Pressgang Productions

Oxford's contemporary theatre specialists

THE LOVE OF THE NIGHTINGALE by Timberlake Wertenbaker OLD FIRE STATION THEATRE, OXFORD 27 NOV-1 DEC 2001 Box Office 01865 297170

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Does God Play Maths?

"Physical laws should have mathematical beauty"

-Dirac

For hundreds of years, scientists have used and abused the language of mathematics in an effort to describe the behaviour of the physical world.  Yet as they succeed in describing one set of facts, they fail to describe another.  It is as though the scientists are trying to fit a carpet that is too big for the room: they push down in one corner, only to have the opposite corner pop up.  The greatest breakthroughs in physical science have come from those with the insight to strip away the layers of complexity; to cut the carpet.  But will the carpet ever be cut small enough?  Can the natural world ever be fully explained with mathematics?  It is staggering that any relationships can be established between the two.  Mathematics is so abstract, built on concepts with no physical meaning, like number and proof.  Yet this intangible systems seems to hold reign over the natural world.  Insights in one branch yield insights in the other; a bridge has been built between the unbridgeable.

In Tom Stoppard’s Arcadia Thomasina Coverly embodies a fundamental human drive – the desire for knowledge.  A scientist is born and not bred; she is the child who always asks “why”.  Thomasina demonstrates this instinctive curiosity early in the play, asking about Fermat’s Last Theorem.  Born in 1601 to a leather merchant, Pierre de Fermat became a legal clerk by trade.  (It is his lack of formal training that earned Fermat the nickname “Prince of Amateurs” among mathematicians.)  Fermat read widely, and was fond of annotating texts.  In 1637 he left a marginal note in Diophantus’ masterwork, Arithmetica that has intrigued and puzzled mathematicians for more than 350 years.  He stated that “it is impossible to separate… any power above the second into two powers of the same degree”.  In more modern notation, we say that the equation xⁿ + yⁿ = zⁿ has no solution in whole numbers if n is greater than two.  (Any GCSE mathematics student will recognise the case when n equals two as Pythagoras’ Theorem.)  This problem owes its fame in no small part to the fact that Fermat claimed he has a “truly marvellous” proof of the theorem, but gave no details.  This was the famous “note in the margin”, which Thomasina claims was simply “a joke to make you all mad.”  Did Fermat really have a proof?  Unlikely.  British mathematician Andrew Wiles finally published a correct proof in 1994, but this could not have been the proof Fermat may have had.  Wiles’ proof took the simple problem of Fermat’s Last Theorem, and solved it using complicated twentieth century techniques.  Fermat’s problem was so simple, so smooth, that to get a grip on it it was necessary to add many layers of complexity.  Wiles had to work, in Thomasina’s words, “outward from the middle of the maze.”

Thomasina’s interest in mathematics flourishes later in the play as she turns her attention to geometry.  In trying to describe the mathematical equation of a leaf she talks of the creation of a “new geometry of irregular forms”.  The geometry to which Thomasina refers is fractal geometry.  In this, a set of instructions, called an algorithm, is applied to a number or a shape.  These instructions might be “square the number and add one”, or “copy the shape, rotating the copy through 45 degrees”.  The same set of instructions are then re-applied to the resulting number or shape, producing a new one.  The repetition of this process is called iteration, and fractal geometry deals with the behaviour of shapes and numbers under iterated algorithms.  Thomasina carries out a few iterations in her notebook.  But as Valentine demonstrates, modern computers can take the process to new levels, iterating an algorithm thousands, or even millions of times.  It is the self similarity in this process that allows it to model nature so accurately – each branch of a fern leaf is like a miniature copy of the whole leaf.  Thomasina questions Septimus before making her discovery, asking if “nature is written in numbers”.  All we can be sure of is that mathematical patterns crop up again and again in the natural world.  Numbers are certainly written in nature.

The idealism of the young Thomasina becomes apparent when she claims that “if you could stop every atom in its position and direction… then if you were really really good at algebra you could write a formula for all the future.”

This concept – the deterministic universe – was destroyed earlier this century with the birth of Heisenburg’s Uncertainty Principal.  Werner Heisenburg showed that the greater the certainty with which you know where an atom is, the lesser the certainty with which you know how it is moving.  In order for Thomasina’s “formula for all the future” to be calculated you would need to know exactly where every atom was, exactly which way it was moving, and exactly how quickly.  Impossible, said Heisenburg.  Simply by observing an atom you are bouncing light off it, altering its velocity.  This fundamental principle has led to the study of quantum mechanics – describing the behaviour of atoms in terms of probability rather than certainty.

Thomasina’s formula would also rely on Newton’s equations being the fundamental rules determining how all objects behave.  Thomasina challenges herself here, asking “is God a Newtonian?”  In 19^th century England, when copies of Newton’s Principia adorned every coffee table in the country, this might have been considered sacrilege.  But Newton’s laws are simply a theorem; a scientific model that Newton formulated to best model the facts he had.  In the last century, Newton’s laws have been superceded by another model: Einstein’s relativity.  Is this scientific theory the magna carta for the behaviour of the universe?  No.  Will another, more accurate model replace it?  Almost certainly.  And with each cut of the carpet, we find ourselves closer to a solution.  However, it is my beliefs that as close as we may get to that solution, we will never attain it.  The laws that govern our universe are simply too wonderful to be laid down in the language of mathematics, and too abstract to be understood by any human being.

As Arcadia draws to a close, and disorder takes hold of Sidley Park, Thomasina makes her most startling observation, the second law of thermodynamics.  As immortalised in the Flanders and Swann lyric “heat won't pass from the cooler to the hotter/ you can try it if you likes but you'd far better notter,” the second law states that heat must flow from hot to cold, and that whenever work is done energy is irrecoverably turned into heat.  In reality discovered jointly by Clausius in 1850 and Kelvin in 1851, it is the second law of thermodynamics that means we need energy to power our fridges as well as our cookers.  The directional nature of the law means that it acts as a one-way sign for the universe.  If time were to be reversed all physical laws would still hold, with the exception of the second law of thermodynamics – heat would be seen to flow from cold to hot.  The second part of the law is the most alarming.  If all energy must be irrecoverably be turned to heat, the universe is destined to become a featureless desert, a fraction of a degree warmer than absolute zero.  “The improved Newtonian universe must cease and grow cold.”  Thomasina first notices this consequence when stirring her rice pudding.  The jam must become irrecoverably mixed in – “disorder out of disorder until pink is complete.”

All of Thomasina’s ideas concern simplicity and complexity in the physical world and the way it is described.  Newton’s laws, fractal geometry, the second law of thermodynamics, even Fermat’s problem, all attempt to tell us something about the way the world works.  By increasing the complexity of the problems – by adding more structure – we attempt to get closer to the truth.  Sadly, but inevitably, we fall so far short.  Do the answers lie in simple equations like that of Fermat’s Last Theorem, or in vast, complex formulae like Thomasina’s “equation of the future”?  Perhaps time will tell.  But through this uncertainty one thing becomes clear: Mathematics and Science are in their infancy, whereas Nature is fully matured.  Perhaps it is time we learnt to walk before we can run.

HARRY SMITH

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