Arcadia

Tom Stoppard's Arcadia is a carefully balanced play that resists summary description. It is no more ``about'' the history of science than it is ``about'' landscape gardening, academic infighting, Romantic genius, or carnal embrace. But Stoppard is interested in all these things and expects his audience to share and enjoy his enthusiasms. In this talk, I'll describe part of the foundation of ideas on which his play is constructed, with particular emphasis on science and mathematics.

About half of the scenes in the play are set in 1809 and 1812, in England. When intelligent and well-informed people of that place and time thought seriously about the physical universe, they relied on Isaac Newton as their guide to its nature and structure. Newton died in 1727, but his overwhelming influence on the thinking of scientists and other intellectuals in England and France continued for a century after his death.

There were several reasons for this influence. One is that the explanations provided by Newton's theory were strikingly general, accurate, and simple. Newton was the first to formulate a satisfactory theory that accounted for the elliptical motions of planets around the sun, the straight-line motions of objects dropped from high places near the surface of the earth, and the parabolic motions of objects thrown into the air. Previously, scientists had supposed that there was some fundamental distinction between heavenly bodies and terrestrial objects that made them move in different ways; Newton's laws applied to both.

The physical universe, in Newton's system, consists entirely of material objects in motion, attracting one another gravitationally and occasionally colliding and rebounding, like billiard balls on a frictionless table. The mechanics of their interactions can be summarized in a few concise equations, and these laws, once understood, are simple. The reason that they were not obvious in Newton's time is that to formulate some of them one needs mathematical ideas and notations that Newton had to invent -- the ideas of the differential calculus. But by the time in which Arcadia is set these ideas and notations were already commonplace among people who had scientific training.

Newton's laws are simple. The equations that one derives from them have exact solutions. If you observe the position of, say, a newly discovered comet for several nights, you can solve the equations and compute its orbit for months and years into the future, or (working backwards) figure out where it had been months and years earlier, with nearly perfect accuracy. Newton's laws are deterministic; they purport to establish, with the certainty of mathematics, necessary connections between the present state of the universe -- the positions and velocities of the bodies that compose it -- and the states of the universe at other times.

This determinism posed some conceptual problems for the philosophers and theologians of the Enlightenment:

But it also led to the expectation that the physical universe could be completely understood through the classical mathematics that Newton developed -- through what an off-stage source in Stoppard's play refers to as ``good English algebra'':

This striking conception of the universe as the material instantiation of a mathematical formula was propounded by several writers in the early nineteenth century -- most memorably by Pierre-Simon Laplace, who wrote (in his Analytical Theory of Probabilities, the first part of which was published in 1812):

Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective positions of the beings which compose it, if moreover this intelligence were vast enough to submit these data to analysis, it would embrace in the same formula both the movements of the largest bodies in the universe and those of the lightest atom; to it nothing would be uncertain, and the future as the past would be present to its eyes.

Laplace is completely confident that the physical universe has an intelligible structure. This confidence is a characteristic of Newtonian science in the Enlightenment. Proposing an epitaph for Newton in about 1740, Alexander Pope wrote:

Nature and Nature's laws lay hid in night:
God said, Let Newton be! and all was light.

From our perspective, the implication that Newton was God's official interpreter of Nature seems arrogant and overblown, and modern wits have parodied Pope's couplet:

It did not last. The Devil, shouting ``Ho!
Let Einstein be!'' restored the status quo.

The structure of the classical mathematics that sustained Laplace's confidence was deductive. The serious study of mathematics at that time began with the construction of proofs in plane geometry; to prove some proposition that had never been proved before was the highest accomplishment and greatest glory of a mathematician. In Arcadia, one of the principal characters in the 1809 and 1812 scenes is a thirteen-year-old genius named Thomasina Coverly. Her tutor, trying to keep her busy and challenged for an hour, sets her a problem of classical mathematics -- finding a proof of Fermat's ``last theorem'':

Serious students of mathematics in 1809 knew this proposition well. It appeared in a problem collection that has been reliably dated to the third century. The name of Pierre de Fermat is associated with it because, in his copy of the Latin translation of this ancient problem collection, he wrote a note in the margin of the page on which the proposition is stated: ``I have discovered a truly remarkable proof which this margin is too small to contain.'' However, no one knows what this proof was or whether it was correct, and the problem remained unsolved until very recently. In fact, the modern proof (by Andrew Wiles, now of Princeton University) was completed a few months after the initial performance of Arcadia. (The 1994 proof cannot be anything like the line of reasoning that Fermat was thinking of; it is rather indirect and makes essential use of mathematical results and methods that were unknown and practically inconceivable in Fermat's time.)

Wiles devoted eight years to the construction of the proof. Here is how describes the moment at which he completed the last main step:

I had this incredible revelation. It was the most important moment of my working life. Nothing I ever do again ... it was so indescribably beautiful, it was so simple and so elegant, and I just stared in disbelief for twenty minutes, then during the day I walked round the department. I'd keep coming back to my desk to see it was still there -- it was still there.

This is a modern expression of the classical mathematician's attitude towards his subject: There is nothing more beautiful than a simple, complete, deductive proof of a previously unproven result.

But when one takes Laplace's universal formula seriously and starts to consider what such the formula that describes everything would really look like, one begins to realize that the elegance and concision of classical mathematics, as exemplified by Newton's laws and Fermat's proposition, are misleading. Stoppard initially approaches this idea comically:

There is a serious point behind this patter, though. The natural world includes many things and phenomena that are far too irregular to be described by the kinds of equations that Newtonians preferred to think about. Simple formulas yield the ellipses of planetary orbits, the hyperbolas of comets, and the shapes of manufactured objects -- their graphs are smooth curves, or curves that become smooth if viewed at a sufficiently large scale, or at worst curves with well-defined cusps or corners:

The mathematical description of natural objects of ``irregular forms'' calls for curves that are not smooth at any scale, but have cusps and corners everywhere. Such curves are now called fractals. Only in the last quarter century have they been studied extensively. In the nineteenth century they were completely unknown.

Some fractals are pure abstractions, but others are surprisingly good models of irregular forms in nature: trees, ferns, the skylines of mountains and the contours of coasts.

Only in the last quarter century has it been technologically possible to study fractals in any but the most superficial way. One needs computers to investigate iterated algorithms and to depict fractals that model irregular forms in nature. A curve that simulates the appearance of a fern might require the plotting of one hundred thousand points -- in other words, the solution of a hundred thousand equations.

A single graphic image of a fractal may depend on as many numerical values as a pencil-and-paper calculator could compute in a decade:

Iterated algorithms model many apparently irregular things in nature: the numbers of goldfish in a pond (the population in one year is one of the conditions that determines the population in the next year, in a feedback relationship -- low populations increase, too-high populations decrease, seemingly irregularly), predator-prey populations, the erratic intervals between the drips from a slowly leaking faucet. The people who study them are engaged in a kind of mathematics that is partly empirical and exploratory rather than deductive, partly proof and partly visualization and modelling:

The implications for science and mathematics are revolutionary:

As Valentine points out, iterated algorithms often exhibit ``sensitivity to initial conditions.'' A tiny imprecision in the measurement of the current state of the world is magnified by the operation of the algorithm -- the feedback mechanism of iteration -- into large differences in the predicted state of the world in the future. Laplace's imagined formula is rendered useless by the fact that in practice the quantities are not known with perfect exactness. If we sat down with the formula in hand and tried to calculate whether it would rain on the garden party three Sundays from now, we would find that the answer would depend critically on variations of the speed and direction of the wind so tiny that they could not be measured or detected even in principle.

Another thing one discovers when one tries to fit everything into the Newtonian view of the world is that, at Valentine's middle scale, time is not as symmetric as Newton's laws seem to imply. In Newton's system, time has no implicit direction -- it is equally easy to determine where a planet was on some past day or where it will be on some future day. Time is notionally reversible; the billiard balls colliding on the frictionless table behave the same way whether one follows them forwards or backwards through time.

The asymmetry of time in the physical universe was scarcely noticed until about Thomasina and Septimus's period. For instance, in 1812, scientists were just beginning to understand the nature of heat, the way it distributes itself through matter, and the principles on which steam engines operate. Joseph Fourier presented a monograph ``On the Propagation of Heat in Solid Bodies'' to the Paris Institute in 1807; the characters in Arcadia, like most European scholars, became acquainted with it only after it was published as the winning essay in a prize competition conducted by the Institute in 1811. Sadi Carnot's book On the Motive Power of Fire, published in 1824, contained the first rough version of what later became known as the second law of thermodynamics.

(Incidentally, I think that Carnot was one of Stoppard's principal models for Thomasina Coverly. He was born almost exactly one year after Thomasina; he was recognized at an early age as a prodigious and original thinker. Unfortunately, he died untimely young, of cholera, at the beginning of what surely would have been a brilliant scientific career. On the Motive Power of Fire was his only published work.)

In a closed system that contains both hot and cold bodies, the hot ones cool off and the cold ones warm up. Unlike the collisions of billiard balls, this physical event doesn't go equally well in reverse; lukewarm bodies do not spontaneously separate into hot ones and cold ones.

Valentine: You can't run the film backwards. Heat was the first thing which didn't work that way. Not like Newton. A film of a pendulum, or a ball falling through the air -- backwards, it looks the same. ... But with heat -- friction -- a ball breaking a window -- ... it won't work backwards. ... You can put back the bits of glass but you can't collect up the heat of the smash. It's gone. ... The heat goes into the mix ... And everything is mixing the same way, all the time, irreversibly. (pp. 92-93)

Probability works against it: just as there are more ways for a deck of cards to be mixed by suit than separated into suits, so that shuffling almost never results in spontaneous separation, so random collisions of fast-moving and slow-moving particles tends to move extremes of both kinds towards the middle, not the other way around.

In any physical event in which energy is not added to the system from outside, some heat is dissipated irrecoverably. So, for instance, a steam engine cannot recover during the one part of its cycle all of the energy it produced during another part, even if the pistons are not connected to anything external and are therefore ``doing no work.'' The second law of thermodynamics implies that you can never extract as much energy from a system as you put into it; the physical events that seem to be counter-examples to this proposition always turn out to involve some alternative source of energy, not taken into account in the original description of the system, that compensates for the dissipation of heat.

The physical universe is a system to which it is by definition impossible to add energy from some external source. So the second law of thermodynamics implies that in the fullness of time everything runs down. The universe will end in ``heat death'' -- a state in which heat is almost evenly distributed throughout the universe, with only random, small, transient variations. In this state, almost no physical events occur that are in any way useful or interesting -- without temperature differences, there is no way to control or direct the energy in the heat.

In the play, Thomasina reads Fourier's monograph and draws this unexpected and frightening conclusion. She tries to explain it to her tutor, who cannot understand her diagram of the operation of the steam engine or its relevance to the ultimate fate of the universe:

So she tries to describe the implications of the idea without mathematics, and the tutor objects:

And, you know, in a way he's right -- if you leave out the mathematics and the discipline of observation and experiment, science does appear, to some observers, to be a kind of story-telling, and the history of science to be a succession of increasingly strange fabulations, inconsistent with one another and remote from human experience. In one of the scenes in Arcadia that is set in contemporary England, Bernard Nightingale, lecturer in English literature at the University of Sussex and aspiring authority on the life and works of Byron, expresses this view:

Let me see if I can present Stoppard's ideas about why scientific progress matters. It is indeed a defect of Newton's system of the world that it explains the movements of the planets but not the shape of a leaf or the stirrings of the human heart, and no muttering about ``infinities where we cannot follow'' can satisfy the universal human desire to know:

Hannah: It's wanting to know that makes us matter. Otherwise we're going out the way we came in. ... If the answers are in the back of the book I can wait, but what a drag. Better to struggle on knowing that failure is final. (p. 37)

Admittedly, the advancement of knowledge is itself irregular, marked by occasional reverses -- Stoppard uses the burning of the great library at Alexandria as a symbol of them. Thomasina thinks of this ancient disaster with a deep and personal sense of loss:

And, because we are often fortunate enough to learn from those we follow, each generation can be more clever and selective in choosing what little it can carry. We have amazing tools of understanding. As mathematicians, we are not afraid even of infinity! One of the greatest accomplishments of nineteenth-century mathematics was to make tractable and consistent the concepts of the infinitely large, infinitely small, and infinitely long. They are, in a sense that we now know how to make precise and demonstrable, extensions or limiting cases of finite quantities. As Thomasina says, responding to Septimus's attempt to make infinity into a divine mystery --

Moreover, the knowledge that we acquire through science is not as remote from human experience as Bernard supposes. He thinks that the universe has nothing to do with him; science, in his view, deals only with the very large and the very small, so that he can dismiss it as irrelevant. If the study of the physical universe had ended with Newton, or with Aristotle, he might have a case. But we want to know more than Newton's laws and classical mathematics can teach us. We want to know the shapes of ferns and mountains. We dare to learn the truth even when observation and experiment and mathematics entail a story that, told in human terms, is utterly tragic -- the view that eventually everything passes away, everything breaks, everything falls apart:

Time flows irrevocably in the direction of heat death.

And yet the mathematics that leads to this melancholy image of the world at the end of time also generates the beautiful and intricate patterns that describe the manner of its passing:

Valentine: See? In an ocean of ashes, islands of order. Patterns making themselves out of nothing. I can't show you how deep it goes. (p. 76)

Because of the progress of science, we have strong evidence for a tragic view of human life -- that in the long run youth and genius and beauty and life itself all run down and dissipate into randomness. But science also gives us new ways to perceive and value these delightful attributes of the world as it is now, to walk in the beauty that the universe presents to us who know and love it.

This document is available on the World Wide Web as

http://www.math.grin.edu/~stone/events/Arcadia/

created November 6, 1997
last revised November 14, 1997

John David Stone (stone@math.grin.edu)