Inside every idea is the seed of its own undoing. This is obviously quite depressing but it’s also interesting – it’s where artists live. ‘What’s the painting that can’t be painted?’ they think. ‘What’s the book that can’t be written?’ Beneath this the metaphysicians take over: ‘What’s the word that can’t be said?’ And before that, long before, we might imagine a Palaeolithic figure tripping on the greenest shoot of this idea, finding himself blinking into the great conceptual abyss: ‘What’s the thought that can’t be thought?’

This is admittedly a fringe problem, unhelpful for the task of spearing mammoths or finding the freezer aisle. But it becomes much more immediate when we hit on an idea which is made *entirely* of it, which is by definition indefinable. Things get truly hairy when this ineffable thought is also one which everyone everywhere seems to have, as if we’ve been built to contain an unresolvable error, a kind of philosophical auto-destruct.

This is troubling, but, being a perverse sort of species, we just worship it until it seems normal. We might call our Palaeolithic man’s new idea the Beyond, but it’s better called infinity. He reached infinity merely by taking small blocks of language and shunting them around in what is essentially a procedure of logic, and it’s useful to bear this in mind when thinking of the large and magical things that here emerge. We might imagine him rephrase his thought – ‘What’s the thing which I can’t see or hear, but which is nonetheless a thing?’ – and come up with gods.

Since this logical shunting is so simple, the idea of infinity emerged wherever there were people, and it surely needed expressing in richer forms. There was no notion of science or maths to frame it, but there was paint, clay and limestone. An ancient symbol of infinity is a snake devouring its own tail to form a ring. This is called the ouroboros and it pops up in Plato’s *Timaeus* in Greece, statues of Shiva in India, carvings of Quetzalcoatl in Mexico, burial trinkets in ancient China and Tutankhamen’s tomb; in Scandinavia, Sumeria, Japan, Rome.^{1}The ouroboros signifies infinity as it does interrelated chin-strokers such as ‘cyclicity’ and ‘collapsing duality’, so I am of course condensing the many connotations of this old and conveniently global symbol. The reason for the appearance of specifically this image all over the ancient world could be vaguely posited as some slow progression, perhaps from Egypt outwards; its occurrence in the pre-Columbian Americas, however, is well beyond my meagre powers of research and explanation, not to mention quite eerie.

**Crass Generalisation #1**: mathematicians think infinity is used in shamefully careless terms even by physicists. One can guess from this at how they view its use in the humanities and, the horror, by artists.So infinity had a home in the supernatural and its vehicle was art, but eventually there appeared a second runaround, less sexy but much more practical: mathematics. We might wonder, then, what the relationship between the arts and maths is insofar as they express infinity, and whether there is a mutual appreciation.

**Crass Generalisation #2**: ask an artist what they think of the broader relationship between the arts and maths and they will drawl of a beautiful reciprocity, an ouroborean circle-jerk; ask a mathematician and they will shrug before getting back to work.

But let’s not forget that it’s artists whose antecedents captured infinity so powerfully in the ouroboros, coiling it around the more analytically minded generations to come. Perhaps there really is something of a cycle here, if we look close enough.

This essay will attempt to look close enough. The first part will take the form of a case study, charting a (necessarily selective, somewhat breakneck) history of the infinite, asking what infinity has done for the arts. The second part will consider the relationship between the broader disciplines, including science, and the nature of influence, asking the more productive question: what can the arts do for infinity?

Contents

## 600BCE–1CE: Pythagoras to Aristotle

Our story begins with another of the great arts-maths overlaps, the golden ratio, as revered by the number-cult of the Pythagorean Brotherhood. Like all ancient Greeks their maths was essentially built of geometry, rooted always in what could be drawn, the visual. The Brotherhood loved shapes and was quick to link the numbers underpinning them with aesthetic quality. Pythagoras determined that the key to beauty lay in proportions, this exemplified by the golden ratio, which can be found in everything from your arm (the ratio of your forearm to your hand) to the Great Pyramid of Giza^{2}Pythagoras had visited Egypt and it’s perhaps no coincidence that the earliest known examples of the golden ratio, of mathematical infinity, and of the ouroboros are all from there. The ancient Egyptians had a god of infinity and believed the world to be a hollowed-out sphere in an endless, surfaceless sea. The god sometimes has the head of a frog (all the better to swim the cosmic abyss) and is called, somehow appropriately, Huh. (the ratio of its slanted height to half the base’s width), and this was a guarded secret.

The genius of the Pythagoreans was that they took the highly practical count-with-your-fingers maths of the time and unstuck it from the material world. They did this with a new and rigorous approach of posing and testing theorems, propping them with axioms, logic – but they tied this together with mysticism.^{3}I’ll forgo the customary list of the Brotherhood’s eccentricities as, although by all accounts a loon, Pythagoras was probably one of the smartest and most influential humans in history. Even if he did religiously codify his hatred of beans. What’s hard for us to grip is that it may have been exactly the Brotherhood’s mysticism, their non-material, essentially religious approach which allowed them to take maths into the realm of abstraction we associate it with today – a realm we like to see as governed by reason and therefore as distinctly profane. For them the natural numbers (as in whole, non-negative numbers) were endowed with special qualities, the neat relationships between them demonstrating completeness, perfection, and therefore godliness. ‘God is number’ became the mantra. That is, whole numbers and ratios, collectively called ‘rational numbers’.

The cataclysmic event may have come in a few forms: in using their master’s famous a^{2}+b^{2}=c^{2} to find the length of a square’s diagonal; in pondering the square root of 2; even in their beloved golden ratio. What’s certain is that, when calculated fully, each of these things yields a number made of non-terminating, non-repeating decimals, a number *which does not end* and which, lacking completeness, could blow their religion to bits. One of the Brotherhood found such a number (known now as ‘irrational numbers’, which include pi), though we don’t know which number it was. The few Pythagoreans party to this discovery swore an oath of silence, but inevitably someone blabbed and, depending on the account, was strangled, drowned, struck down with foul sorcery, or ran away.

As this idea of mathematical endlessness was spreading, Zeno of Elea came up with his famous paradoxes. The most well known is that of Achilles and the tortoise, wherein the fastest runner in Greece oddly challenges a tortoise to a race but soon finds that, as his distance to the tortoise halves, and the remaining distance halves again, and again, he is always nearing the tortoise but can head only for an infinite point, never catching it. This is within the bounds of logic, it seemed, yet does not reflect observed reality, so it deeply troubles the assumption that logic is inseparable from truth.

It was a triple suplex of philosophical scepticism, reducing the grand edifice of logic (on which maths is built, let’s not forget) to mere linguistic sleight of hand, and it’s possible that Zeno did it just to annoy people.

Which it did, and none more so than Aristotle. Of his many refutations of the paradox, the biggest was to hold up its core claim, ‘there are infinite points between A and B’, and draw a distinction between the *potential* infinity implied by the statement’s logic and the *actual* infinity which he claimed in practice is not there. This is to permit ‘potential infinity’ as a means to an end, a conceptual tool, while keeping the unnerving ‘actual infinity’ safely out of the picture. It doesn’t resolve the paradox but sort of sterilises it. This notion dominated the understanding of infinity for the next two thousand years.

## 1–1500: Augustine to Nicholas

In the fourth century Augustine of Hippo says, in what sounds like the set up to a low-level drug deal, that we can’t know infinity directly but we can know this other guy, God, who in turn knows infinity. In the eleventh century, Anselm of Canterbury says that God is perfect in that he is ‘a being nothing greater than which can be conceived’, here playing with the idea of limit in a way common to the subject since Zeno.^{4}And which makes a monkish precursor to the ‘times infinity plus one!’ A-bomb of primary school mathematics. That would be my reading, without wishing to denigrate the clever saint, though my most ballbreaking old philosophy lecturer published and tutored (at length) on Anselm, and at this I hear his beard catch alight. In the thirteenth century Thomas Aquinas says that no earthly thing can be infinite because this would mean there was no ‘beginning’ point at which the ‘first cause’ (or God, himself infinite) got the universe going. The theme here is that Aristotle’s strictly off-world infinity has been transposed to God.

Now in Spain, 1280CE, Moses De León publishes the Zohar, the foundation text of Kabbalah.^{5}He claimed the Zohar was written in the second century, though he probably wrote it himself. At the heart of Kabbalah are the ten sefirot, each representing a human virtue, a body part, a colour, and together making a mind, a body, a brilliant, colourless light – the ‘ein sof’. This translates to ‘unending’ and is of course God, infinite and therefore unknowable in his entirety but knowable when broken into the ten aspects. ‘Ein sof’ starts with the Hebrew letter ‘aleph’, first letter of the alphabet and of the Hebrew word for ‘one’ (‘echad’), and so is deemed of primary significance in Kabbalah, which is riddled with codes and associations like this. As with the Christian theologians, God is still the ‘answer’ to what infinity is, but what differs here is the complex mathematical and numerological systems laid over it. Numbers are creeping back in.

Early Renaissance. Nicholas of Cusa is sailing back from a Papal mission to Constantinople when he has a hallucinatory vision of the infinite. He claims it has gifted him the power to understand the ineffable and explain it to others. Strongly mystical, he was fancied a pantheist and believed in many possible worlds. Central to his thought was that nature is structured by opposites living within each other^{6}Many gods, many worlds, merging opposites: all ideas found in Hinduism, though there’s no link to be seen between Nicholas and eastern religion. Such challenges of continuity are a historian’s nightmare and storyteller’s dream, as we’ll see in Part 2 when we look at the problems of influence. and therefore God’s infinite nature must also exist within humankind. This was important because it brought infinity back down to earth, and in a way that somehow didn’t rile any bishops.

Giordano Bruno was another mystic churchman who made significant contributions to maths and science, similarly claiming that there are an infinite number of earths, suns and universes. Though for this he was, in a somewhat inconsistent moment from the church, burned alive.^{7}In this period scientific minds were born into a world where the only place for them was the church, in the case of Bruno leading to a life of misfit wandering and fiery death. But it’s too easy to say that medieval Christianity didn’t know what to do with people like them because truly neither do we. Nicholas suggested before Copernicus that the earth is not the centre of the cosmos and described the motions of the planets before Kepler – astonishing insights, yet he reached them by almost entirely mystical means, by which I mean magicky guesswork. What are we to do with the fact he was right?

## 1500–1700: Galileo to Leibniz

Ironically it was the church which ushered in the dissolution of religious infinity. In the early 1600s Galileo Galilee began using his newly developed telescopes to strengthen the Copernican theory that the earth moves around the sun, not vice versa. The church, very much on the vice versa side, made him sort-of-recant his theories and live out his days under house arrest. Being restricted to his home meant large-scale experimentation was no longer possible and he delved further into pure maths. There he developed the idea of one-to-one correspondence, which was to have enormous repercussions on the understanding of infinity.

When we count we are, bit by bit, amassing a pile of imaginary things, labelling each new iteration of that pile with a number. This is to adopt a metaphysical correspondence between things, lots of ones, and a singular notion of their quantity. All basic stuff, though we rarely need to think about it directly. Galileo developed the idea that you can make a similar correspondence between numbers and other numbers. Consider the natural numbers. Now consider, say, all the even naturals, and correspond them so 1 (the first natural) is matched to 2 (the first even), then 2 to 4, 3 to 6, etc. The natural numbers are of course infinite, so with this correspondence in mind the even numbers must be equally infinite because they continually match up. This means that, while we might intuitively expect there to be half as many even numbers as natural numbers, there are actually just as many.

Galileo was allegedly so shocked by this seeming derangement of number that even he, smiter of received ignorance, suspended all investigations into infinity. But it had now been looked on with the modern eye, its elements separated and mathematical depths glimpsed, and others would follow.

Later in the seventeenth century, Isaac Newton and Gottfried Wilhelm Leibniz made discoveries in the study of mathematical change and measurement, each claiming different routes to what would eventually become calculus. Both, however, made use of infinity as a way of predicting rates of change using infinitely small measurements. For Newton this was the temporal ‘fluxions’ and for Leibniz the quantitative ‘infinitesimals’. This happened at a time when all the applied sciences were confronted with continuous phenomena they needed to resolve and it was quickly influential. Its foundations, though, were philosophically problematic as they took infinite quantities (like the point into which Achilles recedes) as synonymous with the whole numbers they are *near* (like the tortoise). Achilles may be very near the tortoise, but he is not the tortoise.

## 1700–1800: The romantics

Assorted spleens were vented at the imprecision of this method, but irritatingly all of the calculations done with it seemed to work. This induced a small conceptual crisis in all that calculus touched, which was many areas. That includes the arts, but, like the problem of ineffability was made liveable simply by worshipping it (as in, religion), the crisis of calculus was absorbed into the arts’ articulation of it. Philosopher Edmund Burke formulated the idea of the ‘sublime’ directly from the infinity in Newton’s calculus, this forming a cornerstone of the romantic movement. For the romantics, who needless to say had a lot of feelings, the sublime was a boundless enormity and beauty, a natural state of the mind, but it was also disturbing, at its heart that certain instability. Infinity in early calculus was less about what it *was* and more about what it *did*, its effects. The romantics switched this from calculations to emotions.

William Blake wrote: ‘If the doors of perception were cleansed every thing would appear to man as it is, Infinite. For man has closed himself up, till he sees all things thro’ narrow chinks of his cavern’.^{8}Aldus Huxley would later name *The Doors of the Perception* from this line, having famously eaten some hallucinogenic cactus and found infinity in his trousers. ‘Those folds in the trousers – what a labyrinth of endlessly complex significance!’ Samuel Taylor Coleridge wrote that Gothic architecture, the ornately complex, dark and godly style, ‘is infinity made imaginable’. His poem ‘Kubla Khan’ and its dreamlike deep-consciousness stanzas, ‘Where Alph, the sacred River, ran / Thro’ Caverns, measureless to Man / Down to a sunless Sea’, reads as a man in direct communion with the sublime (and a lot of opium).

## 1800–1900: Cantor

Come the nineteenth century, the steady re-emergence of Platonic infinity^{9}That is, the possibility of ‘actual’ infinity existing as mathematical objects in the world as typified by the idealism of Plato, who believed that every worldly object aspires to an ideal, perfect version of itself, which is more basic to reality than those objects we perceive directly. This is in contrast to the Aristotelian view of actual infinity being but an impossible aspiration, as mentioned. The history of infinity can be read as a seesawing between these two ends of the philosophical playground. Spoiler: most of today’s mathematicians are at the Platonic end regarding infinity and ‘maths objects’ in general. Though the idea that engaging in maths is to assume a metaphysical position, that the philosophy *precedes* the numbers, is something that (Crass Generalisation #3) gives mathematicians the willies so they tend not to think about it. is building into the most substantial period in the subject’s history. The undisputed king of this time, and perhaps of infinity as a mathematical subject, is Georg Cantor, who was referred to at different times as Russian, German and Danish, Christian and Jew, and smeared as ‘scientific charlatan’ and ‘corruptor of youth’ (Leopold Kronecker) whose ideas were a ‘grave disease’ (Henri Poincaré).

Cantor’s approach was to avoid thinking about numbers directly, instead treating them as groups, or sets. Everything around us can be organised into sets and sub-sets, and this, Cantor said, is the most basic stuff of maths. He used it to think about concepts, in particular types of numbers – like Galileo had before. Since numbers are infinite he was working with sets of infinite size, and these combine in different ways. The set of all real numbers is made of all irrational numbers (those with endless, non-repeating decimals) and all rational numbers (those with finite decimals); within that, the set of rational numbers is made of integers and whole numbers; further in, the set of whole numbers is made of natural numbers and zero. We see the sets shrink concentrically as the definition tightens. This manipulation of groups, known as set theory, is important because it’s the basis of perhaps the most central discovery in the history of infinity.

Knowing the set of all whole numbers to be infinite but countable (in the sense that, though we could not literally count them all for lack of time in the universe, we nonetheless know how we would: by continually adding one, therefore they are ‘countable’), Cantor imagined the set of all real numbers, which contains every number with a decimal point, including those non-terminating irrationals like the golden ratio and pi. He envisaged the numbers in any order and endlessly on top of each other in a list, decimals extending ever out. An infinity of infinities, therefore – but in theory ‘countable’. He then marked a diagonal line from the first unit of his list, making a new number out of that first unit of the first number, the second unit of the second number, the third unit of the third number and so on. He took this diagonal number (which must already appear somewhere else in the list) and changed it in simple way. It can be anything as long as the same change happens to all units. For our example, let’s change all even numbers into 1 and all odd numbers into 2.

This means that the set of all real numbers is not countable, because letting the numbers increase in the normal way (adding 1) would not ever generate this new diagonal number. The set of all whole numbers remains countable, while both sets, real numbers and whole numbers, are infinite.This new diagonal number is now different at every point from the original number. Any potentially matching number elsewhere in the list must deviate from this new one at the point where this new diagonal intersects, since the simple rule has modified it, so this new real number must be different from every other number in the entire list. Even though the list should contain, in theory, every real number.

Therefore the infinity of the set of real numbers is ‘bigger’ than that of the set of whole numbers, and so is a different type. Having demonstrably different types of infinity means there must be an infinity for there to be different types of. This means, for Cantor, that infinity is real. Not just a useful idea but something truly observable. He used the Hebrew aleph symbol (ﬡ) to denote the different kinds of infinity he found.

This represented a major shift in how infinity was used. For Cantor, a religious man who believed that God was directly guiding his understanding of the infinite, it was not just a technical achievement but a spiritual and emotional one. He saw these latter aspects as part of the maths. How, then, to express the powerful intellectual tremors he caused?

## 1900–2000: Borges

The very singular brand of fantasy practised by Jorge Luis Borges often uses mathematical and scientific concepts as its hidden foundations. He had an almost pathological reflex to the infinite and he knew his Cantor. In his short story ‘The Book of Sand’ a mysterious Bible seller appears with a book where the same page can never be found twice; where pages are numbered in wildly large sums and in no order; where the first and last page can never quite be separated from the covers. It’s the book that can’t be written – the infinite book.^{11}One might say that the Book of Sand’s pages mimic the numbers in the set of all rationals (since the rationals are long but finite, like the contents of a page are bound by the page’s size), and in an especially highfalutin mood one might say that the combinations of decimals in each number must be relatable to the combinations of letters on a page, and therefore, just as the infinite book contains every book ever written, the set of all rational numbers also contains, in simple code, every book ever written.

In the story, the narrator, who is a fictional Borges, excitedly buys the book and hides it ‘behind some imperfect volumes of the *Thousand and One Nights*’.^{12}Of which Borges (the real one) wrote about obsessively. It is an infinite book, the tale of a young queen telling night-long and bewilderingly nested stories in order to stay her decapitation by a king. On night 602, Borges notes, ‘the king hears of his own story from the queen’s lips. He hears the beginning of the story, which includes all the others, and also – monstrously – itself. Does the reader have a clear sense of the vast possibility held out by this interpolation, its peculiar danger?’ Indeed. You may like to hear that Borges wrote this in his apparently quite esoteric column in *El Hogar* – Argentina’s answer to *Tatler*. But in his infatuation with the book he soon sees no one else, thinks of nothing else, and realises that the book is an obscenity, a ‘nightmare thing’ which is devouring him. This awesome, intoxicating and ultimately terrifying feeling is one of the senses of infinity which permeate Borges’s work, here in its fantastical objects which in turn illuminate the fantasy in our own everyday objects. The Book of Sand, for instance, is in the hands of a Bible seller and bought in exchange for a Bible – that book which is equally infinite but in *interpretation*, endless routes to the old infinite God, and everywhere else besides.

In another story, ‘The Aleph’, a poet writes an enormous and grandiose poem about all the regions of the world, which Borges-the-narrator mocks. But it emerges that the poet, Daneri, has in his cellar a three-centimetre-wide glowing object called the Aleph which contains, ‘without superposition and without transparency’, all the points in the world – everything – and he has been using this to over many years write his globetrotting poem from direct experience. That the poem had been deemed poor and pompous by Borges provides the obvious reading that infinity can’t be even faintly captured in art. To become aware of infinity, it suggests, is to become aware of the depth of one’s inability to describe it, even to remember it, and therefore the infinite is inherently sad.

And yet the poem is successful while narrator-Borges’s work flops, and Daneri is named after *Da*nte Alig*eri* for having gone on the Aleph’s world tour, like narrator-Dante tours the afterlife in *The Divine Comedy*. Why name the pompous Daneri after Dante, whom real-Borges idolised, and make the poem a success? Because the truly pompous character is Borges. It’s a self-critique of his own inability to write infinity and the thing which infinity for him represents.

The ghost of the story is Beatriz, the narrator’s dead lover, named after Beatrice, narrator-Dante’s guide through Paradise, in turn named after Beatrice Portinari, real-Dante’s lifelong obsession who died young. In ‘The Aleph’ the narrator’s palpable longing for Beatriz is in the end what gives infinity its mournful shape. What this suggests is that Borges, whose stories contain almost no women, and who rarely mentioned the idea of love in all his writings, associated infinity, about which he wrote constantly, with love. These things were joined for him in their beauty, awe and inherent, inevitable sense of loss. In his way he wrote about love endlessly.

## 1900–2000: Sierpiński to Mandelbrot

Back in the realm of maths, the twentieth century saw infinity get visual. During the First World War, mathematician Wrołcaw Sierpiński conceptualised the Sierpiński gasket, a pattern of linked regressing triangles. This has been retrospectively dubbed a fractal, which is an infinitely complex and self-similar shape or pattern, something which is equally complex at different scales.

As Sierpiński was developing his triangles, fellow mathematicians Gaston Julia and Pierre Fatou were using a transformative process to create simple mathematical rules which, applied to a point on a plane (or a dot on a piece of paper), would generate another point in a certain place. The transformations were treated as iterative functions (the repetition of a mathematical action many times over in order to ‘build up’ an object) and repeated many times. Through the initial given rules, patterns began to emerge in the dots, underlying and complex orders which had been hidden in the rules, waiting to be visualised.

In the 1970s Benoît Mandelbrot was researching fractals at IBM, using some of the world’s most powerful computers to iterate the same rules a colossal amount of times, more than Julia and Fatou ever could, and produce what came to be known as Julia Sets – swirling baroque shapes, both organic and jewel-like. Mandelbrot wondered what connected all the different Julia Sets. He devised a map of the rules which make them, where each rule has a location, and marked a point wherever there lay a rule which would result in one kind of Julia Set (‘connected’, referring to the type of swirl produced), and left it blank if it was another kind (‘unconnected’). The computers iterated this process many times over and the results were printed out.

What emerged looked like the impact of a horse-fly up a windscreen. It is also one of the most beautiful things ever witnessed. The Mandelbrot Set, which is the sum of all themselves-beautiful Julia Sets, is a shape with infinitely complex edges. This means that, with a computer, one can zoom endlessly into the edges of the shape for a vertiginous stack of galaxies to unfurl psychedelically around, revealing more Mandelbrot Sets brimming equally with points.^{13}Mandelbrot Set viewers can be found online and if you haven’t seen the M Set then it’s very important that you go exploring its edges right now. Any visual artist who is not in revelatory awe of the M Set should probably find another hobby. This from a collection of simple formulae.

The Mandelbrot Set is of course highly abstract, but the underlying order to which it alludes can be seen in surprising places. Mandelbrot was also an engineer and so had a mind for applying his maths to the world.^{14}That is in stark contrast to the prevalent force back in European maths in the ’60s and ’70s, led by a collective of mathematicians in France who met and published under the fictitious persona ‘Nicolas Bourbaki’. They favoured only completely abstracted maths with no relationship to the lived world. Pictures were forbidden, so you can imagine how they saw fractals. Curiously they may have had a strong influence on the avant-garde Oulipo group of experimental writers, but that’s another story (or turtlenecked in-joke masquerading as a story). Thinking in these terms he described fractal geometry as fundamentally about smoothness and roughness (or really *complexity*, like those infinitely complex edges). Smooth is signified as 1 and rough as 2, so values between can be ascribed as to an entity’s complexity. Mandelbrot gave the coastline of Britain a complexity of 1.26, and a typical cloud a rougher value of 1.35. The Mandelbrot Set, it was proved in 1991, has a value of 2.

Fractal geometry is a way of generating startlingly accurate-looking representations of all sorts of real-world phenomena, from the head of a cauliflower to the blood vessels in your hand (and your whole body, down to the rhythm of your heartbeat) to mountain ranges and the mouth of the Ganges, so there is the strong impression that it describes a fundamental organisational principle within nature. More than just physical entities, even behaviour can be described fractally: the ebb of crowds and stock markets, African traditional architecture and village structure.

## 1900–2000: Wallace

Artistically this is a tough act to follow. It helps that the more abstract patterns revealed by fractals are not entirely new to the eye, being pre-empted in Hindu mandalas, Islamic tiling, little Sierpiński triangles running across church walls, MC Escher’s tessellations, even (and perhaps all these examples descending from) the simmering half-seen complexities produced by closing your eyes. It doesn’t help, however, that the M Set, without the touch of any artist, is visually more astounding than all of these things. It is truly the painting that can’t be painted. So the market on visual recursion may well have been cornered, but what of recursion on a more mental level, like those fractal behaviours?

David Foster Wallace’s 1996 novel *Infinite Jest* was stated by the author, following a clairvoyantly astute interview question, to have been structured like a Sierpiński gasket. The interviewer, Michael Silverblatt, puts it:

It occurred to me that the way in which the material is presented allows for a subject to be announced in a small form, then there seems to be a fan of subject matter, other subjects, and then it comes back in a second form containing the other subjects in small and then comes back again as if what were being described – and I don’t know this kind of science – I said to myself this must be fractals…

Wallace had a serious interest in maths, studying modal logic at university, and this carried into an attraction to the underlying operations of stories, which can be plotted along familiarly mathsy lines: the Aristotelian rise, the Fichtean curve, Freytag’s pyramid.

Beyond affirming Silverblatt’s reading, Wallace did not elaborate on exactly how he devised a fractal narrative, so theories abound on *Infinite Jest*’s hidden workings. Following heavy editing the novel in the end became more a ‘lopsided’ Sierpiński gasket, according to Wallace, but self-similarity remains a thematic undercurrent.

Why do this? ‘It seems to me,’ says Wallace in that same interview, ‘that so much of pre-millennial life in America consists of enormous amounts of what seem like discrete bits of information, and the real intellectual adventure is finding ways to relate them to each other and to find larger patterns and meanings – which of course is essentially narrative’.

The millennium’s flood of information, in which we are all now adrift, was reduced by Wallace into those recursive thematic droplets. A fractal could aptly show this because, like the info-flood, its ‘chaos is more on the surface’, as he said, while ‘its bones are its beauty’ – those bones of the underlying order of things, the simple formulae.

Note, crucially, that this approach is not designed for mathematicians or merely those who might pick up on it. If fractals reflect a deep organising principle of nature, then maybe that same principle, buried in a novel, will make it, in his words, ‘taste right’, will make it resonate with something we feel happening around us but can’t articulate, revealing the unspoken core of things.

## The nature of influence

On the above history’s first viewing it may have appeared that, since infinity’s yarn-ball was unravelled into its respective strands – maths and art – the influence has been distinctly one-way. If we’re to instead think about how art could be influencer, rather than influenced (and of course not in any political sense but rather as something which gives and shapes), then we might wonder if there are examples of this elsewhere.

André Breton, pope of surrealism, drifts across our view, through a New York flea market with painter Max Ernst. They are explaining their art movement’s juxtaposition of culturally disparate imagery and how it does this in order to illuminate universal affinities. With them was Claude Lévi-Strauss, who absorbed these ideas to form structural anthropology, the anthropological school concerned with deep pan-cultural structures of behaviour. Later, Lévi-Strauss was looking for the underlying structure of the marriage system of isolated indigenous populations in the South Pacific, specifically how such small groups seem to avoid or minimise incest. The problem was presented to mathematician André Weil, who developed an approach using algebraic group theory to solve it. Weil went on to become a founding member of the influential Bourbaki movement in maths.

Now, what if we expand the maths side of things to include science? Renaissance architect Filippo Brunelleschi (of the Duomo’s dome fame) developed linear perspective as a way of creating three-dimensional space on a two-dimensional plane, bringing the ‘vanishing point’ to painting, the idea of an infinite spot into which angles recede, and revolutionising art. Mathematician Girard Desargues used this in the 1600s to solidify projective geometry, in which points of infinity are used to extend or project standard Euclidian geometric space. This became the basis for topology, the study of spaces where shapes can bend and warp, and that made way for Riemannian geometry, a specific switching of Euclid’s postulates which results in some deeply peculiar shapes. From this we got tensors – objects which describe the functions of the coordinates of space – which Albert Einstein needed for his great theory of space-time, General Relativity, which gave us nuclear power, GPS systems, and even the old cathode ray tube TVs, perhaps to watch a nice documentary about Florentine art on.

So the chains of influence are there, and especially if we broaden our parameters. If on one side of influence we have all kinds of formulating and systematising knowledge, why not broaden the art side to all of the subconscious, the ‘non-rational’, the mystical? Consider the I Ching, ‘book of change’, ancient Chinese tool of divination and guide for living, high in mysticism. Its complex system of symbols, each made of lines broken and unbroken, inspired Leibniz to develop binary numbers, zeroes and ones which can represent all other numbers. Mathematician George Boole used binary numbers to create Boolean algebra, an area of mathematical logic in which the truth values ‘false’ and ‘true’ can be expressed as ‘bits’ of information designated 0 and 1. This became the essential language of all computers.

What about the same start and finish but by a different route? The I Ching was a favourite of psychoanalyst Carl Jung, who saw its obscure oracular method (throwing little sticks about) as a way of exploring unconscious, non-causal connections between events. He was inspired by this in defining ‘synchronicity’, the idea that seemingly unrelated events can share common meanings through an underlying non-causal framework of existence, this occasionally popping through to us in the forms of coincidence, déjà vu, the uncanny. His friend, physicist Wolfgang Pauli, became taken with this idea while looking for patterns in atomic matter. He discovered an underlying non-causal pattern he named the Exclusion Principle, which states that no two electrons in an atom can have the same quantum numbers. This is a foundational idea of quantum theory, giving us super-quick quantum computers where the bits can be both 0 and 1 at the same time.

## Problems

That is all a lot of fun, those names and ideas tumbling past, but the problem with highlighting a chain of influential individuals is that it’s rather like finding points on the infinite number line: between any two points there will always be another, between that and the next there will be another, and so on. The above connections, in order not to quickly domino their way into encompassing everything ever done by anyone, are reduced to the point that they are essentially just ‘great men’^{15}And it is distinctly men. There was a gifted and influential mathematician in Alexandria in the fourth century called Hypatia – a woman. She was killed by a mob, possibly because they thought she was a witch. stories, ignoring the everyday ephemera of interactions and happenstance which history, and indeed life, are largely made of. But, seeing as it’s the discipline of history which is now at issue (and perhaps always was), such stories can tell us useful things about pinning down influence. Some basic problems:

**a) Parallel connections**. It’s been claimed that Leibniz took inspiration for binary numbers from a different kind of divination, Sub-Saharan geomancy. As in all divination, it’s necessary to randomly generate material from which the interpreter will invent a story. In geomancy, the fortunes are told through drawing lines in the sand according to a close-guarded method which creates (just like the stick-dropping method of the I Ching) an ingenious pseudorandom number generator which converts the lines to a base-2 system – or binary numbers. It’s supposed that this made its way to Europe and Leibniz via the Moors in Spain. Are binary numbers from this or the I Ching? Or both?

**b) The pinch of biographical salt**. Some claim that Cantor, of Jewish ancestry and choosing to symbolise types of infinity with the Hebrew aleph, took his mathematical inspiration on infinity directly from Kabbalah, the ein sof, etc. Amir Aczel makes this argument in *The Mystery of the Aleph: Mathematics, the Kabbalah, and the search for Infinity.* Perhaps a case of a connection being more enjoyable than it is substantial.

**c) Subjective stimulus**. The Bhagavad Gita, philosophical Hindu tale of a prince going to war under the guidance of Krishna, was used by Weil as a defence in his trial for conscientious objection during the Second World War. He argued that the pursuit of mathematical truth was his true dharma, or role, and not fighting. Meanwhile physicist J Robert Oppenheimer, director of the Manhattan Project and fan of the Gita,^{16}Infamously uttering the Hindu god’s line from the tale as he witnessed the first ever atomic bomb test in the New Mexico desert, ‘I am become Death, destroyer of worlds’. explicitly used it as a moral basis for his part in the development of the atomic bomb, saying that his role was that of the soldier, the prince Arjuna, and as such he had to have faith in the powers above him. It’s hard to think of two more contrasting uses of the same artwork.

**d) False dichotomy**. Johannes Kepler set out to prove that the harmonic intervals of the musical scale, when seen as mathematical ratios, could be matched to the velocities of the planets’ orbits when viewed from the point of view of the sun. Amazingly, he did find an observable correlation. So is this a textbook incidence of arts influencing science? No, because Kepler never saw music as anything other than the science of sound, a body of natural laws to be discovered – just like Pythagoras before him, who first found that the harmonies could be understood as ratios.

**e) Linklessness**. There is pleasing similarity between: Kabbalah’s ein sof when drawn in its circular form, the ten sefirot as sphere’s of reality curved by God; *The Divine Comedy*’s universe of equally concentric spheres, bound by angels extending infinitely towards God; and topology’s Riemann Sphere, in which all the lines of a two-dimensional complex plane can be mapped in a one-to-one correspondence to a sphere sitting at the plane’s centre – except the plane’s ‘edge’, which folds up into an infinite point on the top of the sphere. (Credit goes to Aczel for those connections.) But how might this have actually worked? Was Dante aware of Kabbalah, and had Bernhard Riemann read Dante? Or were there other steps in between, such as Dante somehow inspiring Brunelleschi in creating vanishing points which in turn linked on to Riemann as described above? Or are there no connections at all?

That last one is indeed a doozy. It begs the question – at what point does influence truly begin? History’s incalculable domino effect of ideas implies a chicken-and-egg situation regarding art and maths/science. The above history of infinity captures this in the sense that, the further back you delve, the more intermingled the two sides are. No act of creation exists in a vacuum and creativity itself is common to all. Whether chipping away the granite to reveal a sculpture or the numbers to reveal a formula, it’s in essence the same act.

## Making things

So, what of our initial question? How might artists contribute to the development of infinity, or of maths and science in general, rather than their more secondary role of enunciating the majesty or gloominess of each step made by others? If all sides are joined by a common creativity then a semantic journey to its centre, like Dante and Virgil moving through the tightening rings of heaven (or hell, if preferred), may help.

Creativity, the busy-handed practice of things, is cored with inspiration. But the problem with inspiration is that it can’t be forced. This is known acutely by artists. You can’t plan for being inspired, but perhaps, given that inspiration is arguably driven by the shifting of ideas from one context to another – reappropriation, our next ring in – it’s possible to plan for being in an environment from which the shifts can be made, such as one that’s new and compelling. But what is the mindset of reappropriation? If it remains tricky to pre-empt a particular shift then the mind must be relaxed, indiscriminate; one must embrace the innermost circle: association.

In the centre, as ever, is infinity. Association is the all within the one, everything linked directly to everything else; it can be the serene infinite God or the jittering 4am associations of the Morning Star. It’s a broad church. Either way, what we’re trying to describe is the rawest stuff of ideas colliding together in the mind. This is where surrealism has its home stadium, where dreams occur. Dmitri Mendeleev famously conceived the periodic table in a dream; Srinivasa Ramanujan said his mathematical ideas were passed to him directly by the goddess Namagiri while he slept; August Kekulé discovered the ring-shaped structure of the benzene molecule while daydreaming into a firegrate. He said he saw an ouroboros.

Psychedelia is also there with a busy and mostly efficient embassy. Francis Crick is alleged to have been on LSD when he came up with the double-helix structure of DNA.^{17}Alleged, never confirmed. Although the claim is parroted in many sources, and Crick and other scientists at Cambridge at the time were known to have taken small amounts of acid for inspiration, the journalist who finally put the DNA-while-tripping theory to him was told ‘Print a word of it and I’ll sue’. Not cool, Francis. Not cool. What’s significant is that, of the endlessly varied things people might think or feel while on a trip, the one almost universally recalled sense is that of the infinite, of the relatedness of normally disparate things. Likewise religious reveries, those shamanic trance-parties at the feet of theology’s citadel, preceding it, are all about association.

## The trap

Madness is there, perhaps the purest expression of infinity-as-association. This chimes with a traditional and convenient conception of madness as a sort of overload, connections firing where we would do better to see none. But here lies a trap. Not in the sense of some genuine link between infinity and madness, as we are told repeatedly there is, from Rabbi Ben Zoma, alleged in the Talmud to have gazed irretrievably into God’s divine light, to accounts of Cantor’s later years. The trap is in losing our senses to the strange romance of this idea and falling short of making association into something productive.

Inevitably Cantor’s mental illness and lonely death came to refigure his work as if it was always somehow about these things, as if it was infinity itself which ‘drove him mad, as it had mathematicians before him’ – as the blurb to Aczel’s book puts it. For his belief in the connection of spirituality with maths, and the certain mathematical leaps this made him comfortable to take, contemporaries dubbed him a mystic and a fool. Attacks grew more intense and personal from the academic community, especially from his old mentor Kronecker, which he found particularly crushing. His illness was likely from stress, persecution from colleagues, and failure to prove the Continuum Hypothesis, his late-life obsession.^{18}The Continuum Hypothesis is Cantor’s intuition that, between the set of countably infinite integers and the set of uncountably infinite real numbers, there are no other hidden types of infinity. Those two sets must logically be side by side, he thought, but could never prove. This came to be superseded by other obsessions during his decline, including his attempt to prove that Shakespeare’s plays were written by Francis Bacon. Seeing this hunger to prove a theory as a sort of literary proxy for what he could not achieve with the Continuum Hypothesis is not difficult, though probably the only proof it instilled in anyone’s mind was that of his madness. In his circles there would have been few surer signs than a mathematician turning to literary theory. He developed chronic depression and was in and out of sanatoria for the last twenty years of his life, where eventually he died impoverished.

It’s a further boon to the dramatists, though, that the Continuum Hypothesis was to be finally dealt with by Kurt Gödel. He showed through his two Incompleteness Theorems that no mathematical system (which is made of axioms) can demonstrate its own consistency without adding more axioms, and a system encompassing those new axioms would then need more axioms from beyond the system, which would need more and so on infinitely. This means that no self-contained system, like the Continuum Hypothesis, can ever be considered complete, which is to say it can never be proved.^{19}It would later be shown by Paul Cohen that *neither can it be disproved* (which I have no qualms at leaving on an ellipsis)… It’s a boon to the dramatists because Gödel also declined into serious mental illness, starving himself to death in 1978.

The irony, notes Wallace, is that this age-old connection of infinity with insanity is just what Cantor’s work showed as false – in its neat lifting of infinity’s veil, making it for the first time clear and usable. But the age-old narratives survive for a reason. Wallace, who wrote his own non-fiction history of infinity and who themed one of the great novels of the twentieth century, *Infinite Jest*, on fractals, would take his own life in 2008 after a long fight with depression.^{20}The deeper and pleasureless irony is that Wallace’s work, which is about finding sincerity amid the world’s hardwired estrangement, is forever in danger of suffering the same macabre distortion he saw so clearly of Cantor’s work. A passage in his history of infinity, for instance, takes one of the biggest problems of knowledge and logical truth – that we must live according to the principle of induction and presume things to happen merely because they have happened before – and explains it through the description of a paralysing fear of getting out of bed in the morning, due, he says, to the impossibility of knowing that one wouldn’t fall between the floorboards and endlessly down, because its not having happened before is, logically, no guarantee at all. We should not let this illustration, which was intended to be ridiculous, become chilling in light of Wallace’s depression. We should not let Wallace’s or Cantor’s ends disfigure their work. But this passage about a genuine terror of getting out of bed – familiar to those who describe depression – shows how the driest rumination on logic can turn into all-consuming emotion for those, like Cantor, who are blessed to be able to identify life’s unsolvables and cursed to be unable to ignore them.

This narrative, which Wallace tragically could not help but become part of, is a powerful but firmly literary understanding of mental illness. Not to be confused with a clinical understanding. There is an undeniable intellectual terror to the infinite, but this is surely a very simple fear, perhaps the root of all fears – the unknown, the what-you-know+1.

## Work

Association, meanwhile, is part of our sense of infinity, but not as some mind-splitting fissure down at the pit of consciousness, looked on by a cursed few. It’s not the incandescent Garden which sends rabbis mad or the abyss opening up on Cantor’s writing desk. The associative act can actually be a bit boring. It is the classroom, the library book. It is the *work* in educating oneself. That is exactly what artists can do: educate themselves in maths and science, topology, courtship habits of molluscs, take themselves into a place of new stimulus which they might reappropriate to wherever else in whatever crackle of inspiration. Many do, but not enough. Artists understandably like to learn about art, but there can lie the infinite regress of self-reference, tethered to nothing while moving nowhere.

Surely we’ll not see another Pythagoras or Bruno or Kepler, a polymath mystic lunatic who doesn’t know where the metaphysical crisis ends and the broken toaster begins, charging headlong at everything. Economies today specialise us into ever tighter roles. But if we can learn the ideas that have gone before, we can each be a point at which an existing idea morphs from one state to the next, a tie in the net which in the end catches something greater.

One might ask who the next Cantor will be, but one might better ask what she will have listened to, looked at, what was the story she read as a child which in some quiet way came out in every strange and world-changing equation she would go on to write. We might ask who the story’s author was and what parallel education in mathematics glints from beneath its words, perhaps making the relationships between numbers into the relationships between characters, and so for that little girl making sums into jokes, heartbreaks, victories.

One might ask, as if we weren’t congenitally aware, how the icon of the ouroboros so vividly ensnares the contents of infinity, doing so long before those contents were broken down and rationalised. As if with a motherly hiss keeping them safe until the time came. One *should* ask what similarly fertile links we might make between artistic and scientific truths.

Doing so may be difficult, or time-consuming, and may of course amount to nothing. Most likely, though, it will have effects which are too subtle and complex to perceive directly but which are nonetheless there, and positive. This is the happy vagueness we refer to when we say ‘for the greater good’.

And as we make the long hunt for ways to switch ideas between contexts, in the hope of finding commonalities between those contexts and the different regions of experience, we may begin to doubt ourselves. But doubts are essential. All of the achievements described here were fired and improved by them. If we had no doubts about the world then we would have no need for art, or maths, or the infinite.

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