People are carrying these gadgets around with them—a mobile phone exploits quantum tunnelling, but how many people could actually explain what quantum tunnelling is? A chasm is opening up between people’s experience of the world and the sort of inner sanctum of the priesthood who decide how it works. It doesn’t seem like a healthy state of affairs. It’s almost a form of activism to say, well, maybe we can bridge that gap. I suppose I would see it as trying to close up a wound. That’s my personal stance. Robin doesn’t see it like that. He sees it more as an exciting possible synthesis. But what would it be like if you could suddenly download the totality of quantum physics, relativity theory, nanotechnology, microbiology— all of that knowledge? If it could just become some sort of downloadable package and everyone could have it? If suddenly everyone was walking around with all of that knowledge, all of the sciences in their head, how would they relate to the world?

rm: So we thought that what we were trying to do this week, or rather, the problems that we were engaged with, could be seen as an index of this much wider social chasm that Matthew described. And this is something which has informed the way in which the volumes of Collapse developed. In Volume 4, a great deal of the notion of Concept Horror came from thinking about the way in which we know things about science and we can abstractly accept philosophical statements, and yet we don’t really integrate them into our experience of life. And so the question was: If you could actually absorb what, say, quantum physics is telling you, what kind of experience would you be having of the world? The suggestion being that there the people who have given the best speculative portrait of what the world would look like if you could actually absorb these incredible theoretical insights are writers of weird fiction and horror. And then thinking about this role for artists in producing dramatizations was instrumental in preparing the fifth volume of Collapse, in which Conrad’s piece appears. That volume was called The Copernican Imperative, meaning the idea—the inverse of Concept Horror, in a sense—that there are theories and structures that science reveals to us which we can’t possibly have any experiential grasp of, we can’t see or feel even in some transduced form, and that we have to simply accept living in a world structured by the utterly unexperienceable. We have to accept that there is no reconciliation between our intuitive way of being in the world and what science tells us about reality.

Those are two polarised positions, but I’d like to ask Conrad a key question that emerged from this week: whether you see that kind of reconciliation between the abstract scientific image of the world and our everyday experience or our sensory experience, or to put it another way, whether you see this process of bringing these very abstruse systems back into natural language, as part of your task as an artist. cs: One of the things that’s been…not exactly disheartening, but something I’ve been talking to Matthew more about, is that we had this discussion about space and time and the spacetime continuum, and Matthew was quite clear that I had a very misguided idea about what it was, in that my idea of it came from reading a lot of popular science. Matthew has been a little bit discouraging in the sense that he said these are all metaphors and the only way to really understand it is to understand the mathematics behind it, to read that mathematics.

mw: Popular science tries to make very difficult ideas accessible— and that’s part of the problem I’m grappling with in writing a book which sort of falls into that category, even though my motives are different. Things have to be simplified. If you could explain relativity theory in very simple language to anyone, then you wouldn’t bother with Einsteinian tensor mechanics or Riemannian geometry to do it. People aren’t purposefully trying to be complicated. You need to do that in order to really make it clear. So if you want to draw it down into something people can talk about, you use metaphor and visualisations and you have to sort of bend the truth a little bit. The problem is that people can then read that and go away thinking they’ve understood something. In fact, they’ve taken something from it but their understanding is perhaps…. cs: I think ‘gleaned’ is a good word. mw: Yeah, ‘gleaned’. So because Conrad’s an artist, it’s not really a problem, you can glean from sources of inspiration and turn these into artworks which aren’t attempting to be scientific models. There’s one piece by Conrad that I’m particularly enthusiastic about. It’s called Loop System Quintet and it illustrates very beautifully a set of musical ratios in terms of a set of five interlinked kinetic sculptures. Each one’s got a rotating arm with another arm rotating at the end of it and they’re geared in certain ratios, with a light at the end so that the light is travelling along a seemingly very chaotic path, but if you take time-lapse photographs you see these very beautiful, what are called torus knots, these sort of woven patterns. I’m very familiar with that in terms of both music theory, because I’ve played music in the past, and mathematics. To me, that’s a very successful dramatization of a scientific idea, but where does that differ from a simple model? There’s no failure in that. You talk about failure a lot. That didn’t fail. That succeeded. So what happens when one of your artworks actually succeeds? cs: As an artwork, I don’t think that piece is a failure. I think that piece works quite well as a scientific model, in the sense that it was

like the animal multiplication table: I’d built a machine and I was quite naive and didn’t really know what the effect was going to be when I put these specific bevel gears into this machine and turned them all on linked to one another. I took a photograph of the first time I turned it on and it created these beautiful patterns…. mw: You didn’t know what you were going to see? Conrad Shawcross, Loop System Quintet (2005). Image courtesy of the artist. cs: I knew they would cause a trace but I had no idea it would create all those complex knots…. mw: Is that part of the art, that you didn’t know what you were going to get? If you’d already known that, things would have been different?

cs: I don’t know if it’s part of the art, but it’s part of the process. It’s similar to science in the sense that the experiment sometimes offers things you didn’t expect—and often some of the most striking discoveries have been by accident. mw: So in the artwork, rather than the concepts themselves, you’re dramatizing the process of discovery, or reliving it…. cs: But I think that in something like Loop System Quintet, while it does demonstrate some of the things you’re talking about, a lot of quite arbitrary, quite aesthetic, subjective decisions have been made: to make this thing with one arm half the length of the other, and it’s all made of oak…. In terms of my latest work, I’m actually more hesitant about using such spectacular materials, but it was the way my systems were conceived at the time. As a viewer, I don’t think it’s successful as a demonstration because, if you’re the viewer, you’re confronted by this thing and there are so many ways to interpret it. mw: You just see all the things spinning around... cs: And you don’t know that the trace is there until you see the photograph on the wall. There’s this strange dichotomy between being a part of it and being at a distance from it and seeing the order. There are a lot of different levels on which it can be viewed, and if it was clearer and more succinct as a demonstration it would be less successful for me. But yes, it does intrinsically just take these basic ratios of waves to each other, the things we hear and which sound beautiful to us and which we’re moved by, and I was very interested in why our brains find such particular order so enigmatic or so moving. And I was trying to look at those in an objective way by representing that power in a visual way. A kind of synaesthesia. audience member: I was going to say, in answer to your concern about ‘Is it art or is it science?’, I think it’s a bit like asking if a photon

is a particle or a wave. To me, the two endeavours are attacking reality with different tools, some of which are poetic and visual, and some of which are abstract and sequential, and getting something out of it which is new, and that’s the element that is so exciting about anything that is successful. When you speak about failure, I think what you’re talking about is imperfection—that it’s a success if there is an imperfection, because the imperfection allows room for progress. Once you’ve got perfection, it’s boring. It has to be imperfect, so you can move on, so there’s more. cs: I quite like watching really bad movies because they show you how difficult and how problematic it is to represent. It’s not the same if you see something that is seamless. Seeing things fail is actually an incredibly effective way of looking at how complex the world is. rm: It breaks things open. cs: Yeah. audience member: The sort of science that Matthew engages with is the pursuit of the few, really—of people who are highly specialised— but the rest of us just have to go with it and believe you, and I think a lot of contemporary art, for many people, is the same: you have to just go with it and trust those artists that they are pursuing a line of enquiry or an investigation or experimenting or playing—whatever their practice is. I was just wondered if that had cropped up: the notion of belief. I personally think art is a belief system, especially contemporary practice, and I feel the same way about science because I feel alienated from it, I don’t have the capacity to understand, so I have to believe when people tell me about these things. mw: There are several interesting questions embedded in that. Firstly there are the scientists themselves believing something, the scientists and mathematicians who know what they’re doing, believing in certain foundational truths. The idea of science is partly

to strip away belief systems to try and get to the truth, but you always have to believe, at least, that the world has some sort of consistency to it. There’s no point looking for laws of physics if the universe could just suddenly decide it’s going to change them at any moment. So you have to believe in some sort of consistency through time. And also there are certain things, subtle things, which science uses because it tends to reduce everything to models which can be quantified. There’s something called the law of large numbers, which I can’t get into, but it’s a very subtle, hidden belief system within experimental science which no one really wants to talk about. It seems to work so everyone just leaves it alone, but I’m quite interested in interrogating it. But that’s not really what you were talking about. You were talking about a general public believing that scientists are actually correct in what they’re doing and that they’re not just taking you down a blind alley. Similarly, it is true if you go into a gallery and you see a piece and maybe you read the press release, you might think, maybe this is a genuine pursuit of this deep idea that I’m being presented with, or maybe this person’s just…having a go, really. Making it up as they go along. In the case of science, you can go and do a maths degree, you can go and do a physics degree, you can learn it for yourself and get some verification of what you’re seeing. There is also a whole hierarchical structure of peer review and publications and things, so someone who’s just making it up in mathematics is going to get shown up very quickly. You can’t get away with that for very long. There could be some super-specialised area where only about two or three people know what they’re doing, where maybe, for funding reasons, they really are making it up. But I doubt it. But in the arts, there isn’t that similar structure. Someone could, if they were sufficiently confident, just make up a whole story about what they’re doing. It’s hard to get any verification that they are really doing what they say they’re doing. rm: Nevertheless, there is some sort of selective process at work. We did touch on this a little bit. We talked about how there’s a

scientific institution and so when we talk about ‘science’ we can’t just talk about that in terms of one lone scientist in his laboratory. We have to talk about this institutional structure and its history and its archival process. And we tried to think about whether there is an equivalent in the art world. Is it simply the art market? Is it art history? cs: There is a relay of ideas in the art world. I’m not particularly one who is that interested in making work in reference to the work of earlier artists, in the sense that I’ve always been interested in these scientific or philosophical areas instead, and I haven’t tried to locate myself in that context. But a lot of art justifies itself in terms of taking that process further—whether it’s seen as a process of abstraction or a process of political engagement. There are huge movements and there are ways of tracking that chronology. And people do take up the gauntlet and are heavily influenced by people before them. rm: But I think the test of whether it works can only ever be applied retrospectively. One can’t really tell which of the artists working today will be judged to have added something to that sequence and will be incorporated into its timeline, its archive. mw: The passing of time filters that out somehow. Whereas science is doing that in real time. audience member: But there is a peer review system in operation within the arts. mw: But it’s not really working to any kind of rigorous consensus that can be pinned down. audience member: It would be science then, though, wouldn’t it? mw: Yeah, exactly. It isn’t science. There are always going to be personalities involved, there are going to be political, emotional,

economic factors. If those are involved in science then that’s just corruption. That’s just wrong. But in the arts it’s inevitable. cs: But in science there are exceptions. I mean, Babbage is a very good example of something that was never filtered in the right way. It was a hundred years before his technology, his vision, was understood. mw: But nobody at the time could prove that it was flawed theoretically. It’s just that no one at the time could see whether he was really onto something. rm: Nevertheless, there are contingent cultural factors which affect the way that scientific ideas are taken up, which can often retard scientific progress. mw: Oh yeah. The current of which areas of reality get investigated by science is not perfect. There isn’t some sort of noble, pure investigative process which humans are engaged in for the benefit of all. Science has largely been driven by the desire to develop better weapons systems, to be honest. A lot of mathematics is driven by the desire to better understand ballistics, firing cannon balls, and then you get into wanting to build nuclear weapons and things. So the history of where science has gone is pretty impure. But whether or not a particular equation or theory stands up is another matter. That’s something that everyone, in principle, can look at and agree about or disagree about…. rm: In a sense, science just is the effort to extract knowledge from its genetic origin, and to cement it into an freestanding axiomatic or systematic form. mw: Yes, and it can never quite do that. It’s always trying to lift itself out of those contingent, arbitrary aspects and exist in a pure, archetypal way.

rm: Whereas, I guess, art is always interested in bathing itself in its history and reassessing its past and its meaning at every moment. mw: In principle, science is quite happy to destroy its own history completely. If a new idea comes along that completely supersedes everything else then, in a completely non-sentimental way, everything else before it is just gone. That’s it. Nobody’s going to say, oh, I used to really like that. [Laughter] It doesn’t matter. The idea is to get to the truth, and art has a different agenda. The one thing we kept coming back to is that we, Conrad and I, have different motivations and different agendas. We can talk to each other and we can talk around these issues and possibly work together and create things together, but there is always going to be that gap: that art isn’t science and science isn’t art. rm: That seems a perfect way to draw this discussion to a close. This has been…I’m certainly exhausted after this week, I think we all are, but it’s been wonderful to relive the week’s events here and to explore what’s come out of it. I’d like to thank Conrad and Matthew for being here. They’ve both been enthusiastic and gracious participants in this experiment, and it’s been a pleasure to be a moderator and a participant, and sometimes the audience as well, sitting back and watching this process unfold. 1. M. Watkins and M. Tweed, Secrets of Creation, Volume One: The Mystery of the Prime Numbers (Dursley: The Inamorata Press, 2010). See <http://www.secretsofcreation.com>. 2. R. Kaplan, The Nothing That Is: A Natural History of Zero (Oxford: Oxford University Press, 2000); S. Plant, Zeros+Ones: Digital Women and the New Technoculture (London: Fourth Estate, 1998). 3. D. Tammet, Born on a Blue Day (London: Hodder and Stoughton, 2006).

The Residency Urbanomic Studio, 8–12 March 2010—edited transcript Editorial Note As this publication is intended as a document of research in progress, it was important to present here the dialogue recorded during the period of the residency project, which, even when selectively transcribed, yielded nearly eighty thousand words. The editorial process reflected the complex of meanings around ‘Redactions’ recited at the beginning of each publication in this series. As always I made it my editorial prerogative to compensate for the occasional opacity, hesitations, and infelicities of spontaneous dialogue, and to streamline and enhance the text where it seemed necessary. Especially in terms of Matthew’s explanation of his work, there were returns to the same topics across the space of the week, and expositions of the underlying mathematics did not appear in logical order of priority. Perhaps these passages could have benefited from being drawn together and compacted. But it was more important to convey a sense of the trajectories of these various threads of thought, from the theoretical to the technical to the pragmatic, as they made their tangled way through the week, being dropped only to be picked up again later as the group tried to find a way to twist them together. With few exceptions, then, all of the transcribed parts have been left in sequential order, digressions and lateral drifts included. The sometimes comical juxtapositions to which this gives rise are not an artefact of editing but were part and parcel of the process itself. The halting, probing, frustrated movement of research is represented without attempting to smooth over or hide the marked shifts that occurred during the week, between enthusiasm, disorientation, longueurs, fraught meta-commentary on the project’s progress,

waves of activity, studious construction, and outbursts of collective delirium. rm conrad shawcross: In terms of trying to get something concrete coming out of the week, I was thinking we could create an axis in this space, but instead of creating a sequence of numbers where each unit is the same length, we could distort it so that each length of a wall is the next prime number, so its actually concertinaed towards one end. So the room is not a continuous space-time, the room is completely warped, so that corner is much more dense. matthew watkins: Like in those Escher drawings with the big fish in the middle and the little tiny fish at the edge. In the hyperbolic plane he’s working with, they’re actually the same, but the sense of scale alters as you move closer to the edge. I can already see ideas there, if we try to use three-dimensional space. Not so much chopping things up into a sequence of prime number lengths, but using a logarithmic scale. Because when you rescale everything logarithmically, prime numbers cease to spread out in that way and they’re normalised; there’s still randomness there, but you normalise the spread of the series—which is epitomised in the spiral, basically, the way the spiral unfolds. We could do something with that, possibly. Number theory tends to seem inaccessible to people, and one reason is because they see logarithms and they run away, that’s why I try and use spirals to make them visual for everyone. But yeah, if we could touch on that using some sort of visual means…it’s difficult to imagine using three dimensions, there’s not really any need for a third dimension. Just about everything I’m concerned with at the moment you could do it in two dimensions cs: I suppose that’s where I fit in. mw: Yeah we don’t want to add on a third dimension just for the sake of it, ideally it would have some sort of purpose. robin mackay: In any case, we shouldn’t feel so early that we have to decide on making something.

cs: The only thing I’d say is if we’re going to make something we’d need to make sure we can get hold of the materials. mw: If we could just put it not completely to the back of our minds, we might just wander around aimlessly. rm: The whole project is couched in terms of research and preproduction, to explore the way in which work is produced, so there’s no urgency or responsibility to present something finished. cs: No, sure, it’s just a way for me, I’m a visual thinker, not that I’m desperate to make a sculpture, it’s just a way of seeing things. paul chaney: This is the territory we’re looking for, theory and visualising theory: you just want to get your hands into something that’s tangible, and Matthew and Robin are wanting to get their brains into something intangible, but they might be two ways of expressing the same thing, and this is where we’re trying to get to. rm: It would be a good idea to bring in some materials, so they can just be on the table…. mw: When I was doing festivals, I had a lump of Blu Tack, some cocktail sticks, some string, and a bag of beans, and if anyone came up and asked me anything, I’d just use what was available, or wave my arms around. pc: I remember a ‘Sacred Geometry’ workshop you ran in a tent, and all you had was a piece of A4 paper and a magnetic Learner sign from a car that you used as a straight-edge and a right-angle. You did a whole geometry lecture using an L-plate. mw: It was the most popular thing there, it almost started a riot there that day…in the end me and a few others were trying to build an icosahedron out of Blu Tack and cocktail sticks while everyone else was arguing. rm: If we get some mouldable wire, we could quickly… mw: And a bag of kidney beans, anything you can count with…

pc: Small cable ties or something… mw: Maybe a bit of plasticine. cs: Or Play-Doh. If we had some Play-Doh and some really thin sticks of wood, some dowelling…something we could create a structure with…, // rm: I can’t really remember how the idea for this project came about, but looking at the title Secrets of Creation, it works on two levels. Firstly the conversation I had with Matthew some time ago,1 addressing the idea that, in the distribution of prime numbers, there’s some kind of mysterious not yet understood principle which may help us get closer to the reasons why mathematics ‘works’­, that is, the ‘unreasonable effectiveness’ of mathematics, and its connection with physics. The other level involves looking at how artists use concepts from other disciplines, and create tangible experiences out of ideas that may be quite abstract. What happens in that process, and how does the outcome relate to the strictly determined concept an artwork is drawing on? It’s similar in philosophy—philosophers often use scientific concepts in strange ways to do something that may be speculatively stimulating, but can’t be anchored back in the rigorous scientific context it comes from. In one sense you could say that simply amounts to misuse, but could this same process of ‘dramatization’ of ideas into visual or material objects or experiences also serve to communicate those concepts with people who would otherwise have trouble grasping it in its original context? mw: There are two slightly different things going on there. In philosophy they’re not trying to help their readers grasp the concept, they’re just using it as a platform from which to say something else. Whereas an artist could potentially could be helping you to grasp the concept. rm: Although it’s never purely explanatory in that context either.

cs: There’s also the question of motive. The process of making art is often the process of the artist trying to grapple with an idea that they may not be able to articulate at first. It’s through that process that you become more aware of what you’re trying to say. That process doesn’t necessarily always take into consideration the viewer or a specific idea that needs to be conveyed. Actually there is a sort of unpredictability there, it’s never purely educational, for the viewer, it’s often to do with your own inner processes, it’s more about the journey of the artist. For example, when I first made a rope machine I was really preoccupied by the process of rope making. I’d seen a TV programme about it and I became really possessed—I really need to make this machine. So I set about learning how to make it. It was only really through that process that I realised this was very much about the perception of time on a human level. But even having realised it was about that for me, I’m quite hesitant to force that upon people. It works on many different planes, and it’s not necessarily important that that’s what I got from it. The difference between what you say about philosophy and art, particularly sculpture, is that they are reliant in a very empirical way on a metaphor, a structural metaphor, for a lot of their ideas; whereas mathematics is much stronger in a sense: it doesn’t rely on metaphor, it just relies on its own language to describe things. For a non-mathematician it’s impossible to get into that world, I guess.

Conrad Shawcross, Chord (2006). Image courtesy of the artist. Photograph: Alex Delfanne. mw: Well, that’s the question being posed here. The question, and I don’t know how closely you want to stick to it, is about the extent to which art can allow non-experts to grasp abstract ideas. So that’s very specific use of art, and that may not be the artist’s motivation. rm: If the answer to the question is that that isn’t what artists do, fine; in trying to do it, the artist will end up doing something else. mw: Conrad, have you ever made something whose hope was to enable the viewer to grasp an abstract idea? cs: I think that there are these elements in them, but they aren’t rigorous academic approaches to something. And I accept that they are physical renderings of ideas that aren’t necessarily rigorous or mathematical. With Loop System Quintet, I’d done a lot of work with lights, and that particular work was spurred on by reading things, and

just gleaning some basic ideas that the essence of matter, rather than being particles, was a loop that was folded on itself, a complex knot. And the work was just about, in a simple way, immediately responding to that and taking what I wanted from it, as an excuse to build this series of pieces. It had a relationship to that exciting idea, that moment when you think that things aren’t particulate, they’re wave-based, everything’s a vibration, in multiple dimensions, far more than we can perceive. And taking that, using it as a springboard for a whole body of work, I wasn’t necessarily concerned with the fact that it may not be entirely correct, it’s just responding imaginatively, as a child would, to those ideas. Artists or poets will often hear something or observe something and then there will be a cognitive slippage, An idea that you’ve taken on board, but then see it in very different ways, it’s not necessarily rigorous but it’s to do with absorbing the ideas and processing them and forcing them out again. At the same time, it’s not about being disrespectful to the core ideas. I think in terms of the envisioning of the invisible, there is something similar in what I do to what a lot of scientists do: they’re trying to see the invisible, to see what’s there in a different way and represent the real in a different way in order to create objectivity. rm: Often with Urbanomic. I work with a relay model of collaboration, where one collaborator can receive material from the other and, in the safety of their not being around, they can come up with all sorts of associations that can be relayed back, and will be received as something unexpected, and can then spark off further ideas. In that model there’s a separation, whereas here we’re going back to a more traditional idea of collaboration, and I can see the potential dangers lying in wait, particular dangers related to this question: as you say, you don’t want Matthew to think you’re just taking his ideas and misusing them in a completely arbitrary way… mw: I don’t mind at all what an artist does, The word Conrad used was ‘disrespect’, I’d add an element of caution to that—it’s not as if I feel some sort of respect is necessary to use these ideas…

rm: But for it to be productive on both sides, what would be ideal would be for your own view of what you’re doing could be shifted as well, so that something would emerge in the middle. It can’t simply be mathematically rigorous, because that would negate the whole point of doing it; but at the same time we need to find a way of guiding it so that it holds fast to what’s important in the work. What I think is interesting about Matthew’s book is that it’s an earnest attempt at communication, and Matthew, you’ve really abandoned all the apparatus of academic mathematics, it’s written in the first person, uses images and metaphors freely. But it’s very much not ‘Prime Numbers for Dummies’, because it also covers really speculative elements and questions about how these issues relate to human psychology and to society. And I think those questions are very important as well. mw: That’s something that would be more traditionally associated with an artist. rm: Conrad’s piece Binary Star seems to me a good example of this process of dramatization works, in that it models an actual phenomenon, a binary star system, but the physical existence of the work makes a difference, actually seeing the play of the shadows, understanding the way that it relates back to the idea of oneness and these preprogrammed cultural ideas which come from the fact that we only have one sun, understanding that they are in fact contingent. Being in the space and experiencing it makes a difference.

Conrad Shawcross, Binary Star (2006). Image courtesy of the artist. mw: Going back to the question, to get an idea of what you’re reaching for here, the Binary Star piece was probably the best one to look at. What you’ve just described, I can connect that to the question in at least two ways. At one level, if I went into the gallery and saw that piece, and I read the press release that explains a little about binary star systems, at least that they exist, at some level, seeing this three-dimensional thing in motion, the overall experience would allow me as a non-expert to grasp the abstract notion of a binary star system, but at the same time, the experience of going into the gallery might also allow me to grasp the abstract idea, the problem of unity, and the effect that having a single sun has had on our perception of the world. That’s another set of abstract ideas, which is distinct from astrophysics. cs: Partly because of the bad reviews I got for that particular piece, in retrospect I felt that if was too close to just being a scientific model. It may have just been the fact that I called it Binary Star, that was just pushing it too far, it was just a huge old-fashioned orrery.

mw: Was it designed to reflect a binary star system accurately? cs: Yeah, it was two ellipses, they hooked around each other. Also there was a computer-controlled system, they had sensors that accelerated toward each other, and once they slingshotted around each other they would decelerate. mw: So there was some effort to make it accurate? cs: Yeah there was, it was not a pure model, but it was attempting to represent that and put someone in front of that piece and to have a phenomenological experience of it. I was trying to create this piece, put it in the context of a gallery, and whether you had a philosophical or mathematical or a purely physical relation to it…the show was a general thing, it was called ‘No Such Thing as One’, with an obvious reference to religious ideas, a monosolar system for a monotheistic religion; it was really looking at what the world would be like if you had a two-sun system. But in retrospect, just calling it Binary Star rather than some more pretentious title contributed to making it seem a bit too reductive. But then at the same time I am excited by scientific models, and by the decontextualisation of them as objects. At the moment I’m doing a residency at the Science Museum, and the power of taking an artwork and something out of the Science Museum, placing them in a different context, those object have a powerful potential in those new contexts. So a simple model can be quite interesting in a gallery context. rm: The cosmologist or physicist might work with binary stars but they’re never going to make a huge model of one that fills a room…. So there’s a sort of unnecessariness or uselessness about the way the model’s made that changes things completely. mw: They’d just use a computational model because that’s what you do now, it’s easier and its more accurate. rm: That would strip out everything extraneous, of course, although you might add some of it back in if you want to demonstrate the model to others.

This seems closely related to what Matthew says in the introduction to the book about quantitative and qualitative views of number, in the sense that a mathematician wants to talk about a quantitative reductive sense of number where it doesn’t matter what the number 6 ‘feels’ like or what it might be psychologically or culturally associated with—that’s got nothing to do with the number ‘itself’. For instance, in analogy to the binary star system, what would it be like if we had a base 6 numbering system, how would that alter our cultural sphere? That’s just not an interesting question for a mathematician’s point of view. Perhaps we can get something out of the tension between those two ways of looking at number. cs: When I was doing Loop System Quintet, each of the pieces expresses a ratio, and so they are called, for example, 9:8, but I was just thinking of saying this what 9 is to 8 could be imagined this as a drone, a chord, It’s the idea of representing something aural in a non-aural way, a kind of synaesthesia, the sound has been objectified, it’s the idea of looking at something whose effects could be quite subjective and emotional, but looking at it objectively, envisioning information in a different way. And that’s similar to what you’re trying to do in the book, envisioning information in a way that’s separate from the discipline and medium that created it. But what I wanted to look at is what 9 is to 8 separately from any other number, what those two things become when they’re just alone together. mw: They can never be alone together, because they can only be what they are in terms of the sequence—you can’t just have 9 without reference to the previous eight integers that led to it… cs: You’re not saying that the others don’t exist, but you’re putting it a different way around, you create this complex dynamic that is entirely produced by the opposition of these things, defining them in relation to each other rather than in terms of the number series. Each one has its own character. rm: In the overlap of the sphere of psychology and number, you’re often talking about a powerful and pervasive structures—in

capitalism, the .99 thing—£4.99, £9.99. There’s no rational justification, it’s ridiculous, and yet it’s powerful. mw: It obviously works. cs: I remember as a kid saying to my mum, why is it .99? She said it’s just to get people to buy it, and I thought it was the most stupid idea… mw: I think there’s a small subset of people who prefers to be honest and just sell something for a tenner. But we could also talk about hotel operators in the US, and how they won’t have a thirteenth floor. And that’s not because they have any superstition about it themselves, it’s just straight-up capitalism—those rooms aren’t going to be occupied as often. So these psychological relationships to number have penetrated the sensible world. 13 is also to do with the moon—there’s a relationship between 12 and 13 which has to do with illuminations in the solar year. So there have been various theories that 13 is basically to do with the shift from a lunar-based consciousness to a solar one. cs: My grandmother is a Catholic, and there were 13 of us one Christmas—which meant I had to sit on my own! I was the eldest kid and I had to sit on a separate table, my mum had a real row with her about it. It was tough! mw: There are obviously external cultural reasons why people have that thing about the number 13, whereas .99 clearly is because it’s just less than a hundred. So it’s not like there’s anything intrinsic in the integer 99 which resonates with any part of the psyche, it’s relative to other things. Whereas 7—if you survey people about their lucky numbers, 7 is predominant, and you can relate a lot of it back to religious symmetry. But I’m interested to know whether there is some kind of invisible understructure of the number system. cs: Until recently I thought most societies in the world had always had a 7 day week, but I was wrong about that. But have calendars always been based around the moon?

mw: As far as I know, I mean, it’s such an obvious way to divide up time, cs: You could have seven four-day weeks. mw: You could. The lunar cycle divides into quarters quite nicely, because you’ve got the moon, the half-moon, the full moon, there’s no obvious division into seven there. But I don’t know, I haven’t looked at the history of the week. It’s a strange one, people just seem to take it for granted that things go around in a seven day loop, but they don’t ‘really’—most people haven’t thought about it, it just seems like intrinsically that’s how things are, they don’t see it as an external imposition. cs: I wondered whether there had been any societies recently in the world that have had a different cyclical week, because you could find yourself in a situation where one person had a seven day week and another had a five day week, and it would take a while to come back in phase…could there be two people whose Mondays never fell on the same day? mw: The Judeao-Christian and Islamic calendars are all lunar, they’ve all got the seven day week, and of course they’ve influenced the vast bulk of global civilization. A ten day calendar was used in France, decimalisation, during the Revolution. cs: The same as the metre. mw: There was a performance artist who was living in decimal time for a while, in public… rm: Doing it as an individual performance is very different from building a culture. After a year of living decimally as a collective, would it begin to seem natural to say, it’s duoday again, and for duoday to have a particular feel that everyone acknowledges. mw: Like everyone going out at the weekend, this tide of sociality. //

pc: Physics often uses metaphorical language, but it’s empirical. A metaphor in art can be allowed to mean a different thing for every person, whereas in science and maths, when you’re using a metaphor to describe an abstract concept, it has to be understood, it has to mean the same thing, you have to be on the same page. mw: So we’re looking at that grey area in between? cs: With a lot of sculpture it’s very dependent on the actualisation— like the rope, the rope is woven, time is woven, we have all these kinds of metaphors for time, it’s taking the words back, its almost like these metaphors coming home, coming back to the original processes that created them: a ship ploughs the waves, but it comes from the plough, and you’re taking it back home, it’s become linguistic, and it comes back into the real world, to show that etymology of its origin. kenna hernly: In Western culture the fact that there’s a fear of 13, is quite interesting because it’s quite abstract, although there’s cultural Judaeo-Christian heritage that goes into it. In Chinese the number characters are all auspicious, based on the fact that it’s a tonal language, so what number words are in another tone has a significance. 4 is also death, so 4 is a really bad number, and so all multiples of 4 are also bad. Also, a lot of swear words are a combination of numbers. So to call someone a 3 and 8 is really bad. mw: This is the kind of thing that a very dry and rigid thinking about mathematics would dismiss, and this is everything I’m hinting at in my first chapter about qualitative number. So any feelings that anyone has about any number are completely subjective and if you get any cultural consensus about what the qualities of a number are, it has some arbitrary historical set of reasons, it’s nothing rooted in the number system itself. I also try to leave open the possibility that there is some mysterious psychic interface between the substructure of number and the mind. But it’s very hard to talk about, because you have all these cultural things that are arbitrary and accidental.

The most intriguing indications that there is something going on in it come from autistic ‘savants’ like Daniel Tammet2 who are able to articulate their experience of the number landscape as something really quite tangible. And there are experiences of synaesthesia and some neuroscientific research which shows that there does seem to be more going on than just arbitrary sets of associations, but it’s difficult to talk about. Because even though two people with autism may have similarly extreme experiences of number, very different to what most of us do, they may have entirely different feelings about what ‘their’ numbers are—one might see a number as a certain colour or texture and the other might see an entirely different number or texture. And that would be grounds to dismiss the objective reality of such a perception. At the same time, it seems to me that, there’s something beneath that. rm: So some people are able to grasp explicitly some aspect of number that we’re always dealing with, but they’re able to express it in perceptual terms? mw: I think it goes deeper than that. It’s as If we’re only seeing the heads of the numbers poking up above the surface, as these distinct individuals. And for whatever neurological, psychological reason that we don’t understand, for them the threshold’s been lowered, and they can see the understructure. This is just speculation, but that’s how it seems to me from the way that they express it, within the framework of language and the concepts that are available, which aren’t adequate because culturally we don’t deal with numbers like that—they have to attempt in the best way they can to describe an experience that’s not shared. rm: As if we are able to learn to handle them by rote, but not really able to penetrate…. mw: …seemingly, to the core of what number is. But that core is not in any way knowledge to them… cs: When I was doing a bit of basic linguistics, I realised that actually its very flawed to try to talk about language in language; you have to

at least do it in another language, try to talk about English linguistics in French or something, to produce some kind of objectification, because otherwise you’re just trying to analyse a medium using that same medium. And in terms of what you’re saying, you need to talk about these things which are very essential, break down what numbers are by using something else, a different way of looking at it, maybe a system that doesn’t yet exist. mw: It hasn’t evolved yet. Maybe other cultures have had it and we never recognised it. Or possibly it’s all just neurons misfiring, and there’s nothing there, and I’m just a romantic crackpot! cs: Do computers have the potential to look at these things? mw: No, I don’t think so, not computers as we now conceive them. Maybe if you had a vast kind of neural net/AI thing, well beyond anything we can now imagine, but even then… rm: It’s not something you could get by just analysing more finely, it would be a major shift in perspective. mw: Yeah, you’ve got libraries and libraries full of detailed mathematical works that everyone can agree about, and that’s not getting you a millimetre closer to this. And part of the problem is that there’s no agreement about it, you can’t have that level of consensus; so mathematicians might be vaguely interested, just as anyone might. But they have nothing more to say about this than anyone else. It’s neuroscientists, really, who lead the way, even though they don’t know what they’re dealing with either. But one of the reasons I know about this is that the prime numbers come up quite regularly, and this is one of the reasons I think these people are tapping into something. There’s something that’s sort of quasi-objective, I don’t know if that’s the right term—but it’s not their own private imagination: it’s that a lot of them seem to identify prime numbers as qualitatively different. Tammet says they’re smooth like pebbles, and that they’re like signposts; so if he’s lost, he’s drifting off to sleep, just cruising round in number world, he’ll see a prime number, and it’s like when you get lost walking around in the evening

in an area you’re unfamiliar with, and then you see a road sign—oh, thank god for that, that’s the way back. It has this sort of friendly, familiar quality. And that’s what a lot of the others say—well, there are very few people with this sort of condition who are also sufficiently articulate to talk about it, it’s been more psychologists and psychiatrists drawing things out of people. But the prime numbers seem to be a common theme, and that would indicate that these people are seeing something…. pc: Do you think that—you briefly touched on it earlier—in the early stages of human cognition when we began to count things, but the system of doing things abstractly with number hadn’t yet arrived, that might that have had a synaesthetic origin, involving a kind of direct emotional response to groups of objects? rm: That’s the question of abstraction: say you’re living in a group of people in which the most important things is to count your livestock, and you only have between 5 and 10, you only have so many numbers you’re dealing with, so there’s likely to be more opportunity for qualitative differentiation. But when we’re dealing with numbers all the time, counting all sorts of things at different scales, then I think that tends to fade. mw: When you read about number associations, whether it’s just people’s feelings, or historical cultural anthropological data, it’s always the first few integers, because the thing is, they just keep going! So it’s far more likely that someone of any culture is going to have something to say about the number 11 than it is that they have some feeling about the number 16285…! You’re less likely to have any experience of that number. If you had eleven objects on the table, some people could just look at that and tell you its eleven, it has an eleven-ness about it. But there’s a very blurry threshold after which it becomes less evident. To come back to the autistic experience, I mean, it’s awful having to invoke probably very inaccurate Hollywood films, but Rain Man, you know, some research went into that, I’m sure. There’s a scene where he’s got a box of toothpicks, and all of them fall on the floor, and he just says ‘246’. I’m

assuming that’s based on some evidence of people with similar conditions being able to ’feel’ the quantity based on a qualitative impression, and that’s just the same thing we might do with the smaller numbers, but more efficient. But I’m always hesitant to immediately gravitate to a causalmechanistic explanation. The classic thing that people with these conditions are able to do, the classic ability, is that you give them a date, say May 9th 1302, and they say ‘that’s a Wednesday’. For some reason a lot of people with similar conditions like to do that, and they’re very adept at it. Of course the sceptic types, people who make it their business to iron out anything remotely unexplained in the world, will say, they can’t know it without working it out, there’s an explanation. I’ve seen attempts to say that there are shortcuts and tricks involved and it’s really not that remarkable or special—an attempt to bring it back to something we ourselves can relate to and can do, but just a bit souped-up. My natural tendency is more to think, no, this seems to indicate that there’s a whole other way of seeing that these people are tapping into. But I’m aware that, as with quantum mechanics, there can often be a romantic attempt to attribute mystery to something, to hang it on something. rm: I think this is the right response, to try to remain nonjudgemental and open-minded but at the same time to be vigilant against that romanticising tendency. mw: I want to re-enchant the world, I want that in my world, but I’ve also seen the dangers of just attributing it to anything. cs: Also, in the history of mathematics, new discoveries haven’t primarily come from number crunching but from intuition, from new perceptions, from conceptual leaps. rm: Mathematics doesn’t come from number crunching. cs: The discoveries come from hunches and intuition. And there are a lot of things that happen by mistake, I mean obviously, people stumble across things—there are definitely strange intuitions.

mw: I agree with you, I wouldn’t dispute what you’re saying, but there’s also—and I always bring this up when people mention this— yes, it is amazing the way it works, these sort of thought experiments, and this is basically what we’re driving at. But there’s this projection of Einstein, for instance, as the absent-minded, benevolent scientist—you know, he’s just become a cartoon character now, and the bright school kid gets called Einstein. Most people couldn’t tell you anything about him , he’s become a mythical character. And what a lot of people don’t realise is that the theory of relativity was built as just the next layer of a very tall tower of concepts. He used Minkowski’s four-dimensional spacetime manifold, he couldn’t have done anything without Riemann’s geometry and Minkowski’s tensors. He just had that extra supplement…. cs: Are there visual renderings of those kinds of manifolds? mw: Well, because they’re four-dimensional, in three dimensions you can only make shadows or slices of them to get a sort of feeling, it’s a bit like you’re just looking at a silhouette of something. cs: Because I suppose for me the challenge of trying to represent those four-dimensional objects is a real key element for me, trying to convey that, because obviously it’s always going to involve a device or some of way of trying to occupy a three-dimensional space. Those sorts of devices are really what I’m trying to get at, taking the things that are invisible or imperceptible, and trying to create a small window into the possibility of that world. mw: There are different reasons for doing that. In mathematics and physics, you have structures which are four- or higher-dimensional, where even a three-dimensional perspective drawing is never going to give you the whole thing. There have been attempts to render these things visually. For the researchers who are into the detail, to be able to get inside and play with them and see what happens if we move that there, it is quite useful to have a visual model. But then there’s the idea of artistically making it accessible to somebody, just to get a feel for it—a good example would be a YouTube video of a

rotating hypercube that somebody’s animated. So it’s a fourdimensional cube, but its animated, moving in a three-dimensional space. But it’s not at the same time—I mean, what it actually is, is a group of pixels on a two-dimensional screen, but if you had the right materials and a lot of money, you could make a three-dimensional version of what we’re seeing in perspective there. But even that wouldn’t be a hypercube rotating, it would be a silhouette of a fourdimensional hypercube rotating. rm: In the Critique of Pure Reason Kant wants to say that there are forms of our experience which aren’t purely logical. The logical categories are the forms of thought but time and space are nonconceptual, they are the shape in which sensation is given, and in a sense they are arbitrary—they’re just what we’ve got, or if you prefer, what we’re trapped inside. And he gives the example of left and right hands, where there is no conceptual difference and yet there is a difference in terms of space. So, it would be nice to show Kant a four-dimensional rotation of a hyperglove turning inside out…. But to sum up this thing about numbers and the qualitative and the quantitative: for me, you don’t need to reduce numbers to psychology, but you also don’t need to dismiss psychology. Based on the idea that what we’re interested in is not just working mathematics, we’re interested in reality at large, then this is a question about how various cultural and symbolic systems interlock with reality. We have symbolic systems that we consistently apply to reality, and another way we interface with reality is through the imaginary, through ‘looser’ associative mechanisms. For me there is an interesting opportunity here to get into an intermediate territory. mw: The interzone has been touched on already, the qualitative/quantitative number thing which I’m fascinated by, I don’t know how that is and whether we’ll be coming back to that, we’re still working out what the boundaries are. Because that’s not really about an abstract idea and its representation in art, is it. But it is about the problems of linking abstract conceptual thought to that inspirational or creative layer of being.

// rm: I also wanted to explain this thing about dramatization, it’s a concept of Deleuze’s which I’m misusing, but I think it’s useful. Dramatization is also of course a theatrical notion: that you need to stage ideas in order to communicate them. But there are further levels to it. Schematically, in the traditional idealist or Platonist, model we would have an ideal realm or intelligible world—the world that we inhabit when we think abstractly, and then the world in which we live. It’s in the ideal world that there ‘exists’ (in some problematic sense) the ideal sphere of which all of the oranges in front of you are inexact instantiations. They are copies—just as any circle you discover in nature, or draw on paper, is a copy of the ideal circle. So we have this relation of copying or instantiation between the ideal and the real. The idea of the virtual and real involved in dramatization displaces that tradition of ideal and real. There is no relation of resemblance, there is no actual instantiation that resembles, inexactly, its ideal counterpart. The virtual consists entirely of relations of difference, differences of potential if you like, and a certain configuration of forces or differences, when instantiated in reality, for instance, could produce a sphere. A soap bubble is not spherical because there is an ideal sphere of which it is a copy, but because equal forces are pushing in all directions, its an instantiation of a certain relation of forces. And the sphericalness of an orange is a different dramatization of a similar virtuality. mw: The orange dramatizes the relationship. rm: Rather than instantiating an archetype, right So you’ve got this virtual problematic relationship—problematic because it’s a sort of tension that needs to be ‘worked out’, and it’s dramatized in actuality, in a specific material and a specific circumstance. And of course depending upon what material this relation is actualised in, it may create something else, maybe a pear…there is not necessarily such a straightforward relation of copying or resemblance.

Of course in mathematics there are all sorts of situations where you have one equation that can produce very different outcomes depending on how it’s applied. mw: You can have the same basic abstract description with very different outcomes. rm: And then, in twentieth-century mathematics a lot of the great advances have come from one person working in one field and another working in another field, and some two things suddenly interlock—there is a way to look at this in which they suddenly look like the same structure, and that’s a structural revolution in mathematics—rather than it being about quantities or numbers, it becomes about structures. mw: Yeah, you have increasingly broad structures that subsume everything that came before into a single structure, and you get some extraordinarily abstract, like category theory, the Bourbaki group are really into this categories which can subsume almost any other type of mathematical structure, speaking very loosely, and I think the next one up is schemes, sets of categories, and then there are stacks…it gets extraordinarily abstract. Grothendieck was the man who did a lot of this. He went to live in the Pyrenees, and ended up as this reclusive weirdo talking about the devil, I think he’s still there actually…he just became a kind of mountain mystic. Of course most mathematicians don’t really have time for all that, they’re caught up in the details of what they’re doing. But they all recognise its validity… rm: The point is that, between the virtual and its dramatization you could say that, because there’s always a kind of twisting. The same as if you look at the extreme Dawkins idea that you may as well do away with the actual animals, it’s all about DNA, competition between selfish genes, this is obviously flawed because the mutations all happen in biologically instantiated creatures, so there’s actual feedback, it’s not just, as in Plato, one way from the ideal to the real, there’s interaction between the two. The way in which forces are dramatized in time and space has an effect on what types

of systems exist, and that creates new virtualities that will be redramatized. All of that is at a very abstract level, But as well as applying to mathematical work, it also applies to what the artist does, because if the question is how you move from an abstract concept into an actual object, it’s again this twisting that is really interesting. I don’t see it as a kind of rigid barrier, I see it more as a series of gradations, so you tentatively actualise a problem to make it slightly less abstract, and it becomes twisted in the process of dramatization, which leads you to make further decisions that further affect the actualisation…. mw: At which point does it arrive in physical reality, though? rm: I don’t know that things ever arrive entirely in physical reality, do they? There’s always a virtual component that clings to them. If something were absolutely realised and actualised, we wouldn’t be able to register it at all. We understand reality through concepts. The mathematician-philosopher Albert Lautman says there’s a kind of gradation between virtual and real, and he says mathematics in every case is the closest we can get to the problematic kernels of reality, these knots of virtual difference—but in fact there are ultimate ideal forms behind them which are inherently ungraspable. For Lautman it seems mathematics is the most acute dramatization possible. In that way you can understand the history of mathematics as a continual and interminable progression and unfolding of virtualities that structure everything, Perhaps what you were talking about can be mapped onto this: we have numbers, and then you were imagining this kind of understructure we can’t quite grasp… mw: Yes, only a few people with unusual brains can ‘get’ closer to it and report back, but… cs: Dorothy Hodgkin talked likened her work in X-ray crystallography to trying to work out the structure of a tree by looking at its shadow. But it’s more interesting than Plato’s cave, because this is a

metaphor for our own position: we’ll never be able to fathom the full complexity of a tree from a silhouette. For me that has been a really beautiful, really useful metaphor. Again it’s this idea of the real being gleaned from limited amounts of information, and just seeing around that corner somehow, Science is its sort of a blind way of seeing the world, cosmology or mathematics are a blind way of seeing, they rely on stumbling in the dark, inferring in other ways. rm: In mathematics you could say there are a set of techniques for discovery, and it relates to this question of twisting in dramatization: you have these very abstract relations being realised in various forms, and what mathematicians tend to do is to develop techniques of turning, manipulating, manoeuvring things in these abstract spaces until they are ‘untwisted’ enough to trace them back to the virtual. Would you agree? mw: Physicists, as you’re describing, try to take the shadows and reconstruct the reality, stumbling around in the dark using symbols. But mathematicians aren’t trying to explain the world, they’re concerned with this strange space that they have access to. So in what sense can a mathematician be thought to be bringing the virtual down into the real? rm: No, the other way around—the mathematician is probing reality in order to discover the virtualities that are dramatized in it. And for me what’s magical about mathematics is when it offers an explanation for a set of different things in the world that makes them ‘the same’, but not in this pseudomorphic Platonic sense, but in this intricately twisted way: if we can think them from what seems to the layman like a peculiarly contrived technical point of view, these very different things in fact instantiate the same virtual relations. cs: Isn’t it the opposite, in a way, that mathematics just aggressively explores itself, and it’s just happened then to apply to the real world and we’ve realised that it is extraordinarily relevant to everything? So it’s actually a self-immersed discipline—not completely, but its

progression is not necessarily responding to the real, it’s not necessarily finding inspiration in the real world. rm: But it has to…not from the empirical, maybe, but… mw: Mathematics and physics weren’t even considered separate disciplines until relatively recently. The idea of the pure mathematician is a fairly modern one; and the history of mathematics has been periodically informed by the need to survey land or develop weapons, for instance. cs: But does it need the real world to do its work? mw: Well, it doesn’t need it but... rm: If the real encompass everything that we can experience, then obviously mathematics comes from the ‘real world’ in some sense mw: You’re not just talking about the world of maths here, you’re talking about reality, and that encompasses not just everyday objects but thoughts and subtle layers of experience. cs: There are mathematical discoveries, but we don’t know whether they’re dependent on reality or experience… rm: Empirical origin and ontological priority are two different things. It could be that, as a mathematician, I drank lots of cofffee today and that’s why i came up with this theorem. But it may be that the theorem ends up helping us discover something incredibly profound about the structure of the caffeine crystal—the theorem isn’t ontologically dependent on the coffee…if it’s true then its true. But I think we should try to get back to prime numbers, since we have Matthew here! mw: I’d like to talk about what I suppose is the philosophical area I’ve arrived at through what I’ve been doing with prime numbers. The thing that’s motivated me. And it’s to do with the innovative and problematic naure of multiplication. I think there’s definitely source material here for an artistic work, as well as a philosophical problem that hasn’t been looked at, that I’m aware of.

We tend to think of multiplication and addition as being on an equal footing. When you go to school, you learn to add first, of course, but before you know it you’re multiplying too, the symbols × and + are almost the same, on the calculator you’ve got the plus and the times button; and most people think of them as living in the same category of thing. But I’ve gradually come to realise that they’re wildly different categories of thing. Multiplying is iteratively adding. To add numbers you can visually imagine yourself taking a certain number of steps so for 3+7, you take three steps in a certain direction, then take seven steps, and you’ve moved a distance comparable to 10 steps. But to illustrate 3×7, you have to take three steps then three steps again, and do that seven times. The notion of time is suddenly being brought into the picture in a way that it wasn’t with addition. Of course, time was in there before, but it’s a different kind of time. When you take three steps and then seven steps, you’re counting the same thing the first time; 1, 2, 3, and the second time: 1, 2, 3, 4, 5, 6, 7. So the 3 and the 7 play the same basic role, it doesn’t matter which one you do first, you’re counting three steps plus seven steps. Now, when you illustrate 3×7, you take three steps, then you take three steps, then you take three steps…. Now, what you’re counting the first time, with addition, is steps. What you’re counting the second time is the act of taking three steps. You’re now counting numbers, 3×7 is also sometimes shortened to ‘three sevens’; ‘three sevens are twenty-one’. Well, the concept of threeness was seemingly brought down into our experience so that we could count biscuits, people, sheep, pebbles, and so on. So you’ve got this psychological apparatus that can filter sense data, recognise things as similar, and then project this notion of threeness onto them. That seems to work quite well, most cultures seem to have grasped it to some extent, even if some haven’t got as carried away with it as others. Everyone counts, and counting is unproblematic, relatively speaking. But when you introduce multiplication, the things you’re counting suddenly shift from some objects in front of you, to numbers. Numbers are the things that are doing the counting. So when you start counting numbers, something strange has happened.

The best metaphor I’ve come up with is feedback, in the sense that, if I have a microphone and an amplifier, the basic intention of that system, as anyone left to play with it can deduce, is that you make a noise into the mic and it comes out of the amp louder. Now if I just happen to be messing around and I wonder what happens if I put the microphone in front of the speaker, I’ll find that I get this unpredictable but quite unpleasant sound, but you can play with it and eventually do something quite interesting, you can make a piece of sound art if you really want to. You get feedback when you’ve taken the system and applied it in a novel way it wasn’t intended for: the set-up is meant to amplify an external source, not itself. You get this unexpected feedback that’s endless in its potential richness. And similarly with a video camera and monitor: the video camera’s there so you can see something over there on the screen. But what happens if you put the camera in front of the screen, you get this turbulent fluid flow effect, and no one can predict exactly what it’s going to do. When someone designed video cameras and monitors they weren’t planning that. The act of counting things is like the microphone-speaker or video camera-monitor apparatus: numbers allow us to scan the world and extract something from it—it’s a tool. But when we take that apparatus that’s scanning the world and wonder what would happen if we were to turn it around on itself…. It wasn’t intended for that, but this is what the Ancient Greeks did, they started quantifying and enumerating aspects of the number system itself. So when I say how many sevens go into fifty-six, I’m asking a very different question than if I ask how many biscuits are on that plate. The thing you’re enumerating is the numbering system itself. And the act of multiplying and dividing are basically two sides of the same coin—as soon as you can multiply you can ask, how many of these does it take to multiply into that. It’s the answer to the same question in the opposite direction. Now, its perfectly possible to deal with the system of counting numbers without multiplication and division, just adding and subtracting, taking steps along the line and back. You can add and subtract forever and you won’t really find anything very interesting. In

fact, it will lead you to believe that the number system is this featureless expanse where each number is just another fencepost in an endless picket fence that stretches to infinity. But as soon as you have this notion of, what if we start counting numbers themselves, how many of this number will fit together to make that number, then you have multiplication, and as soon as you have multiplication the prime numbers ping into being. They weren’t there before. If you’ve only got addition, they’re not there. So that novel application of the number system applied to itself suddenly brings with it something like the opening credits to Doctor Who, something you would never have expected, something swirly and weird and kind of intriguing. The Ancient Greeks basically pulled that down into the world by creating this feedback loop and, as a result, a thousand years later we’ve got library shelves full of utterly incomprehensible-looking analytic number theory textbooks, with all the Greek letters, and subscripts and superscripts, and—it looks horrible, a strange splurge…it’s beautiful if you understand the symbolism, but…Where the hell did that all come from? One minute you’re just counting livestock, then suddenly you’ve got Riemann’s explicit formula. And as far as I can tell it’s endless, this stuff just keeps unfolding, you can keep watching it forever and it’s always going to keep doing something new, you can keep playing with the angle of the camera and finding new tricks. cs: I made something when I was eighteen which started just from playing around with computers. And I ended up going down quite a dark avenue with this machine—I got involved in some dodgy nightclubs in London, and actually got completely ripped off by this gangster. But…I was doing the same thing, playing around with video feedback loops. I created a system with a camera, then you had your monitor over there. I found out, as you probably know, that it was much better to use a black and white monitor rather than colour, And you’ve got these different parameters: the angle of camera to the screen, you can zoom in—and you get to a certain point where you go beyond the horizon or whatever you want to call it, you go into that feedback system.

But I added another device to this, which was a quadroscope of mirrors. So I had a four-sided and an eight-sided thing that would create a virtual sphere from the signal coming out, and you’d look right into this thing. Apart from just the visual effect of having this fan of mirrors in two planes, creating the illusion of a sphere, this would also create a positive feedback system because you’d have more information sent back into the camera every time. So it was a kind of generative feedback system that would always produce more information than it started with. It would become very stable, the whole sphere would oscillate and if you moved it sideways then you would have intense fractals generated, and you’d reach this point where you’d zoom in and change the angle to the monitor and it would feed out around the screen. It would wobble and shake then this line would join with that one, and it would go into this incredible new phase. There was another device, a laser pen that you’d shine onto the screen, and you could play with the most subtle things, create a dot here and it would open out.

mw: Then you could switch the pen off and... cs: You could just watch it gradually calm down to a stable state, then you could turn the pen on again. So, you’d have the master system in one place, then that would also feed information to a secondary monitor, a much bigger monitor, that would be near the dance floor or something. So everyone in the main room could see without being in front of it, the whole system, like a floating orb . It was called the Floating Orb of Fractal Chaos! mw: That’s certainly a dramatization of the notion of a feedback loop. But the feedback of the number system onto itself would be harder to

dramatize in that sort of way, because it’s the role of time that’s difficult. To have a feedback system you need to have time dimension. The feedback we’re talking about with the number system doesn’t require time in the same way, but it requires something like time, it requires almost a kind of virtual time for it to happen in…. Look at the act of counting something. I might be able to look at that plate and reduce it to four—I could quite easily convince myself that I could just look at that and see there are four biscuits there, and that I have just taken in the fourness in a single act. But in fact, as far as I can work out, unless perhaps they were arranged in a square or something exceptional, if you just chucked a handful of pebbles onto a plate, however small the number and however quickly you register the number, there is some part of your sensory apparatus that is ‘touching’ each one. You know, a child will point: 1, 2, 3 4. Some part of you has to see each one, otherwise it doesn’t get counted, there has to be some registration that it’s there. So if you could stop my brain and take the videotape out and stretch it out and see the timeline, you would see that I recognised the first pebble, then I recognised the second, and you’d find five separate little spacetime events. You’ve just stepped out of reality into the editing room, so you’re not in the spacetime continuum any more, you’re in a sort of Godcapsule looking at the tape—but you’re counting things in the time continuum, saying okay, that’s Matthew recognising a pebble, there’s another instance of Matthew recognising a pebble…. So you’re counting instances rather than pebbles. Now, when I multiply, if I carry out the act of 3×5 in any way whatsoever, like when I do three rows of seven dots, which is quite a good representation of 3 x 7—I’ve just made three rows of seven dots, I’ve filled seven bags with three stones, or whatever I’m doing that represents multiplication, if I then take out the tape and go into the editing suite, what I see are seven instances of something more categorically evolved, in the sense that, in the previous situation, I was in the editing suite looking at myself recognising a pebble. Now I’m in the editing suite looking at myself recognising groups of three,

seven times. So I’m counting seven bits of spacetime that have been recorded, and in each of those I am counting three of something. So I’m in the editing suite counting seven again, as I was before, but within each of those seven I’m counting myself counting. Effectively, I then have to stop the tape and step out of the editing suite into the meta-editing suite, to look at myself looking at the tape and say, okay there’s me recognising three things… So there’s something about stepping outside the time continuum, but you step outside that again when you multiply. And all the weirdness about the prime numbers is something to do with the number system being applied to itself and it’s something to do with time being compounded with itself. I think the mystery of number, the this something we can’t quite get to about number, which I’m probing for in this trilogy, and I can’t say more than this, I think it is tied up very much with our failure to understand the nature of time. cs: Is that a similar problem to where you stare into your own eyes in a mirror, so that you perceive yourself inside yourself and your own looking? It’s the Narcissus moment, that you’re having to create an infinite feedback of yourself in your own head, and you have this example of…it’s like the origin of consciousness. rm: Multiplication is the narcissism of number? And multiplication and addition have different temporalities? The other thing is that the process you described, Matthew, is like the ur-operation of mathematics, it’s what mathematics does, turns operations into objects so that you can then carry out operations… mw: …on operations, yeah. It’s a classic thing you do all the time, making function spaces. A function, the word itself…it’s something you do to something else, it’s something that does something. So a function maps one set of mathematical objects, usually numbers, onto some other set. But then you can define the space of all possible functions that have some set of qualities, and you can then create an abstract infinite-dimensional space every point in which is a function. Then you can have a function on that, a functor, or an

operator, and you end up with this kind of endless tower of turtles, with functions of functions of functions, and then categories, stacks and schemes. And that’s exactly what’s going on, you’re taking the operation, the thing that does something—and this idea of something doing something implies that you’re falling back on the idea of time, your experience of time, in order for something to be ‘done’ like that. A function can be represented as a graph, it’s all there at once, in the Platonic realm where it’s supposed to exist serenely, where nothing’s happening, it’s just all there. And yet we use this word, this function maps this to that—it’s doing something, and usually to explain it to people you use arrows or you’ll talk about this going there. I’ve made that really explicit in the book, where you have funny little creatures knocking golf balls around. Because that’s the best way for people to get their heads around it: to physicalise it, to bring it back to something they can experience. So yeah, then the function gets sort of ossified into an entity, nothing’s ‘happening’, it’s just sitting there at a point. But when you step outside that and apply functions to functions…this is a very interesting grey area—there are probably philosophers who have talked about this, but generally the problem is that they don’t know enough maths to say anything meaningful. But there is a reliance on ideas of time that we draw from our experience, and you can’t say certain things mathematically without falling back on it. rm: The way in which we describe mathematical operations necessarily relies on intuitive notions of time, yet those notions of time don’t really enter into the mathematical formalisms? mw: No, so if you asked someone where is mathematics, where does it live or whatever, then, if they were able to answer that at all, they would generally describe it as ‘timeless’. Mathematical theory evolves culturally, but the mathematics itself is held up as the ideal of something utterly timeless and unchanging. And yet most of the ideas that have evolved have come about through applying notions of time.

And then there’s dynamical systems theory, which is a branch of mathematics that has come out of people studying fluid flow, aerodynamics, hydrodynamics. And as usual, once the theory gets sufficiently interesting you’ll get pure mathematicians who will just study the theory without any reference to a flowing liquid, they’ll just be studying a dynamical system entirely in mathematical terms. And then they can start studying a dynamical system in six dimensions, which is something you couldn’t make in ordinary space, but it follows the same set of consistent rules and you get points of stability, unstable regions, and things like that. But in talking about a dynamical system, it’s unavoidable that there’s a time dimension—it involves something happening. And what’s extraordinary is that Alain Connes, a French mathematician in the sixties, was classifying particular group of type III algebras. He was looking at noncommutative algebras, total pure mathematics, it had nothing to do with liquids or fluids or anything happening in space, he was trying to classify type III algebras in the same way a botanist might classify certain types of orchids…but he found that certain of these type III algebras have a dynamical structure. If you interpret one of the dimensions you’re looking at as time, then they seem to be flowing. It’s not time, it’s something else, but unavoidably it has this uncannily strong resemblance to a dynamical system even though it’s a completely static, abstract algebraic structure. So you have an algebra that’s flowing—but flowing in an abstract time dimension. Connes also did some work on the Riemann zeta function, for a while he thought he was on the verge of solving the Riemann hypothesis, it doesn’t look like he has, but he moved things forward, and that also involved type III algebras. And the more I’ve looked, the more dynamical systems theory seems to be informing number theory. And no one understands why that should be. I mean, of all mathematics, number theory is the most atomic and timeless—and yet people who are studying things that are swirly, moving, flowing, in time, are (to put it very crudely) able to tell you something about number. So, I can’t draw any conclusions from that except that it’s strengthens my sense that the reason we don’t fully understand

number is linked to the reason we don’t fully understand time. rm: Maybe the question of this project then is posed the wrong way around. We’re asking how we can dramatize these abstract mathematical concepts, where dramatize means bringing them into time and space. But since mathematics itself has arguably arisen from our examination and manipulation of things in time and space, the number system already had time secreted in it somehow—and so, no matter how far you go into abstraction, that time will sometimes leak out, and you’ll have these moments where, working through some material that is supposedly totally abstracted from our experience, you’ll suddenly see it work in terms of time. mw: Yes, we can’t do anything without time, we can’t think without it. So our conception of number…I mean, the archetype of the simplest introduction to the number system is a child going ‘one, two, three, four’. Something happening through time. Looking at marks on paper is a step beyond that. And even then, you might say, we can get away from time, we can represent the number 10 as a triangle of dots. But for me to recognise it as ten, even though I might be able to do it very quickly, if you get into the editing suite and slow down the tape, I’m still registering in time, I’m doing that ‘one, two, three’ like a child. So you can’t escape from the time thing. rm: Kant again—you can never perceive any object without an unconscious process of synthesis: objective experience is first of all synthesis in time—what he calls ‘inner sense’. And one of the sources of Deleuze’s notion of dramatization is what Kant calls ‘schematism’—the mechanism which somehow (even he admits it is ‘hidden’ and ‘mysterious’) allows us to associate bundles of sensedata with pure concepts—to associate an imperfect drawing of a triangle with the pure concept of triangle, for instance. mw: But the word ‘dramatization’ in the way you’re using it, does it necessarily apply to something that occurs in time, or could it be a static model? Could a non-kinetic sculpture be a dramatization?

rm: Yes, even something that’s static is still in time, even if it’s not using it explicitly. cs: I’ve always thought of mathematics as this ubiquitous thing that’s not like human language, it’s not something we’ve invented, it’s just something we’ve discovered. Whatever system, whether it’s nature or humankind or martians, it’s all reliant on that ubiquitous structure that underpins everything. Whereas what you just said is that mathematics is our own way of responding to the real world. Is mathematics a language, or is it something that is there and that we’re uncovering? mw: This is a big debate, is maths ‘out there’ and we’re uncovering it, or are we constructing it as we go? Platonists, social constructivists, and intuitionists, are the three main schools who take different positions on that. I’m somewhat removed from this debate. I think possibly the question’s being asked in the wrong way. But there is a certain type of dismissive and reductive person who annoys me, someone who, whenever you present them with something that you think is quite remarkable and outside the sphere of normality, even before they’ve registered it they’ll tell you ‘that’s just…’ and they’ll try to bring it back into what they already know about. I get this with quite intelligent people. And it is important to defend against tendencies that go the other way, but that annoys me. These characters will always fall on the side of, maths is merely a construction, something we’ve invented to allow us to do this and that. And if you say, hang on a minute, what about the Mandelbrot set, I mean, who put that there…? This is Roger Penrose’s exhibit A in The Emperor’s New Mind. He’s basically arguing that computers’ can’t ever think. And as part of that argument, which is quite lengthy, he holds up the Mandelbrot set and says, how did that get there, who put that there. I’ve still met people who’ve said, you’re looking at something that to you looks remarkably complex and beautiful, but actually it isn’t —you just think it is because your brain is conditioned by this set of categories, and so on. Okay—but I feel I have the ultimate weapon to shoot down that argument, which is the fact that the underlying number system, once we introduce multiplication, the

prime numbers are there, whether you like it or not, Everyone can agree about where they are, and yet they make a very irregular looking pattern—again, you could say, it only looks irregular because…. But underlying the irregular pattern there turns out to be a set of waves, which nobody had noticed until 1869, And the frequencies of those waves then open up another question: Where did they come from? And then in the seventies it was shown without a shadow of a doubt that the frequencies of those waves collectively have all of the characteristics of a spectrum of vibrations of some kind of dynamical or physical system. Beyond a shadow of a doubt, the statistics are very convincing. Mathematics doesn’t usually rely on statistical evidence for something , but this is like someone finding all ten of my fingerprints on something—you know, come on, it’s no chance occurrence. So mathematicians are convinced now that the waves underlying the distribution of prime numbers are vibrations, in some sense. But as Marcus de Sautoy says in his book, ‘it’s now evident that the Riemann zeta zeros are frequencies of vibrations, we just don’t know what’s doing the vibrating’.1 cs: I did read your interview in Collapse 1, but can you explain, what are the zeros?. mw: The word ‘zero’ is shorthand. When you have a function, the function maps elements of one set to other elements of that set. In this case we’re talking about a particular function—the Riemann zeta function—operating on the complex plane. And like the real number line, the complex plane has a zero where the real and imaginary axes meet. Now, take any function on the line or the plane. One of the first things you want to know about the function is, which point does it map to zero? Because if you know which point maps to zero, you can deduce an awful lot about that function. The Riemann zeta function acts on the complex plane—meaning that every point on the complex plane other than one, which is a singularity, gets mapped to other point. The way I intend to visualise this in the second volume— and it’s a real shame I have to use golf, but it works really well!—is to use an infinite two-dimensional golf course to represent the complex plane. And the only hole on the entire golf course is zero, where the

axes meet. So, you can put down your golf ball anywhere on the complex plane, and then you summon this golfing spirit that hits it to another point. And if you put it down it in the same place it always ends up in the same place. So that represents the function. The question is: Where can I put this ball so that it’ll end up in the hole? Which locations on the complex plane will get mapped by the zeta function to zero? If I know that, then I can deduce a lot of things about the zeta function. And in fact, one of the early questions about the distribution of prime numbers, the prime number theorem—which says that the prime numbers are distributed according to a logarithmic rule—the proof of that relied on isolating all the zeros in this narrow strip. Imagine on the complex plane you’ve got zero, where the horizontal and vertical axes meet, then you’ve got one, the real axis, which is a singularity. There is a vertical strip between zero and one, called the ‘critical strip’, Now, Riemann himself proved using fairly straightforward methods that all the zeros—all the points at which you can put down your golfing spirit and he’ll hit the ball and it will end up at zero—have to lie on that strip. He also proved that they all lie in mirror-image pairs across the halfway line. He also showed that if absolutely none of them lie on the edge, on the boundary, then the prime number theorem is true. And it took another hundred years for it to be proved that none of them lie on the boundary, Riemann himself believed that they all lie on this line down the middle, exactly on that line, the ‘critical line’. And he actually calculated the first of them and they all were on that line. He said ‘I have reason to believe they all lie on this line’—and that’s what’s known as the Riemann hypothesis. The idea is that, the more you can narrow them in, the more you can restrict them to a narrow strip, the more you can say about the distribution of prime numbers. And mathematicians almost universally believe they will all lie on the line, but no one’s been able to prove it, it’s been a holy grail of mathematics since then. cs: And if you were to find one that doesn’t, the exception to the rule?

mw: It would throw all of mathematics into disarray. What would that number be? The thing is, if there’s one off the line, there would have to be four. Because it turns out they lie in mirror image pairs across the critical strip and across the real axis. So what you’re really looking at is the heights, and they pair off. The first few look completely random, no one knows why they are where they are. Then they start to squash together. And by now, they’ve calculated billions of them. So the really interesting thing that was discovered in the seventies is that someone thought to do a statistical analysis on what data had been gathered about these heights, this mass of numbers. With this kind of data set, physicists would naturally think of running some type of number-crunching process, to see if they could find anything. A pure mathematician doesn’t really think like that, generally. But Michael Berry at Bristol University had been studying quantum chaology systems. Quantum mechanics and chaos theory don’t usually have much to do with each other. A lot of people confuse them and think that since they’re both weird and interesting and modern they must be very closely related—they’re not, not at all really, except there’s one place where they do meet, and that’s the study of quantum chaology, where you’re looking at the relationship between certain quantum systems and related dynamical classical systems where you can spot the signatures of chaos. It’s a very specialised area of physics. And one of the tools that used to study dynamical systems and to deduce things about the underlying chaotic potential is the spectrum of energy levels. You get spectra of all types, you get spectroscopy used in chemistry, you can analyse starlight into a spectrum, and so on; a spectrum will always return a set of quantitative values, frequencies—or energy levels, as you get in dynamical systems. Michael Berry’s thing was looking at the spectra of dynamical systems and using statistical tools to analyse the data they provide to deduce something about the underlying system. He helped develop some tools and applications for this purpose. Berry happened to read something, or go to a lecture, it was just a number theorist talking to a bunch of scientists in very general terms,

a bit like I’m doing here, about what we know about prime numbers. And he just got curious about how that set of data of the zero heights looks a bit like an energy spectrum. So he ran some analyses on it and it turned out that it was—again, it’s like finding all ten fingerprints, it just is the energy spectrum of some unknown vibrating quantum chaological system. cs: Chaological? mw: Chaological, okay. A quick run through quantum chaology then. So you’ve got the classical realm and the quantum realm, Newtonian physics and quantum physics. But the boundary between them is interesting, and there’s something called semi-classical physics which explores that boundary. One of the things that characterises QM is that you have a Planck length or a Planck constant—you have this granularity whether you’re looking at energy, time, or space, which you don’t have in the classical realm. And it’s measurable, the Planck constant is tiny but you can measure it. For some reason theoretical physicists started wondering, what if we pretended the Planck length was different than it is. What effect does that have on a quantum system? In every quantity you’re dealing with, whether it’s distance or energy or time you have a Planck whatever, there’s a Planck constant from which all those can be defined. And you can mess with the that theoretically. The Heisenberg uncertainty principle guarantees that you can’t measure below that scale effectively, your measurement becomes increasingly inaccurate as you approach that scale, there’s a horizon beyond which you can’t really say anything. cs: But theoretically there are things smaller than it? mw: Oh yeah, things exist below the Planck scale, but it’s not just that our equipment isn’t good enough to see smaller than that, it’s that in order to get any information about anything going on at that scale you would have at the very least to bounce a photon off it, that’s the most sensitive measurement you’ve got. Now, if you bounce a photon off a biscuit, it doesn’t really have much effect. If

you bounce a photon off something small enough it sends it flying, so where was it? By measuring it you’re interacting with it, you’re affecting it; you can take that into account, you can pull that into your process of measurement, but the problem is, the smaller it is, the smaller distances or energies or whatever that you’re measuring, the more you’re going to mess with it, and the less you can then deduce about where it was or what it was doing, so you end up with this limit where you hit a kind of zero information point, and the Planck scale has to do with that. So, semi-classical physics involves looking at messing with the Planck scale to see how it affects your quantum system, and in particular letting the Planck constant tend to zero. And the point is that, when it hits zero, when you end up with a Planck constant of zero, you end up back in the classical realm. The classical realm is basically the quantum realm with a nonexistent Planck constant. But semi-classical physics is about letting it approach it and seeing what happens. Now, every quantum mechanical system has what’s called a classical underlying counterpart, and that is achieved by simply letting the Planck constant tend to zero. So you’ve got, say, some sort of set of particles in a magnetic chamber resonating and interacting, its a quantum-mechanical system, but you can actually mathematically see what happens when you take the Planck constant to zero, and then you end up with a model of a classical system that is somehow related to it, its counterpart. Often they’re framed in terms of billiard models, a kind of electrical billiard table or something, things bouncing off things, the classic examples of classical physics. So every quantum system has an underlying classical counterpart system, and similarly any classical system can be quantized—sort of, it’s very difficult…but effectively, you can create these pairings of classical and quantum systems. Quantum chaology is looking at that interface. And the main question in quantum chaology, at least as far as I know, is to say: chaos exists in the classical realm, chaos theory is entirely concerned with the classical realm. Once you get down into the quantum realm, your mathematics becomes entirely linear,

whereas classical physics, chaos theory, deals with nonlinear systems. The nonlinearity sort of gets smoothed out in the quantum realm. So there’s normally no communication between quantum physics and chaos theory. But since every quantum system has an underlying classical system and vice versa, we can do the following: say you’ve got a classical dynamic system, you’re not going to tell me what it is, but I’m able to see what the quantized version is like, I have access to the quantum-mechanical counterpart. How can I tell from what I’m looking at, from the information I have, that your classical system is chaotic rather than integrable (non-chaotic)—that has to do with periodic orbits and sensitivity to initial conditions? So, if you give me the quantised version of your classical dynamical system, how can I determine whether or not it was chaotic? Are there signatures, is any of the chaos somehow encoded in the structure of that quantum counterpart system? Berry and his team and others elsewhere showed that yes, actually you can determine whether or not the original system is chaotic, by looking at the spectrum. Because the classical system is to some extent characterised or determined by periodic orbits. In the quantum system, the closest thing you’ve got to that is the energy spectrum— a set of resonant frequencies or energy levels that it flips between. The hydrogen atom would be the simplest example. Quantum mechanics is a particularly effective in some ways, in many ways; a hydrogen atom is a photon and an electron, you can calculate the spectrum of that, it’s fairly straightforward, using quantum theory. But as soon as you add another electron, say, take that to a helium atom, you can’t solve the equations, it’s too complicated. But any atom or similar configuration of particles will have a set of discrete energy levels which it jumps between, in discrete but irregular jumps. So a quantum system can be characterised by its spectrum. Normally when you think of a spectrum you’re thinking of something like the spectrum of light from a distant star, you do that with a lens and get a set of levels. But essentially you’re just looking at a big set of numbers: if you put starlight through a prism you’d get a set of levels, and an atom will do the same, the configuration of an atom will give you a set of energy levels. You have emission spectra and

absorption spectra, but essentially you’re dealing with a spectrum of energy levels. You can run a statistical analysis on one of these spectra: you’ve got a set of numbers, and you can look at correlations, the gaps between them, you can look at average gaps, you can renormalise them as they tend to bunch up according to a certain rule, for various predictable reasons, so you can sort of stretch them out to compensate for that squashing, and then you can look at the average gap between them. You can collect all sorts of statistics, just as you could on a population or something like that. And it turns out that the statistical analyses of these spectra provide you with a set of tools to determine whether the system you started with, the classical system, was chaotic or not. So the chaos is somehow coded into that initial spectrum. So that’s what quantum chaologists do—and the only reason I’ve even heard of this stuff is because it ended up getting linked to number theory, otherwise it would just be a super-obscure branch of physics. But as I said, Michael Berry, who was concerned with this sort of thing, went to a lecture about number theory, something he wouldn’t have ever learnt about in a physics education, you just don’t —you know what a prime number is but you probably wouldn’t be able to tell anyone what the prime number theorem said—and this set of zeta zeros looked so much like a spectrum that he subjected it to that same kind of analysis that you would subject a physical energy level spectrum to. And it’s not simply is it chaotic or not; with those statistical tools you can deduce just how chaotic it is, effectively—what’s called its Lyapunov exponent. So the spectrum is able to tell us that somehow the distribution of prime numbers, this set of zeta zeros, is a chaotic system that has a set of very specific characteristics of certain dynamical systems, one of which is that it is time-irreversible. Some systems are time-reversible, meaning that if you run them, they look the same backwards and forwards in time: if you change the t variable to -t they follow the same rules. If I bounce a billiard ball around a very low friction table and you film it, and I play the film backward, you wouldn’t know whether the film was forward or

backward, whereas if I drop an egg on the floor and it breaks, if I show you that film you can be quite sure which way the film”s running. So it turns out if you have magnetic steel balls bouncing around on a billiard table, that’s a time-reversible system, but if you apply a huge magnetic field to the table, then that is time-irreversible, in other words from the film you can see where it’s heading, because the magnetic field is exerting its force. So, chaotic systems can be characterised as time-reversible or -irreversible—but there’s a set of other very specific technical classifications that chaologists have been working on. So we now know that underlying the number system, via the prime numbers, we have a set of waves that regulate the prime numbers via their frequencies, detected as a set of energy levels of some kind of mysterious as-yet-undiscovered chaotic time-irreversible dynamic system…and no one knows where the hell it came from. Nobody suspected that it was there—and this is a complete reversal of the normal relationship between maths and physics. This is not mathematicians providing models for physicists to work with, this is Michael Berry thinking, hmm, that distribution of prime numbers might work as a model for this particular type of dynamical system. It’s totally the other way around, it’s a physicist coming over saying look, I’ve got this model and it appears to explain what’s going on with your prime number system.3 This is just...turning people’s heads inside out, I don’t understand why there isn’t more discussion of this really, because it’s the strangest thing I know about the structure of reality, that this number system that we supposedly invented—going back to the original question of whether we’re inventing or discovering it—well if we invented it, then what the hell is that about? Where did that vibrating thing come from? There’s no question that we invented the number system. But if it takes us there, we’re looking at his very strange reflection of our own consciousness or something—part of me thinks that this quantum mechanical chaological dynamical systems stuff has got something to do with consciousness, because number and time these issues are so closely tied to our perception. And it’s quite scary in a way:

What exactly are we looking at here, this mysterious dynamical system? Everyone asks me, what would happen if you made one…. rm: That’s what I was going to ask, is there any comparable system that it’s like? mw: Yes, as I said, it exists in a certain category of dynamical systems. Michael Berry’s actually said, yes, I believe one day someone’s going to make one of these in the lab, and it’ll be great— we’ll be able to calculate the energy values and there they’ll be.. Basically they’re looking for this thing called a Hamiltonian—a dynamical system can be almost entirely characterised by this particular operator called a Hamiltonian, all of its characteristics can be deduced by applying different mathematical analyses to the Hamiltonian. And there have been all sorts of stabs in the dark at producing the Hamiltonian that will do that, and Berry and his colleague Don Keating have got really close but not quite, and then other people have messed with it. But it’s almost as if there’s something they’re missing. And so I don’t really know, I imagine a computer model would be more likely, rather than a bunch of articles in a resonance chamber or something—I’m not sure what these systems are like, I’ve read about them but they’re usually presented in mathematical structures, so I haven’t hung around in a lab watching anyone do this. A lot of it is done with computer modelling, I would guess, rather than building one. But Berry has talked about building one in the lab and how it would lead to a totally new kind of physics. Basically it’s going to be an oscillator, it’ll be something that’s vibrating. And in classical mechanics you have the simple harmonic oscillator, the pendulum on a string, which is the sort of basic unit from which all other oscillators can be mutated and combined. And Berry thinks that this oscillator will play the same role in chaos theory and will lead to a totally new type of physics and technology. Again, it’s slightly scary, I’m not sure we should be messing with that! rm: Like a mathematical version of the Large Hadron Collider, it could create a whole new numberverse?

mw: Anyway that’s where that’s at. But that original question as to whether mathematics is a human creation, my take on it is that Penrose’s example of the Mandelbrot set is good evidence, We did not put that there, that’s not simply us applying our cognitive apparatus to the world that’s out there; we don’t really know what it is, and we’re uncovering it. This to me is the ultimate nail in the coffin of the constructivist argument that mathematics is just the product of social and psychological forces. rm: There’s this broader region where mathematics as we know it still retains characteristics of our contingent approach to it and the specificities of the way in which we discovered it, but it’s like an overlapping of two worlds. There’s something about human cognition, something to do with human brains, that enables us to carry out certain activities and manipulations in the world, and those somehow give us access to these spaces of pure mathematics, but we can never quite get there, so mathematics is always tainted, or coloured, depending on how you want to look at it, by the path we took to it. I find it interesting how, referring back to the diagram of the real and virtual, it’s almost as if you’ve gone all the way into the virtual and discovered something there that belongs to the other side…right at the core of mathematics you discover a quasi-physical system. But again it comes back to this question of time, doesn’t it, because if it’s a physical system then what is the time parameter? mw: Yes exactly, that’s the big question—we’re talking about another kind of time, a second time dimension, or something of which physical time is merely a shadow. I don’t know. rm: Do you think that relates back to this question about multiplication? mw: Yes, because, as I said, the prime numbers aren’t even there before you multiply. So you’ve got that narcissistic application of the number system to itself, and that generates this feedback, and this is getting into the physics of that feedback, really.

rm: That seems like a really fruitful place to focus, because it’s at once a really simple question—what happens when you do multiplication, and that might lend itself to visualisation in a certain ways—but at the same time it seems to be the key to this extremely ramified and complex set of structures...but ultimately it’s all about feedback mw: It takes you back there, yeah. There’s also the image of the logarithmic spiral, just to bring it back more into the realm of something that could be artistically dramatized, The reason why there’s a spiral on the cover of my book isn’t just because it looks cool, the spiral is absolutely central to the visual understanding of what’s going on in it. And it’s there’s obviously a whole tradition of spirals being used in religious art and sacred mystical expression…everyone likes a spiral. Hippies like spirals, people who take drugs get linked to spirals, Even dadaists were messing with spirals on rotating discs. So it has some kind of universal resonance which I’m aware of, I haven’t got time to explore everything here but I’m aware that and I’ve noticed it in people myself. But the most direct way of explaining the distribution of prime numbers is through the logarithmic spiral, because it visually represents the interface of addition and multiplication, and it’s that friction that gives you the prime numbers. The two images that I present of how you can relate to numbers— and I’m a bit biased I supposed, I’m going to try to avoid saying ‘this one’s good, this one’s bad’, its more like one has been overdone and one has been neglected—there’s the boring approach to number, which is that 1, 2, 3, 4, 5 picket fence number system, and then there’s the one where you treat numbers in terms of their prime factorizations, and where every number has a different, unexpected internal structure. And in that system, the numbers don’t necessarily have to be seen as a sequence. Or, at least, you can disengage them from that sequence and they float freely—I present them as being almost like molecules, these clusters of prime factors.

And this is where the feedback starts, this is where the unexpected irregular patterns start to appear, and it’s the friction between the additive—the picket fence view of the number system is an entirely additive view—and the introduction of multiplication, which is this innovation that leads to all the feedback and weirdness. As soon as you introduce multiplication, the idea of division becomes accessible, and the idea of indivisibility is then a part of that dimension, and then the notion that the numbers that aren’t indivisible can be divided systematically into a set of factors which is unique and inarguable. So there’s this tension between the additive view of number and the multiplicative view. The additive view is barren and featureless, while the multiplicative view has this sort of richness and unpredictability to it. But the fusion of those can be understood best in terms of the spiral, because the logarithmic spiral visually conveys multiplication. If we take a line out from the centre, it crosses the line infinitely many times of course, but always at the same angle. This line hasn’t got any scale at the moment, but if we introduce one, let’s say if the distance from the centre to the first crossing is 1, just an arbitrary measure, then the next crossing in terms of the spiral, let’s say it’s 5.2, that means the next crossing will be 5.2 × 5.2, the next crossing will be at 5.23, so 140 point something. The powers are represented by the crossings of the spiral, so every time you move to the next crossing, you’re increasing the number of coils —one coil, two coils, three coils—but the distance you are from the centre is being multiplied. // cs: Does one exist? Can you prove its existence? mw: How could you prove the existence of one? cs: I don’t know, I’d like to argue that it doesn’t, but I’ve been called naive for doing that! rm: You can define it…. mw: If you’re working in mathematics or physics, you have to accept certain definitions from which to work, and there’s also maybe a

misunderstanding here of the meaning of ‘proof’, which is a word that gets used in mathematics in a very particular way, and also gets used in legal situations of course. And it gets used by philosophers, but it’s a very different kind of thing: when a philosopher proves something, another will come along and prove the opposite, whereas in mathematics you can’t do that! Someone who should know better was asking me the other day whether the theory of relativity had been proven. I just thought, surely you should realise that you can’t prove anything about the physical world. So there’s a lot of confusion about proof, and to prove the existence of one, that would depend on what you mean buy ‘exists’ and how you define ‘one’. cs: Is it just an idea? mw: It’s a kind of cluster of ideas that are joined somehow. ‘One’ had meaning within the number system. If you haven’t got a number system then there’s not really a need for it cs: Does it exist ‘atomistically’—if you go down to the level of the neutron and the quark, is there some essential singularity? mw: Well, there does seem to be a point at which you reach a sort of singular state of things. One interpretation of the fact that , for example, every electron appears to be absolutely identical—unlike a pack of biscuits, which all seem to be identical but if you look more closely they’re not, its an illusion of identicalness that’s the result of mass production, and similarly in biology when you look at organisms that seem identical. We’re quite used to that, that there are similar things that are in fact different. But when you get down to the subatomic particles, they are utterly indistinguishable. And I think there is something in Heisenberg’s uncertainty principle that says that, if two electrons stray too close together, then not only do you lose track of which one’s which—you certainly can’t stick a label on them, and they have no distinguishing characteristics, all measurable quantities are going to be identical—it actually becomes meaningless at some level, it’s not even a epistemologically relevant

question any more which one is which, they are utterly interchangeable and indistinguishable. So I seem to remember reading years ago that someone has suggested that it’s most useful to just think that there’s only one electron—there’s just one, that jumps about through time and space and manages to be everywhere it needs to be. So you have this idea that everything’s made of atoms, plural, and each atom has a certain number of electrons. But in fact it’s just the same electron being everywhere. It’d be the same for any other article—if you isolate a particular species of subatomic particle, it’s not like you can have two that are different, one slightly heavier than the other, it would appear that they are all utterly indistinguishable. In fact, I remember someone visualising it as a spiral with a line going through it, which touches the spiral at many points. Each of those points would be what appears to be a separate electron, whereas in fact the electron is the spiral itself. So at that level you could say there really is one—there is one of each particle. And possibly if you got deeper into string theory you could say each of those particles of which there are one are in fact all the same particle being looked at from a different perspectives. That would take you back to something like the monad—everything’s just a single unity that’s being refracted through different perspectives. But mathematicians would never concern themselves with the existence of one, as such. If you said that to them, they’d just ignore you and talk about something else. It’s not that it’s a stupid question, it’s just that you can’t stick anything to it. Just the word ‘exist’ is so problematic, I’d just try to avoid it. Two is another question—the nonexistence of two, in the right context, is an interesting question. cs: Why? mw: If you’re talking about two from an arithmetic point of view, then it’s just defined as the successor of one. I don’t know if anyone’s read the bit in Secrets of Creation about the Peano axioms, but you start off by saying that there are these things called numbers, a general category, and the first axiom is that there’s something in that

category, and we’ll call it one. Then the next axiom is that every number has another number associated with it called its successor, and we tend to think of that as the number that comes ‘after’ it. So the number one has a successor— we can deduce that from the axioms—and we can give that the name two, and it’s distinct from one. Since two is a number, it must have a successor, and you can call that the successor of two or the successor of the successor of one, or you can just call it three. So you can see what happens. Then there are two more axioms to make sure there are no bifurcations or parts of the number line that don’t connect up. So here one is defined axiomatically—we’re told there are such things as numbers, and we’re told that there are some. Two is the first nontrivial number, really. If you’ve only got 1, you can’t count anything, everything is just what it is. But with two…I’ve got two pistachios here, I’m effectively perceiving a chunk of spacetime, taking my sensory data and breaking it down, identifying this bit of spacetime and seeing commonalities in the structure, and then that’s backed up by experience and language and an understanding of plants, berries, fruit, seeds, mass production—I mean, you can easily have two forks, like this, and they would seem identical, and you can understand that they’ve been manufactured for a reason and they’ve come from the same source —so it all seems fairly uncontroversial. But in fact, if you were to take some being who was completely unfamiliar with our way of seeing the world and show them the forks, they might not see these as having anything to do with each other—they might focus entirely on the differences between them. If you were the size of a microbe, these would be like wildly different planets, with different features. But we reduce the world into discrete objects we find it useful to enumerate and name. So we have this concept of a pistachio, and we recognise these as two pistachios. But in fact there’s no twoness there at all, the twoness is us mapping this ideal pistachio onto that and onto this. But okay, we have to define what we’re talking about, try and come up with a definition of pistachio, then when someone puts another object down on the table, I can evaluate it according to the definition and decide whether or not it’s a pistachio, and if so, add one to my

count. But suppose they just give me this: Does a pistachio without a shell count as a pistachio? We have to argue about definitions. If they give me a perfect model of a pistachio made of different materials, or if they give me something that’s identical to a pistachio but has a different genus or something but you couldn’t see the difference…or what about a genetically modified pistachio? In order to count anything, you have to have a category for the thing you’re counting. To get this across I often say to people ‘count everything in this room’—that immediately makes you realise you can’t count unless you know what you’re counting, even if I say ‘every object in this room’. And in order to have a category, the category needs to be defined, and we have to consult our definition. Even if it’s a really precise definition right down to the DNA, someone somewhere could engineer a mutant object that was right on the border between pistachio and non-pistachio, and you could argue about it forever— because all definitions are in natural language, they are always a string of words, and are always open to interpretation. So twoness intrinsically relates to our way of breaking things up into categories. But in order to have two of anything, they have to be absolutely identical. Otherwise every thing is its own unique miracle. Why have we been lulled into thinking that there are two of anything? There are two processes that have reinforced this. One is biology, in other words living things that reproduce themselves, that have babies and make copies of themselves. So that the individuality of each tree gets stripped away and it gets reduced to an example of a category. The world we live in is full of soil and seeds and DNA, there are always examples of similar objects. And the other one is mass production: you can get a box of this or that, and they’re all sufficiently similar. So it seems like the world’s full of similar objects, and it’s perfectly normal to count things. It’s really an interesting exercise to look at the world instead as either just a unified whole or as a world made of fragments, each of which has a completely different story and history. The fact that I can plant this pistachio and grow a tree that produces even more of them, makes that even more remarkable, rather than less so.

pc: That goes back to the time question too: you might be able to argue that that’s one thing now. But if you planted it, it could potentially turn into millions. mw: Yeah, you also have to separate it out into an isolated chunk of space time in order to talk about it at all. And in fact, the bigger picture is that that’s part of a process. If you saw that pistachio in the spacetime continuum, it would be an elongated object which would join up to the tree…if you took a cosmic time-lapse photograph of that, you’d have the whole history of it as a connected fourdimensional object, which would join it back to its tree, and that tree would be a four-dimensional continuous object that would go back to the original pistachio that got planted, which would go back to its tree. So that is connected in four dimensions back to the first pistachio ever planted, and back to a pre- or proto-pistachio, and back to the first thing that ever lived. // mw: Any philosopher that attempts to talk about…whether it’s string theory or the zeta function…chances are they’re going to not know what they’re talking about, because it’s a full-time occupation to understand those things rm: I question that. I’d say, in the case of things like relativity theory, scientific advances that are truly important, there is always a conceptual shift that can be understood without understanding the technical details. You can understand the significance of quantum mechanics without doing the equations. You can understand the conceptual shift that is involved, don’t you agree? mw: Well no one entirely understands QM, some quantum physicists would say anyone who claims to understand it clearly doesn’t! But conceptual shifts…it’s easy to talk about them but it’s easy to be vague about them. You can’t be vague in mathematics. rm: This seems very pessimistic from a cultural point of view, to resign yourself to having a mathematical priesthood, and encourage

non-experts to say, I don’t understand it so I’ll just have to give up. Isn’t that a dangerous path? mw: I’m not promoting that, but just pointing out the reality of how much time it takes to understand even one area of science and be able to say something meaningful about it. It didn’t used to be like that, people used to be able to have a broad understanding of science, an overview of what was known, and to be able to interpret it philosophically. And it’s just the ridiculous proliferation of thought has led to this situation. It’s an interesting time, it’s a crisis situation, you’ve got the UK government appointing a Professor for the Public Understanding of Science, but there’s a huge gulf between the popular understanding of how the world works and what the official view is. People are trying to bridge the gap using art or television or popular paperbacks, and none of them are really doing it. rm: What would it mean to close the gap? mw: Well I think this is what you’re trying to ask—its this ‘grasping by non-experts’, isn’t it? rm: But aren’t you saying that you mistrust anyone who claims to have done so if they haven’t put in the time necessary to understand the underlying mathematics? mw: That it can’t be possible. Well, in a way I’m working against what I’m saying, in that I’ve written a book that allows anyone who can do a sudoko puzzle to understand the basics of the Riemann zeta function. I think it would require developing new languages and new sets of visual data and metaphors and things. So you could almost develop a science of explaining scientific ideas, it would be a different discipline. I mean, if I were busy doing research on analytic number theory I wouldn’t have had time to write a book like Secrets of Creation. I’ve put all my energy into trying to explain very difficult ideas to my friends because I really think they should understand them. That process in itself is a form of research into this question.

But I guess it’s coming from the point of view of someone who does grasp the underlying mathematical ideas. But what role can art have in this? From what you’ve said before, Conrad, you’re taking nuggets of abstraction and binding them together and throwing them out into the world to see what ripples it produces, there’s a fanning out, different threads of people taking it in different directions. That’s perfectly valid, there’s no problem with that. But this idea of dramatizing abstract ideas, if people are grasping it then there should be some sort of consensus or streamlining of understanding. It’s not as if everyone can take it off in their own direction, as if their own interpretation is perfectly valid. But equally we’re not talking about just illustration or producing models, which his why it’s hard to grasp what we’re talking about…. If Conrad were merely an engineer, and I said I’d like you to illustrate this mathematical idea, and we worked together and made a machine that did just that, and that was all it was, is that art? Or is that just a mechanical illustration? You could take illustrations from my book and make them into computer animations, and then into sculptures, you’re just extending the illustration. So we might be talking about a new discipline that isn’t art, but a new type of activity. cs: Binary Star is actually a good example, because I made a piece that was pretty much an actual model, but my intention was not just to make a model: there was also the choice of materials, the unwieldiness of it, the archaic feel of it, all of those came into it because I was trying to challenge some of the philosophical preconceptions rather than just creating an orrery. There were a lot of other things that entered into the process of my making it. So, for example, if the Science Museum said, we want to make a binary star model, that would be different, the motives would be different. rm: Is it that you let yourself be influenced by the process of making, whereas if you were commissioned to make a model, specified in advance, then you wouldn’t be affected by the materials? I remember talking to one of the assistants, Andy, who works with you, and he was talking about the craft element to this making. He said that he remembered when his dad used to tinker with his

motorbike in his garage and take bits off it and re-oil them, make improvements and clean it, he said he felt very much like he was engaged in that kind of bricolage activity, making the parts for one of these works, and that a lot of it was to do with a tactile relationship with materials and actually having a kind of empathy with the machine and feeling with the machine, so it wasn’t as if everything was on the drawing board, all finished, and then you went ahead and made it, it’s a more involved process. cs: Yeah, but at the same time my aesthetic is quite rigorously sculptural. The recent pieces are quite aggressively trying to follow a dogma of functionality…so that defines its aesthetic. And I like doing that because when people are confronted with these objects they feel this quite strong rational presence, and each piece is backed up by rational meticulous well-designed mechanisms, but beyond that they’re quite irrational. So a certain suspension of disbelief is maintained. Rather than it looking madcap, I want it to have that sort of authority, but without reason. mw: If you had some people at the Science Museum who were given this non-artistic task of building a model, they’d still have to solve certain problems in terms of how to make it, they’d still get the input from their own aesthetic, everyone’s got one, and different people would produce things that might all fit the bill, they might all do exactly what it said on the commission, but they’d all have a different feel to them. So for me it’s hard to see where the boundary lies—I agree with you, it’s all about motive, but you know there’s going to be some seepage of art when someone’s making something, at whatever level. Really, since people started making stone axe heads there have been ‘unnecessary’ artistic elements. But I was going to ask a question here; how would an aesthetic affect what we’re talking about—how does aesthetics affect this issue of the grasping of abstract ideas. I mean, is it relevant? cs: I think that if I rendered my works badly they wouldn’t look like real scientific objects, and I really try in my workshop to take as much in-house as possible and to become expert with the lathes and

mills and really make sure they have an authority—not just some outsider madcap contraption made of old bicycle wheels, but something that’s properly engineered. So whether you come across it at the Science Museum or in an art gallery context, or if you were to find this thing in a scrapyard or in an Oxfam, this mechanism, you might not know what it was for, but in an archaeological sense you would assume that thing to be authentic, to have been made for a particular reason. mw: Okay, but imagine three different people making the same binary star model, you with your aesthetic and your motives, then someone who’s much more wacky and whimsical, then some boring engineer who just wants to get the job done…and all three of those are examples of something, but is there any predicting how well each one would allow the non-expert to grasp the abstract idea underlying it? cs: I think it depends on the context of showing, whether it was showing in the Science Museum alongside another orrery, or shown in an art gallery with my work, or found in the scrapyard, it depends on their context. I don’t think I would necessarily be happy with sharing space with the zany bicycle wheel one, I think it would be a quite distracting way of representing the kinds of ideas I’m trying to represent. A well engineered one with stainless steel and titanium, minimal and functional, or whatever, it could be extremely good in terms of the form really following the function. But it’s the context that matters. I love the aesthetic of the machine, and those early pieces were in particular very engaged with that aesthetic. But actually, I moved away from making with wood, and part of that was precisely to do with this. I got lumbered with this idea, which became a kind of lazy journalistic idea of my work, that I was this kind of Heath Robinson contraption artist. And it really annoyed me, it wasn’t what I was doing at all. Partly because these things were just sketches. But I was really trying to do something subtler than that: they weren’t just made for the sake of making contraptions. They were very much about time, and one of the only ways to convey time is through

movement, and maybe that’s quite basic—there should to be other ways of representing it, I’m constantly trying to think of different ways —but kinetic movement is a very useful way of doing it. I’m always shying away from identifying as a kinetic artist, because it isn’t just about movement, or creating a spectacle of whirling wheels and pulleys—it’s hopefully more than that. That’s why, from those earlier quite complex aesthetically rich pieces, I’ve stripped it down more so the function is laid more bare and there isn’t so much theatre to it, it’s more about the process, the function. pc: More generally, Conrad’s really thinking about the implications of every choice that he’s making with how the thing looks, and that’s what makes it an artwork rather than something else—the depths to which one goes to carry out one’s intention. // mw: We’re learning this already: Conrad isn’t trying to illustrate scientific ideas. And scientists wont be interested in aesthetics…. cs: I think that’s why Robin put us together: you’re not in that academic scientific world, You’re attempting to get these ideas across, and I am as well, we have a similar intent in a way, yours is more rigorous than mine, but you are interested in aesthetics in the sense that your book is relying upon an aestheticisation of these ideas. You are to a certain extent relying on metaphor and illustration to convey ideas. mw: That’s true, there’s definitely visualisation. cs: They’re metaphors whether they’re actualised in drawings or not. mw: But is a metaphor artistic, or is it just a metaphor? rm: It’s a mechanism for moving from one to another thing, and the resources of metaphor are contained in the cultural baggage we inherit, rather than any symbolic system. A function is a way of getting from one thing to another, in the system of mathematics; a metaphor is a way of getting from one thing to another in the system of the imaginary.

mw: But then do mathematicians rely on metaphor in any way? rm: Once an axiomatic system is established, you don’t have to rely on anything other than its axioms, theorems and definitions. But for me axiomatic systems aren’t so interesting. Once you’ve made it into a machine, yes, it functions. The more interesting question is how is mathematics made, what’s alive in mathematics, and I think, yes, metaphor must play a part in that. cs: Metaphors can be used just purely to explain something, using something familiar to explain something less familiar. So if someone’s never been to the ocean, you say of a boat that it ploughs the waves because it’s something to do with their culture of ploughing the field. But a metaphor can also be used poetically, on a more artistic level, a much more poetic level. I actually take metaphors that have been used before—philosophers often use structural metaphor as a means to describe something or to illustrate an idea, whereas I’ll take something like say, in epistemology you have the two metaphors of the building: the building relies on the assumption of certainty, so the ground is your symbol or metaphor for certainty or stability. Then the other is Neurath’s boat, all repairs have to be done at sea, and in order to for you to stay afloat you have to be prepared to repair your boat with all the existing bits of timber in it. It’s more romantic, everything’s in flux. mw: Whereas my metaphors can actually fail, can miss the point. pc: That is where people get confused about art and science, looking for poetry where there shouldn’t be poetry and vice versa? The culture of institutions trying to bring art and science together is actually one of the big problems... mw: I think that’s really important. Where’s the motivation, whose idea is it to do this, and why? Why does this ‘art+science’ stuff get funding? My feeling is that there’s a specific sector involved, hi tech, biotech, and so on, that they want to rebrand the image of science, and so they commission things that people find pleasing or fun or cool or sexy. I think that might be where a lot of it’s coming from, but

then there are other people joining into that current that have other agendas. And it’s often highly unsatisfying. There was a project I saw where a photographer got a commission to do something about genetics. And they took these photos of genetics labs, beautifully taken, beautiful light and everything. They gave a presentation talking about the scientist in his lab coat as a sort of priest and…it was like clutching at straws, like the artist was not really sure what they were expected to be doing. There was some interesting stuff to take pictures of, and then create this…. pc: It’s about the other isn’t it, exoticism? mw: Yeah, it’s not really about science, even the artist doesn’t really understand what it’s meant to be about. And then a poet had written poems using the letters from the DNA sequence, CTGA, in different permutations, poems with four lines starting with those letters, which was quite clever but…you know, it didn’t help anyone understand genetics. cs: It could have been any four letters. mw: Exactly, and a lot of this stuff feels like that. There’s money out there but, I don’t know, are there really any artists around who are interested in getting to the essence of these things? cs: There are a lot of artists who are doing really interesting work. My remit at the Science Museum isn’t some sort of corporate conspiracy to get people to fall in love with science or with art. My role is much more to do with challenging people’s preconceptions about science. mw: To me that’s a really valid thing for an artist to be doing, but that’s not focusing on particular ideas within science, that’s looking at the idea of science. cs: As part of that residency, there’s a piece that I just made called Celestial Meters. It looks at the Versailles metre, it was all conceived with a ten hour clock during the French Revolution, an attempt to find

a more democratic decimal system for France, and there were lots of debates at the time as to what to use as this absolute, this measure of one, whether to use a pendulum that took a second to swing, but that would be reliant on time, and also the gravitational field changed at different places in the world so it wasn’t ever going to be the same. So they decided to use the earth as the measure, but they got it slightly wrong, because of technology. This would make a great screenplay for Monty Python, because there were these two scientists who set off, one going north and one going south, armed with sextants and compasses and all the equipment of the time, to measure the circumference of the earth. But they were still being shot at by French royalist resistance who were determined to maintain the foot, the pied. So they were doing this very sublime intellectual project and being shot at—no, stop, we’re trying to measure the size of the earth! So there’s this crazy idea of science, something that seems very low down the list of priorities. They went slightly wrong, but the metre is supposed to be this intrinsic thing, a sector of the earth from the equator to the north pole through Paris. The decimalisation of the meter is useful, but there’s no coherent meaning to it; at least a foot, even though it’s in twelfths which isn’t so useful, has some kind of practical relationship to our body. mw: No one thinks in metres, There’s a reason those measures have been around for thousands of years, yards and feet aren’t really that far off. cs: I think it would have been a lot easier if they’d just made the metre the average height of a man or woman—if a metre was around 6 foot it would have been a much more useful scaling system, it needs to refer back to ourselves, it would be just as democratic. mw: There was a campaign against metrification by some quite serious-minded people in the UK, led by Jean Michell, the antiquarian I mean, I always thought the metric system was great, as against this old-fashioned system with arms and furlongs, things being divided by twelve and twenty, which just seemed really stupid

and hard to understand. But his argument was that it encodes really ‘ancient wisdom’. Conrad Shawcross, The Celestial Meters: Venus . Image courtesy of the artist. cs: I’ve just inherited a whole set of threading tools all in imperial threads, 1/364th and so on, I inherited it from this engineer, they just can’t possibly work in metric. Fractions are actually easier in that context, when you go below one it’s actually appreciably easier to use fractions rather than decimals. Anyway, so The Celestial Meters are a series of nine stainless steel rods, at the end is a milled flat that has the name of one of the planets. The rods range from eighteen centimetres to nine and a half metres long. Earth is 98.7cm long, which is the correct metre.

mw: And they’re all a ten-millionth of a sector of that planet? cs: But they’re all units of one in their own right. So it comes back to this idea of there being no such thing as one. mw: The unit is arbitrary. cs: But they’re all their own absolutes in a sense. In the Science Museum this could very much be an intervention. Not presented as ‘this is an artwork’, but that you encounter something that appears to be authentic. And the real problem is where you draw that line: you draw people in, there is a sense of a suspension of disbelief, but you don’t actually want to con people into thinking this is a real historical artefact, so at some point you have to allow that to be a realisation, or at least to make sure that, if they bother to spend the time looking at it, they realise that it’s not. mw: In what sense is it inauthentic? cs: It’s not a real scientific object, it’s fake history. It was shown in the measurement gallery, and when you think of that story of the origin of the metre, the metre is such an everyday thing, you think it’s ubiquitous and has been around forever, whereas in fact its this precarious thing that doesn’t really mean anything, I find it quite a philosophically useful story to show how precarious and arbitrary the things of our everyday lives are. But it’s there as a philosophical springboard for people. So it’s useful to have it in the context of a museum, in a collection concerned with measurement. mw: That’s probably the best example yet: you have a creation that, without having to be too explicit, allows people to grasp the arbitrary nature of units of measurement, which is actually quite difficult to ‘get’, people think these things are just built into the world. They might think about it, but this makes it inarguable, in a way—so you’ve done it…but it’s not a highly abstract idea, it doesn’t require experts to grasp it. cs: But they’re also quite austere, these objects, so unless people are willing to hear the story, they’re like, what the hell is this, is it a

drain rod or something? mw: So people would wander into the room, and is there any information? cs: Yes there is, there was a press release and things and if they choose to, the viewer can engage with that. And that’s kind of the point of the gallery context. Because in that context people are quite lazy, and they don’t have the same responses. If you came across that rod in a local charity shop and it was all polished, beautifully made, and it just displayed quite evidently the amount of time and care that had gone into making it, and it had an inherent value just because of the technical skill involved in making it, but you had no explanation for it, then as an artist I’m excited by that, finding that sort of thing in that decontextualising environment. Because you then really have to put your archaeological cap on, really try and work out what that form is and what its function is. And as in archaeology, you have to be careful about your assumptions. // mw: Okay, the fundamental theory of arithmetic. I’m happy to give a quick pistachio tutorial…it’s pretty basic stuff that you learnt at school.

Often the numbers are presented as a rather homogeneous featureless sequence telescoping off into infinity, in which each number is defined entirely in terms of its place in the sequence, of being the one that goes after the one that comes before it, and before the one that comes after it. So you wouldn’t think there was much to discover there. And since we are conditioned to think of numbers in that way, as soon as we start counting, most people just see numbers as this very arid, featureless necessary tool whereby we count and measure things but where there’s very little to be discovered beyond that. Now, what’s interesting, as I was saying earlier, is that when you apply the number system to itself and you start counting numbers rather than sheep or biscuits—so if I’ve got this idea of three and I can ask how many threes it takes to make fifteen. I’m counting numbers rather than counting other things—

which isn’t really what this thing was designed for. But you can do it, and the ancient Greeks noticed this. So what we’ve got is multiplication, which is converse to the notion of division—basically, as soon as you’ve got the idea of multiplication you’ve got the idea of division. Now, if we forget the sequence and we’ve just got a pile of shells, the first thing we do is we count them, and we have sixteen. Now, if we wondered, what is it about this sixteen that makes it different from other numbers, one of the things we can do is to divide it into two piles of eight, or eight piles of two. But in order to be able to talk about that, we need to be able to talk about multiplication. Here we have two representations of the number eight, so we have two eights. As I was trying to explain before, that’s a very novel use of the word ‘two’: you’re counting an entirely abstract thing. If we now take one away, we can make five piles of three or three fives. Take another one away, we can make seven twos or two sevens. This making of equal piles is the first stage of factorisation. And now, if we remove one more, then we have thirteen. And the point is-­ —and it’s not about opinion or culture or how hard you try—there ‘s no way at all you can make equal piles out of that number. And that’s something that makes thirteen stand out from fourteen, fifteen, and sixteen.

Taking away another one, we have twelve and we can make two piles of six. If we remove another one, and we have eleven, again there’s nothing we can do with that. And again, with seven, I don’t care how argumentative someone is, they can’t make multiple equal piles. Of course, they can make seven piles of one if they really want to, but you can make piles of one out of any number. With five you can’t do anything, and then two, the same. So, 2, 3, 5, 7, 11, 13: without any reference to our system of numbers or base 10 or any kind of mathematical concepts, these seven numbers have marked themselves out as distinct in some way. Now, I’ve already written these numbers on some other shells, so that rather than these representing a single unit like they just were—I should really have used a different object—each represents one of those numbers that couldn’t be divided up equally. We have 2, 3, 4, 5, 7, 11, 13, and I’ve gone further, I’ve added 17, 19, 23, 29, 31. All

of these numbers, if you were to make piles of this size you couldn’t equally subdivide them. So the Ancient Greeks were the only culture that made a point of noticing this—at least, that we have any record of. No one else seems to have even considered the matter, which is bizarre, I would have assumed that every culture that had any mathematics at all would have noticed it, and you’ve got he Chinese, the Indians, the Mayans, the Egyptians, they all had sophisticated surveying maths, they had a reasonably high level that could allow them to do all sorts of things, but they didn’t bother looking at which numbers were divisible, or if they did they didn’t think it was worth noting down, but that would be quite strange, it seems. And I do wonder whether the only reason it seems a normal thing for us to look at is that it’s this strange Ancient Greek headspace that we’ve inherited, that thinks of taking something like a number system and applying it to itself. Anyway, what we’ve got on the shells here are the prime numbers. Now, the fundamental theorem of arithmetic, which is the most important thing we know about the number system, says that every number can be uniquely built by multiplying prime numbers together. And that there’s only one way of doing this. And this was proven by Gauss in the late 1700s, although it is suspected to have been more or less known by Euclid, but he never got around to proving it. So a number is either a prime number, or it can be got to by multiplying prime numbers together. So we can think of two, it’s already there. I’ll deal with one later. Three’s just three. Four, as we saw, can be built from two piles of two. So we add these two twos together, like a molecule, and that’s four. Five is already a single prime molecule, if you like; six is two times three, seven is already prime, eight is 2×2×2—there we’ve now got the first ‘cluster’ as I call them, the first counting number that needs more than a pair of prime numbers to make it up. There’s no way you can multiply anything else together to get eight, that’s the unique factorisation. Nine is 3×3, ten is 2×5, eleven is just on its own, twelve again requires more than two shells, we need 2×2×3, Thirteen is already a prime number, Fourteen is 2×7, Fifteen is 3×5. The point is the uniqueness of this, at this stage it might seem self-evident that they’re unique. But

obviously, the counting numbers continue forever. So if you think about that, there are going to be numbers that are so big that you can’t ever write them down, there are numbers that couldn’t be represented in the available space in the physical universe…in fact, that’s most of the numbers—all numbers beyond a certain point are too big to fit into the universe. cs: If you want to take something in the tens of thousands, say, is there a procedure for factorisation? mw: There are a few tricks you can do to speed the process up. Basically, to be very thorough about it, with each number you go up to the square r0ot and check all possible divisors. But there’s no formula you can put in to check whether it’s a prime number or not. rm: this is key to how cryptography works, isn’t it? mw: It’s based on the fact that there’s an asymmetry in the difficulty of two inverse process. Two big prime numbers, about a hundred digits each, can be multiplied together on a computer very quickly, but factorization takes an awful lot longer. If someone worked out a way of calculating prime number factorizations more or less instantaneously, then there’d be some major problems, everybody’s secrets would be revealed! Now, as I said, you could have a number that was millions of digits long, and even though you may never ever know what its factorisation is, we can be mathematically certain that there is a unique set of prime numbers that can be combined to create it. cs: I feel sorry for all the even numbers, they’re ruled out from the start! mw: Yes there’s an interesting point there, because people often come to the conclusion that there must be something very special about two because it’s the only even prime number. Whereas in fact, it’s just that two eliminates all of its multiples—but so does three, so does five. Now, we have this word ‘even’ meaning divisible by two— and if it’s divisible by two it can’t be prime. But if we were to introduce the word ‘threeven’ or something to designate all numbers

divisible by three, then likewise, three’s the only threeven number that’s a prime…and we could do that for numbers divisible by five or nineteen or whatever. rm: Do we only privilege evenness because we’re bilateral? mw: There must be some reason, and that may be to do with it. pc: What if we were starfish? rm: Then we might have fiven numbers. mw: Now if I get out my calculator and multiply all these prime numbers I have here together, it would yield a fairly large—by normal standards—counting number, but we could be sure that there is no other combination of primes that could be combined to generate that number. Now when I say unique, it’s not to do with the order you state the factors in. If I say 2×3×11x11, that’s the same as 2×11×3×11 or 3×2×11×11. Each one of these molecules made of prime atoms is, basically, a bowl of fruit: if I ask what’s in the bowl of fruit and you say two bananas, a pear and an apple, it’s the same bowl of fruit whichever order you list them in. Hence the image of these clusters is quite useful, because it’s not clear which order you would list them in. So we can think about the prime numbers as atoms—this isn’t something I’ve invented, a lot of people have talked about them as the ‘atoms of arithmetic’. And all the possible molecules you can create by combining them generates the entire set of counting numbers, and there’s only one recipe for each of these. That’s the fundamental theorem of arithmetic. What’s interesting about this is the seeming irregularity of the factorisations. cs: When did you learn about this at school? mw: I think we did it when I was about eight or nine, I don’t think they even teach you about the fundamental theorem any more though. cs: It never had this more epic grand connotation to it, it’s a shame.

mw: It’s taught in an incredibly boring way. Mathematics education has a lot to answer for. But then you have to think, why is everyone being taught it in the first place, where did that come from? If you have to teach every single young person in the country maths to a certain level, then you’re not going to have enough inspired maths teachers to do it. So, you have this one image of number where you’ve got this thing called one, there’s something that comes after it, another thing comes after it, and they’re just blank featureless things that just process with no apparent interest. But if you look at two as a factorisation, three as a factorisation, immediately you see there’s some irregularity going on: from two to nine you’ve got prime, prime, non-prime, prime, non-prime, prime, non-prime, non-prime. The primes stand out from the non-primes because they consist of a single shell. But the number of factors also varies—you’ve got one, one, two, one. Everyone thinks, what’s the pattern? But there isn’t one, at least not in the sense you think there is. You could continue on down that line for years—and this is all illustrated in the book—and each number, you factorise it, and it could produce a single shell or it could produce a hundred thousand shells. One number you might break open, see what’s inside it, and there’s just two prime numbers, another there’s fifteen, the next one’s a single prime, the next one could have more prime factors than you could fit pistachio shells in the universe. You’d have to imagine these pistachio clusters the size of planets, the size of galaxies, most numbers would look like that. The numbers that can be represented by clusters of pistachio shells that would fit inside the known universe are an infinitesimally tiny proportion of all the numbers there are. And yet, even though we can never know about more than the tiniest splinter of the beginnings of the number system, because we don’t have enough time and space to represent or know about them or do anything with them, all of those numbers that just go on and on forever—and this is the only kind of infinity we have any kind of practical experience of—we can be absolutely certain that there is a unique recipe for each one, each one will break down into prime numbers.

And the position of prime numbers, even though they look irregular, there is a statistical pattern there, which is to do with the spirals I showed you earlier, and beyond that, the fluctuations of the statistical pattern of the number of prime numbers encountered at any stage along that sequence can be estimated by counting how many times you cross the spiral. Imagine the spiral going through zero, it would go through 1, through e, which is about 2.7, e × e which is about 7.29, then e × e × e, which is a little over 20. Now if you walk along and count how many times you cross the spiral, you divide where you are by how many times you cross the spiral. That will give you a good estimation of how many prime numbers there are up to that point—sometimes there will be more, sometimes there will be less, but it’s a relatively good approximation. But it’s the more or less, the fluctuations—sometimes you’ll get great clusters of primes, sometimes you’ll get great arid deserts without any—its the fluctuation of the actual behaviour of the primes around the norm that is interesting. And that’s where the waves I showed you come in —the error, the deviation of the thing from its mean behaviour can be entirely captured with those waves. rm: Going back to the idea of this as a form of representing numbers for a moment, what you seem to suggest in the book is that this straightforward or simple or standard way we’re taught numbers as equal units has an affinity with all sorts of cultural phenomena which you associate with capitalism, with mass production, and you suggest that if we were to think of numbers in this ‘molecular’ way instead, using the primes as these atoms out of which numbers are composed, and it becomes this unpredictable landscape with interesting features, its not so monotonous at all—you seem to suggest this would shift the way in which we saw the world. Bringing this back to the question, I’m wondering whether an artistic dramatization could get across that kind of shift in a way of thinking. mw: I think in this case it could. rm: So what would the world look like if this was how we were taught to conceive of numbers at school?

mw: It wouldn’t look like it does now: if we’d have started from this point of view, we wouldn’t have ended up where we did. And if we were to promote this way of looking at things now, I don’t know how it would affect things at all. But our understanding of reality revolves so much around breaking things down into countable objects and then counting them, that if the number system itself were exposed to be something far more chaotic and unpredictable than we thought it was, that’s at some level going to resonate in the collective psyche. I don’t know—I’m interested to know, and that’s why I’m pushing the ideas out. The overall aim of what I’m doing is to show people that the number system isn’t what they thought it was. Not that it’s interesting and cool and that they should learn about it, but just that it’s not what they thought it was, and if anyone claims to know what it is, they’re deluded. We’re dealing with something completely mysterious and unknown. Because as soon as you start doing this, just dividing up shells into equal piles, if you follow that where it leads, you end up with what I was showing you earlier with the Riemann zeros and spiral waves and quantum chaotic oscillators— that’s where it will get you. No one knows what we’re dealing with, and yet we’ve built a world around it. Questioning that foundation on which we’ve built things could lead to some kind of shift in how we see everything—but it’s all speculation, I don’t know what the effect would be, I’m just prodding the world with my writings to see what it might be. As for ‘dramatizing’ this, I was at one time thinking of a sculptural installation, partly inspired by the artist Pip Youngman in Somerset, who made what’s called the Somerset Space Walk along the Bridgewater to Taunton canal. He talked about when you’re a kid and you get a book about the solar system and it shows you the relative size of the planets, and how it has to distort space to squash them all into one page, so you get the sense that Jupiter is a lot closer to you than it really is. It might then explain that they’ve had to rescale it, that these are not the real relative distances. But he found this unsatisfying, so he got a commission from the Somerset County Council to install, along several miles of canal, this huge golden sun at one end, and then you walk for a few miles and there’s a tiny little

pea-sized thing that’s Mercury, then a few more miles and its Venus, and then Earth, which is a little bigger—and then it’s miles and miles and eventually you get to Jupiter which is considerably bigger than Earth. But It’s all to scale, and with this physical experience of walking, you get some sense of the solar system you’d never get from an astronomy book. Pip Youngman, Somerset Space Walk (1997). Photograph by Pam Goodey. So my idea was to create a sculptural walk: to find some very boring piece of urban architecture, a new wall that was being built or something, that went on and on for miles, and to install within that sculptural pieces that would represent this sequence of factorisations, but in a nonexplicit way—it wouldn’t actually explain it all, there would just be coloured spheres joined together like models

of atoms and molecules, but they wouldn’t be labelled 1, 2, 3, 4, 5, they’d just be different colours. So it wouldn’t be obvious what you’re looking at, you’d be walking along and there’d be a yellow ball then a red ball, then two yellow balls, then a green then a red…and so anyone who had a curious mind would start thinking, What’s the pattern here? And they might work it out or they might not, or word might spread. It would be a way of physicalising it and making it real for people. I worked out how much it would cost and how difficult it is to get permission to do these things, and the idea went out of the window. But it connects to the trial and error process I’ve been working with in explaining mathematical ideas to nonmathematical people, which is basically what I’ve been doing for years, trying to get my artist-type friends to understand what I’m doing. You try it enough and you find that some ideas work better than others. And what I have been arriving at is this idea that you have to engage the parts of the brain that are tied in with the motor functions. So for example I describe adding numbers in terms of walking a number of steps: in the book, I describe the staircase function that counts the number of primes by actually building towers of blocks. Now, of course, you can take a bunch of kids out and actually do these things, but I think that, even if you read it, even if you just imagine you’re walking along this endless road, if you’re reading that and understanding what you’re reading, at some level that part of your brain is being engaged. In the same way that, if you’re reading a novel and it says ‘he clenched his fist’ you’ll do the same, often, or your breathing can speed up, or you can grimace—in order to empathise with what’s going on, some part of the brain becomes engaged that actually does the action. So, that’s been my experience, and as you’ll see this in the book there are a lot of visualisations of walking along a line, because in some way the person who’s read that and understood it has done it in some way: it goes into the brain as if you’d done it yourself physically. rm: You used the same word Andy used when he was talking about working with Conrad—he said that you developed an empathy with

the machines. With abstract concepts, it’s not obvious how to build that empathy, and that’s the kind of movement we’re looking for: how to build an empathy between perceptual habits, motor skills, spatial awarenesses that are intuitive and something far less concrete—so that someone can in some sense ‘experience’ them. mw: From my side, finding ways of pushing this out, whether you call it art or not, my experience has been that you want to engage the visual and the motor cortex. And actual walking or building, or at least the imagining of that, is something I’ve discovered through trial and error that seems to be useful in this area. rm: This also reminds me of Neuro-Linguistic Programming, which is essentially a theory about how people process reality through language. But since it has an idea that the body is an informationprocessing engine, some practitioners use exercises that involve understanding how you mentally structure your representation of a given situation, and then setting it up physically, actually laying it out in a room, on the floor as a grid or some kind fo shape, and making you literally walk through your mental map of a situation. You then inhabit the problem with your body, which makes it tangible and gives you different resources to address it. cs: Like those memory experts who have memory palaces they can walk through, and store knowledge in. mw: Yes, the ancient art of memory is all based on space. cs: One thing that came up yesterday, and which I was thinking would be a really good challenge to set ourselves, was that both in my practice particularly and in art in general and science to a certain extent also, you have a real problem with how you represent more than three dimensions, how do you get that point across. You have to employ some kind of metaphor or allegory, by definition it can’t be a straightforward representation. It’s a challenge for me to try and tackle this kind of thing where you need to put in more information than is available in three dimensions, to find some kind of window into that multidimensional space.

mw: It’s not really an issue with the concepts I’m working on, but in a more general sense of the dialogue, the question Conrad is raising is a valid one. If I were just a general representative of the scientific establishment I would approve of that, because there’s a huge popular misunderstanding of the whole notion of dimension: people tend to imagine something really mystical, teleportation or something…. But you can describe the dimensions of a piece of furniture, it’s nothing special. I’ve come up against that a lot when trying to explain things to people, they’ll say well, if time’s the fourth dimension then what’s the fifth dimension? And you’ll have to scrape away all these layers of misunderstanding and get back to something…they’re usually disappointed because it’s not as sci-fi as they would like. rm: Isn’t this to do with what happens with Riemannian manifolds, the idea that you take the idea of dimensionality and separate it from the spatial dimensions we live in, so dimensionality becomes something more abstract. There’s nothing special about threedimensional space. mw: No, and people struggle with that, if you look at the maths leading up to Riemannian manifolds—I mean, you don’t even need them, you could just talk about n-dimensional Euclidean space. cs: Is this the same Riemann? mw: It’s the same man, but there’s no connection—as yet. Riemann only wrote one paper on number theory in his life, the rest of the time he was working on a lot of different things. But in some ways his work on Riemannian geometry is how he is best known. But as Robin said, he isolates the idea of dimension—although it wasn’t just him, it was a process that was going on at the time—if you take the idea of 1, 2, 3 dimensions and can represent them symbolically, then there’s nothing to prevent you from adding more dimensions and representing them too. We can’t see what they might look like, but we can calculate their volumes or measure the angles between their sides, all of these things are entirely possible.

But for a lot of people, mathematics begins to look like something out-there and mystical when you start talking about higher dimensions. They just go all Star Trek, and it’s a bit disappointing to have to deflate people and say, no, it’s just a set of four variables. But it seems to function as a dividing line between people who can grasp abstract mathematics and people who probably aren’t going to be able to, this notion of dimension. It’s also the way the word has been used—people say ‘a whole new dimension’ of something, a metaphorical usage of a very precise word which everyone seems to feel quite comfortable with, but which leads to quite a lot of confusion when you take it back into the scientific realm. But Conrad’s way of dealing with the fourth dimension largely involves time, which is readily available to a sculptor—and also you have motion. But there is this misunderstanding when people think the fourth dimension is time. No, you can have all sorts of fourdimensional structures and one of them is the spacetime continuum. But you could equally well use colour, you could use temperature, you could use electromagnetic intensity, volume of sound, there are a lot of different parameters you could evoke as a fourth dimension. rm: Once you think of three-dimensional objects with qualities, you’re already thinking of higher dimensions. And that’s how the manifolds work, anything with parameters you can measure? mw: Not quite, there’s nothing to be measured in a Riemannian manifold. It’s a good point, actually, because you can grasp the abstract notion of a metric, and the fact you can change metrics. cs: What’s a metric? mw: It’s a mathematical rule whereby you can define meaningfully the notion of distance between two points in a space. A metric has to satisfy three different criteria. One is that taking any two points in a space gives you a non-negative number, a zero or a positive—you can’t have two points at a negative distance from one another. The difference between two points is greater than the difference of a point from itself, and d(x,x), the distance from x to x, has always got to be 0, that’s the first rule. The second rule is that d(x,y) is the same

as d(y,x), that’s the symmetry rule—it’s the same distance to walk to the shop as it is to come back. And then the third rule is triangle inequality: If I go from x to y, then I go from y to z, the total distance d(x,y) + d (y,z) is always greater than or equal to d(x,z). In other words, it’s never quicker to go indirectly. Any set of rules for assigning positive numbers to points in that way, which satisfies those criteria, is a metric. Now, the metric we use is the Euclidean metric, that’s the familiar one. If you’ve got two points on a ruler, 3.5cm and 4.2cm, you just subtract the smaller from the larger number and that’s the Euclidean distance in the onedimensional context. If you’ve got two points on a piece of graph paper, you can take your ruler and measure them but there’s a difference in x coordinates and a difference in y coordinates, but using the Pythagorean theorem you get a Euclidean distance between the points; and the same in three dimensions just by extending the Pythagorean theorem. And so we tend to think that simply is distance. We were only familiar with that one metric, until topologists came up with the idea that metrics are arbitrary. And the best example of a non-Euclidean metric that people can grasp is the hyperbolic metric used by Escher, in the circular etchings or woodcuts that he did, the circles with fish and birds, where you have large forms interlocking in the centre and small ones at the edge. They’re pretty much identical in shape but different in size, and there’s a curvature too. Even though its a disc, it’s a representation of a two-dimensional space that extends infinitely, called the hyperbolic plane. And the metric is defined in a very different way on a hyperbolic plane, such that those tiny little fish out on the edge of the disc are in fact the same size, they have the same area, the distance between their wingtips is the same as for the big ones in the centre. To look at it you’d think not. But that’s because you’re still stuck in the Euclidean metric. The metric in the hyperbolic plane takes account of how close you are to the edge: the closer you are, the more the metric starts to dilate. There was one particularly good bit of pop mathematics I read where someone described it as like living on a big discworld where it’s really cold out on the edge, and much warmer in the middle. And the only material

you have to make rulers of is a material that shrinks really rapidly with a drop in temperature. So as you get out to the edge, your ruler’s getting smaller and smaller. You’re measuring a fish and finding it’s one and a half units long, the same as the one in the middle. But your metric—your ruler—has changed, that’s the point: the ruler responds to where it is in space. And remember, so as long as the metric satisfies those three rules—and the hyperbolic metric does—it’s valid. There are an infinite range of metrics, that’s just a familiar one. cs: It’s basically like when I was talking about distorting the room by changing the scale of the unit, creating a sculpture that would become hyperbolically distorted toward one end. rm: Is it the same principle that describes how space is curved around a black hole? mw: Well, then you’re getting into curvature, which is different. They’re related, but in differential geometry you have very specific definitions of curvature. Given a two, three or four-dimensional manifold there are several different types of curvature you can define. There are torsions and other measures. Basically, relativity, for the first time, puts time more or less on the same footing as the other three spatial dimensions—it’s just a dimension that you can measure, and you’re going to weld the time axis onto the three spatial axes. But they aren’t quite the same, because there’s what’s called a signature. Whenever you have multidimensional space, you have these axes, or if you want to look at it in that way, four coordinates, and you can assign a plus or minus value to each one. So given a four-dimensional space you can have a signature of +−−−, ++−−, etc., And that determines the metric. Basically, in a totally Euclidean four-dimensional space you’d have a set of all points x, y, z, t, and to get the distance between two points you’d take the difference in the w coordinates, square it, the difference in the x coordinates, square that, etc., add them all together and take the square root. Now, depending on the signature, you don’t add them all together like that: you can add or subtract

them. So the signature used in special relativity is a +++− signature, with the minus corresponding to the time dimension—the time dimension has a different signature to the other, spatial dimensions. So when you measure the distance between two spacetime points, it’s √x12+y12+z12−t12. So we’re not looking at Euclidean fourdimensional space. You have Euclidean geometry, which is the familiar flat one, then you get the curved geometries, the elliptic or hyperbolic geometries —and that’s what we appear to be in, in this spacetime reality. Time and space aren’t interchangeable in relativity, because time has a negative signature. But they’re more interchangeable than they were in Newtonian thinking, because suddenly time has to be introduced as another axis in this four-dimensional space. And you can get rotations of this four-dimensional spacetime in certain contexts, you get strange effects where things you’d usually associate with extension in the spatial dimensions suddenly turn into temporal displacements. It’s over my head but, because you’re playing with a geometric structure, Hence all of the problems of interpretation in relativity theory. Then in general relativity, where Einstein integrates gravity and mass, that warps the spacetime continuum so that the metric is actually changing from point to point, so depending on the presence of mass you end up with an elaborately curved fourdimensional manifold. And that’s actually defined as the curvature, which can be precisely quantified in terms of this metric. rm: So according to the metaphor you mentioned before, wherever there’s matter there would be cold spots where your measuring instruments would change. cs: And can time exist without space, are they mutually dependent? mw: I’m not really qualified to talk about that. Unfortunately, people struggle to find a language to talk about these things. cs: It’s not unfortunate, it’s interesting. mw: The unfortunate thing is that the languages we’ve got are inadequate because we’ve evolved our language, our linguistic

categories, whatever you call them, to deal with earthbound existence. cs: For me that’s where it becomes really interesting, because that’s where abstract ways of working can be valid. mw: But does it concern you that you might end up being completely misguided—you might end up misunderstanding something as a result of those inadequacies of language? cs: I’m trying to understand things to the best of my abilities, but I know there’s a glass ceiling because I’m not trained in this. rm: It’s possible for a scientist to run the equations on a computer and to set out a hypothesis but even they struggle to give an interpretation in natural language. But for me this is a driving question, as in the Copernican Imperative volume: What is the meaning of ‘understanding’ in that case, if you can’t bring it back into experience or language, once something becomes so abstruse that it can’t become integrated into human culture? Relativity is a good example of this, in that relativity itself, the actual theory, hasn’t been absorbed into human culture whatsoever. What have become a part of popular culture are strange tangents from it and bad takes. And maybe that’s an interesting process to look at, the failure in absorbing scientific ideas back into culture. mw: There’s a desire to absorb it. People want to ask me about this stuff, and obviously you have a whole SF industry which is usually confusing people because it’s not generally driven by a coherent understanding. So, everything from that, to Stephen Hawking’s attempts to write a book about what he’s doing, have generally led to the public getting more confused. And I don’t know if that can ever be improved upon, without changing the education system so everyone’s learning tensor mechanics when they’re thirteen. cs: I’m optimistic that these things filter through. I mean, there was a point when it was completely inconceivable for the general public to be able to understand that the world is spherical and that we aren’t

the centre with the sun rotating around us. It was just impossible to grasp as an idea, intuitively difficult to handle. rm: But again, I question whether we really have absorbed even that into common consciousness. But how do you measure this absorption? If you ask someone, is the earth spherical or not, they’re going to say yes, because at some point they have been told this. But it’s a different thing to ask whether I experience myself as living on a spherical body (and what that would mean, whether it’s even meaningful at all, whether it would have any impact). Holding a commonly-sanctioned belief is different from it penetrating into the way you experience the world—just as we were talking about how the world would be different if we started with prime atoms rather than picket fences, and just as Matthew described the Solar System walk and the way it taps into the body, into the motor system. In many ways we do still experience the world in the same way people did in the Middle Ages, I think. mw: Unless you’re an astronaut or an airline pilot, and your daily life has some sense of that, then yes, you haven’t really internalised it. It almost has to be internalised in a bodily way. If you’re flying around the planet all the time, then your body is doing that, and it ‘gets’ it. Whereas if you’re just reading about geography, it gets it in a different way. If you sit watching the night sky for long enough, and hold your concentration long enough to just see a little bit of motion, suddenly you get this feeling of this enormous rotating sphere…you kind of grab hold of it, thinking you’re going to fall off. And it’s a completely different experience, it happens at a different level. Although I hadn’t made the connection, I suppose that’s what I’m trying to exploit. cs: There is a popular idea that mathematicians, the Einsteins or the Stephen Hawkings of the world, have a sort of epiphany, where it’s not just a crunching of numbers, but at some point there is this Matrix-like experience where they suddenly see things in this very different way—an epiphany through mathematical education.

mw: But that may well be because they’ve got to a point where they’ve internalised mathematics and the relationship of mathematics to the world. So when they see an equation, they can sense the implications of where it’s going, and the world feels and looks different. Whereas if you just show someone the theory… rm: I’m not even sure whether it’s true that a scientist who’s working on these things necessarily achieves such internalisation. That’s why I think the role of the artist, and the process of dramatization, can be interesting—because you can work on these processes. I remember when we interviewed Roberto Trotta, the cosmologist working on dark matter, I’m not sure whether we put it in the printed interview but there’s a point where we asked, all these amazing thoughts you’re communicating to us, how do they affect the way you see the world. And he just says, ‘you can’t live physics’—you just carry on your life like a normal person. And that’s part of the question of dramatization for me, how can we live these things, how can we induce that moment where it feels like you’re about to fall off the earth. If these ideas are true, if they’re about reality, then how do you let them enter into the way you live in the world—or, as per the ‘Copernican Imperative’, is that a hope we have to jettison entirely? mw: There are definitely a lot of theoretical physicists who are doing something like playing a very elaborate game of chess, and getting very good at it, and then going home to live in an entirely different reality, with the partitioning of theoretical work as just one little part of your world, and then there’s your family life, and so on. That to me is unsatisfying, I feel like it all ought to be integrated, but maybe, as Trotta says, you just can’t. cs: Where do you take off your scientists or mathematicians hat and put on your own hat, where does intuition come in for you, or your personal belief? From all the data you’ve gathered over the years, do you arrive at your own theory of everything? mw: My own theory of everything? cs: Yeah.

mw: No, I’m not a physicist, I’ve picked up bits of physics, that’s about all. cs: You are, in the sense that you’re looking at the structure of the world and how it works. mw: As John Michell says, ‘my own chosen attitude is total confusion’—and that’s fine. It’s important to say what you don’t know. I don’t have a theory of everything. I’ve got a table full of bits of jigsaw puzzle, but I don’t think there’s enough there to make a picture. And people keep coming up with new bits. rm: It’s important also to add here that this personal predicament, which I really recognise—about wanting to integrate and not wanting to simply say, okay I’ve left my thinking back at the office and I’m going to go back to my life—is not simply personal, it’s an index of something that’s socially significant. Namely the fact that the vast pool of knowledge that has been built up over the modern era simply hasn’t been integrated into the social sphere. It has had massive effects on the social and is embedded within the social in the form of technology… mw: …and entertainment… rm: …and modes of technological governance. So it’s inflicted on society rather than integrated. mw: There’s a contradiction, just at the level of the mathematics that underlies the technology and the statistical analysis that’s used in governance. We were sitting in the pub last night and people were saying oh, you’re a mathematician, and talking about their childhood experiences—I hate maths, we had to learn logarithms…there’s all this negative emotional feeling around it, which seems unhealthy, there’s a total lack of integration. If you’re going to build your social infrastructure around the obsessive utilisation of numbers then surely people should feel comfortable around them.

rm: Hence the problem of ‘grasping’ or internalisation, and the question of whether art can play a role in that process cs: Also in terms of those epistemological structures that support everything, the idea that those things are built in, they’re actually precarious, all these structures that we use in our lives. Whether that’s scientists or its on the more general level of an average person, there’s a gulf. mw: There ‘s a huge gulf between the view of the average person in the world and the complexity of the device in their pocket, which is built on an understanding they have no grasp of. So if we’re going to address that, that seems very fruitful, because it does seem to be that there’s a real need to reintegrate. Lots of people have talked about it and theorised it, but to me it’s pretty apparent that this gulf is opening up between people’s understanding of the world and how the world is run, and that’s leaving people feeling fairly lost and alienated. cs: There’s a gulf, but this is progress on the basis of things that someone knows, rather than no one knowing. mw: Maybe they don’t ‘know’, they just have models. rm: I’m not subscribing to a denigration of ‘Western society and technology’. But there are things that I’ve found in common between Matthew’s work and Alain Badiou’s work in contemporary philosophy, particularly the book that I translated, Number and Numbers.4 Badiou opens it by putting the problem in this way: the primary problem isn’t that our societies are governed by numbers, but that even as we complain about it, we don’t even know what numbers are. And in a sense that’s the same thing you’re saying, Matthew, even if your solution is very different. For Badiou, mathematics is ontology, it is the best way we have of describing what is. His approach is to say that only when we understand what number is, and that it is essentially monotonous and uninteresting, like Being, can we begin to understand what can’t be captured by number—the Event. But although Matthew thinks, on the contrary, that there is something

interesting in number that we haven’t grasped yet, the suggestion is the same: that unless people understand what number is, and by extension have a grasp of the mathematical realm, they will be in thrall to it. mw: So if we’re going to look into how you could bridge that gulf and you could integrate the cutting edge of what we understand about the way the world is into a common culture, then what’s the role of the artist? You have the theorist or physicist who’s working out their quantum theory equations, then going home, forgetting about it, and eating their dinner. And then you’ve got the person who’s working in the supermarket, maybe watching a popular science documentary but not really getting it. And they’re not communicating. Can the artist act as an intermediary. And if the artist is going to do that then doesn’t the artist also have to understand the subject matter? And therefore they’d also have to be a scientist. And time constraints don”t allow that…. cs: For me, I don’t think the point is necessarily trying to get exact forms across. They are abstract, there’s never going to be a way of understanding something that’s just way beyond our experience, or our scale.

Conrad Shawcross, Continuum (2005). Image courtesy of the artist. rm: That’s defeatist isn’t it? Why shouldn’t you be able to bring into experience something that’s outside? cs: I’m just dubious about the idea that we’ll be able to, I don’t know, experience the big bang or…. Let me show you this piece, Continuum. This came after building the rope machine, which was this quite metaphorical, allegorical piece. I made Continuum in the National Maritime Museum. It’s on the Greenwich meridian so it’s dealing with the history of timekeeping. And having looked at the structure of rope which is a cyclical structure with a strong linear element to it, a way of perceiving time as both a line and a cycle, I wanted to make a model that was more mathematical and less allegorical. So this is just essentially twelve coils made of one connected piece. It’s made from birch ply, from thousands of pieces of wood bolted together. As it moves you get the sound, a real sense of the bolts shifting, creaking, the whole thing shifting, and yet it doesn’t go anywhere. It’s essentially a big torus.

mw: Why Continuum? cs: It was a reference to time. But it was a connected piece, it has this sense of having no beginning or end, but it has these twelve cycles that could be like a twelve-hour clock or the twelve months of the year, so it had a reference to the psychological conditioning of the human perception of time. I wasn’t trying to say this was a model of time, but all of these things were very much in my mind, the idea of looking at and perceiving time, the year, the idea of what it was to rotate around the sun. So it hopefully conveys some of the problems associated with the perception of time. But it’s not a model, it’s not scientific but it has this guise of being quite a rational object, something that wouldn’t be out of place in the Science Museum. mw: You’re absorbing scientific ideas and then you create this as a response to your internalisation of parts of those ideas. And then other people come into the Museum and have a psychological response to what they’re looking at, and possibly there’s some literature they might read. But everyone’s going to get a different response. There’s a possible distortion of the signal. Your aim isn’t what I was just talking about, being an intermediary—that would be a simplification of what you’re trying to do. cs: I think the scale of this is really important, it’s actually about twelve feet high, so it’s a very physical experience. mw: Yeah it’s definitely going to leave some sort of impact. There’s a sense that something is continually feeding through itself, it’s quite similar to the rotating hypercube in that sense. I’m starting to get this idea of a new kind of discipline that would involve artists and scientists, maybe that’s what we’re doing here. But in order to facilitate that integration, you need to broaden the idea of art into something bigger. For some reason I was thinking about this massive craze for fractals in the early nineties, at a time when computer graphics had got to that stage, and there were a lot of different elements going on: rave, people on drugs wanting to look at far-out images. But basically lots of people saw fractals, not having seen them before.

They didn’t know what they were, the mathematics behind them weren’t explained. And there they are, looking at something that resonates with images of nature, but in a way they haven’t seen before. And it possibly allows them to integrate something of the higher mathematics that is infused in that world, without understanding it theoretically. It wasn’t an artist that generated that, it was just a programmer. But that would be an example of how that integration might be furthered. rm: That ability to be tripped out by fractals is a product of intensive research by the military-industrial complex over the entire twentieth century, and there is a weird social irony in consuming this object in that way. mw: You could imagine another history where people would have arrived at those images by some other route. But it interests me because sometimes, when those images were circulating, I would sometimes just send them to people who had no idea what I was working on for my PhD, and say ‘this is what it looks like’— something for them to see, without having to study the equations. And to me that’s the best I can do: it looks a bit like that. I read at least one professional mathematician having a rant about the oversaturation, saying that, you know, there’s nothing particularly special about these, there are a lot more beautiful mathematical objects than this, and it’s just because there are two-dimensional colour images that people are getting so excited about them. And what I felt was that this involved a psychological insecurity in mathematicians, because they’re not all necessarily particularly mature people psychologically, they take comfort in knowing something that other people don’t and having this theoretical knowledge. When some of that gets blown out into a vivid image and people go wow!, it can feel like its an incursion into your territory. I think that may have been part of the motive, even though some of what this person was saying might have been true. But you are pushing out something that leaves people with a sense of awe and beauty. And when a brain sees a fractal, it may actually

cause some kind of neural response whereby that mathematical structure is registered at some level. cs: Our brain has such a need to find patterns, But that’s exactly what I was doing with the musical thought in Loop System Quintet. I was just looking at music theory, and how reassuringly the things we find beautiful in music are these absolute relations, and dissonances are quite displeasing. Your brain is processing that information and somehow enjoys it and finds the order in there. To represent that in another way is very interesting, the visualisation of that information when it’s processed by your eyes rather than your ears. Have there been any studies of the harmonics of prime numbers? Because obviously you’ve got 1:2, and 2:3, but then 5:3 and 7:3 are dissonant intervals. mw: You’re touching on something really interesting there. It’s not so much that you get good results if you use prime numbers, more that the prime numbers, or primeness—in fact what’s called relative primeness—plays a very significant role in music theory. With the torus knots that you’re generating, what you may have noticed is that the pairs of numbers you’re using have no common factors. There’s no point making a 6:4 torus not because it’s going to be the same as a 3:2 torus knot. On torus knots generally, in my undergraduate dissertation I was studying these things called lens spaces, which are two tori glued together in a kind of impossible way, so I had to look into knot theory back then. And basically you end up with these spaces which are denoted L(p,q) where p and q are whole numbers, like 3:2. And the definition at the beginning is that a lens space is L(p,q) where p and q are relatively prime. The best way to understand ‘relatively prime’ would be this: the number 430 and the number 14 are not relatively prime because they share a factor, they both have 2 as a factor, whereas 430 and 77 are relatively prime. You can break down any two numbers into factors and then ask whether there is an overlap. If there is, if you’ve got 7 here and 7 there, if you made a torus knot out of those numbers, it would be the same as another torus knot you could get.

I haven’t gone that deeply into music theory, but prime numbers have come up throughout that, and it’s because of this same idea of relative primeness. I’ve never checked it out, but apparently there is a way of quantifying how pleasing or dissonant an interval would be —why does a 3:2 sound more pleasing and a 7:3 slightly less pleasing? cs: I know that the ear offers quite an exciting possibility of a more objective way of processing information than the eye. There’s much more subjectivity to visual taste and aesthetics, but with the ear there are very clear responses. mw: The ear can identify information in a way we don’t really understand. There’s some fascinating stuff by metrologists about the possibility that the brain is harnessing some sort of quantumtheoretical structures, you know, looking at the possible origins of consciousness, and the suggestion there is that the reason why humans all over the planet can more or less agree on pleasing musical intervals—although there are subtle distinctions—is a byproduct of the fact that our brains have evolved in order to do this thing with phase-locking, which is a factor in all perception. But if you quiz people—Does that 17:3 sound more pleasing that than 11:4?—you might get some differences of opinion, but presumably if you surveyed enough people you’d start to find there are trends. And there is some suggestion that those can be understood in terms of prime factorisation. rm: You do also talk in the book about the idea that you could listen to the Riemann function, but it sounds pretty bad. mw: Yeah, it sounds like a low rumbling white noise. Musical metaphors have been proliferating since 1868 when Riemann discovered that, basically, taking the seemingly random sequence of prime numbers could be, you could peel away these layers of approximate structure, and you end up with a very precise error function, and then that could be subjected to Fourier decomposition and expressed as a clean and precise overlay of waves. Those waves have frequencies, this is a harmonic analysis, these waves

would be called harmonics in that sense. But they’re not sine waves, they’re these spiral waves. Matt Tweed, the illustrator of Secrets of Creation, came up with the idea of this crescendoing trombone—if you’d got a bowl or a bell you’d get this kind of sound. So those are the harmonics of the prime numbers in a sense, but that wasn’t what you were asking about. rm: You could encapsulate a great deal by making it into a question of music, saying, the primes are ‘playing’ this sound, why doesn’t it sound harmonious to us? There’s some kind of twisting or refraction that goes on in the number system. The harmony is there, but we have to somehow unpick, work through, all the convolutions it’s embedded within and work out where the order is in this apparent unpredictable lack of pattern and harmony. mw: Why doesn’t it sound nice to us? rm: Why doesn’t it sound pleasing—if it has an intrinsic numerical beauty and elegance, as you’d expect, then you’d think it would sound good, right? //

Matt Tweed, spiral wave trombone, illustration from Secrets of Creation vol. 1. cs: I’m trying to justify why I’m making three-dimensional objects, and you’re saying, all we need is a two-dimensional graph to do this. But when I’m envisioning a musical chord, I don’t work in two dimensions in my head, maybe you do because you’re used to working in a two-dimensional plane. But in synaesthetic musical experiences, I will envision something as a three-dimensional. It’s a fleeting thing, I can’t see it clearly, it’s a something I’m trying to envision, but it s a three-dimensional object. mw: But relating that back to the three-dimensionality of your torus knot, the problem then becomes, if a guitarist plucks two strings, your torus knot has no problem in representing it in three dimensions. If the guitarist plucks three strings then you need another dimension, you need to make a hypertorus in four dimensions…. cs: Yes, I get that it’s a very subjective decision. mw: The torus knot’s a really beautiful way to represent a ratio. cs: But I make very arbitrary decisions about how it’s made. And it was a surprise to see how complex it was. I’d made one that was just a 2:1, it had a 2:1 gear on it. But it ended up with quite a pilgrimage because I couldn’t afford to get these gears cut, I had to go to China myself to get them cut. I went down a road in Shanghai and there were just gear shops everywhere, and I got them cut—I had a translator otherwise it would have been impossible—brought them back in my suitcase. I had 140 gears cut, brought them back, and built the machine, and then these mathematical visualisations came from them. mw: That wasn’t the intent? cs: Well, I didn’t know what the intent was, but I was taking something that was very intrinsic, and it was something that was a kind of keystone of music, so I knew that something was going to

come out of it. But they were all driven by one motor, and I hadn’t really thought about the common denominators, how they would all come back in line so that everything was new again and in phase, and then go off and disperse along their own paths and then come back. That was really rewarding, and it felt quite scientific in a way, although it just came from hands-on experimentation with things. // rm: You could say it’s a problem of natural language as opposed to refined symbolic language. In one sense it’s ridiculous to think that one could bring the refined symbolic language of higher mathematics back down into natural language without exposing it the whole climate of anthropological and cultural baggage, association, metaphor…it’s an absurd idea. Especially if you broaden ‘natural language’ to include gesture and the motor habits we were talking about. mw: It loses its fidelity or resolution or something. Otherwise there would never have been a need for the symbolic language in the first place. It’s not like people are just trying to be purposely complex and incomprehensible. rm: And yet I don’t think you can envision these symbolic languages as being completely severed from natural language, because it’s come from the same source. mw: But you’d have to backtrack historically and reconstruct it. If you take a mathematical statement which has been compressed into a line of symbols and you unpack it, you can turn it into an essay that quite precisely says the same thing in natural language, so it’s all about compression. And when you decompress it, it fills so much space you can’t deal with it. Say a college want to commission a book on the Riemann hypothesis because there’s some entrepreneur who’s offered a million-dollar prize for it. They want to know what it is because they think they might have the chance of winning all this money. So they commission a mathematician to write a paperback, and it leaves you

with this feeling that you’ve understood something that you haven’t at all, because he’s had to do that decompression. The same information but with a lesser signal to noise ratio—you’re translating something in such a way that, you know, that book would have to be extremely long in order to be accurate. So something has to be lost along the way. There have to be compromises made. Either you need to learn the symbolic language, or you need a very long attention span, or you need a book you can sell. Those constraints mean that people always get an oversimplified version of things. This idea of almost conning people into thinking they’ve understood something, it’s not helpful at all— people watch a documentary or read a book and think they’ve understood string theory. And then they tell people about it who then think they maybe have understood something. rm: Again, is ‘to understand’ to be able to manipulate symbols in the appropriate way, is it to achieve an integration into your experience of the world, or is it to have a historical sense of where these ideas came from and how and why they developed? mw: The first one is the one that’s generally used, symbolic manipulation. The second is the one that I don’t think that can be achieved. Is there some sort of poetic bridge that could be built that would somehow combine some element of informative historical context with this sort of sublinguistic or translinguistic visceral bodily motor-function-based experience where you’ve got some sense of what you’re dealing with and you integrate it into something? It shouldn’t be ruled out as a possibility, but it involves a new form of creativity, really. rm: Does art also perform this work of distillation, concentration, or compression? If you’re talking about conceptual and postconceptual art, you are often dealing with objects that in one experience telescope a whole history of art up to that point and add a new gesture to it. mw: But the difference is in decoding it, it can be unpacked in a very objective way…there’s a clear framework whereas if you look at a

Duchamp piece which is a culmination of everything that went before, you couldn’t deduce that....it’s fuzzier. That’s where the chasm between these things is. rm: There are certainly some cases where you could say artists are making very specific propositions, even if there is no calculus. Of course, the viewer’s experience of that work obviously brings in many other things, it takes place in that context of natural language. mw: An artwork has a history to it and also associations. Whereas if you’re looking at a mathematical question, it might be that the symbols have associations, you’d have a certain feeling about an equation—but the whole point is that you bracket that out, that doesn’t have any bearing on it. Suppose some sort of new brain scanning technology were to emerge whereby you could teach someone a concept and you could take scans before and after and actually see what happens when that concept is grasped, it’s held, that person’s successfully understood that, this person hasn’t. We’re nowhere near that, and it may never come to that. But in that case there may be a role for a kind of artist figure who finds ways to just switch those connections on, bypassing all the technical stuff. So during the seemingly nonrational experience of looking at painting or sculpture or hearing sound art or whatever, something will have happened and they will have the same neural configuration as someone who’s studied a certain set of concepts—that may be possible but I don’t know what that would be like. cs: But the last thing I want to do is for everyone to come out of that room with the same brain configuration! The conversation that can be had afterwards about the collective experience, but it’s doing different things in your brain. It’s not putting all of the dots in one place. mw: No, you can’t measure the success of one of your pieces based on the effect afterwards. cs: And also, the more brazen and clear the idea is in the work, the less it’s art, it’s just a vehicle, a model, rather than having this

possibility and poetry and this space where interpretation enters, and that’s really important. The Science Museum is filled with all these archaic methods of doing things that have been superseded by more efficient ways. The difference engine is an incredible object, loaded with all this pathos and history of Babbage’s life, the face that he didn’t manage to make it in his lifetime. The speaking clock is another incredible complex machine, all of these discs, cranks, and pulleys, and all there is at the other end is a voice telling you the time…. It’s doing the same thing that your watch or phone does now, but this whole sense of how complex time management is gets lost in the effortlessness of those devices. These older models really put you back into the complexity… rm: Rather than a black box. cs: Yeah, they show you just how complex it really is, and their failure, or their redundancy or their overcomplexity, their prototypical nature, is a really good way of confronting someone with these problems that are overridden by the fluency of newer technologies. rm: So you’re trying to expose that same kind of thing in your works? This seems very related to what Matthew was saying: a set of mathematical symbols is also like a black box in which is contained an entire history which, once you open it up and start unpicking it, unwinds into a huge epic saga. That’s all boxed up in efficiently coded symbols. And quite rightly, for scientists to be able to be fluent and to manipulate them. But there’s another side to it, there are all these workings inside which I guess as a mathematician you don’t even need to know. Mathematics textbooks just go straight into the symbols, they won’t introduce concepts at all because they don’t really need to, they certainly don’t need to introduce the history of concepts—you don’t find the historical story of differentials, for instance, in a calculus textbook. mw: No, although that might be the best way to teach them.

rm: You go straight into manipulating the symbols without ever opening up that black box and seeing how we got from natural language, gesture, manipulating physical object, to… mw: No, that’s not the mathematician’s concern, and there’s not really anyone whose concern it is, it’s a sort of in-between world. There are historians of mathematics, which is related but not really the same thing. It’s similar in computer programming, you can remember writing things in machine code, and then you got these assembly languages and higher-level languages. And now you look at what people are doing when they’re coding, and they’re just playing with dialogue boxes. Some people who are coming into it now just assume that that is the level at which things work. Whereas we know that there are all these other layers. rm: You could just take the view that’s just how it’s done now and, as with mathematicians, that compression means we can get more done, and do new things. But while perhaps not useful as such, to understand what goes on inside the black box is important. mw: It can be, but not for everyone. Some people don’t need to know, they can build wonderful applications without needing to. We’re naturally the kind of people who want to know about it, and want to ask where it comes from. But it’s about compressing more and more code into smaller and smaller units so you can achieve more with less effort. rm: This same personal predicament of wanting to integrate, and the social problem of the fact that knowledge is commodified, packaged up into units which there is no need to inquire into, you can just link them together and use them—it’s analogous, at least, to commodification. There is participation in complex processes and machines without any insight into what’s going on beyond the surface level. mw: Like the iPhone and the billions of things you can do with it, but how many of those millions of people know anything about what’s actually going on inside it?

rm: You’d have to be totally schizophrenic to be thinking about that all the time…. mw: You couldn’t, there’d be too much to take on. rm: When you see Conrad’s work, the dimensions of the works and the way they’re made, that does elicit a sense of wonder and somehow involves taking technology and science out of their box and laying them bare into these strange machines. cs: I don’t want to be associated with the ‘mad’ thing—if you go to the Science Museum, there are so many objects you could describe as ’mad’ because they now seem a redundant crazy way of doing things, but they were quite rational at the time. // rm: You have the Riemann zeta function which approximately describes the prime distribution, but you’re interested in the deviation or the error, and one way of understanding it is by decomposing it into a series of waves. So then you have this concept of the ‘music of the primes’. But why do you then move onto spiral waves, what’s the significance of the spiral wave? mw: Those are the waves into which that irregularity decomposes. They decrease in frequency and grow in amplitude. This is not how the ideas emerged, I’m not interested really in telling that story, others have done that, I’m interested in getting a deeper understanding. In Riemann’s explicit formula for the distribution of the primes, he was he was able to express the prime counting function, the ‘staircase’ that counts primes, in terms of a path integral in the complex plane. This infinite path, which you have to study to the limit, encompasses all the zeros of the Riemann zeta function. So anyway, the result of all this mathematical manipulation was that this function was expressible in terms of this rather nasty looking formula which has got an infinite sum in it—the sum over the zeta zeros, in other words for each zeta zero there’s another term that gets added on. And each of those terms that get added on for each

zeta zero, or technically speaking each pair of zeros, is what a mathematician would call a logarithmically rescaled sine wave, or sine function. So you’ll have something like the sine of k log x. And what’s going on there is that, rather than just the sin of kx, which would be a sine wave, where k would regulate the frequency. You have a fixed frequency, a wave that crosses the axis at regular intervals, and the frequency is just measuring how often it does so. That’s a sine wave. With a logarithmically rescaled since wave it would be the same kind of thing, but it gets stretched. And the logarithm is doing that, is basically squashing it at one end and stretching it at the other, in the same way a logarithmic spiral unwinds as it moves outward. Now, the functions being added on in the sum are actually in a form that leads to amplitude growth at the same time as logarithmic stretching. These functions haven’t even got a name, but there are a whole family of these. There are two things you play with—alpha, which controls how rapidly the amplitude growth opens up, and k, which controls how fast the wavelength stretches. Here, the alpha and the k are controlled by the position of the zeta zero. This is a slight simplification, but basically the further up you go in the plane, the more stretching there is, and it’s the left-right position, the horizontal location, which controls the amplitude growth. Now, the point is, these show up in the explicit formula, and number theorists would just work with these just as functions. I was interested to know what they looked like, so I graphed them and they looked like this, quite beautiful. And I don’t know why it occurred to me to look at it, but it just struck me that these things seem to relate to spirals in a way that’s analogous to the way that a sine wave relates to a circle. So the classic way to generate a sine wave is to just go round and round a circle, then add your horizontal and vertical position, so if I plot my vertical position as I’m going around a circle against time, then I get a sine wave; if I speed up the rotation I get a higher frequency, if I use a bigger circle I get a bigger amplitude. So, I don’t know why I looked at it but it just seemed to me that this had something to do with spirals. And I messed around with the equations for an algorithmic spiral and this is what I found—I

don’t think anyone’s bothered to do this before, and it wouldn’t help an analytic number theorist to know this, it would only help a nonspecialist. rm: You’re dramatizing! mw: Yes! Imagine you were travel around a logarithmic spiral at a fixed number of coils per hour—so you are accelerating, because each coil gets bigger. But basically, if you had an arm that was going around at a fixed rate and telescoping out with the spiral, then rather than just going down between two fixed points, your vertical position is expanding out. And if you plot that against the exponential timescale you’ve got something shooting off to infinity at an exponential rate. So if you plot this thing going round at a fixed number of coils per hour, you get these, I’ve just called them spiral waves because no one’s called them anything. The explicit formula says that the staircases that show you where all the prime numbers are can be expressed as a given formula plus this infinite sum over all the zeta zeros of these spiral waves. So we’ve got this staircase graph that shows you how, as you proceed along the number line, how many new primes you find at any given step, and it’s irregular. But if you zoom out from it you can see very clearly that it’s suggesting that prime numbers are distributing around this average behaviour. Effectively, the difference between the actual behaviour of the prime numbers and their average behaviour—the deviation—looks like a messy noisy signal, but the point is that it grows more accurate. Basically you’ve got a staircase which looks like a diagonal slope, you zoom in and see that in some places the staircase is above that line, in some places it’s below the line. In some cases, if you go far enough out, it can be quite a long way below. But the further you go out the more the percentage error is dwindling. Now, wherever it’s below the line, the difference gives you a negative number, where it’s above the line the difference gives you a positive number. So this deviation function describes the areas where there are more primes than there ought to be, where the prime count is in excess of its norm.

If the Riemann hypothesis is true this thing stays within certain boundaries, if it’s false it might stray and you’ll get huge fluctuations in the distribution of primes. But when you look at this deviation signal—I use the word ‘signal’, it looks like a signal—when you look at it, because its growing, you wouldn’t expect it to submit to a Fourier decomposition, because a Fourier decomposition ends up with sine waves of fixed amplitudes. The fact that its growing ties in with the fact that, when you break it down into waves, the waves themselves are growing. And from the explicit formula you can deduce that the difference between the actual prime count and its deviation, in other words this thing that’s measuring how far above or below the diagonal staircase the actual count is, this signal, can be expressed as an infinite sum of these spiral waves. Distribution of prime numbers expressed as a step function. π(x) is the number of primes less than or equal to x. Riemann deduced that, and this is so extraordinary that it led Enrico Bombieri, the Princeton mathematician, to say that the fact the distribution of prime numbers can be so accurately represented in a harmonic analysis like ‘an arcane music and a secret harmony’. But it’s entirely deducible from Riemann’s systematic investigation into how you could use complex analysis to capture the behaviour of the prime count function.

There’s a beautiful animation of how when the waves are added it approximates the prime count more and more closely.5 You can see how the yellow line changes as the spiral waves are added. What you’re seeing in yellow is a curve, but if you add infinitely many spiral waves, one for each pair of zeta zeros, you get exactly the blue staircase. Even having added just 250, you can already see where the verticals are, so you can tell where the primes are. A stage in the approximation of the prime ‘staircase’. So that’s why the deviation can be broken down into spiral waves. It’s partly to do with the fact that the prime numbers spread out: just the fact that two is a prime number means that all of the even numbers are gone, then since three is prime all the multiples of three are gone. So it’s no surprise that the further on you go, the more sparse the primes become. That’s a fairly obvious thing, you find

less primes the further on you go. You can quantify that, you can quantify the extent to which they spread out, it turns out that it’s a very simple logarithmic rule—which again is not surprising, since logarithms relate addition and multiplication. rm: When you mentioned ‘signal’ I was thinking about this entire thing in terms of signals analysis—as if you were someone working in SETI and suddenly got this signal on the screen, these waves. And you’re trying to understand what kind of source it could come from. At first you say, well it’s more or less a diagonal line. Then you begin to concentrate on how it deviates from a diagonal line. mw: Yes, the actual information contained in it is in the deviation. rm: Exactly. So then get to the stage where you just get rid of the underlying and you just have the deviation. You analyse it into an infinite series of waves. We’re listening to some cosmic orchestra and we now have defined a set of instruments each of which is producing this kind of infinitely expanding wave. If we had discovered that each instrument was producing a certain kind of wave with recognisable characteristics, we could say they’re playing a very harp or a timpani or something. So the question I would ask at this stage if I was receiving this signal is: What sort of instrument produces this wave? mw: A trombone. rm: A trombone? mw: Matt drew a bunch of angels playing trumpets because that’s a culturally present image, but really it should have been trombones. rm: Then if we go to this figure of the spiral, I’m trying to imagine this, grasp how it works. How would you make a machine that produced this spiral? cs: The problem is that you’ve got an exponential system, which isn’t sustainable.

mw: Yes, you’d be out of the gallery within a few minutes. And there is a problem in creating any spiral related thing using gear systems. cs: The only way I can see, and it’s a little complicated to make, is if you had an arm…and somehow I’m actually thinking of a secondary arm, you would put a thread onto this driveshaft that actually opened up exponentially…. mw: Oh yeah, brilliant. cs: You put a thread on that, you fix the rope, you put a nut on there, it travels on there, and as it rotates that relationship is constant. If that rotates, that’s a fixed axle. mw: Can you make that kind of thing? cs: This is incredibly complicated because a lathe is by its nature locked into fixed constants. rm: So you’d need one of those to make the lathe in order to make it. cs: A chicken and egg thing. You’d have to do it digitally, because the lathe relies on a linear rotational feed. You’d have to get a computer-controlled laser to cut it. mw: Then once you’d done that you’d have a purely mechanical device, but it would be sourced back to digital. One footnote here is that I like these spiral waves and I find a lot of people seem to respond to them really well. A more traditional way of explaining the same thing might be to say that this signal has an exponential growth in it, so let’s just scale it, put it through a simple logarithmic transformation to squash it and flatten it out, and then it would be susceptible to a traditional Fourier analysis, break it down into a set of sine waves and then they have frequencies. So some people have said, you’re just mystifying it with these spiral waves, just strip it back to traditional sine waves. But the reality is, you know, it really is lots of these added together, and the rescaling and renormalisation would be the complicated bit.

rm: To hear these spirals would be interesting—what sound does a spiral make? mw: Michael Berry, the quantum chaologist I mentioned, programmed his early home computer to suck all the zeta zeros in and make it generate a load of waveforms and see what it sounds like. And it sounded like a total cacophony. He said, I’d need a lot more zeros before you could really hear it…I need more data! And then the last thing I heard was that Marcus de Sautoy went to interview him for his book and had a listen to it—this is almost twenty years later—and said that it sounded like a low rumbling white noise. Part of the problem of listening to something like this is, first of all, the deviation function goes on forever. So you can only ever hear a tiny infinitesimal splinter of the beginning of it, you can only listen to the opening chord, really. And also there’s no timescale. What timescale do you measure the signal on? I mean, you can grid that horizontal axis according to seconds or milliseconds or years…and if I listen to a Beethoven early string quartet speeded up a thousand times, I might not have quite the same response to it. There’s no way to know what speed to listen to it at. Of course you can mess with that, and he’s tried different speeds. But you’ll never get away from the fact that you will only ever hear a tiny excerpt of the very beginning. // cs: So what do you think , tomorrow we really need to decide whether we’re going to try and realise something. What I think is that you guys should make something and then I’ll respond to it. rm: It’s not important to make some kind of finished work, but we want to show something, so we want to have some kind of visual memento of the process. mw: Feedback? Could we set up a video feedback loop? Then I could talk about multiplication as a form of feedback, talk about looking in the mirror—we’ve got a theme there. cs: We could easily make a quadroscope with mirrors.

mw: I think one of the most worthy ideas that has come up here is this association between the way the world is operating, the way people feel about it, and this horror of the logarithm, for example, when someone is quite happy to get their iPhone out and exploit the most elaborate technology imaginable. And that disconnect…which is somehow, I feel, unhealthy or dangerous…I imagine this as a form of subtle activism attempting to bring these things back together. Another thing I’ve found about maths, and again I noticed this last night, is it’s the same narrative for everyone: I used to like maths until…. Kids like it when they can understand it, if they can do it, they think it’s great. But as soon as it gets to a certain point, and that point is different from everyone—I hate maths, well actually I used to quite like it up until we did…algebra, long division, quadratic equations, whatever it is, they suddenly don’t understand what’s going on any more and of course you’re not going to enjoy being expected to solve an equation when you don’t understand what on earth it’s about. But as long as you keep the kids on course with that senses of wonder, everyone loves maths. The education is structured so, it’s like this train that just keeps rolling, and if you can hold on and get to the end of the line then you’ve succeeded, but most people end up falling off at some stage and they just get left behind and once the train’s gone it’s gone. If you’re off school sick for a week, it doesn’t really matter if you miss a week of history or geography when you’re studying South America or the Spanish Civil War or something, because you can just jump back in somewhere else and it all makes sense. But if you miss maths, then the train’s gone and you’re chasing after it for the rest of your school career cursing it. So reengaging a sense of wonder is important. But there’s a fine line between that and this dangerous science worship that Wittgenstein warned about. With this pop-science enthusiasm, this little-boy-playing-with-lego, this is great, everyone should like this, look at all the problems we can solve…there is a complete failure to acknowledge the shadow, which is what bothers me. It causes problems as well as solving them. And there’s that image you put forward earlier of people at a rave looking at fractals thinking they’re

really beautiful but no one acknowledging that its the miiltaryindustrial complex that has allowed these things to come into our visual field. To me it’s important that we acknowledge the shadow of science, and that’s not part of the agenda in terms of popularisation. // cs: Are there a lot of occurrences of prime numbers in nature? mw: Very few, a lot of people would assume there would be, and I would assume there would be, but the one thing that everyone cites when they’re writing popular articles is cicadas. Cicada’s life cycles, that’s really the best example. Seemingly, cicadas have evolved to get their life cycles out of phase with the life cycles of their predators. So if you’re on a twelveyear life cycle, anything with a two, four, or six-year life cycle is going to be just about ready to eat when you’re hatching, whereas if you’re on a thirteen-year life cycle, then everything else is going to be out of phase with you. So there’s an evolutionary explanation for why a thirteen- or seventeen-year life cycle for cicadas works. cs: That seems like a process of elimination rather than a specific use of primes. mw: Yes, it gravitated toward the primes and then settled into that steady state because it works. Someone actually wrote a computer simulation that involves some imaginary creatures, predators and prey, and leaving generations of these beings to mutate and to expand or shorten their life cycle. They basically took that idea from the cicadas, and simplified it, abstracted out the important parameters, and then ran this computer simulation where these imaginary creatures all ended up gravitating to prime number life cycles. And they were able to find and generate new primes using this biological simulation—it’s completely inefficient, but its an interesting exercise. But other than that, no. One of the reasons I think is because a single prime number doesn’t really have any meaning except in relation to the entire sequence, the entire set of concepts. And

nature’s finite. It’s infinite in its richness and diversity, but it’s finite in terms of how much stuff you’ve got to work with and how much time you’ve got. So there’s no way the entire set of prime numbers can get involved. At first it seems a bit disappointing. If you’re writing a popular article and you want to talk about how important prime numbers are, it’s like, why aren’t they all over nature, then? But it seems like the Riemann zeta function, the single mathematical entity that underlies the entire set, that’s embedded in the very structure of reality at some level—it goes deeper than the contingency of individual biological forms or particular ecosystems or the way that life might have evolved on any particular planet. The point is that the Riemann zeta function’s just out there, oscillating in some strange way. So it’s found in nature, but at a level below what most people would consider nature. So unlike the Fibonacci sequence, which shows up everywhere and the Gaussian distribution which shows up everywhere, individual prime numbers have absolutely no bearing, it would appear, apart from that one example. Other examples may surface that would be comparable to the cicadas, but they’re fairly marginal, really. // mw: So you must have been aware of the ideas of fractal geometry and chaos theory at the time when you made the Floating Orb of Fractal Chaos. cs: Yeah, but it was one of those really exciting things, I’d just got this video camera and I was playing around with it, just learning to use it, then I did the obvious TV thing of zooming in, and I was really hooked, I really felt like I was discovering a totally new phenomena, this quadroscoping. It definitely was an advancement of the feedback system, because instead of decaying and losing resolution each time, it was actually generating, compensating for the data loss. mw: So it was in operation for a while?

cs: It was in operation, yes, it went on tour! mw: Just the name you gave it made me thing that you were probably, were you reading about chaos theory at the time. cs: When I was a teenager my parents gave me the books and I’d been looking into that, that system of how the Mandelbrot set feeds an answer back into the equation, I’d definitely been struck by it as an idea, and it was really into it. But then I was just trying to give this thing a sensational name, and it was quite fun. It just made you smile —we’ve discovered this thing, it’s best described as a Floating Orb of Fractal Chaos! And it’s in my bedroom! I made a more immersive device but it was a bit to much, I had three quadroscopes, you’d have a TV on the end of each of these, and then you’d put your head inside, so you could go up into the structure, and it would be this crystalline infinity. The idea was that I was going to get the whole thing to rotate around your head. But I didn’t have the technology to make this incredibly dangerous lurid device to rotate around someone’s head—a recipe for decapitation. I was really preoccupied by this for ages, until my tutors were like, look, come on this is great but it’s just a phenomenon and you can go down the route of becoming a disco light designer or you can get back to some subtler work…. // cs: But I have this poetic idea that if you think of light as movement, when that movement leaves the body, it’s not a real sense of stillness, but there is this moment when the body becomes completely still, and the electrons stop firing, and the lungs, and there is this moment of stillness, but really it’s infinite movement… mw: From the point of view of fundamental physics, there’s no difference between someone who’s alive and dead! cs: But in terms of my relationship with someone who’s alive, there’s a big difference!

mw: Yes, experientially, psychologically and on a biological level there is, but from the point of view of fundamental physics there isn’t. The idea that when you die, something happens involving fundamental physics…. cs: But the idea that you have this phenomenal experience where you’re travelling through spacetime, of consciousness, at a certain rate… mw: No. No, no, no, that’s where you miss the point and that’s where I really take issue with this, ‘travelling through spacetime’. Nothing travels through spacetime, spacetime is a completely static object, it’s a four-dimensional thing that doesn’t do anything, because time is just like measuring a length. cs: We’re enveloped within it. rm: We’re seeing a ‘slice’ of that static spacetime, though? cs: We’re moving with it concurrently. we’re on the train, so there’s no sense of… mw: There’s no motion in spacetime, nothing moves, to move you need time! cs: It’s a metaphor, but… mw: It’s a very confusing one then! You can’t separate movement from time. Movement is defined in terms of time. You can imagine from God’s perspective, if you were to press the pause button on the universe and everything stops, so you basically just stop time. But to talk about moving through time…there’s a whole linguistic confusion there. cs: So how would you say it then? You just… rm: You yourself have said ‘the flow of time’. cs: We’re reliant on metaphors. mw: What’s this thing about death, I don’t see how death comes into it.

cs: This is poetry, I’m not trying to argue this in theory, but I’m trying to convey the idea that, at this certain point there is this stillness within the body, the idea that the energy of life leaves you and you are at that moment completely still, and your consciousness and everything leaves your body. There is this poetic idea, that I’ve gleaned from these ideas of relativity that, at this moment where we exist at a certain energy level in that continuum, and we’re moving through it for that brief period of seventy years—which for us is everything but it’s just a tiny instant—for a certain time life is strong enough to battle against the onslaught of the speed of time—and then your body gives up, your system gives up, then you’re back into that…incredible speed of light. rm: So all life is a slowing of the speed of light? It’s more like the idea of contraction and decontraction that you get in relativity, which is that as you go faster, like the metrics you were talking about in Escher, there is a contraction. Is that what you’re saying? cs: You could see it that way. But I just had this idea from reading ideas that had maybe been misquoted from Einstein, that there is this idea that space and time are interchangeable and that when you are completely still—if that was possible—you would travel at the speed of light through time. I liked the phrase ‘at the speed of light through time’. [general laughter] I just find it beguiling and…the idea that the speed of light, c, is this one constant that we hold onto, the thing everything fights against. In terms of this history of trying to find an ultimate unit, it’s that absolute that we hope is constant. If you’re talking about what to measure something against, that is the last rock we can cling to, the only rock we have found that has weathered interrogation, and we need that to be true. So anyway… I’m not religious about this, it’s a cognitive slippage, but that’s where it’s interesting, it’s not necessarily a well-researched… mw: No, no, and if you qualify it in that context then…I mean, a poet can read popular science, take from it whatever he or she wants, and express something poetically. And I’m not about to criticise that

or say it’s not valid, because it’s not pretending to be science, it’s not claiming to clarify. It’s just using something as source material. But there’s a blurring that can be confusing to other people, I think. Because if you’re adopting the language of relativity theory, someone might assume that you’re using those terms in the way they’re intended. I can’t really comment on that. If you want to do that, you can, but it’s…I mean, you could wire electronic components together into a sculpture but it wouldn’t be a working circuit, although it might look to some people like one, like it must do something. Well actually no, these resistors and transistors and capacitors have been chosen for their aesthetic qualities. It’s a bit like that. And an electronic technician can’t comment on that sculpture, on its validity. They’re two people playing different games. rm: What does the word ‘like’ mean here? You might say, I read about relativity theory and it seems to me that it’s like this. It’s similar to what you often get when people say quantum mechanics tells us that ‘everything’s uncertain’ or some generalisation like that; and inevitably, that some traditional people ‘already knew’ this. You present a bit of pithy ancient wisdom and insist that it’s ‘like’ what quantum physics tells us. What’s involved in that ‘like’? The origin of these two statements its heterogeneous, the nature of what they’re saying is different: Heisenberg’s uncertainty principle is based on a massive body of theoretical and experimental data, whereas the wise words of the shaman are coming from somewhere else completely. So what does it mean to say they ‘already knew’ this? mw: It appears that people are taking bits of reality and they’re creating what a mathematician would call isomorphism, these things correspond within some framework. But that framework can vary greatly in terms of its rigidity and the rigour of the rules by which that’s being done. So of course there are some dreadful examples of this sort of thing. rm: Just utter pseudomorphism. There are all these different spheres of production of discourse, and if we try to draw them back into this

common pool of natural language and informal discussion, then we end up linking things together that have little to do with one another. I mean, no doubt we don’t understand what the shaman was talking about either! mw: It can be as simple as someone who doesn’t understand what they’re doing looking at something and thinking, that reminds me of that, because it triggers some associations, just because of some particular word. I’ve had that when I’ve tried to explain something to people who are not very receptive. I’ll use some particular adjective or something, and they’ll just grab onto that and go off with it. Do you really want to understand what I’m talking about, or do you want to make up your own thing out of it…? rm: Does that always result in information entropy, can it ever be productive? mw: If you’re working in the sciences formally, a physicist can potentially go to a lecture by a biologist, and the biologist might be talking about some kind of self-organising criticality, and might trigger an idea that can be applied to solid state physics. And there’d be some more formal isomorphism that could form in the mind, and the physicist could go away and investigate that and see if there was anything underlying. You can generate these cross-disciplinary metaphors. rm: So that sort of imaginary slippage can play some role in the scientific? mw: Yes, very much so. cs: I think that a lot of artists are inevitably succumbing to the zeitgeist, we’re picking up this stuff as it filters down into popular consciousness. They’re obviously enriching the culture, these philosophical ideas, and they have a really significant part to play in that sense, just as they do in technological culture. They end up being ubiquitous, whether it’s in your mobile phone or whether it’s present culturally.

mw: there’s a difference, though: someone designing mobile phones would have to have studied quantum mechanics to actually exploit very specific phenomena to get the result. cs: I know, I’m saying it’s affecting everyone’s mind, whether you go to a museum or whether you’re calling your friend, we’re surrounded by the results of these ideas. // rm: I found it productive yesterday to get to that crunch point with the loose talk about relativity: it’s the sharp end of the problem. The process that’s gone on here got us to this point yesterday where there was a tangible tension in the room, between what Conrad was trying to do with concepts, and Matthew’s attempts to discipline them. cs: But also I think the common boat we find ourselves in is that there we don’t have adequate language to describe these things. rm: We talked about how, for the working scientist, there’s no necessary need or will to bring the work down into the realm of natural language, whereas as an artist you’re interested in what happens when you do that. cs: But scientists do need to propagate their ideas and to pass down knowledge and information, so, at some point, even if they’re loath to do it, younger generations of scientists will need to be introduced to concepts rather than just being given huge equations, and they’ll need to be introduced slowly. mw: But it’s gradual isn’t it—you start with simple equations.... rm: The introduction doesn’t happen through natural language, but through axiomatic definition. mw: It’s almost an initiation into another language. But I agree, when you go to your first physics lecture, there have to be some kind of gradual introduction in order to get you in there, It’s an interesting loop. Because a small amount of natural language is used to get you

to the point where you can start using symbolic language, which then gives you a foothold to get deeper into the symbolic language. And eventually when you’ve got far enough up the ladder, two physicists could just be entirely in symbolic language, just writing equations back and forth with the occasional gesture, and there would be communication there. cs: But they need to have an interface with the public too. rm: But do they? This is the social question of having massive bodies of knowledge and insights into reality that are not integrated into common consciousness—even if they are behind the scenes producing equipment that we use, but have little understanding of. pc: What is the motivation? Why would someone go about creating that interface? mw: The motivation to attempt that project of integration could be a sense of discomfort about the way the world is. But that’s not what scientists do, that’s what we’re hypothesizing that some future artists or hybrid artist-scientists or some entirely new class of people might want to do. That’s not the motivation of doing science, that’s the motivation for trying to bridge the gap between science and popular awareness. rm: But if the motivation for science is not purely instrumental but, idealistically speaking, is to discover more about reality, then comprised within that must surely be the will to communicate what you’ve discovered to other human beings. mw: I think originally that was perfectly possibly. Until it became too specialised. Early scientists’ discoveries could be readily communicated, but there’s been this accruing of layers and layers of symbolic language. There is this problem, and yet no one has really addressed it except for some hopeless government panels commissioning people to make a piece of art, to get more people teaching science, to make science seem more cool, these feeble attempts to integrate.

rm: As you said earlier you are in a minority, as a scientist who does feel the need to integrate your work into the rest of your life. mw: Yes, I don’t get the feeling that that’s the norm. And it’s the fact that it isn’t the norm that bothers me and sets me apart further. rm: To go back to what happened earlier, I had a sense that what was going on was that Conrad wanted to draw something out of the concept of relativity to use, he really wanted to solidify this intuition he had that he could use it in a certain way to talk about other things. But at the same time, he was in danger of losing any kind of anchoring to the actual concepts, and then Matthew was trying to say, you can’t talk about it like that, and at one point Conrad was feeling like a bullied child! Matthew was trying to let out as much rope as he possibly could, but at a certain point he would always pull it back and say no, you can’t say that because you’ve taken leave of the concept. cs: I just felt the situation was that there was a period where I was being incredibly naive, and Matthew didn’t have the heart to tell me it wasn’t true! rm: But that was what was interesting to see, that he was being generous up to a certain point, but then he would just say no. mw: What’s interesting is when you threw up your hands in frustration and said sorry but I’ve only got these words to use, I’ve only got this vocabulary, I can’t say this, can’t say that. You’ve been trapped, how can I express this...and that’s the whole problem, there aren’t any words, there are equations, there are tensorial functors…. rm: Nevertheless there is a thread that leads from humans as biological creatures and their natural language to the most exalted heights of theoretical physics: Humans are the ones who do physics so there is some continuity. mw: Yes but it’s a tower of layers and layers of compressed knowledge. rm: It’s not a thread that one individual can retrace.

mw: No, you have to basically have some faith in the integrity of that structure and the authority of the past and the mechanisms of peer review, and the idea that invalid ideas eventually get exposed. For example, Euclid’s proof of the infinity of prime numbers is a wonderfully simple proof that he came out with thousands of years ago, and it’s as valid today as it was then, it’s still taught to students, and no one says, maybe there’s a mistake in it somewhere, let me check that again...If it’s been around that long and that many mathematicians have checked it, then there’s no chance at all. You can never really know with a mathematical proof. What if everyone’s slipped up? There are a few minor examples of how a lot of people have been wrong for a long time. But you have to sort of trust the integrity of everything that’s gone before you. // rm: it is to do with bringing down an abstract concept that’s emerged out of these compressed towers of symbolic language back down into natural language, and the question of what happens when you do that. mw: Is it communication, or is the signal distorted to the point of meaninglessness? The problem with most attempts to bring down those abstract ideas into natural language is that you end up compromising to such an extent that people think they’ve understood something, and that’s probably more dangerous than not even bothering. pc: Do you think that most people do actually have the capacity to understand to that level? You obviously do to some extent because you’re publishing your own book. mw: Yeah, I think anyone who’s got a reasonable level of concentration and the ability to read without any prior exposure to higher mathematics can, if you present the ideas in the right way, grasp them. pc: So it’s not a matter of having the capacity, it’s that they haven’t been given the right tools and shown in the right way?

mw: Yeah, the tools haven’t been developed, so I suppose what I’m doing in this series of books—and it’s not supposed to be art, it’s supposed to be a friendly exposition. So yeah I do think…it’s interesting actually, because I was saying artists can’t do that, but I’m trying to do it. But I’m not an artist. rm: That’s where we got to on Tuesday, wasn’t it, where it seemed you were both doing very different things. But then we talked about the use of metaphors and images, some tools that were common to what you were both doing. mw: I must have some sort of faith somewhere that you can bring these ideas into natural language without too much distortion of signal—where the signal has been so badly distorted that it’s worse than useless, and it creates an illusory sense of understanding. cs: I think what runs through my work is the effort to represent time visually, that’s one of the things I try to do, if you look at a lot of the sculptures I make. I’m trying to represent time, and that’s a key problem in science and mathematics. mw: Well, time doesn’t come into mathematics. // rm: Say you’ve got natural language. So you’ve got words. And words are subject to these different mechanisms that shift them and connect them, metaphor, metonymy, linguistic slippages, they’re expressed in sound so you have those similarities, and all of these things that basically constitute poetics. What happens in science is that gradually you come to use these words in a far more structurally defined way. So in mathematics you talk about points and lines, and initially it’s very intuitive what those words mean, something like a dot, and a pencil stroke on a piece of paper. But by the time you get to twentieth-century mathematics they become defined axiomatically so they have no necessary link to the things those words once pointed to in empirical reality. They’re just defined within a system that’s completely abstract.

So in that case…I’m trying to diagram how words begin to be linked together in this systematic way which has nothing to do with their natural linkages. In so far as you talk about a point and a line in post-Hilbertian geometry, you’re not talking about those things out there in reality. mw: Even though you can trace them back, with a conceptual archaeology. rm: Yes, you can trace them back to their origin, but this is the point where a genetic account and, let’s say, the ontological claims of mathematics part ways. Because you can say there’s a genetic link between drawing lines in the sand in Ancient Greece and the concept of a line in abstract geometry, but that’s of no consequence. The axiomatic system works and has validity and import whether or not it is applied back to lines drawn in sand. It’s at that point you abandon sense and meaning, even if you can have always bring it back and apply it to points and lines in the real world as one model among others. And then the next stage is what Matthew was talking about, this kind of hierarchical embedding or tower-building. And I think a lot of people have the idea that science works by someone coming along and overturning what was there before and erecting a new theory in its place. But what happens is more like, say between Newton and Einstein. Einstein just shows there’s a larger conceptual space of which Newtonian mechanics is a subset. So you have these systems of concepts, and you can box up a set of concepts, and connect other concepts to them in these systems that are highly embedded. And out there on the outer edge you have whatever is the latest addition. mw: Theories that are relatively free-floating but will eventually get subsumed into a bigger embedding. rm: But then what I wanted to say is that even here out on the edge, you’re still initiating the new by borrowing natural language. Like sheaves, where you have this highly abstract object, but the

mathematician thinks, what can I call this, and they’ve used some kind of imaginative poetic intuition… mw: Yes, and you have fiber bundles, and you have stacks and schemes…. rm: And so this is this problematic thing about what happens when you try to reintegrate them. You’re trying to recover meaning from this compressed mass of systematised symbolic systems, but you’re trying to recover it from this outer ‘crust’ which is the latest thing that is available to you, that you can see. Say relativity theory—you’re trying to take time and space, and connect them back to their sense in natural language, but without being able to trace all these links, all this stuff that’s now embedded inside what appear to be the same words. And then you’re resubmitting them to all the slippages and metaphors and poetic mechanisms of natural language. I’m saying, the problem is that we only have access to this most recently concretised outer shell or crust, but that’s been built around a massive existing edifice. The words used may have been borrowed from natural language, but you can’t recover them because they are wired up to all this complex stuff inside, like a black box. That was how the conversation was working: Conrad was saying, I’m looking at relativity theory, I’m using the words ‘time’ and ‘space’ and ‘speed’, and Matthew was saying, no, you have to come back up here and look at how it’s wired up inside, here and here, you don’t understand. What are we trying to designate by ‘understanding’? In one sense having a grasp of that systematic hierarchy and a facility with its symbolic systems, and on the other hand what we’re continually reaching towards here, what we talked about with the motor system and integration. mw: Imagine watching a somewhat disconnected Cambridge physicist with his laser pointer smugly presenting his paper on the fact that the universe is a vacuum fluctuation, as if this is just a formal chess-game. An artist might sit in that lecture hall not

understanding the science but seeing the contradiction that I can see, which was that this person’s state of being is completely out of sync with what he’s supposedly discovered—that in a sense he hasn’t grasped the ideas that he’s presenting to us, effectively. The artist could go away and make some absolutely harrowing, terrifying, profound work of art and then encourage this young physicist to come along to the gallery, and he would then be so utterly shocked and traumatised by this artwork that he would genuinely grasp that the universe was a vacuum fluctuation and go away personally transformed! rm: This is precisely the relation between the two volumes of Collapse, between The Copernican Imperative and Concept Horror. The idea of ‘concept h0rror’ is that with many modern scientific concepts, if you could really grasp them in that visceral way, you would be horrified, and that horror and weird literature and film and science fiction over the last couple of centuries have tried to create dramatizations in order to do that. Whereas The Copernican Imperative is suggesting, maybe it’s impossible for us to do that and efforts at reintegration can only be refractory to our understanding of the universe. mw: There’s this disconnect now with the vast majority of physicists who are dealing with things that, if their implications settled in, they’d really be changed in the same way that perhaps long-term meditation or exposure to traumatic events might change someone. Reintroducing trauma to physics…. rm: These discoveries and modes of thought are a part of human culture, but human culture’s response is largely to hold all this stuff over there out of the way, because if it leaked out it might cause problems. My position would be, lets let out the weird stuff out—or in —and see what happens. mw: But the quantum-mechanical understanding of the universe, relativity theory, if all of these things were fully integrated into the human nervous system, if you could download it all, if someone designs some new application where you can assimilate a whole

scientific world view, would that leave the world incapacitated, would it leave the world looking more beautiful, more wonderful, would it change your motivations, would it cause you to just go off and create new religious forms? You can’t know. But an artist can certainly address—which is different from the question of dramatizing abstract ideas—an artist could and possibly should address the dissociation, the chasm that’s opened up between the concept and he state of the mind of the person who’s supposedly grasped it. pc: What do you recognise in the semantics of the word ‘grasp’? mw: It comes back to the physical, doesn’t it? To do with the hand. But the etymology is from ‘to steal’... I suppose a lot of words get extended from motor function to something more conceptual or abstract. // mw: In multiplication the number’s going from being a number to being an operator, it’s becoming a verb, something that does something to something else. rm: But I don’t understand the sense in which you could say that ‘happens’ unless that capacity is already present in number—we were talking about this question of whether it’s already there or not. mw: You know, if you invented a system with a microphone and an amplifier, feedback is ‘inherent’ in that but it may not even have occurred to the person who designed that technology that it would happen, it’s an unintended consequence. But that’s assuming that number is just a technology, a human creation. rm: So multiplication is not an innovation, really? mw: It’s more like a hack. rm: No, it’s a Hendrixing! Instrumentalising that accidental property. mw: But actually, when you say five fiving itself—it works better when you use a different number. It could be the same number, but that’s a special case.

rm: What kind of a verb is it? What is fiving? If what five does is to five, then what does that consists in? It depends what theory of number you have what kind of verb it is. Does it consist in collecting together all the things that are five, or… mw: Then you’re going back to the definition of five as the set of all things that have five in them. Whereas to five something, when you’re using the verb, then you’re meaning to conjoin five copies of something thereby creating a greater something. So you could say, if you’ve got a beach full of emeralds… rm: But you can’t define it circularly—you can’t say that to five is to conjoin five… mw: You can. rm: You’d have to define it in terms of its precursor: to five is to four and then add another one. mw: That’s what the Peano axioms do, and that takes care of the numbers as nouns. The natural numbers aren’t nouns, but they’re not verbs either, I’m just using the words loosely. But if I say that to five some pebbles on the beach is to collect five pebbles and put them together... rm: But you’re still calling on the concept of five. mw: Oh yeah, we’re not attempting to define the number, we’re just talking about what happens when you take the concept and apply it in a novel way. How you define numbers is a separate matter. rm: But if to five is to conjoin five things, then that was already there, there’s no innovation involved, because the innovation lies in fiving numbers. So it was already a verb? mw: Okay, if I five some peanuts, if I take some peanuts and put them end to end so we have five peanuts, here we have the possibility of a representation of number. But if I five some twos, what we have there is not a form of nuts, that chain of peanuts is in a different category than a peanut, whereas when I five these twos,

then we suddenly make a leap whereby we say—and we tell children this is perfectly natural—we say, five twos ‘are’ ten. By conjoining twos, which are numbers, we arrive at another number. Whereas by conjoining peanuts, we don’t arrive at another peanut. But we can join twos to arrive at another number. And that’s the innovation. That’s the feedback. rm: So all I was saying was, the innovation doesn’t consist in making five into a verb, because it was already a verb. The innovation consists in applying that… mw: …to number. Yeah, exactly. rm: In that sense, is it really feedback? Because you’re applying five as a verb to two as a noun. mw: The reason I arrived at this moment of feedback, maybe a misguided notion of feedback, was from looking at audio and video feedback, and then looking at what the role of number is prior to the introduction of multiplication, assuming that people counted things and perhaps combined numbers using simple addition before it occurred to them to multiply. So, prior to multiplication, number was used to enumerate concrete similar objects. These all belong to the category of peanuts, so I can assemble, I can take a handful of peanuts and count them. I can talk meaningfully about five peanuts. So five, the notion of five, could be applied to any type of object of which there are at least five in the world—effectively, any sort of object of which there could be multiple copies, there could be five of them. But when I say five twos, something strange has happened, because I have this cultural operating system, as I call it, this mechanism that searches the world looking for similarity and then enumerates things according to these categories, and it’s never going to see a two anywhere, there are no ‘twos’ anywhere in the world. But if you turn it around on itself, so you’re pointing the camera back in at the monitor—and again, most metaphors fall apart if you subject them to too much scrutiny, but by turning the operating system around on itself, by turning the instrument back on its output, you’re counting something that’s not in

the world, but in your mind or in the shared framework that allows counting to occur. Two is a shared concept that allows you to count two sheep, five is a concept that allows you to count five peanuts. To count five twos, what would that be? There aren’t five twos . But if I represent two in some way, a pair of peanuts, and I say there are 2 peanuts, lets repeat that 5 times—again, there’s always an element of time, or stacking things together end to end, which means we have to involve time because each one gets placed after the other. If you’re explaining it to a child you could simply say, okay multiplication is like adding something a certain number of times, put the peanuts down five times and let’s count them and we can see there are ten peanuts. People might think, you’re just making a very obscure point with no bearing on anything. The reason I’m doing this, as you know, is that if you don’t introduce multiplication you don’t have primality, and primality opens a whole can of worms, so maybe this is where the worms come from. cs: The monkey nut…it’s two bundled up into a single one. It was just staring us in the face all the time. We were using this as an example. mw: I could say each one was two—yes, it is the perfect representation of two, but that was unintentional. Some have only one kernel in, though. rm: Deviants. mw: So maybe feedback isn’t a good way to go at all. rm: No, I think it is. mw: It is feedback, although there’s no time as such, although time comes into it again, because you say five times two, i.e., how many times do you look at two nuts. cs: But this is the nut to represent all nuts. mw: Peanuts are actually legumes aren’t they?

cs: Yeah, leguminous cosmology. rm: How could you create a machine for fiving? Maybe if you have a lens that splits up the light coming into it into five separate beams. Then maybe you could apply something like Conrad’s Orb feedback mechanism to that. cs: What’s the best mathematical definition of a peanut containing two…an example of a singularity that is actually two? rm: Dyad? mw: Yeah that’s a good word, monads and dyads. Mathematicians don’t worry about things like that. What are you thinking, the arrow is the feedback thing? rm: [Drawing] sometimes you have to do the colouring in to do the… you’d need a set wouldn’t you, a set of lenses…but then if it each was a single one, you wouldn’t get a sense of it applying to itself. Say you have from one to five… mw: I think I know what you mean. So you have a lens that splits the image so you have five peanuts. Then the idea is to put something behind which has got a threeness in it. Yeah we’re getting there now. And then you could move one behind the other, you could then five the three. Do you work with optical materials, apart from the fractal orb of chaos, Conrad? cs: Well, light bulbs. mw: Lenses and refraction and that kind of thing? Do you feel comfortable with that kind of material? cs: Yeah. I think so. mw: I think we’re hitting on an idea you might be able to pick up and go with. cs: I’ll just finish my transvestite peanut sketch. Maybe it’s not even a boy. It’s wearing a hyperbolic skirt.

mw: So you could have a series of these lens assemblages, each of which was a number and they could then be applied to each other. There’d have to be objects behind them, otherwise you wouldn’t see anything at all. But you could have a monad, you could have a unity, put the monad behind the five lens, you see five monads. So your lens splits unity into fiveness, the idea of multiplicity. And then the monad could be any object, it could even be someone’s image…like one of those fun-house mirrors. rm: It doesn’t quite work, because the idea is that number is employed as an operation, whereas there, the number is the image the lens is producing, and the operation is the lens splitting the light. mw: Yeah. rm: They don’t quite coincide. But it’s close. mw: If you had…so you place a peanut behind the lens and you see five peanuts. rm: There should be a pistachio. mw: So you have this idea of multiplicity, you can have more than one peanut or pistachio in the world, a replication of it. But if you then replace that with, if you were to put down three pistachios, you’d get three lots of five, hence multiplication is being demonstrated. But again that’s just a simple demonstration. rm: But I was thinking there’d be a second piece to this, there’d be a three as well. mw: So you’d put a single peanut through both of them, split it into an image of three that would then split into three lots of five, and you’d see fifteen. rm: But still the operation of splitting and the thing you’re splitting aren’t quite coinciding as numbers do when you use multiplication. mw: That may just be a problem, numbers are such an abstract thing that...

rm: Isn’t there a way of making an object that’s both an operation and a thing that’s operated on? cs: So you’re taking an original peanut and putting it through lenses? Could you use a video camera and then take that feedback system but take it to five monitors, which then, you’re splitting an RF signal, it’s getting weaker, you’re splitting and you’re creating a number each time. Because each time you’re filming the screen with another camera. rm: Then you put the five screens back in front of the camera again? cs: This is the peanut, the original. You’ve got your camera filming the peanut, which is there, and then that signal goes to the monitor there. And then that goes to three monitors. So you branch it off. mw: So you’re seeing three copies of the same peanut. cs: Then you have three cameras, you can get cheap security cameras. rm: The interesting thing about doing it in this way is that, in the original version, you’ve got this actual peanut to start with, which is kind of an embarrassment, because you want numbers, you don’t want peanuts. Whereas here you can hide the initial peanut, you can just say you’ve got three number-images, and that’s where you start, and from there you can feed it back into itself, multiply it. So you’d have one camera on three of them, and then that camera would feed to five, you’d have three times five. cs: But you might end up with a distant memory…this crackly image of the peanut. rm: Yes, they become more abstract. They lose their specificity as objects, that’s good. cs: How many are you going to use? mw: We were just thinking about multiplying a pair of numbers, so you wouldn’t lose much in just a two-step thing, I mean, you could theoretically have a whole chain of these—but then that’s moving

away from the abstract idea we were originally trying to communicate. So if you split the signal into three, then split to five. cs: Why five? mw: Just to show how 2 × 5 is a novel approach to reality. // mw: We’re not sure this works yet. The idea is, we’re trying to illustrate the idea that multiplication is categorically different from addition because it is the number system applied to itself. And therefore it’s something akin to feedback, although it seems that if you try to stretch that metaphor too far it breaks, because feedback usually takes place in physical time. cs: This isn’t really feedback is it, it’s not affecting the original, it’s a xeroxing, a gradual copying but with less information. mw: Yeah, it has something in common with feedback, but not the thing you might think of. rm: It’s more like self-awareness, like the thing Conrad was saying about looking into the mirror. mw: That is feedback in the sense that it keeps changing…it adjusts, it’s not just a one-time thing that happens. rm: A loose sense of self-awareness. cs: So how do you want to branch it, do you want to go from one thing to three monitors? mw: And then each of those monitors would split into five. cs: Into five? Wouldn’t it be better to show what three does to itself? mw: That’s one specific case, it’s not what three does to itself, it’s what numbers do to numbers. mw: As in, ‘we’re going to nine it’. Two and three would be the smallest we could get away with, because we need two different numbers.

cs: So we’ll do twoing a three? mw: We could do threeing on a two, or set it up so that you could reverse it and show that you’d get the same thing. cs: So how many times are we going to do it, how many generations? mw: Oh, just two generations, if you start multiplying even more numbers then it just complicates things. cs: But sculpturally I’d quite like to take it further. So if we started with a two and then threed it…. But we’re going to start with one, in a sense. mw: Yeah, you always have to start with a one, the monad that gets split. cs: But I think we should use the dyad as an example of a monad, I think this dyad is nice…. rm: There’s no way of getting across in this that you’re having the system operate on itself. mw: There is, in a way, because if you had one number on a trolley…so basically, move that, then we have something that illustrates threeness, three peanuts showing up on the screen. This idea of multiplicity, of threeness being...it doesn’t really matter what the object is, the threeness is not about that. And then it would have to wheel around, and now we’re looking at twoness being threed, but what’s in front of the camera is no longer a peanut, it’s a two. You won’t really see two peanuts separately, you’ll see two peanuts each of which is on a monitor. rm: The monitor itself works, because it’s like a peanut in the monitor is a peanut counted as a unit. The imaging of something equals its being counted. mw: Now I think what we might want to do here is try and introduce a contrast with addition. Because the point we’re making here is not that 2 × 3 =6, but that multiplication is categorically different from

addition. So we want to find some way of showing how 2 + 3 works too. rm: Wouldn’t that just be one camera, one monitor, and just have three of those next to each other, with it being obvious that there’s only one input, one output…. mw: Ummm, let’s see. rm: This is the essential minimal counting operation in this system isn’t it—that you’re taking something and putting it into a frame. mw: Oh no, I know how it works. These camera-monitor complexes are movable, so one can be placed in front of the other to make the point about looping. But also they can be moved next to each other, so you’d place this here, and you’d place both cameras on there, and then you’d have two peanuts and three peanuts—a total of five peanuts, You take this, you move it behind it, and you could set the whole thing up in a way where it wasn’t at all obvious that that was intended, but there’s this clever mechanism whereby—what happens if we put that behind that? And you can even roll that round here so you could three two and you could two three. Same thing. rm: Now you have two states that you want people to see, so that involves moving at least one of the things round, so that’s going to be the crux of how to make it work. mw: There should be an element of surprise in it, I think, in the sense that you’re looking at something that seems obvious, two and three together, you get five. And then there’s this surprise of, what happens if we do this, or what happens if we look at it from this angle —it might just be changing angles. rm:You could have one static and one that moved. mw: There’s something promising here, but what do you feel about this, Conrad? cs: If I’m honest, this might be much less interesting but I just need to sketch it, what I would have done, the way I would have done it if

I’d been given this task, I would have got a camera in front of each one, and then that would have gone to three, then that would have given us six monitors of single peanuts...which may not be right at all, but I’m interested to see. rm: I can see how that would make more sense, in a way. cs: That’s just how my brain works, the idea of branching off, as a way of piecing the singular out into the multiple, maybe it becomes a double, it becomes two rather than one. rm: Your way of doing it, you maintain this impression that you have this ultimate simply unit of counting which is the camera and the monitor. Where you’re taking something and counting it—In Badiou’s language that’s the ‘count-as-one’. mw: I think we’ve arrived at the correct representation of number. pc: What would happen if you swapped those out completely? If you had the three there and the two there? mw: You’d just get six again, that would demonstrate the commutative property, which would be something else. mw: I think it’s very important that we’re able to also compare it to addition, that’s the most important thing. Oh right, 2 × 3=6, they’ve spent a week working out how to show us that…. It’s more the fact that you’re shown something where you think the camera points at the peanut, and then there’s this shift that occurs: you walk around it, there’s a shift of perspective, or something moves, or is removed, and suddenly the camera isn’t pointing at a thing, it’s pointing at the output of another camera. So there has to be a point where they’re both pointing at things, and a point where one’s pointing at another’s output. rm: They can’t be concentric then, they need to be alongside one another at some point. mw: Yeah, I think so.

rm: It has to be the same complexes, but you’re seeing them working in a different way. First as an object, and then as an operation. mw: Yeah. I think we have the kernel of a valid argument…the execution is now the problem. The other thing could be mirrors that moved. rm: How might mirrors work? mw: Don’t know, it’s just one way of moving optical information from somewhere to somewhere else, that’s all. rm: Doing it with mirrors could be economical. cs: It doesn’t feel so good with mirrors. Video is the best system. rm: You’re also adding another element which, in the analogy, is not clear: What is the mirror? But you would get this whole thing about self-awareness and Narcissus. But I much prefer this idea of them being side by side and then moving—it’s more dramatic and feels like it’s to do with orientation, gesture. mw: Imagine some sort of complex system of tracks and motion… rm: It doesn’t need to be that complex. You would have a motor that sent it back and forth. pc: Does it need to have be motorised? mw: I think there’s going to be a really simple version of this that will occur to us if we keep messing with the idea. rm: Another thing about doing lenses or mirrors is that it may constrain where you have to stand to see it working properly. mw: There’s actually something about cameras that is more in keeping with the idea of something doing something, as opposed to just being a clever optical trick. And this idea of multiplication as an innovative technology. rm: Yes, as a machine.

// cs: Do you understand time? mw: No, I don’t think anyone with my cultural operating system really can. cs: Then what cultural operating system would then be placed to understand time? mw: I think the Ancient Chinese one clearly had a very different way…I don’t think you can say one’s better than another. cs: But you are saying one might be better to understand time intrinsically rather than just pragmatically? mw: Yes the Western model is a timeline…it’s very effective for our purposes but it also restricts our perception. And for us to say that is what time is… To get to the real deep essence of what time is, you have to go completely annihilate all ideas about human culture, about the solar system, and try and get to something that’s more fundamental. And there have been attempts—because of issues in recent quantum theory and to some extent relativity theory, and even some things that have popped up in parapsychology, people have been pushed to develop two-dimensional time models, fractal time models, p-adic and adelic models, using wildly different types of number systems. Not just to be novel or wacky, but to say that we actually don’t understand time, something’s missing in our understanding here, so what happens if we model time as this other kind of space. I’m quite interested in alternative time models, I’ve got a bibliography I’ve compiled on them, there’s a surprising amount of material out there, some of them are failed attempts and some are possibilities. I mean, there are many layers here, there’s a cultural psychological experience of time, just a simple matter of how things speed up and slow down—time flies when you’re having fun, these things that everyone recognises, the psychological perception of time. Again, that doesn’t come into the province of physics but it’s interesting.

cs: And also just the fact that as we get older time will perceptibly pass faster. mw: But I’m interested in time independent of human perception, I suppose. And how there’s a problem in the timeline. And, in fact, as a French mathematician said, even the line is mysterious, we don’t even understand the line, in the sense that the real number continuum similarly gets mapped onto time, and so variable t is assumed to be going through the real numbers, but in fact the real number continuum was an arbitrary construct as a completion of the rational numbers. It was discovered in the late nineteenth century that you could complete the rational numbers in alternative ways, padic number systems which fold the rational numbers up into these weird looping structures that you can’t really represent. So…if we are to model time as a line then we have to understand what a line is, we have to understand what the number continuum is—and we don’t. So, anyway. Me orbiting this cluster of ideas and confusion around this, at the same time as being interested in the fundamental nature of number and the mysteries thrown up once you use multiplication, this threshold you cross, which seems to have something to do with time. And then at a deeper level there are those spiral waves I showed you—the idea that there’s some kind of vibrating, oscillating, unknown quantum-mechanical structure underlying the number system, which is hugely problematic in terms of our model of reality. But as Robin immediately picked up on, the question then arises, in what kind of time is that happening, if something’s oscillating or vibrating, in what time is that occurring? We’re not talking about historical time, we’re not talking about clock time, we’re talking about something outside that. It’s very difficult to bring these things together into one solid conclusion. // cs: Do you think there’s some way we could get the peanut to rotate?

pc: I knew you were going to say that! Yes, it would add a nice dimension, wouldn’t it. mw: This is where we get into the artistic subversion of the pure idea. Which I’m not opposed to at all, but, you know, that’s really moving away from the idea of an unchanging unity, a featureless unity. But it’s not a peanut at all. Even lurking behind the unity there’s this duality, this dyadic form, so I quite like that. cs: Aside from it being a useful idea, does singularity exist? mw: It really depends on what you mean by ‘exist’, and that’s a huge problem. cs: Is there such a thing as one? mw: Words like ‘exist’ and ‘is’ are problematic. As I said a few days ago, in terms of our experience I don’t see any problem with one. It’s when you have two that it becomes more difficult. cs: Are the primes ultimate elements, then? mw: Prime numbers are supposed to be unsplittable, but atoms were supposed to be unsplittable and somebody worked out that actually, no, they’ve got bits. So perhaps some day there will be a meaningful mathematical context within which you can split prime numbers. You can’t factorise them, but there might be some extension of the notion of multiplication that allows you to break open the number 37 and realise it’s composed of bits of proto-number or something. I mean, this is sci-fi thinking, a mathematician would just look at me like…. But I’m open to the idea that perhaps lurking inside the idea of one there might be something else. rm: Badiou has this concept of the ‘count-as-one’ which implies that there aren’t any ones in the world, there isn’t unity, everything is multiplicity—but there is the operation of counting as one. mw: But you wouldn’t count something as one unless you had numbers. It’s only with the introduction of multiplicity that it occurs to you to count more than one of something. If you only had one then there’d be no point, everything would just be what it was.

rm: He’s basing his analysis on set theory—so to get started, we count the void as one…but anyway, I thought, in that case, that we need an extra stage here, where you count the peanut as one, before you get into the addition and/or multiplication stage. Because the peanut is an object, but it has to be counted as one in order to be drawn into the numerical world. mw: Right, because the peanut isn’t a number as a peanut—it’s only on the screen that it becomes a number. cs: It goes through a threshold. It becomes symbolic. rm: Which reinforces the idea that it’s not unity here, outside the camera, because it’s got two kernels. mw: Paul, is it too late to order another one of each? Robin just came up with a philosophical reason why we need another monitor and another camera. It seems like now we’re going to start with a peanut with a camera and monitor, the objection Robin raised, which is good I think, being that the peanut has to be initiated into the symbolic world of number before you can start counting with it and multiplying it. pc: It’s like a ceremony. mw: In a way, yes. We could turn this into a ritual. rm: It also accentuates the idea of the peanut not being a unity. You have to count something as one before you can start doing number with it. mw: For example you could have a football team, even thought a football team is composed of eleven people, you can count it as a team. cs: Every group of animals has a different name for a group, and they’re all singular: a flock. mw: Yes, ‘a’ flock has a multiplicity hidden inside it. We’ve got a whole vocabulary of those words.

cs: There are numbers like this that don’t involve the decimal system, that probably go back far beyond numbers, the names for groups of animals. In fact there’s probably a name, using animal groups, for every number from one through to fifteen or something. You could swap out the numbers for them. It goes back to that ritual thing, this idea that numbers came out of ritual—but they also came out of animals, suddenly possessing animals. So, they’re singular words describing multiplicity—in the same way that two is a brace. A singular of boars.6 A lease of hawks. rm: When you have a word like ‘brace’, it’s more obvious that you’re connecting together several items as a single thing. cs: Yeah, calling a group of three animals by a name makes it much clearer what numbers are, they’re singular words for describing plurals. mw: Some cultures have different counting numbers, they’ll have a different number for five when applied to animals and when applied to inanimate objects or people. cs: But it’s interesting how ordinary words, to pluralise them you just add an ‘s’ to the end, but some words change entirely when they’re plural. mw: Whereas sheep don’t do anything. cs: So there’s some interesting stuff there, how language deals with singularity and plurality. rm: Let’s examine what the difference is between the word ‘two’ and the word ‘brace’… mw: One only applies to a specific type of thing. rm: It’s not only that, these specific words are more…they have an earthiness to them, a thingness. Their function is obviously to gather several very specific things into one, whereas number words, we don’t really think about what they’re doing.

mw: No, that’s one of a number’s possible uses, but that’s not all it is. I suppose its’ a further process of abstraction, if you’ve got different words for four this and four that, then there would be things you couldn’t do. cs: It would be quite interesting in terms of this construction, to avoid the use of three and two, and to just confront people with these words instead…. rm: Do we have a word for three? cs: Yes a lease of hawks is three.7 rm: So what we’re showing is a lease of braces? mw: But you can’t have a lease of braces, because you can only have a lease of hawks and a brace isn’t a hawk. rm: But isn’t that the point you’ve been trying to make with numbers too? The strangeness of doing that? mw: You can’t multiply these things, basically. cs: You end up with some sick hybrid animals—half rabbit half boar. rm: But that does show you that, when you do multiplication you reach this high level of abstraction where in principle you can’t be dealing with two rabbits, three hawks and a boar, because… cs: …the product of them is a sick impossibility. rm: Effectively, the Frankenstein animal is a good representation of just how crazy the general idea of number is. mw: So when we’re doing this, let’s say this is a bull, a single bull, this is conveying a brace of hawks or whatever, and this is a lease… can we get a hawk? cs: I think that’s it’s important that we can do a sequence of things in front of people, not just a conversation. You start off with your example of singularity, then you get to the point where you get these

completely impossible hybrids. I think if we can demonstrate this using three animals as a starting point… mw: I can’t see how the animals are going to get hybridised. rm:Well, either you demonstrate that you can’t do it, or you just do do it, and demonstrate why that tells you something interesting about number. cs: You have to do it as an enactment. What I’m suggesting is that you have a singular, a twoer and a threer. That point where you’ve only got one is the boar. And then you’ve got the second stage which is a brace of rabbits, then you’ve got the third which is the hawks. So you’ve got these generational stages you’ve got pig, rabbit, boar. In theory you have to start with a boar becoming a rabbit that then becomes a hawk in terms of its definition. But obviously we’re reproducing the image of it. So if you start with a boar, it’s going into an area where it should be transformed into something else. So we’re getting this singular appear in the very place where it’s represented by the two, the two is represented by two hawks rather than just being the number itself… mw: Okay. So if we were to put a single model pig in front of the camera, then we have a pair of pigs on the first iteration. cs: Yeah, which should be rabbits. mw: There’s not a name for it—you never get two boars together, supposedly? You couldn’t rule it out. cs: It’s a brace of boars. I think what you’re arguing is that... mw: At the moment we’ve got three number words that apply to different kinds of animals, but there’s no reason there couldn’t be an animal that had a word for one of them, two of them, three of them— so that would move us away from this idea of animals mutating each other, which might be distracting from the actual point being made, I don’t know.

cs: I think there is this Platonic idea of a number, of a two, and a representation of that two as a familiar kind of thing for us, we just see it as a symbol. When you take it back and use the word ‘brace’ for the number two, you find that your boar is coming through this, being teleported electronically through this screen, into the realm of two, into the virtual world. It’s in the Platonic world of two, but it’s actually appeared as a boar when it should be a rabbit. It’s found its way into this impossible place, when it should be… rm: When you multiply one hawk by two, you get rabbits? cs: It sounds crazy but… mw: I think there’s the beginnings of something there, but it could confuse people, in that it’s not that rabbits are the only thing there can be two of. rm: It’s a kind of reductio ad absurdum of how numbers that apply to everything, to any object indifferently, and even to other numbers. If we’ve reached the point where we can hold onto the validity of the model and do something really bizarre with it, we have to give equal effort to both. cs: With my work, I’m trying to create a model that doesn’t work. That’s actually much more interesting. For me, the tool for explaining what we’re trying to get at, okay it’s complex and rotates around and does exactly what you want, but it’s failing to be a real springboard for this epistemology of numbers. rm: Matthew is just concerned that his original question isn’t lost, which is the idea that multiplication does something strange to the number system. mw: And this model, without any reference to animals, illustrates that, but it’s lacking in any artistic interest. I’m listening to what you’re saying. My thought would be, okay let’s find an animal for which there’s a name for one two and three, don’t mix up the animals. And make the simple point that, let’s say, if you could have a brace of rabbits and you could have a lease of rabbits, say, you still can’t

have a brace of leases, because a lease isn’t a rabbit, it’s an amount of rabbits. But that’s a simple point that you can make without reference to the animals anyway. So I think I’m just going to give you some more rope to… rm: Conrad is also showing how bizarre it is that you can abstract number from specific things at all. cs: You always learned at school by adding up cakes or whatever, but when you start multiplying them together, it’s a really strange thing, the specific objects start to disappear. rm: Like when you were saying, Matthew, that some cultures have different words for three of something than for three of something else. If you said to them, multiply that by that, they would say, how can you possibly do that? They’re heterogeneous. // mw: Somebody has to be there to move it around if it’s going to be demonstrated. Otherwise you’ve got a static installation. Unless you use electrical switching. It could be done with just a handle…it could be set up so you can do it without causing any damage, you just release something and it swings around, then you let it go and it swings back. If there isn’t something like that, then it just looks like a load of screens—we’re drifting further and further from the original mathematical idea. pc: So you want to make something that functions? mw: Well, I can’t see the point of all this wiring if it doesn’t. If you want to do a demonstration were you hold things up and show people, that’s fine. But an exhibition can’t involve that. kh: Conrad, you’re not drawing one that moves over there are you? cs: I’m doing a multiplication table. I suppose I’m more interested in…how you can’t boar a hawk, it’s just more difficult, I’m interested in the impossibility of this one.

mw: Having a moving device doesn’t at all compromise what Conrad’s doing. pc: The question is, on the table here, during the exhibition, is this going to look like an attempt at an artwork? kh: It is, I don’t think it’s going to come across. I think it’s going to be really difficult—We’re going to have to explain it to everyone all week. rm: We have visual evidence of the whole process we went through here...and then halfway through, Conrad challenged the rest of the team to create a model to respond to…and he has. You can tell by the context that it’s something that’s part of a process, not a finished work. cs: Matthew, can you fill this [multiplication table] in? mw: I don’t really know what I’m looking at, to be honest. cs: Just imagine that you have to do it, there’s no choice. mw: Okay, I’m drawing a leasebracehawk. This will be used to blackmail me some time in the future. cs: Is it a bracenut times a leasenut? mw: Well, there’s commutativity involved in multiplication, so… cs: Leasenutbracenut. mw: Leasebutt…lease…braceleasenut. cs: Shouldn’t the nut come twice? mw: It shouldn’t do…I mean if you have six nuts, the nutness is still…but what’s the difference between a leasebraceboar and a braceleaseboar? Since this is the first prime number…. Are you sure the fumes from the magic markers aren’t affecting our brains? Magic marker, magic mushroom…. Okay, that’s a boarboar. cs: Good.

mw: So a boarnut is just a peanut, it’s just a different way of talking about it. And a boarboar and a boarboarboarboarboarboar would be the same wouldn’t they. You can prefix any word with ‘boar’ and it makes no difference whatsoever. cs: So a leasebracehawk is equal to a nutnutnutnutnut? mw: Ummmm...no, because a nut can never be transmuted into a boar. You can’t get from one to the other. pc: There is some concern that this is pointless, making this model. kh: Yeah, I’m just worried that making the model and the diagrams are the same thing really. cs: No, I think it’s going to be an exciting thing—you’ve got the addition and the multiplying, which is absolutely fine, but I’m more interested in the impossible one, we can put in as our input different things—so we put in the hawk, or the boar… mw: So we have a prop whereby we can talk about the distinction between our approaches, for me it’s a clear representation, whether it’s rabbits or boars makes no difference. cs: But as soon as you actualise it, it will be different. I think we should build it, then put other inputs in at the secondary point, to see if you can start messing with it—put a rabbit next to a hawk, see what that does to our system…. kh: Okay, I see, I see. I was just concerned that…. pc: So yeah, what is the status of lunch? Or shall we have a snack? kh: I think everyone’s probably had enough nuts. cs: At the moment I’m using ‘boar’ instead of ‘singular’. Because singular can refer to other things. Maybe we just need to invent our own words. Robin, can you think of a singular animal, at the moment the system is collapsing because of this. Dodos? rm: Jabberwocky?

// cs: It’s quite abstract, you have a leasebracehawk. A leasebracehawk is a hawk, you put the hawk here, and here, you get a leasebracehawk. When you see six hawks on a screen you can describe it as a leasebracehawk. rm: But my question is, how is that going to take place? Initially we were thinking it would just be a machine and people would come and see it. cs: At the symposium it can exist in different guises. rm: But the original idea was that you’d have these things side by side and then one in front, and it would demonstrate the principle. cs: Yes, I’m just more interested in the more impossible philosophically challenging version. rm: But the fact that you’ve got these words means you don’t have to, because these words are that, that simple state, and then you’re putting them into multiplication and showing the things that happen, in a way you’ve made it more elegant...but it makes it harder work for the viewer. This is also like the idea of molecular numbers that Matthew was talking about a while ago: when numbers are factorised, they are these clusters of different elements that don’t homogenise, and that’s what’s happening here. The brace lease and boar are the atomic components. cs: The nominal primes. rm: This compression of it makes the object simpler but it means the thought processes are much more involved, twisted, compressed. cs: We could maybe do another prime. Maybe the first ten primes, using animal names…we should apply to the Arts Council to open a petting zoo for prime animals, where the secrets of the universe could be revealed by getting them all to multiply in some sick way.

mw: The idea of working with words which capture a multiplicity as a single entity is promising in terms of a philosophical discussion. But what’s spreading out from it doesn’t necessarily lead to a greater understanding. It’s spreading possibilities and ideas and associations, and that’s the artistic process, and then there’s me trying to draw it back into a framework of rational discourse. rm: Where will that process end, if at all? pc: Let’s see if we can get this to work then…. mw: That’s it. If Conrad thinks its ugly, that’s perfectly valid, he’s an artist and he can critique at that level, but then he’s disengaged from it as an artwork anyway, so it doesn’t really matter, It’s a model, that’s the whole point, it’s not an artwork. But it’s funny, it’s almost a kind of exploration of the space between the artwork and the model. Because you’ve got a piece of sculpture—even if its just a plasticine rabbit! rm: Well I suggest we make this up as barely as possible, not on a pedestal as a finished work but just as one part of what’s been done. mw: Just like any other bit, even though it’s more elaborate. pc: Yes, .I think we came to that conclusion—this is just… rm: …a sketch…that took a long time to draw. // cs: So what shall we call it? The track, we’ve now got this track, a road that takes you from addition to multiplication. mw: Thinking of it as a road is quite interesting, because I’d thought of it as being either there or there, not as travelling between them. As if there were a continuum of possible positions between addition and multiplication. cs: It’s a panning shot.... pc: Where’s the screwdriver?

1. M. de Sautoy, The Music of the Primes (London: Fourth Estate, 2003), 280. 1. M. Watkins, ‘Prime Evolution (Interview) in R. Mackay (ed.), Collapse 1: Numerical Materialism (Oxford: Urbanomic, 2007). 2. See <http://www.danieltammet.net>. 3. See M. Berry, ‘Quantum Physics on the Edge of Chaos’, New Scientist, 19 November 1987. 4. A. Badiou, Number and Numbers, tr. R. Mackay (Cambridge: Polity, 1990). 3. <http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/encoding1.htm >. 5. ‘The word boar, as well as being a noun for the male pig, has become over time the accepted term for wild pigs. It was the Egerton Manuscript of 1452 that first recorded the collective noun as a singular of boar, with later authorities adding the term sounder, which became the more popular. The word singular certainly seems odd for a collective noun. C.E. Hare suggests that even though it appears in many lists it is indeed a mistake, quoting the Book of St Albans as confirming its application instead for a single boar of four years or older - the true company term is sounder.’ S. Palin, A Murmuration of Starlings: The Collective Nouns of Birds and Animals (Ludlow; Merlin Unwin, 2013), 79; Dame Juliana Berners, The Boke of Saint Albans [1486] (London: Elliot Stock, 1901), ‘The Companynys of beestys and fowlys’. 6. Or leash—see Palin, A Murmuration, 9.

Matthew Watkins Commentary on the Assemblage This assemblage illustrates a philosophical point which I raise in my book Secrets of Creation concerning the relationship between addition and multiplication. The following is an adaptation of the relevant passage:1 [M]ultiplication is clearly a more complicated matter than addition. Addition is well understood, but as the mysteries of the prime numbers are gradually revealed to you, it will become clear that we don’t fully understand multiplication or, more precisely, that we don’t understand the relationship between multiplication and addition. Louis Kauffman, a University of Chicago mathematician whose wide-ranging interests include the philosophical foundations of mathematics, makes this point: Multiplication is more complex [than addition]. When we multiply 2 × 3 we either take two threes and add them together, or we take 3 twos and add these together. In either case we make an operator out of one number and use this operator to reproduce copies of the other number. The word ‘operator’ is used a lot in mathematics, incidentally—its meaning is very precise, but for our purposes it can be thought of roughly as ‘something that does something to something else.’2 The point that Kauffman is making is that with multiplication, the two numbers involved are playing different roles…. Think about it this way: 2 and 3 are counting numbers. In the context of ‘3 × 2’, what are they counting? The 2 could be counting steps taken along a line, beans placed on a table or marks made on paper. The 3 is counting the number of times this action of taking two steps, placing two beans on a table or making two marks is performed. These are very different kinds of counting. In the case of the 3, the things being counted (something happening twice) all have the number 2 somehow buried inside them. They can all be ‘segmented’ into two similar pieces. These

pieces are the things which the 2 is counting (steps, beans, marks), and these are not ‘segmentable’. They’re basic units, the sorts of things counting numbers are usually thought of as counting. But multiplication suggests another kind of counting—‘the counting of countings’ or counting turned in on itself. This innovation has some very strange consequences, the irregularity of the prime number sequence being just the beginning. In our assemblage, the categorical difference between addition and multiplication is illustrated using complexes of small cameras and video screens. The wheeled platform moves along a track between two positions. The front position illustrates the simple fact that 3 + 2 = 5, with the fixed complex splitting the image of the rabbit into two identical images and the moving one splitting it into three. In effect, one complex is ‘twoing’ the rabbit and the other is ‘threeing’ it. Both are performing the same kind of role, and this is made clear by the fact that the grouping of two screens and the grouping of three screens are positioned beside each other on a line. The back position illustrates the (less simple) fact that 3 × 2 = 6. The fixed complex again splits the image of the rabbit into two, but now the camera on the moving complex is no longer pointing at the same thing as the other camera—it’s pointing at the output of that camera, that is, at the pair of screens, representing the ‘twoing’ of the rabbit. So the role of the 3 in this configuration is categorically different from the role of the two—to put it simply, the 2 is counting rabbits, but the 3 is counting 2s. Rabbits are well-defined physical entities which can be found in the external world (and hence counted), but what are 2s, and where would you find three of them? The original design didn’t include the first stage of the assemblage (which involves a single screen). It would still have illustrated the above point, but the introduction of this seemingly unnecessary element adds another layer of meaning. The single screen effectively brings the physical rabbit into the world of number—it ‘ones’ the rabbit, reducing it from an ‘actual’ rabbit with distinguishing features to a mere token, representing the category of rabbits—the rabbit has been brought from the realm of matter into the realm of number. This ties in with the observation that there cannot really be two of

anything: subjected to sufficiently careful scrutiny, any two ‘similar’ entities will be seen to display differences. The ‘twoness’ is not to be found in the external world, but in the human act of naming, defining and categorising. Only by reducing the rabbit to a numerical token can it be meaningfully doubled or tripled. An unintended consequence of the use of inexpensive cameras and video screens is the gradual loss of image fidelity which occurs as we move from three-dimensional rabbit to a single rabbit on a single screen, to two rabbits on two screens and finally to six rabbits on three screens. At each stage, the rabbit loses a certain amount of its identity. This corresponds nicely to the way that first counting, then addition, and finally multiplication move us progressively further from the entities which are being subjected to these processes. We can imagine moving from, say, a veterinarian having direct contact with an individual sheep (with full awareness of its individuality as self-organising, metabolising, sentient being) to a shepherd counting of a flock of twenty sheep (with partial awareness of the individuality of each sheep) to a large-scale sheep farmer calculating that his ninety flocks of twenty sheep constitute a total of 1800 sheep (with almost no awareness of their individuality). Although the distinction between multiplication and addition may seem a fairly simple and marginal issue in relation to the vast complexity of higher mathematical research, it acts as a starting point for the investigation of what is arguably the central mystery of mathematics—that of how the prime numbers are arranged within the sequence of counting numbers. The primes—2, 3, 5, 7, 11, 13, 17,…—are those numbers which are indivisible. As division is a kind of the inversion of multiplication, the notion of ‘primeness’ only comes into being when we start multiplying, that is, when we apply the number system to itself. Despite the culturally familiar, seemingly innocuous nature of multiplication, it is a conceptual innovation, a categorically different kind of use of number, and the consequences of this can be seen as a kind of ‘feedback’ or ‘interference pattern’ caused by the number system interacting with itself. The consequences of this are still being worked out, with the Riemann Hypothesis (the central unsolved problem concerning the distribution

of prime numbers) having, after 150 years of futile efforts to resolve it, achieved the status of a ‘Holy Grail’ of mathematics. The philosophical implications of these issues for our understanding of time, perception and (perhaps) consciousness itself will be explored in the third volume of my Secrets of Creation trilogy. 1. [This text was written by Matthew Watkins as accompanying documentation for the exhibition at Urbanomic Studio following the residency project.] 2. M. Watkins and M. Tweed, Secrets of Creation, Volume One: The Mystery of the Prime Numbers (Dursley: The Inamorata Press, 2010), 123–24, citing L. Kauffman, ‘Virtual Logic–Formal Arithmetic’, Cybernetics & Human Knowing 7:4 (2000), 93.

Notes on Contributors paul chaney is a self-taught artist and horticulturalist exploring agrarian futurism through long-term engagements with land and agricultural process. From 2004 to 2012 he lived entirely disconnected from public utilities in a self-built cabin and attempted to survive on a two-acre field. Since 2016 he has led End of the World Garden, an off-grid artists’ residency and research platform in a perennial forest garden. He is associate artist at Kestle Barton Gallery in Cornwall, where his ongoing project Lizard Exit Plan explores a post-collapse survival plan for the inhabitants of the Lizard Peninsula. <http://www.paulchaney.co.uk/>. kenna hernly is an art museum educator and researcher living in Washington, DC. She designs and researches ways to support children and adults in learning through art. She is currently a fellow at the Smithsonian American Art Museum. robin mackay is director of Urbanomic, has written widely on philosophy and contemporary art, and has instigated collaborative projects with numerous artists. He has also translated a number of important works of French philosophy, including Alain Badiou’s Number and Numbers, Quentin Meillassoux’s The Number and the Siren, François Laruelle’s The Concept of Non-Photography and Éric Alliez’s The Brain-Eye and Undoing the Image. conrad shawcross is a British sculptor whose work explores subjects that lie on the borders of geometry and philosophy, physics and metaphysics. <http://conradshawcross.com/>. matthew watkins completed a PhD in mathematics in 1994. He is the author of Useful Mathematical and Physical Formulae (London:: Wooden Books, 1999) and of the Secrets of Creation trilogy, both illustrated by Matt Tweed. Since 2000 he has been an Honorary Fellow in Exeter University’s mathematics department and is curator of the online Number Theory and Physics Archive and the related

(but more popularly accessible) site ‘Inexplicable Secrets of Creation’. <http://empslocal.ex.ac.uk/people/staff/mrwatkin/>.