What We Cannot Know: Explorations at the Edge of Knowledge
Marcus du Sautoy
‘Brilliant and fascinating. No one is better at making the recondite accessible and exciting’ Bill BrysonBritain’s most famous mathematician takes us to the edge of knowledge to show us what we cannot know.Is the universe infinite?Do we know what happened before the Big Bang?Where is human consciousness located in the brain?And are there more undiscovered particles out there, beyond the Higgs boson?In the modern world, science is king: weekly headlines proclaim the latest scientific breakthroughs and numerous mathematical problems, once indecipherable, have now been solved. But are there limits to what we can discover about our physical universe?In this very personal journey to the edges of knowledge, Marcus du Sautoy investigates how leading experts in fields from quantum physics and cosmology, to sensory perception and neuroscience, have articulated the current lie of the land. In doing so, he travels to the very boundaries of understanding, questioning contradictory stories and consulting cutting edge data.Is it possible that we will one day know everything? Or are there fields of research that will always lie beyond the bounds of human comprehension? And if so, how do we cope with living in a universe where there are things that will forever transcend our understanding?In What We Cannot Know, Marcus du Sautoy leads us on a thought-provoking expedition to the furthest reaches of modern science. Prepare to be taken to the edge of knowledge to find out if there’s anything we truly cannot know.
Copyright (#u9b12e613-7f53-53ac-b922-016f09fd96da)
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First published in Great Britain by 4th Estate in 2016
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Copyright © Marcus du Sautoy 2016
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Source ISBN: 9780007576593
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Version: 2017-04-01
Dedication (#u9b12e613-7f53-53ac-b922-016f09fd96da)
To my parents,who started me on my journeyto the edges of knowledge
CONTENTS
Cover (#u2781c0ea-fca9-5471-8e08-5211b2fdf6c6)
Title Page (#u81411db6-aaef-5c68-96b6-3bad3640d9b5)
Copyright
Dedication
Edge Zero: The Known Unknowns
First Edge: The Casino Dice
Chapter 1
Chapter 2
Second Edge: The Cello
Chapter 3
Chapter 4
Third Edge: The Pot of Uranium
Chapter 5
Chapter 6
Fourth Edge: The Cut-Out Universe
Chapter 7
Chapter 8
Fifth Edge: The Wristwatch
Chapter 9
Chapter 10
Sixth Edge: The Chatbot App
Chapter 11
Chapter 12
Seventh Edge: The Christmas Cracker
Chapter 13
Chapter 14
Further Reading
Index
Acknowledgements
Illustration Credits
Also by Marcus du Sautoy
About the Publisher
EDGE ZERO: (#u9b12e613-7f53-53ac-b922-016f09fd96da)The Known Unknowns (#u9b12e613-7f53-53ac-b922-016f09fd96da)
Everyone by nature desires to know.
Aristotle, Metaphysics
Science is king.
Every week, headlines announce new breakthroughs in our understanding of the universe, new technologies that will transform our environment, new medical advances that will extend our lives. Science is giving us unprecedented insights into some of the big questions that have challenged humanity ever since we’ve been able to formulate those questions. Where did we come from? What is the ultimate destiny of the universe? What are the building blocks of the physical world? How does a collection of cells become conscious?
In the last ten years alone we’ve landed a spaceship on a comet, made robots that can create their own language, used stem cells to repair the pancreas of diabetic patients, discovered how to use the power of thought alone to manipulate a robotic arm, sequenced the DNA of a 50,000-year-old cave girl. Science magazines are bursting with the latest breakthroughs emerging from the world’s laboratories. We know so much. The advances of science are extremely intoxicating.
Science has given us our best weapon in our fight against fate. Instead of giving in to the ravages of disease and natural disaster, science has created vaccines to combat deadly viruses like polio and even ebola. Faced with an escalating world population, it is scientific advances that provide the best hope of feeding the 9.6 billion people who are projected to be alive in 2050. It is science that is warning us about the deadly impact we are having on our environment and giving us the chance to do something about it before it is too late. An asteroid might have wiped out the dinosaurs, but the science that humans have developed is our best shield against any future direct hits. In the human race’s constant battle with death, science is its best ally.
Science is king not only when it comes to our fight for survival but also in improving our quality of life. We are able to communicate with friends and family across vast distances. We have unparalleled access to the database of knowledge we have accumulated over generations of investigation. We have created virtual worlds that we can escape to in our leisure time. We can recreate in our living rooms the great performances of Mozart, Miles and Metallica at the press of a button.
That desire to know is programmed into the human psyche. Those early humans with a thirst for knowledge are those who have survived, adapted, transformed their environment. Those not driven by that craving were left behind. Evolution has favoured the mind that wants to know the secrets of how the universe works. The adrenaline rush that accompanies the discovery of new knowledge is nature’s way of telling us that the desire to know is as important as the drive to reproduce. As Aristotle articulated in the opening line of his book Metaphysics, understanding how the world works is a basic human need.
As a schoolkid, science very quickly drew me into its outstretched arms. I fell in love with its extraordinary power to tell us so much about the universe. The fantastic stories that my science teachers told seemed even more fanciful than the fiction I’d been reading at home. As it worked its spell on me I consumed every outlet of science I could get my hands on.
I persuaded my parents to buy me a subscription to New Scientist. I devoured Scientific American in our local library. I hogged the television each week to watch episodes of my favourite science programmes: Horizon and Tomorrow’s World. I was captivated by Jacob Bronowski’s Ascent of Man, Carl Sagan’s Cosmos, Jonathan Miller’s Body in Question. Every Christmas the Royal Institution Christmas Lectures provided a good dollop of science alongside our family turkey. My stocking was stuffed with books by Gamow and Feynman. It was a heady time, with new breakthroughs being announced each week.
Alongside reading these stories of the discovery of things we know, I began to get more fired up by the untold tales. What we knew lay in the past but what we didn’t yet know was the future, my future. I became obsessed with the puzzle books of mathematician Martin Gardner that my maths teacher gave me. The excitement of wrestling with a conundrum and the sudden release of euphoria as I cracked each puzzle got me addicted to the drug of discovery. Those puzzles were my training ground for the greater challenge of tackling questions that didn’t have an answer in the back of the book. It was the unanswered questions, the mathematical mysteries and scientific puzzles that no one had cracked that would become the fuel for my life as a scientist.
WHAT WE KNOW
If I look back to the Seventies when I was at school and compare the things that we knew then to what we know now, it is quite extraordinary how much more we have understood about the universe even in the half century that I’ve been alive. Technology has extended our senses so we can see things that were beyond the conception of the scientists who excited me as a kid.
The new range of telescopes that look out at the night sky have discovered planets like the Earth that could be home to intelligent life. They have revealed the amazing fact that three-quarters of the way into the lifetime of our universe the expansion of the universe started to accelerate. I remember reading as a kid that we were in for a big crunch, but now it seems that we have a completely different future waiting for us.
The particle colliders like the Large Hadron Collider at CERN (the European Organization for Nuclear Research in Switzerland) have allowed us to penetrate the inner workings of matter itself, revealing new particles – like the top quark discovered in 1994 and the Higgs boson discovered in 2012 – that were bits of speculative mathematics when I was reading my New Scientist at school.
And since the early Nineties the fMRI scanner has allowed us to look inside the brain and discover things that in the Seventies were frankly not even considered part of the remit of scientists. The brain was the preserve of philosophers and theologians, but today the technology can reveal when you are thinking about Jennifer Aniston or predict what you are going to do next even before you know.
Biology has seen an explosion of breakthroughs. In 2003 it was announced that scientists had mapped one whole human DNA sequence consisting of 3 billion letters of genetic code. In 2011 the complete neuronal network of the C. elegans worm was published, providing a complete picture of how the 302 neurons in the worm are connected.
Chemists too have been breaking new territory. A totally new form of carbon was discovered in 1985, which binds together like a football, and chemists surprised us again in 2003 by creating the first examples of graphene, showing how carbon can form a honeycomb lattice one atom thick.
And in my lifetime the subject to which I would eventually dedicate myself, mathematics, has seen some of the great enigmas finally resolved: Fermat’s Last Theorem and the Poincaré conjecture, two challenges that had outfoxed generations of mathematicians. New mathematical tools and insights have opened up hidden pathways to navigate the mathematical universe.
Keeping up with all these new advances, let alone making your own contribution, is a challenge in its own right.
THE KNOW-IT-ALL PROFESSORSHIP
A few years ago I got a new job title to add to my role as a professor of mathematics at the University of Oxford. It often makes me laugh: the Simonyi Professor for the Public Understanding of Science. There seems to be a belief that with such a title I should know it all. People ring me up expecting that I know the answers to every question of science. Shortly after I’d taken on the job, the Nobel Prize for medicine was announced. A journalist called, hoping for an explanation of the breakthrough that was being rewarded: the discovery of telomeres.
Biology has never been my strong point, but I was sitting in front of my computer screen and so I’m embarrassed to admit I got the Wikipedia page up on telomeres and, after a quick scan, proceeded to explain authoritatively that they are the bit of genetic code at the end of our chromosomes that controls ageing among other things. The technology we have at our fingertips has increased that sense that we have the potential to know anything. Just tap my question into a search engine and the device seems to predict, even before I’ve finished typing, what it is I want to know and provides a list of places to find the answer.
But understanding is different from a list of facts. Is it possible for any scientist to know it all? To know how to solve non-linear partial differential equations? To know how SU(3) governs the connection between fundamental particles? To know how cosmological inflation gives rise to the state of the universe? To know how to solve Einstein’s equations of general relativity or Schrödinger’s wave equation? To know how neurons and synapses trigger thought? Newton, Leibniz and Galileo were perhaps the last scientists to know all that was known.
I must admit that the arrogance of youth infused me with the belief that I could understand anything that was known. If someone’s human brain out there has found a way to navigate a path to new knowledge, then if the proof works in their brain it should work in mine. With enough time, I thought, I could crack the mysteries of mathematics and the universe, or at least master the current lie of the land. But increasingly I am beginning to question that belief, to worry that some things will forever remain beyond my reach. Often my brain struggles to navigate the science we currently know. Time is running out to know it all.
My own mathematical research is already pushing the limits of what my human brain feels capable of understanding. I have been working for over ten years on a conjecture that remains stubbornly resistant to my attempts to crack it. But my new role as the Professor for the Public Understanding of Science has pushed me outside the comfort zone of mathematics into the messy concepts of neuroscience, the slippery ideas of philosophy, the unfounded theories of physics. It has required a different way of thinking that is alien to my mathematical mode of thought, which deals in certainties, proofs and precision. My attempts to understand everything that is currently regarded as scientific knowledge has severely tested the limits of my own ability to understand.
The process of attaining knowledge necessarily relies on our standing on the shoulders of giants, as Newton famously declared about his own breakthroughs. And so my own journey to the edges of knowledge has involved reading how others have articulated the current state of knowledge, listening to lectures and seminars by those immersed in the field I’m trying to understand, talking to those pushing the boundaries, questioning contradictory stories, consulting the evidence and data recorded in the scientific journals that support a theory, even at times looking up an idea on Wikipedia. Although we teach students to question any information that pops up from a Google search, research has revealed that Wikipedia’s accounts of topics at the less controversial end of the scientific spectrum, like the theory of general relativity, are regarded as on a par with accounts in the scientific literature. Choose a more contested issue, like climate change, and the content might depend on what day you look.
This raises the question of how much can you trust any of these stories. Just because the scientific community accepts a story as the current best fit, this doesn’t mean it is true. Time and again, history reveals the opposite to be the case, and this must always act as a warning that current scientific knowledge is provisional. Mathematics perhaps has a slightly different quality, as I will discuss in the final two chapters. Mathematical proof provides the chance to establish a more permanent state of knowledge. But it’s worth noting that even when I am creating new mathematics, I will often quote results by fellow mathematicians whose proofs I won’t have checked myself. To do so would mean running in order to keep still.
And for any scientist the real challenge is not to stay within the secure garden of the known but to venture out into the wilds of the unknown. That is the challenge at the heart of this book.
WHAT WE DON’T KNOW
Despite all the breakthroughs made in science over the last centuries, there are still lots of deep mysteries waiting out there for us to solve. Things we don’t know. The knowledge of what we are ignorant of seems to expand faster than our catalogue of breakthroughs. The known unknowns outstrip the known knowns. And it is those unknowns that drive science. A scientist is more interested in the things he or she can’t understand than in telling all the stories we already know how to narrate. Science is a living, breathing subject because of all those questions we can’t answer.
For example, the stuff that makes up the physical universe we interact with seems to account for only 4.9% of the total matter content of our universe. So what is the other 95.1% of so-called dark matter and dark energy made up of? If our universe’s expansion is accelerating, where is all the energy coming from that is fuelling that acceleration?
Is our universe infinite? Are there infinitely many other infinite universes parallel to our own? If there are, do they have different laws of physics? Were there other universes before our own universe emerged from the Big Bang? Did time exist before the Big Bang? Does time exist at all or does it emerge as a consequence of more fundamental concepts?
Why is there a layer of fundamental particles with another two almost identical copies of this layer but with increasing mass, the so-called three generations of fundamental particles? Are there yet more particles out there for us to discover? Are fundamental particles actually tiny strings vibrating in 11-dimensional space?
How can we unify Einstein’s theory of general relativity, the physics of the very large, with quantum physics, the physics of the very small? This is the search for something called quantum gravity, an absolute necessity if we are ever going to understand the Big Bang, when the universe was compressed into the realm of the quantum.
And what of the understanding of our human body, something so complex that it makes quantum physics look like a high-school exercise. We are still trying to get to grips with the complex interaction between gene expression and our environment. Can we find a cure for cancer? Is it possible to beat ageing? Could there be someone alive today who will live to be a 1000 years old?
And what about where humans came from? Evolution is a process of random mutations, so would a different roll of the evolutionary dice still produce organisms with eyes? If we rewound evolution and pressed ‘play’, would we get intelligent life, or are we the result of a lucky roll of the dice? Is there intelligent life elsewhere in our universe? And what of the technology we are creating? Can a computer ever attain consciousness? Will I eventually be able to download my consciousness so that I can survive the death of my body?
Mathematics too is far from finished. Despite popular belief, Fermat’s Last Theorem was not the last theorem. Mathematical unknowns abound. Are there any patterns in prime numbers or are they outwardly random? Will we be able to solve the mathematical equations for turbulence? Will we ever understand how to factorize large numbers efficiently?
Despite so much that is still unknown, scientists are optimistic that these questions won’t remain unanswered forever. The last few decades give us reason to believe that we are in a golden age of science. The rate of discoveries in science appears to grow exponentially. In 2014 the science journal Nature reported that the number of scientific papers published has been doubling every nine years since the end of the Second World War. Computers too are developing at an exponential rate. Moore’s law is the observation that computer processing power seems to double every two years. Engineer Ray Kurzweil believes that the same applies to technological progress: that the rate of change of technology over the next 100 years will be comparable to what we’ve experienced in the last 20,000 years.
And yet can scientific discoveries maintain this exponential growth? Kurzweil talks about the Singularity, a moment when the intelligence of our technology will exceed our human intelligence. Is scientific progress destined for its own singularity? A moment when we know it all. Surely at some point we might actually discover the underlying equations that explain how the universe works. We will discover the final list of particles that make up the building blocks of the physical universe and how they interact with each other. Some scientists believe that the current rate of scientific progress will lead to a moment when we might discover a theory of everything. They even give it a name: ToE.
As Hawking declared in A Brief History of Time: ‘I believe there are grounds for cautious optimism that we may be near the end of the search for the ultimate laws of nature’, concluding dramatically with the provocative statement that then ‘we would know the mind of God’.
Is such a thing possible? To know everything? Would we want to know everything? Science would ossify. Scientists have a strangely schizophrenic relationship with the unknown. On the one hand, it is what we don’t know that intrigues and fascinates us, and yet the mark of success as a scientist is resolution and knowledge, to make the unknown known.
Could there be some quests that will never be resolved? Are there limits to what we can discover about our physical universe? Are some regions of the future beyond the predictive powers of science and mathematics? Is time before the Big Bang a no-go area? Are there ideas so complex that they are beyond the conception of our finite human brains? Can brains even investigate themselves, or does the analysis enter an infinite loop from which it is impossible to rescue itself? Are there mathematical conjectures that can never be proved true?
WHAT WE’LL NEVER KNOW
What if there are questions of science that can never be resolved? It seems defeatist, even dangerous, to admit there may be any such questions. While the unknown is the driving force for doing science, the unknowable would be science’s nemesis. As a fully signed-up member of the scientific community, I hope that we can ultimately answer the big open questions. So it seems important to know if the expedition I’ve joined will hit boundaries beyond which we cannot proceed. Questions that won’t ever get closure.
That is the challenge I’ve set myself in this book. I want to know if there are things that by their very nature we will never know. Are there things that will always be beyond the limits of knowledge? Despite the marauding pace of scientific advances, are there things that will remain beyond the reach of even the greatest scientists? Will there remain mysteries that will resist our attempts to lift the veils that currently mask our view of the universe?
It is, of course, very risky at any point in history to try to articulate Things We Cannot Know. How can you know what new insights are suddenly going to pull the unknown into the knowable? This is partly why it is useful to look at the history of how we know the things we do, because it reveals how often we’ve been at points where we think we have hit the frontier, only to find some way across.
Take the statement made by French philosopher Auguste Comte in 1835 about the stars: ‘We shall never be able to study, by any method, their chemical composition or their mineralogical structure.’ An absolutely fair statement given that this knowledge seemed to depend on our visiting the star. What Comte hadn’t factored in was the possibility that the star could visit us, or at least that photons of light emitted by the star could reveal its chemical make-up.
A few decades after Comte’s prophecy, scientists had determined the chemical composition of our own star, the Sun, by analysing the spectrum of light emitted. As the nineteenth-century British astronomer Warren de la Rue declared: ‘If we were to go to the Sun, and to bring some portions of it and analyse them in our laboratories, we could not examine them more accurately than we can by this new mode of spectrum analysis.’
Scientists went on to determine the chemical composition of stars we are unlikely ever to visit. As science in the nineteenth century continued to give us an ever greater understanding of the mysteries of the universe, there began to emerge a feeling that we might eventually have a complete picture.
In 1900 Lord Kelvin, regarded by many as one of the greatest scientists of his age, believed that moment had come when he declared to the meeting of the British Association of Science: ‘There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.’ American physicist Albert Abraham Michelson concurred. He too thought that the future of science would simply consist of adding a few decimal places to the results already obtained. ‘The more important fundamental laws and facts of physical science have all been discovered … our future discoveries must be looked for in the sixth place of decimals.’
Five years later Einstein announced his extraordinary new conception of time and space, followed shortly after by the revelations of quantum physics. Kelvin and Michelson couldn’t have been more wrong about how much new physics there was still to discover.
What I want to try to explore is whether there are problems that we can prove will remain beyond knowledge despite any new insights. Perhaps there are none. As a scientist that is my hope. One of the dangers when faced with currently unanswerable problems is to give in too early to their unknowability. But if there are unanswerables, what status do they have? Can you choose from the possible answers and it won’t really matter which one you opt for?
Talk of known unknowns is not reserved to the world of science. The US politician Donald Rumsfeld strayed into the philosophy of knowledge with the famous declaration:
There are known knowns; there are things that we know that we know. We also know there are known unknowns; that is to say, we know there are some things we do not know. But there are also unknown unknowns, the ones we don’t know we don’t know.
Rumsfeld received a lot of stick for this cryptic response to a question fired at him during a briefing at the Department of Defense about the lack of evidence connecting the government of Iraq with weapons of mass destruction. Journalists and bloggers had a field day, culminating in Rumsfeld being given the Foot in Mouth award by the Plain English Campaign. And yet if one unpicks the statement, Rumsfeld very concisely summed up different types of knowledge. He perhaps missed one interesting category: The unknown knowns. The things that you know yet dare not admit to knowing. As the philosopher Slavoj Zizek argues, these are possibly the most dangerous, especially when held by those with political power. This is the domain of delusion. Repressed thoughts. The Freudian unconscious.
I would love to tell you about the unknown unknowns, but then they’d be known! Nassim Taleb, author of The Black Swan, believes that it is the emergence of these that are responsible for the biggest changes in society. For Kelvin it was relativity and quantum physics that turned out to be the unknown unknown that he was unable to conceive of. So in this book I can at best try to articulate the known unknowns and ask whether any will remain forever unknown. Are there questions that by their very nature will always be unanswerable, regardless of progress in knowledge?
I have called these unknowns ‘Edges’. They represent the horizon beyond which we cannot see. My journey to the Edges of knowledge to articulate the known unknowns will pass through the known knowns that demonstrate how we have travelled beyond what we previously thought were the limits of knowledge. This journey will also test my own ability to know, because it’s becoming increasingly challenging as a scientist to know even the knowns.
As much as this book is about what we cannot know, it is also important to understand what we do know and how we know it. My journey to the limits of knowledge will take me through the terrain that scientists have already mapped, to the very limits of today’s cutting-edge breakthroughs. On the way I will stop to consider those moments when scientists thought they had hit a wall beyond which progress was no longer possible, only for the next generation to find a way, and this will give us an important perspective on those problems that we might think are unknowable today. By the end of our journey I hope this book will provide a comprehensive survey not just of what we cannot know but also of the things we do know.
To help me through those areas of science that are outside my comfort zone, I have enlisted the help of experts to guide me as I reach each science’s Edge and to test whether it is my own limitations, or limitations inherent in the questions I am tackling, that make them unknowable.
What happens then if we encounter a question that cannot be answered? How does one cope with not knowing? Dare I admit to myself that some things will forever remain beyond my reach? How do we as a species cope with not knowing? That is a challenge that has elicited some interesting responses from humans across the millennia, not least the creation of an idea called God.
TRANSCENDENCE
There is another reason why I have been driven to investigate the unknowable, which is also related to my new job. The previous incumbent of the chair for the Public Understanding of Science was a certain Richard Dawkins. When I took over the position from Dawkins I braced myself for the onslaught of questions that I would get, not about science, but about religion. The publication of The God Delusion and his feisty debates with creationists resulted in Dawkins spending a lot of the later years of his tenure debating questions of religion and God.
So it was inevitable that when I took up the chair people would be interested in my stance on religion. My initial reaction was to put some distance between myself and the debate about God. My job was to promote scientific progress, and to engage the public in the breakthroughs happening around them. I was keen to move the debate back to questions of science rather than religion.
As a strategy to deflect the God questions I actually admitted that I was in fact a religious man. Before journalists got too excited, I went on to explain that my religion is the Arsenal. My temple is the Emirates (it used to be Highbury Stadium) in north London, and each Saturday I worship my idols and sing songs to them. And at the beginning of each season I reaffirm my faith that this will be the year we finally win some silverware. In an urban environment like London, football has taken over the role that religion played in society of binding a community together, providing rituals that they can share.
For me, the science that I began to learn as a teenager did a pretty good job of pushing out any vaguely religious thoughts I had had as a kid. I sang in my local church choir, which exposed me to the ideas that Christianity had to offer for understanding the universe. School education in the Seventies in the UK was infused with mildly religious overtones: renditions of ‘All Things Bright and Beautiful’ and the Lord’s Prayer in assemblies. Religion was dished up as something too simplistic to survive the sophisticated and powerful stories that I would learn in the science labs at my secondary school. Religion was quickly pushed out. Science … and football … were much more attractive.
Inevitably the questions about my stance on religion would not be fobbed off with such a flippant answer. I remember that during one radio interview on a Sunday morning on BBC Northern Ireland I was gradually sucked into considering the question of the existence of God. I guess I should have seen the warning signs. On a Sunday morning in Northern Ireland, God isn’t far from the minds of many listeners.
As a mathematician I am often faced with the challenge of proving the existence of new structures or coming up with arguments to show why such structures cannot exist. The power of the mathematical language to produce logical arguments has led a number of philosophers throughout the ages to resort to mathematics as a way of proving the existence of God. But I always have a problem with such an approach. If you are going to prove existence or otherwise in mathematics, you need a very clear definition of what it is that you are trying to prove exists.
So after some badgering by the interviewer about my stance on the existence of God, I pushed him to try to define what God meant for him so that I could engage my mathematical mind. ‘It is something which transcends human understanding.’ At first I thought: what a cop-out. You have just defined it as something which by its very nature I can’t get a handle on. But I became intrigued by this as a definition. Perhaps it wasn’t such a cop-out after all.
What if you define God as the things we cannot know. The gods in many ancient cultures were always a placeholder for the things we couldn’t explain or couldn’t understand. Our ancestors found volcanic eruptions or eclipses so mysterious that they became acts of gods. As science has explained such phenomena, these gods have retreated.
This definition has some things in common with a God commonly called the ‘God of the gaps’. This phrase was generally used as a derogatory term by religious thinkers who could see that this God was shrinking in the face of the onslaught of scientific knowledge, and a call went out to reject this kind of God. The phrase ‘God of the gaps’ was coined by the Oxford mathematician and Methodist church leader Charles Coulson, when he declared: ‘There is no “God of the gaps” to take over at those strategic places where science fails.’
But the phrase is also associated with a fallacious argument for the existence of God, one that Richard Dawkins spends some time shooting down in The God Delusion: if there are things that we can’t explain or know, there must be a God at work filling the gap. But I am more interested not in the existence of a God to fill the gap, but in equating God with the abstract idea of the things we cannot know. Not in the things we currently don’t know, but the things that by their nature we can never know. The things that will always remain transcendent.
Religion is more complex than the simple stereotype often offered up by modern society. For many ancient cultures in India, China and the Middle East, religion was not about worshipping a Supernatural Intelligence but precisely the attempt to appreciate the limits of our understanding and language. As the theologian Herbert McCabe declared: ‘To assert the existence of God is to claim that there is an unanswered question about the universe.’ Science has pushed hard at those limits. So is there anything left? Will there be anything that will always be beyond the limit. Does McCabe’s God exist?
This is the quest at the heart of this book. Can we identify questions or physical phenomena that will always remain beyond knowledge? If we can identify things that will remain in the gaps of knowledge, then what sort of God is this? What potency would such a concept have? Could the things we cannot know act in the world and affect our futures? Are they worthy of worship?
But first we need to know if in fact there is anything that will remain unanswered about the universe. Is there really anything we cannot know?
FIRST EDGE: THE CASINO DICE (#u9b12e613-7f53-53ac-b922-016f09fd96da)
1 (#u9b12e613-7f53-53ac-b922-016f09fd96da)
The unpredictable and the predetermined unfold together to make everything the way it is. It’s how nature creates itself, on every scale, the snowflake and the snowstorm. It makes me so happy. To be at the beginning again, knowing almost nothing.
Tom Stoppard, Arcadia
There is a single red dice sitting on my desk next to me. I got the dice on a trip to Las Vegas. I fell in love with it when I saw it on the craps table. It was so perfectly engineered. Such precise edges coming to a point at the corners of the cube. The faces so smooth you couldn’t feel what number the face was representing. The pips are carved out of the dice and then filled with paint that has the same density as the plastic used to make the dice. This ensures that the face representing the 6 isn’t a touch lighter than the face on the opposite side with a single pip. The feeling of the dice in the hand is incredibly satisfying. It is a thing of beauty.
And yet I hate it.
It’s got three pips pointing up at me at the moment. But if I pick it up and let it fall from my hand I have no way of knowing how it is going to land. It is the ultimate symbol of the unknowable. The future of the dice seems knowable only when it becomes the past.
I have always been extremely unsettled by things that I cannot know. Things that I cannot work out. I don’t mind not knowing something provided there is some way ultimately to calculate what’s going on. With enough time. Is this dice truly so unknowable? Or with enough information can I actually deduce its next move? Surely it’s just a matter of applying the right laws of physics and solving the appropriate mathematical equations. Surely this is something I can know.
My subject, mathematics, was invented to give people a glimpse of what’s out there coming towards us. To look into the future. To become masters of fate, not its servants. I believe that the universe runs according to laws. Understand those laws and I can know the universe. Spotting patterns has given the human species a very powerful way to take control. If there’s a pattern then I have some chance to predict the future and know the unknowable. The pattern of the Sun means I can rely on it rising in the sky tomorrow or the Moon taking 28 sunrises before it becomes full again. It is how mathematics developed. Mathematics is the science of patterns. Being able to spot patterns is a powerful tool in the evolutionary fight for survival. The caves in Lascaux show how counting 13 quarters of the Moon from the first winter rising of the Pleiades will bring you to a time in the year when the horses are pregnant and easy to hunt. Being able to predict the future is the key to survival.
But there are some things which appear to have no pattern or that have patterns that are so complex or hidden that they are beyond human knowledge. The individual roll of the dice is not like the rising of the Sun. There seems to be no way to know which of the six faces will be pointing upwards once the cube finally comes to rest. It is why the dice has been used since antiquity as a way to decide disputes, to play games, to wager money.
Is that beautiful red cube with its white dots truly unknowable? I’m certainly not the first to have a complex relationship with the dynamics of this cube.
KNOWING THE WILL OF THE GODS
On a recent trip to Israel I took my children to an archaeological dig at Beit Guvrin. It was such a popular settlement in ancient times that the site consists of layer upon layer of cities built on top of each other. There is so much stuff in the ground that the archaeologists are happy to enlist amateurs like me and my kids to help excavate the site even if a few pots get broken along the way. Sure enough, we pulled out lots of bits of pottery. But we also kept unearthing a large number of animal bones. We thought they were the remains of dinner, but our guide explained that in fact they were the earliest form of dice.
Archaeological digs of settlements dating back to Neolithic times have revealed a disproportionately high density of heel bones of sheep or other animals among the shattered pottery and flints that are usually found in sites that humans once inhabited. These bones are in fact ancestors of my casino dice. When thrown, the bones naturally land on one of four sides. Often there are letters or numbers carved into the bones. Rather than gambling, these early dice are thought to have been used for divination. And this connection between the outcome of the roll of a dice and the will of the gods is one that has persisted for centuries. Knowledge of how the dice would land was believed to be something that transcended human understanding. Its outcome was in the lap of the gods.
Increasingly these dice assumed a more prosaic place as part of our world of leisure. The first cube-shaped dice like the one on my desk were found around Harappa in what is now northeast Pakistan, where one of the first urban civilizations evolved, dating back to the third millennium BC. At the same time, you find four-faced pyramid dice appearing in a game that was discovered in the city of Ur, in ancient Mesopotamia.
The Romans and Greeks were addicts of games of dice, as were the soldiers of the medieval era who returned from the Crusades with a new game called hazard, deriving from the Arabic word for a dice: al-zahr. It was an early version of craps, the game that was being played in the casino in Vegas where I picked up my dice.
If I could predict the fall of the dice, all the games that depend on them would never have caught on. The excitement of backgammon or hazard or craps comes from not knowing how the dice are going to land. So perhaps gamers won’t thank me as I try to predict the roll of my dice.
For centuries no one even thought that such a feat was possible. The ancient Greeks, who were among the first to develop mathematics as a tool to navigate their environment, certainly didn’t have any clue how to tackle such a dynamic problem. Their mathematics was a static, rigid world of geometry, not one that could cope with the dice tumbling across the floor. They could produce formulas to describe the geometric contours of the cube, but once the dice started moving they were lost.
What about doing experiments to get a feel for the outcomes? The anti-empiricist attitude of the ancient Greeks meant they had no motivation to analyse the data and try to make a science of predicting how the dice would land. After all, the way the dice had just landed was going to have no bearing on the next throw. It was random and for the ancient Greeks that meant it was unknowable.
Aristotle believed that events in the world could essentially be classified into three categories: ‘certain events’ that happen by necessity following the laws of nature; ‘probable events’ that happen in most cases but can have a few exceptions; and finally ‘unknowable events’ that happened by pure chance. Aristotle put my dice firmly in the last category.
As Christian theology made its impact on philosophy, matters worsened. Since the throw of the dice was in the hands of God, it was not something that humans could aspire to know. As St Augustine had it: ‘We say that those causes that are said to be by chance are not non-existent but are hidden, and we attribute them to the will of the true God.’
There was no chance. No free will. The unknowable was known by God, who determined the outcome of the dice. Any attempt to try to predict the roll was the work of a heretic, someone who dared to think they could know the mind of God. King Louis XI of France even went as far as prohibiting the manufacture of dice, believing that games of chance were ungodly. But the dice like the one I have on my desk eventually began to yield their secrets. It took till the sixteenth century before dice were wrestled out of the hands of God and their fate put in the hands, and minds, of humans.
FINDING THE NUMBERS IN THE DICE
I’ve put another two dice next to my beautiful Las Vegas dice. So here’s a question: if I throw all three dice, is it better to bet on a score of 9 or a score of 10 coming up? Prior to the sixteenth century there were no tools available to answer such a question. And yet anyone who had played for long enough would know that if I was throwing only two dice then it would be wise to bet on 9 rather than 10. After all, experience would tell you before too long that on average you get 9 a third more often than you get 10. But with three dice it is harder to get a feel for which way to bet, because 9 and 10 seem to occur equally often. But is that really true?
It was in Italy at the beginning of the sixteenth century that an inveterate gambler by the name of Girolamo Cardano first realized that there are patterns that can be exploited in the throw of a dice. They weren’t patterns that could be used on an individual throw. Rather, they emerged over the long run, patterns that a gambler like Cardano, who spent many hours throwing dice, could use to his advantage. So addicted was he to the pursuit of predicting the unknowable that on one occasion he even sold his wife’s possessions to raise the funds for the table stakes.
Cardano had the clever idea of counting how many different futures the dice could have. If I throw two dice, there are 36 different futures. They are depicted in the following diagram.
Only in three of them is the total 10, while four give you a score of 9. So Cardano reasoned that, in the case of two dice being thrown, it makes sense to bet on 9 rather than 10. It did not help in any individual game, but in the long run it meant that Cardano, if he stuck to his maths, would come out on top. Unfortunately, while a disciplined mathematician, he wasn’t very disciplined when it came to his gambling. He managed to lose all the inheritance from his father and would get into knife fights with his opponents when the dice went against him.
He was nevertheless determined to get one prophecy correct. He had apparently predicted the date of his death: 21 September 1576. To make sure he got this bet right he took matters into his own hands. He committed suicide when the date finally struck. As much as I crave knowledge, I think this is going a little far. Indeed, the idea of knowing the date of your death is something that most would prefer to opt out of. But Cardano was determined to win, even when he was dicing with Death.
Before taking his life, he wrote what many regard as the first book that made inroads into predicting the behaviour of the dice as it rolls across the table. Although written around 1564, Liber de Ludo Aleae didn’t see the light of day until it was eventually published in 1663.
It was in fact the great Italian physicist Galileo Galilei who applied the same analysis that Cardano had described to decide whether to bet on a score of 9 or 10 when three dice are thrown. He reasoned that there are 6 × 6 × 6 = 216 different futures the dice could take. Of these, 25 gave you a 9 while 27 gave you a 10. Not a big difference, and one that would be difficult to pick up from empirical data, but large enough that betting on 10 should give you an edge in the long run.
AN INTERRUPTED GAME
The mathematical mastery of the dice shifted from Italy to France in the mid-seventeenth century when two big hitters, Blaise Pascal and Pierre de Fermat, started applying their minds to predicting the future of these tumbling cubes. Pascal had become interested in trying to understand the roll of the dice after meeting one of the great gamblers of the day, Chevalier de Méré. De Méré had challenged Pascal with a number of interesting scenarios. One was the problem Galileo had cracked. But the others included whether it was advisable to bet that at least one 6 will appear if a dice is thrown four times, and also the now famous problem of ‘points’.
Pascal entered into a lively correspondence with the great mathematician and lawyer Pierre de Fermat in which they tried to sort out the problems set by de Méré. With the throw of four dice, one could consider the 6 × 6 × 6 × 6 = 1296 different outcomes and count how many include a 6, but that becomes pretty cumbersome.
Instead, Pascal reasoned that there is a
⁄
chance that you won’t see a 6 with one throw. Since each throw is independent, that means there is a
⁄
×
⁄
×
⁄
×
⁄
=
⁄
= 48.2% chance that you won’t get a 6 in four throws. Which means there is a 51.8% chance that you will see a 6. Just above an evens chance, so worth betting on.
The problem of ‘points’ was even more challenging. Suppose two players – let’s call them Fermat and Pascal – are rolling a dice. Fermat scores a point if the dice lands on 4 or higher; Pascal scores a point otherwise. Each, therefore, has a 50:50 chance of winning a point on any roll of the dice. They’ve wagered £64, which will go to the first to score 3 points. The game is interrupted, however, and can’t be continued, when Fermat is on 2 points and Pascal is on 1 point. How should they divide the £64?
Traditional attempts to solve the problem focused on what had happened in the past. Maybe, having won twice as many rounds as Pascal, Fermat should get twice the winnings. This makes no sense if, say, Fermat had won only one round before the game was interrupted. Pascal would get nothing but still has a chance of winning. Niccolò Fontana Tartaglia, a contemporary of Cardano, believed after much thought that it had no solution: ‘The resolution of the question is judicial rather than mathematical, so that in whatever way the division is made there will be cause for litigation.’
Others weren’t so defeated. Attention turned not to the past, but to what could happen in the future. In contrast to the other problems, they are not trying to predict the roll of the dice but instead need to imagine all the different future scenarios and to divide the spoils according to which version of the future favours which player.
It is easy to get fooled here. There seem to be three scenarios. Fermat wins the next round and pockets £64. Pascal wins the next round, resulting in a final round which either Pascal wins or Fermat wins. Fermat wins in two out of these three scenarios so perhaps he should get two-thirds of the winnings. It was the trap that de Méré fell into. Pascal argues that this isn’t correct. ‘The Chevalier de Méré is very talented but he is not a mathematician; this is, as you know, a great fault.’ A great fault, indeed!
Pascal, in contrast, was great on the mathematical front and argued that the spoils should be divided differently. There is a 50:50 chance that Fermat wins in one round, in which case he gets £64. But if Pascal wins the next round then they are equally likely to win the final round, so could divide the spoils £32 each. In either case Fermat is guaranteed £32. So the other £32 should be split equally, giving Fermat £48 in total.
Fermat, writing from his home near Toulouse, concurred with Pascal’s analysis: ‘You can now see that the truth is the same in Toulouse as in Paris.’
PASCAL’S WAGER
Pascal and Fermat’s analysis of the game of points could be applied to much more complex scenarios. Pascal discovered that the secret to deciding the division of the spoils is hidden inside something now known as Pascal’s triangle.
The triangle is constructed in such a way that each number is the sum of the two numbers immediately above it. The numbers you get are key to dividing the spoils in any interrupted game of points. For example, if Fermat needs 2 points for a win while Pascal needs 4, then you consult the 2 + 4 = 6th row of the triangle and add the first four numbers together and the last two. This is the proportion in which you should divide the spoils. In this case it’s a 1 + 5 + 10 + 10 = 26 to 1 + 5 = 6 division. So Fermat gets
⁄
× 64 = £52 and Pascal gets
⁄
× 64 = £12. In general, a game where Fermat needs n points to Pascal’s m points can be decided by consulting the (n + m)th row of Pascal’s triangle.
There is evidence that the French were beaten by several millennia to the discovery that this triangle is connected to the outcome of games of chance. The Chinese were inveterate users of dice and other random methods like the I Ching to try to predict the future. The text of the I Ching dates back some 3000 years and contains precisely the same table that Pascal produced to analyse the outcomes of tossing coins to determine the random selection of a hexagram that would then be analysed for its meaning. But today the triangle is attributed to Pascal rather than the Chinese.
Pascal wasn’t interested only in dice. He famously applied his new mathematics of probability to one of the great unknowns: the existence of God.
‘God is, or He is not.’ But to which side shall we incline? Reason can decide nothing here. There is an infinite chaos which separated us. A game is being played at the extremity of this infinite distance where heads or tails will turn up … Which will you choose then? Let us see. Since you must choose, let us see which interests you least. You have two things to lose, the true and the good; and two things to stake, your reason and your will, your knowledge and your happiness; and your nature has two things to shun, error and misery. Your reason is no more shocked in choosing one rather than the other, since you must of necessity choose … But your happiness? Let us weigh the gain and the loss in wagering that God is … If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.
Called Pascal’s wager, he argued that the payout would be much greater if one opted for a belief in God. You lose little if you are wrong and win eternal life if correct. On the other hand, wager against the existence of God and losing results in eternal damnation, while winning gains you nothing beyond the knowledge that there is no God. The argument falls to pieces if the probability of God existing is actually 0, and even if it isn’t, perhaps the cost of belief might be too high when set against the probability of God’s existence.
The probabilistic techniques developed by mathematicians like Fermat and Pascal to deal with uncertainty were incredibly powerful. Phenomena that were regarded as beyond knowledge, the expression of the gods, were beginning to be within reach of the minds of men. Today these probabilistic methods are our best weapon in trying to navigate everything from the behaviour of particles in a gas to the ups and downs of the stock market. Indeed, the very nature of matter itself seems to be at the mercy of the mathematics of probability, as we shall discover in the Third Edge, when we apply quantum physics to predict what fundamental particles are going to do when we observe them. But for someone on the search for certainty, these probabilistic methods represent a frustrating compromise.
I certainly appreciate the great intellectual breakthrough that Fermat, Pascal and others made, but it doesn’t help me to know how many pips are going to be showing when I throw my dice. As much as I’ve studied the mathematics of probability, it has always left me with a feeling of dissatisfaction. The one thing any course on probability drums into you is that it doesn’t matter how many times in a row you get a 6: this has no influence on what the dice is going to do on the next throw.
So is there some way of knowing how my dice is going to land? Or is that knowledge always going to be out of reach? Not according to the revelations of a scientist across the waters in England.
THE MATHEMATICS OF NATURE
Isaac Newton is my all-time hero in my fight against the unknowable. The idea that I could possibly know everything about the universe has its origins in Newton’s revolutionary work Philosophiae Naturalis Principia Mathematica. First published in 1687, the book is dedicated to developing a new mathematical language that promised the tools to unlock how the universe behaves. It was a dramatically new model of how to do science. The work ‘spread the light of mathematics on a science which up to then had remained in the darkness of conjectures and hypotheses’, declared the French physicist Alexis Clairaut in 1747.
It is also an attempt to unify, to create a theory that describes the celestial and the earthly, the big and the small. Kepler had come up with laws that described the motions of the planets, laws he’d developed empirically by looking at data and trying to fit equations to create the past. Galileo had described the trajectory of a ball flying through the air. It was Newton’s genius to understand that these were examples of a single phenomenon: gravity.
Born on Christmas Day in 1643 in the Lincolnshire town of Woolsthorpe, Newton was always trying to tame the physical world. He made clocks and sundials, constructed miniature mills powered by mice, sketched countless plans for buildings and ships, and drew elaborate illustrations of animals. The family cat apparently disappeared one day, carried away by a hot-air balloon that Newton had made. His school reports, however, did not anticipate a great future, describing him as ‘inattentive and idle’.
Idleness is not necessarily such a bad trait in a mathematician. It can be a powerful incentive to look for some clever shortcut to solve a problem rather than relying on hard graft. But it’s not generally a quality that teachers appreciate.
Indeed, Newton was doing so badly at school that his mother decided the whole thing was a waste of time and that he’d be better off learning how to manage the family farm in Woolsthorpe. Unfortunately, Newton was equally hopeless at managing the family estate, so he was sent back to school. Although probably apocryphal, it is said that Newton’s sudden academic transformation coincided with a blow to the head that he received from the school bully. Whether true or not, Newton’s academic transformation saw him suddenly excelling at school, culminating in a move to study at the University of Cambridge.
When bubonic plague swept through England in 1665, Cambridge University was closed as a precaution. Newton retreated to the house in Woolsthorpe. Isolation is often an important ingredient in coming up with new ideas. Newton hid himself away in his room and thought.
Truth is the offspring of silence and meditation. I keep the subject constantly before me and wait ’til the first dawnings open slowly, by little and little, into a full and clear light.
In the isolation of Lincolnshire, Newton created a new language that could capture the problem of a world in flux: the calculus. This mathematical tool would be key to our knowing how the universe would behave ahead of time. It is this language that gives me any hope of gleaning how my casino dice might land.
MATHEMATICAL SNAPSHOTS
The calculus tries to make sense of what at first sight looks like a meaningless sum: zero divided by zero. As I let my dice fall from my hand, it is such a sum that I must calculate if I want to try to understand the instantaneous speed of my dice as it falls through the air.
The speed of the dice is constantly increasing as gravity pulls it to the ground. So how can I calculate what the speed is at any given instance of time? For example, how fast is the dice falling after one second? Speed is distance travelled divided by time elapsed. So I could record the distance it drops in the next second and that would give me an average speed over that period. But I want the precise speed. I could record the distance travelled over a shorter period of time, say half a second or a quarter of a second. The smaller the interval of time, the more accurately I will be calculating the speed. Ultimately, to get the precise speed I want to take an interval of time that is infinitesimally small. But then I am faced with calculating 0 divided by 0.
Calculus: making sense of zero divided by zero
Suppose that a car starts from a stationary position. When the stopwatch starts, the driver slams his foot on the accelerator. Suppose that we record that after t seconds the driver has covered t × t metres. How fast is the car going after 10 seconds? We get an approximation of the speed by looking at how far the car has travelled in the period from 10 to 11 seconds. The average speed during this second is (11 × 11 – 10 × 10)/1 = 21 metres per second.
But if we look at a smaller window of time, say the average speed over 0.5 seconds, we get:
(10.5 × 10.5 – 10 × 10)/0.5 = 20.5 metres per second.
Slightly slower, of course, because the car is accelerating, so on average it is going faster in the second half second from 10 seconds to 11 seconds. But now we take an even smaller snapshot. What about halving the window of time again:
(10.25 × 10.25 – 10 × 10)/0.25 = 20.25 metres per second.
Hopefully the mathematician in you has spotted the pattern. If I take a window of time which is x seconds, the average speed over this time will be 20 + x metres per second. The speed as I take smaller and smaller windows of time is getting closer and closer to 20 metres per second. So, although to calculate the speed at 10 seconds looks like I have to figure out the calculation
⁄
, the calculus makes sense of what this should mean.
Newton’s calculus made sense of this calculation. He understood how to calculate what the speed was tending towards as I make the time interval smaller and smaller. It was a revolutionary new language that managed to capture a changing dynamic world. The geometry of the ancient Greeks was perfect for a static, frozen picture of the world. Newton’s mathematical breakthrough was the language that could describe a moving world. Mathematics had gone from describing a still life to capturing a moving image. It was the scientific equivalent of how the dynamic art of the Baroque burst forth during this period from the static art of the Renaissance.
Newton looked back at this time as one of the most productive of his life, calling it his annus mirabilis. ‘I was in the prime of my age for invention and minded Mathematicks and Philosophy more than at any time since.’
Everything around us is in a state of flux, so it was no wonder that this mathematics would be so influential. But for Newton the calculus was a personal tool that helped him reach the scientific conclusions that he documents in the Principia, the great treatise published in 1687 that describes his ideas on gravity and the laws of motion.
Writing in the third person, he explains that his calculus was key to the scientific discoveries contained inside: ‘By the help of this new Analysis Mr Newton found out most of the propositions in the Principia.’ But no account of the ‘new analysis’ is published. Instead, he privately circulated the ideas among friends, but they were not ideas that he felt any urge to publish for others to appreciate.
Fortunately this language is now widely available and it is one that I spent years learning as a mathematical apprentice. But in order to attempt to know my dice I am going to need to mix Newton’s mathematical breakthrough with his great contribution to physics: the famous laws of motion with which he opens his Principia.
THE RULES OF THE GAME
Newton explains in the Principia three simple laws from which so much of the dynamics of the universe evolve.
Newton’s First Law of Motion: A body will continue in a state of rest or uniform motion in a straight line unless it is compelled to change that state by forces acting on it.
This was not so obvious to the likes of Aristotle. If you roll a ball along a flat surface it comes to rest. It looks like you need a force to keep it moving. There is, however, a hidden force that is changing the speed: friction. If I throw my dice in outer space away from any gravitational fields then the dice will indeed just carry on flying in a straight line at constant speed.
In order to change an object’s speed or direction you needed a force. Newton’s second law explained how that force would change the motion, and it entailed the new tool he’d developed to articulate change. The calculus has already allowed me to articulate what speed my dice is going at as it accelerates down towards the table. The rate of change of that speed is got by applying calculus again. The second law of Newton says that there is a direct relationship between the force being applied and the rate of change of the speed.
Newton’s Second Law of Motion: The rate of change of motion, or acceleration, is proportional to the force that is acting on it and inversely proportional to its mass.
To understand the motion of bodies like my cascading dice I need to understand the possible forces acting on them. Newton’s universal law of gravitation identified one of the principal forces that had an effect on, say, his apple falling or the planets moving through the solar system. The law states that the force acting on a body of mass m
by another body of mass m
which is a distance of r away is equal to
where G is an empirical physical constant that controls how strong gravity is in our universe.
With these laws I can now describe the trajectory of a ball flying through the air, or a planet through the solar system, or my dice falling from my hand. But the next problem occurs when the dice hits the table. What happens then? Newton has a third law which provides a clue:
Newton’s Third Law of Motion: When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction to that of the first body.
Newton himself used these laws to deduce an extraordinary string of results about the solar system. As he wrote: ‘I now demonstrate the system of the World.’ To apply his ideas to the trajectory of the planets he began by reducing each planet to a point located at the centre of mass and assumed that all the planet’s mass was concentrated at this point. Then by applying his laws of motion and his new mathematics he successfully deduced Kepler’s laws of planetary motion.
He was also able to calculate the relative masses for the large planets, the Earth and the Sun. He explained a number of the curious irregularities in the motion of the Moon due to the pull of the Sun. He also deduced that the Earth isn’t a perfect sphere but should be squashed between the poles due to the Earth’s rotation causing a centrifugal force. The French thought the opposite would happen: that the Earth should be pointy in the direction of the poles. An expedition set out in 1733 which proved Newton – and the power of mathematics – correct.
NEWTON’S THEORY OF EVERYTHING
It was an extraordinary feat. The three laws were the seeds from which all motion of particles in the universe could potentially be deduced. It deserved to be called a Theory of Everything. I say ‘seeds’ because it required other scientists to grow these seeds and apply them to more complex settings than Newton’s solar system made up of point particles of mass. For example, in their original form the laws are not suited to describing the motion of less rigid bodies or bodies that deform. It was the great eighteenth-century Swiss mathematician Leonhard Euler who would provide equations that generalized Newton’s laws. Euler’s equations could be applied more generally to something like a vibrating string or a swinging pendulum.
More and more equations appeared that controlled various natural phenomena. Euler produced equations for non-viscous fluids. At the beginning of the nineteenth century French mathematician Joseph Fourier found equations to describe heat flow. Fellow compatriots Pierre-Simon Laplace and Siméon-Denis Poisson took Newton’s equations to produce more generalized equations for gravitation, which were then seen to control other phenomena like hydrodynamics and electrostatics. The behaviours of viscous fluids were described by the Navier–Stokes equations, and electromagnetism by Maxwell’s equations.
With the discovery of the calculus and the laws of motion, it seemed that Newton had turned the universe into a deterministic clockwork machine controlled by mathematical equations. Scientists believed they had indeed discovered the Theory of Everything. In his Philosophical Essay on Probabilities published in 1812, the mathematician Pierre-Simon Laplace summed up most scientists’ belief in the extraordinary power of mathematics to tell you everything about the physical universe.
We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.
This view that, in theory, the universe was knowable, both past and present, became dominant among scientists in the centuries following Newton’s great opus. It seemed as if any idea of God acting in the world had been completely removed. A God might be responsible for getting things up and running, but from that point on the equations of mathematics and physics took over.
So what of my lowly dice? Surely with the laws of motion at hand I can simply combine the geometry of the cube with the initial direction of motion and the subsequent interactions with the table to predict the outcome? I’ve written out the equations on my notepad and they look pretty daunting.
Newton too contemplated the problem of trying to predict the dice. Newton’s interest was prompted by a letter he received from Samuel Pepys. Pepys wanted Newton’s advice on which option he should back in a wager he was about to make with a friend:
(1) Throwing six dice and getting at least one 6
(2) Throwing twelve dice and getting at least two 6s
(3) Throwing eighteen dice and getting at least three 6s
Pepys was about to stake £10, the equivalent of £1000 in today’s money, and he was quite keen to get some good advice. Pepys’s intuition was that (3) was the more likely option, but Newton replied that the mathematics implied the opposite was true. He should put his money on the first option. However, it wasn’t his laws of motion and the calculus to which Newton resorted to solve the problem but the ideas developed by Fermat and Pascal.
But even if Newton could have solved the equations I’ve written out to describe the trajectory of the dice, there turned out to be another problem that could scupper any chance of knowing the future of my dice. Although Pascal was talking about his wager with God, there is an interesting line in his analysis which throws a spanner in the works when it comes to knowing the future: ‘Reason can decide nothing here. There is an infinite chaos which separated us.’
THE FATE OF THE SOLAR SYSTEM
If Newton is my hero, then French mathematician Henri Poincaré should be the villain in my drive to predict the future. And yet I can hardly blame him for uncovering one of the most devastating blows for anyone wanting to know what’s going to happen next. He was hardly very thrilled himself with the discovery, given that it cost him rather a lot of money.
Born a hundred years after Laplace, Poincaré believed, like his compatriot, in a clockwork universe, a universe governed by mathematical laws and utterly predictable. ‘If we know exactly the laws of nature and the situation of the universe at the initial moment, we can predict exactly the situation of the same universe at a succeeding moment.’
Understanding the world was Poincaré’s prime motivation for doing mathematics. ‘The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law.’
Although Newton’s laws of motion had spawned an array of mathematical equations to describe the evolution of the physical world, most of them were still extremely complicated to solve. Take the equations for a gas. Think of the gas as made up of molecules crashing around like tiny billiard balls, and theoretically the future behaviour of the gas was bound up in Newton’s laws of motion. But the sheer number of balls meant that any exact solution was well beyond reach. Statistical or probabilistic methods were still by far the best tool to understand the behaviour of billions of molecules.
There was one situation where the number of billiard balls was reasonably small and a solution seemed tractable. The solar system. Poincaré became obsessed with the question of predicting what lay in store for our planets as they danced their way into the future.
Because the gravitational pull of a planet on another planet at some distance from the first planet is the same as if all the mass of the planet is concentrated at its centre of gravity, to determine the ultimate fate of the solar system one can consider planets as if they are just points in space, as Newton had done. This means that the evolution of the solar system can be described by three coordinates for each planet that locate the centre of mass in space together with three additional numbers recording the speed in each of the three dimensions of space. The forces acting on the planets are given by the gravitational forces exerted by each of the other planets. With all this information one just needs to apply Newton’s second law to map out the course of the planets into the distant future.
The only trouble is that the maths is still extremely tricky to work out. Newton had solved the behaviour of two planets (or a planet and a sun). They would follow elliptical paths, with their common focal point being the common centre of gravity. This would repeat itself periodically to the end of time. But Newton was stumped when he introduced a third planet. Trying to calculate the behaviour of a solar system consisting, say, of the Sun, the Earth and the Moon seemed simple enough, but already you are facing an equation in 18 variables: 9 for position and 9 for the speed of each planet. Newton conceded that ‘to consider simultaneously all these causes of motion and to define these motions by exact laws admitting of easy calculation exceeds, if I am not mistaken, the force of any human mind’.
The problem got a boost when King Oscar II of Norway and Sweden decided to mark his sixtieth birthday by offering a prize for solving a problem in mathematics. There are not many monarchs around the world who would choose maths problems as their way to celebrate their birthdays, but Oscar had always enjoyed the subject ever since he had excelled at it when he was a student at Uppsala University.
His majesty Oscar II, wishing to give a fresh proof of his interest in the advancement of mathematical science has resolved to award a prize on January 21, 1889, to an important discovery in the field of higher mathematical analysis. The prize will consist of a gold medal of the eighteenth size bearing his majesty’s image and having a value of a thousand francs, together with the sum of two thousand five hundred crowns.
Three eminent mathematicians convened to choose a number of suitable mathematical challenges and to judge the entries. One of the questions they posed was to establish mathematically whether the solar system was stable. Would it continue turning like clockwork, or, at some point in the future, might the Earth spiral off into space and disappear from our solar system?
To answer the question required solving the equations that had stumped Newton. Poincaré believed that he had the skills to win the prize. One of the common tricks used by mathematicians is to attempt a simplified version of the problem first to see if that is tractable. So Poincaré started with the problem of three bodies. This was still far too difficult, so he decided to simplify the problem further. Instead of the Sun, Earth and Moon, why not try to understand two planets and a speck of dust? The two planets won’t be affected by the dust particle, so he could assume, thanks to Newton’s solution, that they just repeated ellipses round each other. The speck of dust, on the other hand, would experience the gravitational force of the two planets. Poincaré set about trying to describe the path traced by the speck of dust. Some understanding of this trajectory would form an interesting contribution to the problem.
Although he couldn’t crack the problem completely, the paper he submitted was more than good enough to secure King Oscar’s prize. He’d managed to prove the existence of an interesting class of paths that would repeat themselves, so-called periodic paths. Periodic orbits were by their nature stable because they would repeat themselves over and over, like the ellipses that two planets would be guaranteed to execute.
The French authorities were very excited that the award had gone to one of their own. The nineteenth century had seen Germany steal a march on French mathematics, so the French academicians excitedly heralded Poincaré’s win as proof of a resurgence of French mathematics. Gaston Darboux, the permanent secretary of the French Academy of Sciences, declared:
From that moment on the name of Henri Poincaré became known to the public, who then became accustomed to regarding our colleague no longer as a mathematician of particular promise but as a great scholar of whom France has the right to be proud.
A SMALL MISTAKE WITH BIG IMPLICATIONS
Preparations began to publish Poincaré’s solution in a special edition of the Royal Swedish Academy of Science’s journal Acta Mathematica. Then came the moment every mathematician dreads. Every mathematician’s worst nightmare. Poincaré thought his work was safe. He’d checked every step in the proof. Just before publication, one of the editors of the journal raised a question over one of the steps in his mathematical argument.
Poincaré had assumed that a small change in the positions of the planets, a little rounding up or down here or there, was acceptable as it would result in only a small change in their predicted orbits. It seemed a fair assumption. But there was no justification given for why this would be so. And in a mathematical proof, every step, every assumption, must be backed up by rigorous mathematical logic.
The editor wrote to Poincaré for some clarification on this gap in the proof. But as Poincaré tried to justify this step, he realized he’d made a serious mistake. He wrote to Gösta Mittag-Leffler, the head of the prize committee,hoping to limit the damage to his reputation:
The consequences of this error are more serious than I first thought. I will not conceal from you the distress this discovery has caused me … I do not know if you will still think that the results which remain deserve the great reward you have given them. (In any case, I can do no more than to confess my confusion to a friend as loyal as you.) I will write to you at length when I can see things more clearly.
Mittag-Leffler decided he needed to inform the other judges:
Poincaré’s memoir is of such a rare depth and power of invention, it will certainly open up a new scientific era from the point of view of analysis and its consequences for astronomy. But greatly extended explanations will be necessary and at the moment I am asking the distinguished author to enlighten me on several important points.
As Poincaré struggled away he soon saw that he was simply mistaken. Even a small change in the initial conditions could result in wildly different orbits. He couldn’t make the approximation that he’d proposed. His assumption was wrong.
Poincaré telegraphed Mittag-Leffler to break the bad news and tried to stop the paper from being printed. Embarrassed, he wrote:
It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible.
Mittag-Leffler was ‘extremely perplexed’to hear the news.
It is not that I doubt that your memoir will be in any case regarded as a work of genius by the majority of geometers and that it will be the departure point for all future efforts in celestial mechanics. Don’t therefore think that I regret the prize … But here is the worst of it. Your letter arrived too late and the memoir has already been distributed.
Mittag-Leffler’s reputation was on the line for not having picked up the error before they’d publicly awarded Poincaré the prize. This was not the way to celebrate his monarch’s birthday! ‘Please don’t say a word of this lamentable story to anyone. I’ll give you all the details tomorrow.’
The next few weeks were spent trying to retrieve the printed copies without raising suspicion. Mittag-Leffler suggested that Poincaré should pay for the printing of the original version. Poincaré, who was mortified, agreed, even though the bill came to over 3500 crowns, 1000 crowns more than the prize he’d originally won.
In an attempt to rectify the situation, Poincaré set about trying to sort out his mistake, to understand where and why he had gone wrong. In 1890, Poincaré wrote a second, extended paper explaining his belief that very small changes could cause an apparently stable system suddenly to fly apart.
What Poincaré discovered, thanks to his error, led to one of the most important mathematical concepts of the last century: chaos. It was a discovery that places huge limits on what we humans can know. I may have written down all the equations for my dice, but what if my dice behaves like the planets in the solar system? According to Poincaré’s discovery, if I make just one small error in recording the starting location of the dice, that error could expand into a large difference in the outcome of the dice by the time it comes to rest on the table. So is the future of my Vegas dice shrouded behind the mathematics of chaos?
The chaotic path mapped out by a single planet orbiting two suns.
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