On the Erudite Chaos of Tom Stoppard’s Most Complex Play

Hermione Lee Considers the Algorithmic Genius of Arcadia

Thomasina: Yes, we must hurry if we are going to dance. 
Valentine: And everything is mixing the same way, all the time, irreversibly . . .
Septimus: Oh, we have time, I think.
Valentine: . . . till there’s no time left. That’s what time means.

*

Begun in 1991, worked on through 1992 and staged in 1993, Arcadia is a mid-life play. It is written at a time of looking back and looking forward, just as the play looks back and forward. The parallel lines spoken by Septimus and Valentine in the last act hold in one mental space the moment in which we still have time to act, and the prospect that time will in the end run out, for us individually as well as for the universe: “we have time”/ “there’s no time left.” The play is full of anxiety and sadness about time. But it is also a comedy of time, and timings, and plays with time in enchantingly light and suspenseful ways.

Arcadia is a truly original play, and seduced its audiences and readers by being so new and ingenious. The thrill of discovering revolutionary ideas, for the scientists, poets, historians, landscape gardeners and geniuses who inhabit the play, mirrors the ebullient inventiveness of the thing itself.

Time had always been on his mind. It goes right back to his experiments of the 1960s, under the influence of Eliot, with the inexorable ticking taxi meter that measures out Dominic Boot’s day, or Gladys the speaking clock made dizzy by the infinity of time (“Silence is the sound of time passing”), or the early version of Rosencrantz and Guildenstern at the Court of King Lear, ending with Hamlet’s soliloquy: “I have time . . . it will be night soon . . . I have a lot of time.” Out of that came their play, which they spend killing time, stuck in limbo, not knowing their fate, while scenes from Hamlet, in another time zone, keep rushing in on them at fast-forward speed.

Time bumping backwards in curious jolts, in Artist Descending a Staircase; the sadness of lost time cutting across the present in Where Are They Now?; Henry Carr in Travesties talking us back into past time through his fallible rememberings; Hapgood’s particle-like twins operating in two times at once; the see-saw of In the Native State (and then Indian Ink) from the present to the past: all these plays make us think about time. Now, in Arcadia, time is the subject: what is happening to it; how we live in it, not knowing our fates; whether those things which have become “lost to view will have their time again.” Though we must inevitably be lost in time, perhaps time can be conquered, and the past conjured up.

As usual, this was not his only idea. Arcadia is about knowledge, sex and love, death and pastoral, Englishness and poetry, biography and history. Not to mention chaos mathematics, iterated algorithms, Fermat’s Last Theorem and the Second Law of Thermodynamics. It is a play with one set, set in two time zones. It is a comedy with a tragedy inside it. And it is a quest story, which he kept reminding himself, in his notes, to keep in focus: “Simple narrative must be prime. The poet—the critic—the duel—the Suitor—the Garden—the Waltz. The searcher—the quest—the discovery—(and being wrong)—.”

In Arcadia, time is the subject: what is happening to it; how we live in it, not knowing our fates; whether those things which have become “lost to view will have their time again.”

Set at the start of the nineteenth century and in the late twentieth century, it brings together two kinds of revolutions. One is the shift between Enlightenment and Romantic culture. The other is the recent shift between classical science and new ways of thinking in maths, physics and biology. Neither of these happened all at once. Nobody wakes up one morning to find they are suddenly a Romantic poet as opposed to an Enlightenment satirist (Byron was both), and Newton’s laws weren’t instantly replaced by quantum physics. But the play suggests turning points.

Its two time zones, which run in parallel, converge at the end of the play. So do the two strands of the arts and science, which are not opposites, as some people say, but have a great deal in common, and can be equally creative. The cunning beauty and delight of Arcadia is how its ingredients—human, romantic, intellectual, scientific—are meshed together to make a perfect whole.

In Hapgood, quantum physics and Cold War spying were effortfully brought together. In Arcadia, his eclectic reading led to a more rewarding outcome. He told Bobby in 1991 that he had been reading for a play “about the Romantic/Classical temperament, I mean the change between.” He had been browsing in books on Byron and Romanticism in Paul Johnson’s library, and reading about the history of landscape gardening. And he continually followed new developments in science. He said later: “Arcadia came out of the subjects that had been my enthusiasms over years and years.” He started with two thoughts, how to put “chaos” in a play, and how to have a play with one set, moving through time.

His own self-education was itself an example of what fascinated him: how the mind works things out, how knowledge is acquired and put to use. He enjoys writing plays in which thinkers make sense of the world, or apply logic and reason to what seems impenetrable. He loves to set up complications that require acrobatic feats of ingenuity to solve. He likes to show characters wondering if there is any order inside apparent randomness, like Mr. Moon in Malquist, or Rosencrantz and Guildenstern tossing coins. He is interested in people who are obsessed with proving the unprovable, like George in Jumpers, or who are gambling on a completely new way of thinking, like the ridiculous Tristan Tzara, or Joyce the modernist genius, or the stubborn Galileo. How do new systems of thought come into the world, and what kinds of people might think them?

Although he is famously a playwright of ideas, who often says that he starts from the idea rather than from the plot or the characters—or his own life—there are always people mixed up with the ideas. Part of what attracted him to quantum physics was the compelling personal voice of Richard Feynman. When he heard that Feynman had died, in 1988, just after Hapgood was launched, he said: “I don’t think I’ve ever read [an obituary] which caused me such a stab of grief as I felt on reading of the death of an American physicist whom I had never met and whose work was way out of the reach of my understanding.” He knew that it was, fundamentally, “grief for myself”: he had wanted to send him Hapgood as “an object of tribute.” But, more than that, he had wanted Feynman to know that he had tried to cross the “great divide in our culture” between science and art. Reading Feynman had confirmed his view that “science and art are more like each other than unlike . . . [they] are not just like each other, they sometimes seem to be each other.” He called him “an aristocrat in science and a democrat in almost everything else.”

The cunning beauty and delight of Arcadia is how its ingredients—human, romantic, intellectual, scientific—are meshed together to make a perfect whole.

He responded as strongly to James Gleick’s popular 1987 book on chaos theory, which he read soon after Hapgood was done. He was gripped by its account of the new science challenging orthodoxies in maths and physics, and of the solitary, embattled, creative, sometimes unrecognized scientists who brought it to light. “Genius” in science and in art seemed similar to him, and the idea of “genius” is vital to Arcadia.

Quantum physics had appealed to him because it involved doubling and uncertainty, a drastic change from the fixed certainties of classical physics to the realm of unpredictability. Chaos theory, too, involves the relationship between order and randomness, something that had interested him for a long time. He liked the idea of putting something so unlikely and new into a play.

Chaos theory “attempts to systemize that which appears to function outside of any system. It describes a world in which there is chaos in order, but also order in chaos.” Stoppard calls it “a reconciliation between the idea of things not being random on the one hand and yet unpredictable on the other hand.” It appealed to him because it had to do “with the unpredictability of determinism.” In fact, “chaos theory” is a distorting term, because it makes it sound as if it is all about randomness. And for non-scientists, confusingly, it is the same word that we use in everyday speech when we are talking loosely about a state of hopeless disorder. Some scientists prefer to use the term “deterministic chaos,” so as to get in both sides of its meaning, order and unpredictability.

Nor is it just one science. Chaos theory is a hybrid. It mixes together maths, physics, biology, economics, astronomy. It can apply to fluctuating population growth or weather forecasting, turbulence or earthquakes, eclipses or heartbeat patterns, the formation of snowflakes: to any dynamical system where apparent randomness is found to contain order. Grandly put, it is “a science of the global nature of systems.”

“Chaos” got going in the 1970s, so when Stoppard was reading about it, it was still an excitingly new, radical change of direction. In Gleick’s words: “Where chaos begins, classical science stops.” Newtonian, classical science argued for an entirely determined universe. It maintained that given enough information, we can predict future events. The nineteenth-century scientist Pierre Laplace proposed that one could infer from the deterministic laws of the universe a vast entity, or intellect, which could “embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom.”

Or as Thomasina Coverly puts it in Arcadia in 1809, several years before Laplace: “If you could stop every atom in its position and direction, and if your mind could comprehend all the actions thus suspended, then if you were really, really good at algebra you could write the formula for all the future.” But classical science ignored, or dismissed as monstrosities, what Gleick calls “the irregular side of nature, the discontinuous and erratic side.” Galileo (whose distant shadow haunts Arcadia), Newton and their scientific descendants searched for regularity. But that meant “disregarding bits of messiness that interfere with a neat picture.” The new science wanted to ask big questions: “How does life begin? What is turbulence? . . . In a universe ruled by entropy, drawing inexorably toward greater and greater disorder, how does order arise?”

Chaos theory is a hybrid. It mixes together maths, physics, biology, economics, astronomy.

Chaos theory took the messiness and disorder on board. Gleick tells the story: “The modern study of chaos began with the creeping realization . . . that quite simple mathematical equations could model systems every bit as violent as a waterfall. Tiny differences in input could quickly become overwhelming differences in output.” Chaos theory paid attention to aperiodic systems—systems which “almost repeated themselves but never quite succeeded,” like weather, or animal populations. It dealt in non-linear equations, “which express relationships that are not strictly proportional, and generally cannot be solved,” unlike solvable linear equations. Before that, “almost no one in the classical era suspected the chaos that could lurk in dynamical systems if non-linearity was given its due.”

Chaotic systems have three key properties. They are non-linear, they are deterministic, and they exhibit “sensitive dependence to initial conditions.” The general reader knows about sensitive dependence through the image of the butterfly effect: “A butterfly stirring the air today in Beijing can transform storm systems next month in New York.” As anyone knows who has missed their bus to work, or watched a theatrical farce, “small perturbations in one’s daily trajectory can have large consequences.” “Chaos” is a description for the “rapid amplification” of a small difference to “kingdom-shattering proportions.” A well-known nursery rhyme (in Benjamin Franklin’s version) sums it up:

For the want of a nail the shoe was lost,
For the want of a shoe the horse was lost,
For the want of a horse the rider was lost,
For the want of a rider the battle was lost,
For the want of a battle the kingdom was lost, And all for the want of a horseshoe-nail.

Valentine, the twentieth-century computer scientist and biologist in Arcadia, puts it like this, with excitement and delight: “The unpredictable and the predetermined unfold together to make everything the way it is . . . the smallest variation blows prediction apart.”

But chaos is not entirely chaotic. Apparently random effects, it transpires, follow universal mathematical laws. “A complex system can give rise to turbulence and coherence at the same time.” Order in chaos can be modeled in phenomena as wide-ranging as the population growth of grouse (Valentine’s research project) or the vagaries of the stock market. The mathematician Benoit Mandelbrot, one of the strange geniuses in Gleick’s story of chaos, who influences Arcadia, created a picture of reality which showed that “within the most disorderly reams of data lived an unexpected kind of order.”

How can these patterns and order be traced and made sense of? Chaos scientists looked at how the behavior of dynamical systems bifurcated endlessly. As systems respond to variations in input, the graph of their behavior bifurcates, and then bifurcates again, ad infinitum, into a condition of chaotic randomness. But “within those random states, windows of order reappeared.” Mathematicians use feedback loops, or iterated algorithms, to establish patterns of bifurcation. These algorithms—as we learn in Arcadia—would be impossible to work out with pen and paper, even if a genius could imagine them in her head. They can only be done with electronic calculators and computers.

But chaos is not entirely chaotic. Apparently random effects, it transpires, follow universal mathematical laws.

Thomasina was trying to do in the 1800s a mathematical process that was first used in the early 1970s. Stoppard resisted, rather firmly, the frequent suggestion that she was based on Byron’s daughter Ada Lovelace, the mathematician who worked with Charles Babbage and anticipated computer programming, called her methods “poetical science” and died in her thirties. He heard about Ada after Arcadia was written, from Robert May, the physicist turned biologist, one of his main scientific advisers for the play. May worked on population changes in species, and he used “functional iteration” as a way of describing these changes each year through “a feedback loop, each year’s output serving as the next year’s input.” “The output of one calculation was fed back in as input for the next.”

Nearly two hundred years after Thomasina had intuited this process, her calculations come to light. Valentine describes her achievement to a character who knows nothing about physics or maths and so is a useful stand-in for the audience and the reader. “She’s feeding the solution back into the equation, and then solving it again,” he explains.

. . . It’s how you look at population changes in biology. Goldfish in a pond, say. This year there are x goldfish. Next year there’ll be y goldfish. Some get born, some get eaten by herons. Nature manipulates the x and turns it into y. Then y goldfish is your starting population for the following year . . . Your value for y becomes your next value for x.

Any sixth-former doing maths knows about algorithms: the magic is in Thomasina’s anticipating them. As Stoppard says: “You do not have to be Einstein to have the idea of feedback, the idea of the algorithm which just operates on itself. The problem was that there were not enough pencils or paper or time to do it often enough. To do it so many times that the pattern emerges.”

Algorithms can be used to draw the irregular shapes of nature. Fractal or fractional geometry—terms coined by Mandelbrot from the Latin for “broken” and from the English words “fracture” and “fraction”—was a way of measuring qualities that otherwise have no clear definition. Mandelbrot’s example was the degree of roughness or irregularity in a twisting coastline. Fractal geometry enabled mathematics to describe, explore and mirror the unpredictable forms of nature. Gleick explains it eloquently. You used “a shape to help visualize the whole range of behaviours in a system.” You looked at materials like “the fragmented landscapes of an earthquake zone” so as to calculate “the irregular patterns of nature.” You could study “the way things meld together, the way they branch apart, or the way they shatter.”

Fractal geometry enabled mathematics to describe, explore and mirror the unpredictable forms of nature.

Fractal also meant “self-similar.” “Self-similarity is symmetry across scale. It implies recursion, pattern inside of pattern.” Fractal geometry produces detail at finer and finer scales, as in the image of a person reflected in mirror after mirror after mirror, infinitely receding. Mandelbrot liked to quote Jonathan Swift:

So, Nat’ralists observe, a Flea
Hath smaller Fleas that on him prey,
And these have smaller Fleas to bite ’em,
And so proceed ad infinitum.
Self-similarity got into the content—and the structure—of Arcadia.

Thomasina writes in her mathematics primer that she has invented “a method whereby all the forms of nature must give up their mathematical secrets and draw themselves through numbers alone.” Valentine explains how it works: “If you knew the algorithm and fed it back say ten thousand times, each time there’d be a dot somewhere on the screen. You’d never know where to expect the next dot. But gradually you’d start to see this shape, because every dot will be inside the shape of this leaf. It wouldn’t be a leaf, it would be a mathematical object.” Thomasina Coverly has worked out how a mathematical drawing could simulate an apple leaf, and Valentine reproduces the image on his computer two hundred years on. He calls it “the Coverly set,” in tribute to “the Mandelbrot set,” his beautiful, complex computer images of fractal shapes.

Some years after Arcadia came out Stoppard had a public conversation with Robert Osserman, deputy director of the Mathematical Sciences Research Institute in Berkeley. He said then that what fascinated him in the idea of fractal geometry was that “there is a complementarity in the notion that math describes nature, and that nature is following mathematical rules . . . Numbers have a kind of social behavior, they’re not simply tools of description.” Thomasina “understands . . . that it’s not actually the equation in itself, it’s the way you . . . manipulate the equation, the way you pull the solution back into the equation in a feedback loop, which generates shapes . . . That, in the end, is what is moving about the idea that nature is written in numbers. That it’s generative.” It was the parallel behavior of nature and mathematics that fascinated him, just as the parallel creative energy of the arts and the sciences was one of the play’s subjects.

Algorithms were not Thomasina’s only discovery. The past time zone of the play jumps, in a non-linear fashion, from 1809 to 1812. She is thirteen when it starts and nearly seventeen when it ends. When she is thirteen she notices, as a child might notice, that you can only stir the jam in your rice pudding one way: “If you stir backwards, the jam will not come together again.” Yes, says her tutor Septimus, the Newtonian, classical scientist, who at this point knows more than his pupil, time will only go one way, and “we must stir our way onward mixing as we go, disorder out of disorder into disorder until pink is complete, unchanging and unchangeable, and we are done with it for ever.” Thomasina agrees with Septimus: “You cannot stir things apart.” Gleick calls this “entropy explained in five words.”

Three years later, she has become wiser than her tutor. She has worked out that there is no such thing in the universe as an “unchanging and unchangeable” state. Centuries ahead of her time, she has grasped the Second Law of Thermodynamics. She has worked out that “the heat equation . . . goes only one way.” That is, she has foreseen the law of entropy, which tells us that the universe is evolving from order to disorder. Heat, according to the Second Law of Thermodynamics, can only flow in one direction, from hotter to colder. There is no way back in time, no backward swing of the pendulum. The idea that the world’s energy, heat and light will eventually run out and the universe at last be nothing but dark void runs like a sombre drumbeat under the life and vitality of the play.

Algorithms were not Thomasina’s only discovery. The past time zone of the play jumps, in a non-linear fashion, from 1809 to 1812.

Gleick gives an example of “the inexorable tendency of the universe, and any isolated system in it, to slide toward a state of increasing disorder.” He explains—like Thomasina telling Septimus about stirring the jam—that if you have a bath with two separate compartments, one containing water and one containing ink, and you remove the barrier between the compartments so that ink and water mix together, the mixture never reverses itself. This is the part of physics “that makes time a one-way street.” “Everything tends towards disorder. Any process that converts energy from one form to another must lose some as heat. Perfect efficiency is impossible.”

But Gleick does provide some consolation, which affects the mood of Arcadia. What chaos theory shows is that “somehow, after all, as the universe ebbs towards its final equilibrium in the featureless heat bath of maximum entropy, it manages to create interesting structures . . . Nature forms patterns . . . The universe is randomness and dissipation, yes. But randomness with direction can produce surprising complexity . . . Dissipation is an agent of order.”

Stoppard leaped on these ideas with excitement and poured them into his play. As always he relished the technical language of specialist disciplines. “Noise,” for instance, is the scientist’s word for “error,” or “observational uncertainty.” It is “what scientists blame for the inaccuracy of their measurements.” Too much noise, in Arcadia, is what drives Valentine off course in his research. In the play, “noise” becomes a metaphor for extravagant and ridiculous behavior, especially that of the fame-seeking literary don Bernard, a very noisy character. “Trivial” means, for scientists, redundant information that doesn’t lead anywhere or proofs with no value. In the play, it is a telling word for what matters and what doesn’t. Personal relationships and the achievements of individuals, says Valentine, are “trivial” compared with the search for knowledge itself.

__________________________________

Tom Stoppard: A Life by Hermione Lee

Excerpted from Tom Stoppard: A Life by Hermione Lee. Copyright © 2021 by Hermione Lee. Excerpted by permission of Alfred A. Knopf, a division of Penguin Random House LLC. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

Hermione Lee
Hermione Lee
Hermione Lee was president of Wolfson College (2008-2017) and is Emeritus Professor of English Literature at Oxford University. Her work includes biographies of Virginia Woolf, Edith Wharton, and Penelope Fitzgerald (winner of the James Tait Black Prize and one of the New York Times's 10 Best Books of 2014). She has also written books on Elizabeth Bowen, Philip Roth, and Willa Cather. Lee was awarded the Biographers' Club Prize for Exceptional Contribution to Biography in 2018. She is a Fellow of the British Academy and the Royal Society of Literature, and a Foreign Honorary Member of the American Academy of Arts and Sciences. In 2003 she was made a CBE, and in 2013 she was made a Dame for services to literary scholarship.





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On the Erudite Chaos of Tom Stoppard’s Most Complex Play

Hermione Lee Considers the Algorithmic Genius of Arcadia

Thomasina: Yes, we must hurry if we are going to dance. 
Valentine: And everything is mixing the same way, all the time, irreversibly . . .
Septimus: Oh, we have time, I think.
Valentine: . . . till there’s no time left. That’s what time means.

*

Begun in 1991, worked on through 1992 and staged in 1993, Arcadia is a mid-life play. It is written at a time of looking back and looking forward, just as the play looks back and forward. The parallel lines spoken by Septimus and Valentine in the last act hold in one mental space the moment in which we still have time to act, and the prospect that time will in the end run out, for us individually as well as for the universe: “we have time”/ “there’s no time left.” The play is full of anxiety and sadness about time. But it is also a comedy of time, and timings, and plays with time in enchantingly light and suspenseful ways.

Arcadia is a truly original play, and seduced its audiences and readers by being so new and ingenious. The thrill of discovering revolutionary ideas, for the scientists, poets, historians, landscape gardeners and geniuses who inhabit the play, mirrors the ebullient inventiveness of the thing itself.

Time had always been on his mind. It goes right back to his experiments of the 1960s, under the influence of Eliot, with the inexorable ticking taxi meter that measures out Dominic Boot’s day, or Gladys the speaking clock made dizzy by the infinity of time (“Silence is the sound of time passing”), or the early version of Rosencrantz and Guildenstern at the Court of King Lear, ending with Hamlet’s soliloquy: “I have time . . . it will be night soon . . . I have a lot of time.” Out of that came their play, which they spend killing time, stuck in limbo, not knowing their fate, while scenes from Hamlet, in another time zone, keep rushing in on them at fast-forward speed.

Time bumping backwards in curious jolts, in Artist Descending a Staircase; the sadness of lost time cutting across the present in Where Are They Now?; Henry Carr in Travesties talking us back into past time through his fallible rememberings; Hapgood’s particle-like twins operating in two times at once; the see-saw of In the Native State (and then Indian Ink) from the present to the past: all these plays make us think about time. Now, in Arcadia, time is the subject: what is happening to it; how we live in it, not knowing our fates; whether those things which have become “lost to view will have their time again.” Though we must inevitably be lost in time, perhaps time can be conquered, and the past conjured up.

As usual, this was not his only idea. Arcadia is about knowledge, sex and love, death and pastoral, Englishness and poetry, biography and history. Not to mention chaos mathematics, iterated algorithms, Fermat’s Last Theorem and the Second Law of Thermodynamics. It is a play with one set, set in two time zones. It is a comedy with a tragedy inside it. And it is a quest story, which he kept reminding himself, in his notes, to keep in focus: “Simple narrative must be prime. The poet—the critic—the duel—the Suitor—the Garden—the Waltz. The searcher—the quest—the discovery—(and being wrong)—.”

In Arcadia, time is the subject: what is happening to it; how we live in it, not knowing our fates; whether those things which have become “lost to view will have their time again.”

Set at the start of the nineteenth century and in the late twentieth century, it brings together two kinds of revolutions. One is the shift between Enlightenment and Romantic culture. The other is the recent shift between classical science and new ways of thinking in maths, physics and biology. Neither of these happened all at once. Nobody wakes up one morning to find they are suddenly a Romantic poet as opposed to an Enlightenment satirist (Byron was both), and Newton’s laws weren’t instantly replaced by quantum physics. But the play suggests turning points.

Its two time zones, which run in parallel, converge at the end of the play. So do the two strands of the arts and science, which are not opposites, as some people say, but have a great deal in common, and can be equally creative. The cunning beauty and delight of Arcadia is how its ingredients—human, romantic, intellectual, scientific—are meshed together to make a perfect whole.

In Hapgood, quantum physics and Cold War spying were effortfully brought together. In Arcadia, his eclectic reading led to a more rewarding outcome. He told Bobby in 1991 that he had been reading for a play “about the Romantic/Classical temperament, I mean the change between.” He had been browsing in books on Byron and Romanticism in Paul Johnson’s library, and reading about the history of landscape gardening. And he continually followed new developments in science. He said later: “Arcadia came out of the subjects that had been my enthusiasms over years and years.” He started with two thoughts, how to put “chaos” in a play, and how to have a play with one set, moving through time.

His own self-education was itself an example of what fascinated him: how the mind works things out, how knowledge is acquired and put to use. He enjoys writing plays in which thinkers make sense of the world, or apply logic and reason to what seems impenetrable. He loves to set up complications that require acrobatic feats of ingenuity to solve. He likes to show characters wondering if there is any order inside apparent randomness, like Mr. Moon in Malquist, or Rosencrantz and Guildenstern tossing coins. He is interested in people who are obsessed with proving the unprovable, like George in Jumpers, or who are gambling on a completely new way of thinking, like the ridiculous Tristan Tzara, or Joyce the modernist genius, or the stubborn Galileo. How do new systems of thought come into the world, and what kinds of people might think them?

Although he is famously a playwright of ideas, who often says that he starts from the idea rather than from the plot or the characters—or his own life—there are always people mixed up with the ideas. Part of what attracted him to quantum physics was the compelling personal voice of Richard Feynman. When he heard that Feynman had died, in 1988, just after Hapgood was launched, he said: “I don’t think I’ve ever read [an obituary] which caused me such a stab of grief as I felt on reading of the death of an American physicist whom I had never met and whose work was way out of the reach of my understanding.” He knew that it was, fundamentally, “grief for myself”: he had wanted to send him Hapgood as “an object of tribute.” But, more than that, he had wanted Feynman to know that he had tried to cross the “great divide in our culture” between science and art. Reading Feynman had confirmed his view that “science and art are more like each other than unlike . . . [they] are not just like each other, they sometimes seem to be each other.” He called him “an aristocrat in science and a democrat in almost everything else.”

The cunning beauty and delight of Arcadia is how its ingredients—human, romantic, intellectual, scientific—are meshed together to make a perfect whole.

He responded as strongly to James Gleick’s popular 1987 book on chaos theory, which he read soon after Hapgood was done. He was gripped by its account of the new science challenging orthodoxies in maths and physics, and of the solitary, embattled, creative, sometimes unrecognized scientists who brought it to light. “Genius” in science and in art seemed similar to him, and the idea of “genius” is vital to Arcadia.

Quantum physics had appealed to him because it involved doubling and uncertainty, a drastic change from the fixed certainties of classical physics to the realm of unpredictability. Chaos theory, too, involves the relationship between order and randomness, something that had interested him for a long time. He liked the idea of putting something so unlikely and new into a play.

Chaos theory “attempts to systemize that which appears to function outside of any system. It describes a world in which there is chaos in order, but also order in chaos.” Stoppard calls it “a reconciliation between the idea of things not being random on the one hand and yet unpredictable on the other hand.” It appealed to him because it had to do “with the unpredictability of determinism.” In fact, “chaos theory” is a distorting term, because it makes it sound as if it is all about randomness. And for non-scientists, confusingly, it is the same word that we use in everyday speech when we are talking loosely about a state of hopeless disorder. Some scientists prefer to use the term “deterministic chaos,” so as to get in both sides of its meaning, order and unpredictability.

Nor is it just one science. Chaos theory is a hybrid. It mixes together maths, physics, biology, economics, astronomy. It can apply to fluctuating population growth or weather forecasting, turbulence or earthquakes, eclipses or heartbeat patterns, the formation of snowflakes: to any dynamical system where apparent randomness is found to contain order. Grandly put, it is “a science of the global nature of systems.”

“Chaos” got going in the 1970s, so when Stoppard was reading about it, it was still an excitingly new, radical change of direction. In Gleick’s words: “Where chaos begins, classical science stops.” Newtonian, classical science argued for an entirely determined universe. It maintained that given enough information, we can predict future events. The nineteenth-century scientist Pierre Laplace proposed that one could infer from the deterministic laws of the universe a vast entity, or intellect, which could “embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom.”

Or as Thomasina Coverly puts it in Arcadia in 1809, several years before Laplace: “If you could stop every atom in its position and direction, and if your mind could comprehend all the actions thus suspended, then if you were really, really good at algebra you could write the formula for all the future.” But classical science ignored, or dismissed as monstrosities, what Gleick calls “the irregular side of nature, the discontinuous and erratic side.” Galileo (whose distant shadow haunts Arcadia), Newton and their scientific descendants searched for regularity. But that meant “disregarding bits of messiness that interfere with a neat picture.” The new science wanted to ask big questions: “How does life begin? What is turbulence? . . . In a universe ruled by entropy, drawing inexorably toward greater and greater disorder, how does order arise?”

Chaos theory is a hybrid. It mixes together maths, physics, biology, economics, astronomy.

Chaos theory took the messiness and disorder on board. Gleick tells the story: “The modern study of chaos began with the creeping realization . . . that quite simple mathematical equations could model systems every bit as violent as a waterfall. Tiny differences in input could quickly become overwhelming differences in output.” Chaos theory paid attention to aperiodic systems—systems which “almost repeated themselves but never quite succeeded,” like weather, or animal populations. It dealt in non-linear equations, “which express relationships that are not strictly proportional, and generally cannot be solved,” unlike solvable linear equations. Before that, “almost no one in the classical era suspected the chaos that could lurk in dynamical systems if non-linearity was given its due.”

Chaotic systems have three key properties. They are non-linear, they are deterministic, and they exhibit “sensitive dependence to initial conditions.” The general reader knows about sensitive dependence through the image of the butterfly effect: “A butterfly stirring the air today in Beijing can transform storm systems next month in New York.” As anyone knows who has missed their bus to work, or watched a theatrical farce, “small perturbations in one’s daily trajectory can have large consequences.” “Chaos” is a description for the “rapid amplification” of a small difference to “kingdom-shattering proportions.” A well-known nursery rhyme (in Benjamin Franklin’s version) sums it up:

For the want of a nail the shoe was lost,
For the want of a shoe the horse was lost,
For the want of a horse the rider was lost,
For the want of a rider the battle was lost,
For the want of a battle the kingdom was lost, And all for the want of a horseshoe-nail.

Valentine, the twentieth-century computer scientist and biologist in Arcadia, puts it like this, with excitement and delight: “The unpredictable and the predetermined unfold together to make everything the way it is . . . the smallest variation blows prediction apart.”

But chaos is not entirely chaotic. Apparently random effects, it transpires, follow universal mathematical laws. “A complex system can give rise to turbulence and coherence at the same time.” Order in chaos can be modeled in phenomena as wide-ranging as the population growth of grouse (Valentine’s research project) or the vagaries of the stock market. The mathematician Benoit Mandelbrot, one of the strange geniuses in Gleick’s story of chaos, who influences Arcadia, created a picture of reality which showed that “within the most disorderly reams of data lived an unexpected kind of order.”

How can these patterns and order be traced and made sense of? Chaos scientists looked at how the behavior of dynamical systems bifurcated endlessly. As systems respond to variations in input, the graph of their behavior bifurcates, and then bifurcates again, ad infinitum, into a condition of chaotic randomness. But “within those random states, windows of order reappeared.” Mathematicians use feedback loops, or iterated algorithms, to establish patterns of bifurcation. These algorithms—as we learn in Arcadia—would be impossible to work out with pen and paper, even if a genius could imagine them in her head. They can only be done with electronic calculators and computers.

But chaos is not entirely chaotic. Apparently random effects, it transpires, follow universal mathematical laws.

Thomasina was trying to do in the 1800s a mathematical process that was first used in the early 1970s. Stoppard resisted, rather firmly, the frequent suggestion that she was based on Byron’s daughter Ada Lovelace, the mathematician who worked with Charles Babbage and anticipated computer programming, called her methods “poetical science” and died in her thirties. He heard about Ada after Arcadia was written, from Robert May, the physicist turned biologist, one of his main scientific advisers for the play. May worked on population changes in species, and he used “functional iteration” as a way of describing these changes each year through “a feedback loop, each year’s output serving as the next year’s input.” “The output of one calculation was fed back in as input for the next.”

Nearly two hundred years after Thomasina had intuited this process, her calculations come to light. Valentine describes her achievement to a character who knows nothing about physics or maths and so is a useful stand-in for the audience and the reader. “She’s feeding the solution back into the equation, and then solving it again,” he explains.

. . . It’s how you look at population changes in biology. Goldfish in a pond, say. This year there are x goldfish. Next year there’ll be y goldfish. Some get born, some get eaten by herons. Nature manipulates the x and turns it into y. Then y goldfish is your starting population for the following year . . . Your value for y becomes your next value for x.

Any sixth-former doing maths knows about algorithms: the magic is in Thomasina’s anticipating them. As Stoppard says: “You do not have to be Einstein to have the idea of feedback, the idea of the algorithm which just operates on itself. The problem was that there were not enough pencils or paper or time to do it often enough. To do it so many times that the pattern emerges.”

Algorithms can be used to draw the irregular shapes of nature. Fractal or fractional geometry—terms coined by Mandelbrot from the Latin for “broken” and from the English words “fracture” and “fraction”—was a way of measuring qualities that otherwise have no clear definition. Mandelbrot’s example was the degree of roughness or irregularity in a twisting coastline. Fractal geometry enabled mathematics to describe, explore and mirror the unpredictable forms of nature. Gleick explains it eloquently. You used “a shape to help visualize the whole range of behaviours in a system.” You looked at materials like “the fragmented landscapes of an earthquake zone” so as to calculate “the irregular patterns of nature.” You could study “the way things meld together, the way they branch apart, or the way they shatter.”

Fractal geometry enabled mathematics to describe, explore and mirror the unpredictable forms of nature.

Fractal also meant “self-similar.” “Self-similarity is symmetry across scale. It implies recursion, pattern inside of pattern.” Fractal geometry produces detail at finer and finer scales, as in the image of a person reflected in mirror after mirror after mirror, infinitely receding. Mandelbrot liked to quote Jonathan Swift:

So, Nat’ralists observe, a Flea
Hath smaller Fleas that on him prey,
And these have smaller Fleas to bite ’em,
And so proceed ad infinitum.
Self-similarity got into the content—and the structure—of Arcadia.

Thomasina writes in her mathematics primer that she has invented “a method whereby all the forms of nature must give up their mathematical secrets and draw themselves through numbers alone.” Valentine explains how it works: “If you knew the algorithm and fed it back say ten thousand times, each time there’d be a dot somewhere on the screen. You’d never know where to expect the next dot. But gradually you’d start to see this shape, because every dot will be inside the shape of this leaf. It wouldn’t be a leaf, it would be a mathematical object.” Thomasina Coverly has worked out how a mathematical drawing could simulate an apple leaf, and Valentine reproduces the image on his computer two hundred years on. He calls it “the Coverly set,” in tribute to “the Mandelbrot set,” his beautiful, complex computer images of fractal shapes.

Some years after Arcadia came out Stoppard had a public conversation with Robert Osserman, deputy director of the Mathematical Sciences Research Institute in Berkeley. He said then that what fascinated him in the idea of fractal geometry was that “there is a complementarity in the notion that math describes nature, and that nature is following mathematical rules . . . Numbers have a kind of social behavior, they’re not simply tools of description.” Thomasina “understands . . . that it’s not actually the equation in itself, it’s the way you . . . manipulate the equation, the way you pull the solution back into the equation in a feedback loop, which generates shapes . . . That, in the end, is what is moving about the idea that nature is written in numbers. That it’s generative.” It was the parallel behavior of nature and mathematics that fascinated him, just as the parallel creative energy of the arts and the sciences was one of the play’s subjects.

Algorithms were not Thomasina’s only discovery. The past time zone of the play jumps, in a non-linear fashion, from 1809 to 1812. She is thirteen when it starts and nearly seventeen when it ends. When she is thirteen she notices, as a child might notice, that you can only stir the jam in your rice pudding one way: “If you stir backwards, the jam will not come together again.” Yes, says her tutor Septimus, the Newtonian, classical scientist, who at this point knows more than his pupil, time will only go one way, and “we must stir our way onward mixing as we go, disorder out of disorder into disorder until pink is complete, unchanging and unchangeable, and we are done with it for ever.” Thomasina agrees with Septimus: “You cannot stir things apart.” Gleick calls this “entropy explained in five words.”

Three years later, she has become wiser than her tutor. She has worked out that there is no such thing in the universe as an “unchanging and unchangeable” state. Centuries ahead of her time, she has grasped the Second Law of Thermodynamics. She has worked out that “the heat equation . . . goes only one way.” That is, she has foreseen the law of entropy, which tells us that the universe is evolving from order to disorder. Heat, according to the Second Law of Thermodynamics, can only flow in one direction, from hotter to colder. There is no way back in time, no backward swing of the pendulum. The idea that the world’s energy, heat and light will eventually run out and the universe at last be nothing but dark void runs like a sombre drumbeat under the life and vitality of the play.

Algorithms were not Thomasina’s only discovery. The past time zone of the play jumps, in a non-linear fashion, from 1809 to 1812.

Gleick gives an example of “the inexorable tendency of the universe, and any isolated system in it, to slide toward a state of increasing disorder.” He explains—like Thomasina telling Septimus about stirring the jam—that if you have a bath with two separate compartments, one containing water and one containing ink, and you remove the barrier between the compartments so that ink and water mix together, the mixture never reverses itself. This is the part of physics “that makes time a one-way street.” “Everything tends towards disorder. Any process that converts energy from one form to another must lose some as heat. Perfect efficiency is impossible.”

But Gleick does provide some consolation, which affects the mood of Arcadia. What chaos theory shows is that “somehow, after all, as the universe ebbs towards its final equilibrium in the featureless heat bath of maximum entropy, it manages to create interesting structures . . . Nature forms patterns . . . The universe is randomness and dissipation, yes. But randomness with direction can produce surprising complexity . . . Dissipation is an agent of order.”

Stoppard leaped on these ideas with excitement and poured them into his play. As always he relished the technical language of specialist disciplines. “Noise,” for instance, is the scientist’s word for “error,” or “observational uncertainty.” It is “what scientists blame for the inaccuracy of their measurements.” Too much noise, in Arcadia, is what drives Valentine off course in his research. In the play, “noise” becomes a metaphor for extravagant and ridiculous behavior, especially that of the fame-seeking literary don Bernard, a very noisy character. “Trivial” means, for scientists, redundant information that doesn’t lead anywhere or proofs with no value. In the play, it is a telling word for what matters and what doesn’t. Personal relationships and the achievements of individuals, says Valentine, are “trivial” compared with the search for knowledge itself.

__________________________________

Tom Stoppard: A Life by Hermione Lee

Excerpted from Tom Stoppard: A Life by Hermione Lee. Copyright © 2021 by Hermione Lee. Excerpted by permission of Alfred A. Knopf, a division of Penguin Random House LLC. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

Hermione Lee
Hermione Lee
Hermione Lee was president of Wolfson College (2008-2017) and is Emeritus Professor of English Literature at Oxford University. Her work includes biographies of Virginia Woolf, Edith Wharton, and Penelope Fitzgerald (winner of the James Tait Black Prize and one of the New York Times's 10 Best Books of 2014). She has also written books on Elizabeth Bowen, Philip Roth, and Willa Cather. Lee was awarded the Biographers' Club Prize for Exceptional Contribution to Biography in 2018. She is a Fellow of the British Academy and the Royal Society of Literature, and a Foreign Honorary Member of the American Academy of Arts and Sciences. In 2003 she was made a CBE, and in 2013 she was made a Dame for services to literary scholarship.





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On the Erudite Chaos of Tom Stoppard’s Most Complex Play

Hermione Lee Considers the Algorithmic Genius of Arcadia

Thomasina: Yes, we must hurry if we are going to dance. 
Valentine: And everything is mixing the same way, all the time, irreversibly . . .
Septimus: Oh, we have time, I think.
Valentine: . . . till there’s no time left. That’s what time means.

*

Begun in 1991, worked on through 1992 and staged in 1993, Arcadia is a mid-life play. It is written at a time of looking back and looking forward, just as the play looks back and forward. The parallel lines spoken by Septimus and Valentine in the last act hold in one mental space the moment in which we still have time to act, and the prospect that time will in the end run out, for us individually as well as for the universe: “we have time”/ “there’s no time left.” The play is full of anxiety and sadness about time. But it is also a comedy of time, and timings, and plays with time in enchantingly light and suspenseful ways.

Arcadia is a truly original play, and seduced its audiences and readers by being so new and ingenious. The thrill of discovering revolutionary ideas, for the scientists, poets, historians, landscape gardeners and geniuses who inhabit the play, mirrors the ebullient inventiveness of the thing itself.

Time had always been on his mind. It goes right back to his experiments of the 1960s, under the influence of Eliot, with the inexorable ticking taxi meter that measures out Dominic Boot’s day, or Gladys the speaking clock made dizzy by the infinity of time (“Silence is the sound of time passing”), or the early version of Rosencrantz and Guildenstern at the Court of King Lear, ending with Hamlet’s soliloquy: “I have time . . . it will be night soon . . . I have a lot of time.” Out of that came their play, which they spend killing time, stuck in limbo, not knowing their fate, while scenes from Hamlet, in another time zone, keep rushing in on them at fast-forward speed.

Time bumping backwards in curious jolts, in Artist Descending a Staircase; the sadness of lost time cutting across the present in Where Are They Now?; Henry Carr in Travesties talking us back into past time through his fallible rememberings; Hapgood’s particle-like twins operating in two times at once; the see-saw of In the Native State (and then Indian Ink) from the present to the past: all these plays make us think about time. Now, in Arcadia, time is the subject: what is happening to it; how we live in it, not knowing our fates; whether those things which have become “lost to view will have their time again.” Though we must inevitably be lost in time, perhaps time can be conquered, and the past conjured up.

As usual, this was not his only idea. Arcadia is about knowledge, sex and love, death and pastoral, Englishness and poetry, biography and history. Not to mention chaos mathematics, iterated algorithms, Fermat’s Last Theorem and the Second Law of Thermodynamics. It is a play with one set, set in two time zones. It is a comedy with a tragedy inside it. And it is a quest story, which he kept reminding himself, in his notes, to keep in focus: “Simple narrative must be prime. The poet—the critic—the duel—the Suitor—the Garden—the Waltz. The searcher—the quest—the discovery—(and being wrong)—.”

In Arcadia, time is the subject: what is happening to it; how we live in it, not knowing our fates; whether those things which have become “lost to view will have their time again.”

Set at the start of the nineteenth century and in the late twentieth century, it brings together two kinds of revolutions. One is the shift between Enlightenment and Romantic culture. The other is the recent shift between classical science and new ways of thinking in maths, physics and biology. Neither of these happened all at once. Nobody wakes up one morning to find they are suddenly a Romantic poet as opposed to an Enlightenment satirist (Byron was both), and Newton’s laws weren’t instantly replaced by quantum physics. But the play suggests turning points.

Its two time zones, which run in parallel, converge at the end of the play. So do the two strands of the arts and science, which are not opposites, as some people say, but have a great deal in common, and can be equally creative. The cunning beauty and delight of Arcadia is how its ingredients—human, romantic, intellectual, scientific—are meshed together to make a perfect whole.

In Hapgood, quantum physics and Cold War spying were effortfully brought together. In Arcadia, his eclectic reading led to a more rewarding outcome. He told Bobby in 1991 that he had been reading for a play “about the Romantic/Classical temperament, I mean the change between.” He had been browsing in books on Byron and Romanticism in Paul Johnson’s library, and reading about the history of landscape gardening. And he continually followed new developments in science. He said later: “Arcadia came out of the subjects that had been my enthusiasms over years and years.” He started with two thoughts, how to put “chaos” in a play, and how to have a play with one set, moving through time.

His own self-education was itself an example of what fascinated him: how the mind works things out, how knowledge is acquired and put to use. He enjoys writing plays in which thinkers make sense of the world, or apply logic and reason to what seems impenetrable. He loves to set up complications that require acrobatic feats of ingenuity to solve. He likes to show characters wondering if there is any order inside apparent randomness, like Mr. Moon in Malquist, or Rosencrantz and Guildenstern tossing coins. He is interested in people who are obsessed with proving the unprovable, like George in Jumpers, or who are gambling on a completely new way of thinking, like the ridiculous Tristan Tzara, or Joyce the modernist genius, or the stubborn Galileo. How do new systems of thought come into the world, and what kinds of people might think them?

Although he is famously a playwright of ideas, who often says that he starts from the idea rather than from the plot or the characters—or his own life—there are always people mixed up with the ideas. Part of what attracted him to quantum physics was the compelling personal voice of Richard Feynman. When he heard that Feynman had died, in 1988, just after Hapgood was launched, he said: “I don’t think I’ve ever read [an obituary] which caused me such a stab of grief as I felt on reading of the death of an American physicist whom I had never met and whose work was way out of the reach of my understanding.” He knew that it was, fundamentally, “grief for myself”: he had wanted to send him Hapgood as “an object of tribute.” But, more than that, he had wanted Feynman to know that he had tried to cross the “great divide in our culture” between science and art. Reading Feynman had confirmed his view that “science and art are more like each other than unlike . . . [they] are not just like each other, they sometimes seem to be each other.” He called him “an aristocrat in science and a democrat in almost everything else.”

The cunning beauty and delight of Arcadia is how its ingredients—human, romantic, intellectual, scientific—are meshed together to make a perfect whole.

He responded as strongly to James Gleick’s popular 1987 book on chaos theory, which he read soon after Hapgood was done. He was gripped by its account of the new science challenging orthodoxies in maths and physics, and of the solitary, embattled, creative, sometimes unrecognized scientists who brought it to light. “Genius” in science and in art seemed similar to him, and the idea of “genius” is vital to Arcadia.

Quantum physics had appealed to him because it involved doubling and uncertainty, a drastic change from the fixed certainties of classical physics to the realm of unpredictability. Chaos theory, too, involves the relationship between order and randomness, something that had interested him for a long time. He liked the idea of putting something so unlikely and new into a play.

Chaos theory “attempts to systemize that which appears to function outside of any system. It describes a world in which there is chaos in order, but also order in chaos.” Stoppard calls it “a reconciliation between the idea of things not being random on the one hand and yet unpredictable on the other hand.” It appealed to him because it had to do “with the unpredictability of determinism.” In fact, “chaos theory” is a distorting term, because it makes it sound as if it is all about randomness. And for non-scientists, confusingly, it is the same word that we use in everyday speech when we are talking loosely about a state of hopeless disorder. Some scientists prefer to use the term “deterministic chaos,” so as to get in both sides of its meaning, order and unpredictability.

Nor is it just one science. Chaos theory is a hybrid. It mixes together maths, physics, biology, economics, astronomy. It can apply to fluctuating population growth or weather forecasting, turbulence or earthquakes, eclipses or heartbeat patterns, the formation of snowflakes: to any dynamical system where apparent randomness is found to contain order. Grandly put, it is “a science of the global nature of systems.”

“Chaos” got going in the 1970s, so when Stoppard was reading about it, it was still an excitingly new, radical change of direction. In Gleick’s words: “Where chaos begins, classical science stops.” Newtonian, classical science argued for an entirely determined universe. It maintained that given enough information, we can predict future events. The nineteenth-century scientist Pierre Laplace proposed that one could infer from the deterministic laws of the universe a vast entity, or intellect, which could “embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom.”

Or as Thomasina Coverly puts it in Arcadia in 1809, several years before Laplace: “If you could stop every atom in its position and direction, and if your mind could comprehend all the actions thus suspended, then if you were really, really good at algebra you could write the formula for all the future.” But classical science ignored, or dismissed as monstrosities, what Gleick calls “the irregular side of nature, the discontinuous and erratic side.” Galileo (whose distant shadow haunts Arcadia), Newton and their scientific descendants searched for regularity. But that meant “disregarding bits of messiness that interfere with a neat picture.” The new science wanted to ask big questions: “How does life begin? What is turbulence? . . . In a universe ruled by entropy, drawing inexorably toward greater and greater disorder, how does order arise?”

Chaos theory is a hybrid. It mixes together maths, physics, biology, economics, astronomy.

Chaos theory took the messiness and disorder on board. Gleick tells the story: “The modern study of chaos began with the creeping realization . . . that quite simple mathematical equations could model systems every bit as violent as a waterfall. Tiny differences in input could quickly become overwhelming differences in output.” Chaos theory paid attention to aperiodic systems—systems which “almost repeated themselves but never quite succeeded,” like weather, or animal populations. It dealt in non-linear equations, “which express relationships that are not strictly proportional, and generally cannot be solved,” unlike solvable linear equations. Before that, “almost no one in the classical era suspected the chaos that could lurk in dynamical systems if non-linearity was given its due.”

Chaotic systems have three key properties. They are non-linear, they are deterministic, and they exhibit “sensitive dependence to initial conditions.” The general reader knows about sensitive dependence through the image of the butterfly effect: “A butterfly stirring the air today in Beijing can transform storm systems next month in New York.” As anyone knows who has missed their bus to work, or watched a theatrical farce, “small perturbations in one’s daily trajectory can have large consequences.” “Chaos” is a description for the “rapid amplification” of a small difference to “kingdom-shattering proportions.” A well-known nursery rhyme (in Benjamin Franklin’s version) sums it up:

For the want of a nail the shoe was lost,
For the want of a shoe the horse was lost,
For the want of a horse the rider was lost,
For the want of a rider the battle was lost,
For the want of a battle the kingdom was lost, And all for the want of a horseshoe-nail.

Valentine, the twentieth-century computer scientist and biologist in Arcadia, puts it like this, with excitement and delight: “The unpredictable and the predetermined unfold together to make everything the way it is . . . the smallest variation blows prediction apart.”

But chaos is not entirely chaotic. Apparently random effects, it transpires, follow universal mathematical laws. “A complex system can give rise to turbulence and coherence at the same time.” Order in chaos can be modeled in phenomena as wide-ranging as the population growth of grouse (Valentine’s research project) or the vagaries of the stock market. The mathematician Benoit Mandelbrot, one of the strange geniuses in Gleick’s story of chaos, who influences Arcadia, created a picture of reality which showed that “within the most disorderly reams of data lived an unexpected kind of order.”

How can these patterns and order be traced and made sense of? Chaos scientists looked at how the behavior of dynamical systems bifurcated endlessly. As systems respond to variations in input, the graph of their behavior bifurcates, and then bifurcates again, ad infinitum, into a condition of chaotic randomness. But “within those random states, windows of order reappeared.” Mathematicians use feedback loops, or iterated algorithms, to establish patterns of bifurcation. These algorithms—as we learn in Arcadia—would be impossible to work out with pen and paper, even if a genius could imagine them in her head. They can only be done with electronic calculators and computers.

But chaos is not entirely chaotic. Apparently random effects, it transpires, follow universal mathematical laws.

Thomasina was trying to do in the 1800s a mathematical process that was first used in the early 1970s. Stoppard resisted, rather firmly, the frequent suggestion that she was based on Byron’s daughter Ada Lovelace, the mathematician who worked with Charles Babbage and anticipated computer programming, called her methods “poetical science” and died in her thirties. He heard about Ada after Arcadia was written, from Robert May, the physicist turned biologist, one of his main scientific advisers for the play. May worked on population changes in species, and he used “functional iteration” as a way of describing these changes each year through “a feedback loop, each year’s output serving as the next year’s input.” “The output of one calculation was fed back in as input for the next.”

Nearly two hundred years after Thomasina had intuited this process, her calculations come to light. Valentine describes her achievement to a character who knows nothing about physics or maths and so is a useful stand-in for the audience and the reader. “She’s feeding the solution back into the equation, and then solving it again,” he explains.

. . . It’s how you look at population changes in biology. Goldfish in a pond, say. This year there are x goldfish. Next year there’ll be y goldfish. Some get born, some get eaten by herons. Nature manipulates the x and turns it into y. Then y goldfish is your starting population for the following year . . . Your value for y becomes your next value for x.

Any sixth-former doing maths knows about algorithms: the magic is in Thomasina’s anticipating them. As Stoppard says: “You do not have to be Einstein to have the idea of feedback, the idea of the algorithm which just operates on itself. The problem was that there were not enough pencils or paper or time to do it often enough. To do it so many times that the pattern emerges.”

Algorithms can be used to draw the irregular shapes of nature. Fractal or fractional geometry—terms coined by Mandelbrot from the Latin for “broken” and from the English words “fracture” and “fraction”—was a way of measuring qualities that otherwise have no clear definition. Mandelbrot’s example was the degree of roughness or irregularity in a twisting coastline. Fractal geometry enabled mathematics to describe, explore and mirror the unpredictable forms of nature. Gleick explains it eloquently. You used “a shape to help visualize the whole range of behaviours in a system.” You looked at materials like “the fragmented landscapes of an earthquake zone” so as to calculate “the irregular patterns of nature.” You could study “the way things meld together, the way they branch apart, or the way they shatter.”

Fractal geometry enabled mathematics to describe, explore and mirror the unpredictable forms of nature.

Fractal also meant “self-similar.” “Self-similarity is symmetry across scale. It implies recursion, pattern inside of pattern.” Fractal geometry produces detail at finer and finer scales, as in the image of a person reflected in mirror after mirror after mirror, infinitely receding. Mandelbrot liked to quote Jonathan Swift:

So, Nat’ralists observe, a Flea
Hath smaller Fleas that on him prey,
And these have smaller Fleas to bite ’em,
And so proceed ad infinitum.
Self-similarity got into the content—and the structure—of Arcadia.

Thomasina writes in her mathematics primer that she has invented “a method whereby all the forms of nature must give up their mathematical secrets and draw themselves through numbers alone.” Valentine explains how it works: “If you knew the algorithm and fed it back say ten thousand times, each time there’d be a dot somewhere on the screen. You’d never know where to expect the next dot. But gradually you’d start to see this shape, because every dot will be inside the shape of this leaf. It wouldn’t be a leaf, it would be a mathematical object.” Thomasina Coverly has worked out how a mathematical drawing could simulate an apple leaf, and Valentine reproduces the image on his computer two hundred years on. He calls it “the Coverly set,” in tribute to “the Mandelbrot set,” his beautiful, complex computer images of fractal shapes.

Some years after Arcadia came out Stoppard had a public conversation with Robert Osserman, deputy director of the Mathematical Sciences Research Institute in Berkeley. He said then that what fascinated him in the idea of fractal geometry was that “there is a complementarity in the notion that math describes nature, and that nature is following mathematical rules . . . Numbers have a kind of social behavior, they’re not simply tools of description.” Thomasina “understands . . . that it’s not actually the equation in itself, it’s the way you . . . manipulate the equation, the way you pull the solution back into the equation in a feedback loop, which generates shapes . . . That, in the end, is what is moving about the idea that nature is written in numbers. That it’s generative.” It was the parallel behavior of nature and mathematics that fascinated him, just as the parallel creative energy of the arts and the sciences was one of the play’s subjects.

Algorithms were not Thomasina’s only discovery. The past time zone of the play jumps, in a non-linear fashion, from 1809 to 1812. She is thirteen when it starts and nearly seventeen when it ends. When she is thirteen she notices, as a child might notice, that you can only stir the jam in your rice pudding one way: “If you stir backwards, the jam will not come together again.” Yes, says her tutor Septimus, the Newtonian, classical scientist, who at this point knows more than his pupil, time will only go one way, and “we must stir our way onward mixing as we go, disorder out of disorder into disorder until pink is complete, unchanging and unchangeable, and we are done with it for ever.” Thomasina agrees with Septimus: “You cannot stir things apart.” Gleick calls this “entropy explained in five words.”

Three years later, she has become wiser than her tutor. She has worked out that there is no such thing in the universe as an “unchanging and unchangeable” state. Centuries ahead of her time, she has grasped the Second Law of Thermodynamics. She has worked out that “the heat equation . . . goes only one way.” That is, she has foreseen the law of entropy, which tells us that the universe is evolving from order to disorder. Heat, according to the Second Law of Thermodynamics, can only flow in one direction, from hotter to colder. There is no way back in time, no backward swing of the pendulum. The idea that the world’s energy, heat and light will eventually run out and the universe at last be nothing but dark void runs like a sombre drumbeat under the life and vitality of the play.

Algorithms were not Thomasina’s only discovery. The past time zone of the play jumps, in a non-linear fashion, from 1809 to 1812.

Gleick gives an example of “the inexorable tendency of the universe, and any isolated system in it, to slide toward a state of increasing disorder.” He explains—like Thomasina telling Septimus about stirring the jam—that if you have a bath with two separate compartments, one containing water and one containing ink, and you remove the barrier between the compartments so that ink and water mix together, the mixture never reverses itself. This is the part of physics “that makes time a one-way street.” “Everything tends towards disorder. Any process that converts energy from one form to another must lose some as heat. Perfect efficiency is impossible.”

But Gleick does provide some consolation, which affects the mood of Arcadia. What chaos theory shows is that “somehow, after all, as the universe ebbs towards its final equilibrium in the featureless heat bath of maximum entropy, it manages to create interesting structures . . . Nature forms patterns . . . The universe is randomness and dissipation, yes. But randomness with direction can produce surprising complexity . . . Dissipation is an agent of order.”

Stoppard leaped on these ideas with excitement and poured them into his play. As always he relished the technical language of specialist disciplines. “Noise,” for instance, is the scientist’s word for “error,” or “observational uncertainty.” It is “what scientists blame for the inaccuracy of their measurements.” Too much noise, in Arcadia, is what drives Valentine off course in his research. In the play, “noise” becomes a metaphor for extravagant and ridiculous behavior, especially that of the fame-seeking literary don Bernard, a very noisy character. “Trivial” means, for scientists, redundant information that doesn’t lead anywhere or proofs with no value. In the play, it is a telling word for what matters and what doesn’t. Personal relationships and the achievements of individuals, says Valentine, are “trivial” compared with the search for knowledge itself.

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Tom Stoppard: A Life by Hermione Lee

Excerpted from Tom Stoppard: A Life by Hermione Lee. Copyright © 2021 by Hermione Lee. Excerpted by permission of Alfred A. Knopf, a division of Penguin Random House LLC. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

Hermione Lee
Hermione Lee
Hermione Lee was president of Wolfson College (2008-2017) and is Emeritus Professor of English Literature at Oxford University. Her work includes biographies of Virginia Woolf, Edith Wharton, and Penelope Fitzgerald (winner of the James Tait Black Prize and one of the New York Times's 10 Best Books of 2014). She has also written books on Elizabeth Bowen, Philip Roth, and Willa Cather. Lee was awarded the Biographers' Club Prize for Exceptional Contribution to Biography in 2018. She is a Fellow of the British Academy and the Royal Society of Literature, and a Foreign Honorary Member of the American Academy of Arts and Sciences. In 2003 she was made a CBE, and in 2013 she was made a Dame for services to literary scholarship.





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