PREFACE TO THE FIRST EDITION
This book is intended for readers in transition from school mathematics to the fully-fledged type of thinking used by professional mathematicians. It should prove useful to first-year students in universities and colleges, and to advanced students in school contemplating further study in pure mathematics. It should also be of interest to a wider class of reader with a grounding in elementary mathematics seeking an insight into the foundational ideas and thought processes of mathematics.
The word ‘foundations’, as used in this book, has a broader meaning than it does in the building trade. Not only do we base our mathematics on these foundations: they make themselves felt at all levels, as a kind of cement which holds the structure together, and out of which it is fabricated. The foundations of mathematics, in this sense, are often presented to students as an extended exercise in mathematical formalism: formal mathematical logic, formal set theory, axiomatic descriptions of number systems, and technical constructions of them; all carried out in an exotic and elaborate symbolism. Sometimes the ideas are presented ‘informally’ on the grounds that complete formalism is too difficult for the delicate flowering student. This is usually true, but for an entirely different reason.
A purely formal approach, even with a smattering of informality, is psychologically inappropriate for the beginner, because it fails to take account of the realities of the learning process. By concentrating on the technicalities, at the expense of the manner in which the ideas are conceived, it presents only one side of the coin. The practising mathematician does not think purely in a dry and stereotyped symbolism: on the contrary, his thoughts tend to concentrate on those parts of a problem which his experience tells him are the main sources of difficulty. While he is grappling with them, logical rig- our takes a secondary place: it is only after a problem has, to all intents and purposes, been solved intuitively that the underlying ideas are filled out into a formal proof. Naturally there are exceptions to this rule: parts of a problem may be fully formalized before others are understood, even intuitively; and some mathematicians seem to think symbolically. Nonetheless, the basic force of the statement remains valid.
The aim of this book is to acquaint the student with the way that a practising mathematician tackles his subject. This involves including the standard ‘foundations’ material; but our aim is to develop the formal approach as a natural outgrowth of the underlying pattern of ideas. A sixth-form student has a broad grasp of many mathematical principles, and our aim is to make use of this, honing his mathematical intuition into a razor-sharp tool which will cut to the heart of a problem. Our point of view is diametrically opposed to that where (all too often) the student is told ‘Forget all you’ve learned up till now, it’s wrong, we’ll begin again from scratch, only this time we’ll get it right’. Not only is such a statement damaging to a student’s confidence: it is also untrue. Further, it is grossly misleading: a student who really did forget all he had learned so far would find himself in a very sorry position.
The psychology of the learning process imposes considerable restraints on the possible approaches to a mathematical concept. Often it is simply not appropriate to start with a precise definition, because the content of the definition cannot be appreciated without further explanation, and the provision of suitable examples.
The book is divided into four parts to make clear the mental attitude required at each stage. Part I is at an informal level, to set the scene. The first chapter develops the underlying philosophy of the book by examining the learning process itself. It is not a straight, smooth path; it is of necessity a rough and stony one, with side-turnings and blind alleys. The student who realizes this is better prepared to face the difficulties. The second chapter analyzes the intuitive concept of a real number as a point on the number line, linking this to the idea of an infinite decimal, and explaining the importance of the completeness property of the real numbers.
Part II develops enough set theory and logic for the task in hand, looking in particular at relations (especially equivalence relations and order relations) and functions. After some basic symbolic logic we discuss what ‘proof ’ consists of, giving a formal definition. Following this we analyze an actual proof to show how the customary mathematical style relegates routine steps to a contextual background—and quite rightly so, inasmuch as the overall flow of the proof becomes far clearer. Both the advantages and the dangers of this practice are explored.
Part III is about the formal structure of number systems and related con- cepts. We begin by discussing induction proofs, leading to the Peano axioms for natural numbers, and show how set-theoretic techniques allow us to con- struct from them the integers, rational numbers, and real numbers. In the next chapter we show how to reverse this process, by axiomatising the real numbers as a complete ordered field. We prove that the structures obtained in this way are essentially unique, and link the formal structures to their in- tuitive counterparts of part I. Then we go on to consider complex numbers, quaternions, and general algebraic and mathematical structures, at which point the whole vista of mathematics lies at our feet. A discussion of infinite cardinals, motivated by the idea of counting, leads towards more advanced work. It also hints that we have not yet completed the task of formalising our ideas.
Part IV briefly considers this final step: the formalisation of set theory. We give one possible set of axioms, and discuss the axiom of choice, the continuum hypothesis, and Gödel’s theorems.
Throughout we are more interested in the ideas behind the formal façade than in the internal details of the formal language used. A treatment suitable for a professional mathematician is often not suitable for a student. (A series of tests carried out by one of us with the aid of first-year undergraduates makes this assertion very clear indeed!) So this is not a rigidly logical development from the elements of logic and set theory, building up a rigorous foundation for mathematics (though by the end the student will be in a position to appreciate how this may be achieved). Mathematicians do not think in the orthodox way that a formal text seems to imply. The mathematical mind is inventive and intricate; it jumps to conclusions: it does not always proceed in a sequence of logical steps. Only when everything is understood does the pristine logical structure emerge. To show a student the finished edifice, without the scaffolding required for its construction, is to deprive him of the very facilities which are essential if he is to construct mathematical ideas of his own.
I.S. and D.T. Warwick October 1976

"You believe in a God who plays dice, and I in complete law and order." Albert Einstein The science of chaos is forcing scientists to rethink Einstein's fundamental assumptions regarding the way the universe behaves. Chaos theory has already shown that simple systems, obeying precise laws, can nevertheless act in a random manner. Perhaps God plays dice within a cosmic game of complete law and order. Does God Play Dice? reveals a strange universe in which nothing may be as it seems. Familiar geometrical shapes such as circles and ellipses give way to infinitely complex structures known as fractals, the fluttering of a butterfly's wings can change the weather, and the gravitational attraction of a creature in a distant galaxy can change the fate of the solar system.
This revised and updated edition includes three chapters on the prediction and control of chaotic systems. New information regarding the solar system and an account of complexity theory is also incorporated. It is a lucid and witty book which makes the complex mathematics of chaos accessible and entertaining.

Preface
It is 2050, and you are watching Who Wants to be a Billionaire? The contestant is one question away from the jackpot. Up comes his
question: “What is the name of the theory that scientists started developing at the beginning of the twenty-first century, and
which helped the world overcome traffic congestion, financial market crashes, terrorist attacks, pandemic viruses, and cancer?” The contestant cannot believe his luck. What an easy question! But he is so nervous that his mind temporarily goes blank. He starts to consider option A: “They are all still unsolved problems” – but then quickly realizes that this is a dumb answer. Instead, he uses his last lifeline to ask the audience. The audience responds unanimously and instantaneously with option B: “The Theory of Complexity”. Without hesitation, he goes with option B. The host hands him the cheque, and the world has yet another billionaire.
Pure fantasy? Maybe not.
In this book, we will go on a journey to the heart of Complexity, an emerging science which looks set to trigger the next great wave of advances in everything from medicine and biology through to economics and sociology. Complexity Science also comes with the
prospect of solving a wide range of important problems which face us as individuals and as a Society. Consequently, it is set to permeate through every aspect of our lives.
There is, however, one problem. We don’t yet have a fullyfledged “theory” of Complexity. Instead, I will use this book to assemble all the likely ingredients of such a theory within a common framework, and then analyze a wide range of real-world applications within this same common framework. It will then require someone from the future – perhaps one of the younger readers of this book – to finally put all these pieces into place.
Complexity Science is a double-edged sword in the best possible sense. It is truly “big science” in that it embodies some of the hardest, most fundamental and most challenging open problems in academia. Yet it also manages to encapsulate the major practical issues which face us every day from our personal lives and health, through to global security. Making a pizza is complicated, but not complex. The same holds for filling out your tax return, or mending a bicycle puncture. Just follow the instructions step by step, and you will eventually be able to go from start to finish without too much trouble. But imagine trying to do all three at the same time. Worse still, suppose that the sequence of steps that you follow in one task actually depends on how things are progressing with the other two. Difficult? Well, you now have an indication of what Complexity is all about. With that in mind, now substitute those three interconnected tasks for a situation in which three interconnected people each try to follow their own instincts and strategies while reacting to the actions of the others. This then gives an idea of just how Complexity
might arise all around us in our daily lives.
While I was writing this book, I had the following “wish-list” in my head concerning its goals:
1. To provide a book which a wide cross-section of people would want to read and would enjoy reading – regardless of age,
background or level of scientific knowledge.
2. To introduce readers to the exciting range of real-world scenarios in which Complexity Science can prove its worth.
3. To provide the book on Complexity that “I never had but always needed”. In other words, to provide an easily readable yet thorough guide to this important scientific revolution.
4. To provide a book that my kids could read – or rather, a book that they would actually choose to read all by themselves. This is a very important goal, since Complexity will likely become the science of interest for future generations.
5. To provide a book which is just as readable on a plane or bus as in a library. As such, it should also make sense when read in short chunks.
6. To provide a book which provides professional scientists,economists, and policy-makers with a new perspective on
open problems in their field, and to help stimulate new Complexity-based interdisciplinary research projects.
However, as I finish the book and offer it up to potential readers,I realize that the above wish-list can essentially be reduced to just
one item: I would wish that you enjoy reading this book, and that it might provide you with fresh thoughts and insights for dealing
with the complex world in which we live, and which our children will inherit.
There are some practicalities concerning the book’s content and layout which I would like to explain. The language, examples and
analogies are kept simple since the focus of the book is to explain what Complexity Science is all about, and why it is so important for
us all. I therefore avoid delving into too much detail in the main text. Instead, the Appendix describes how to access the technical
research papers upon which the discussions in the book are based, and gives a list of Internet websites containing additional information about Complexity research around the world. Having said this, I won’t pull any punches in the sense that I tackle all the topics
which I believe to be relevant. Part 1 of the book takes us through the theoretical underpinnings of Complexity, while Part 2
delves into its real-world applications. Some of the territory is only just beginning to be explored, with very few answers available
for the questions being posed. From the perspective of other scientific revolutions throughout history this might seem to be par
for the course. However we are not talking about history here –instead, we are looking at work which is emerging at the forefront
of a new discipline. For this reason we will be highlighting where such research is heading, rather than where it has been.
But why should you believe what I write about Complexity? This is a crucially important question given that Complexity Science is still being developed and its potential applications explored. Unfortunately many accounts of Complexity in the popular press are second-hand, i.e. they are typically written by people who have done little, if any, research on Complexity themselves and are instead reporting on their interpretation of
other people’s work. Given the relatively immature nature of the field, I believe that such indirect interpretations are potentially
dangerous. For this reason, I will base the book’s content around my own research group’s experience in Complexity. This has
various advantages: (i) it reflects my own understanding of the Complexity field; (ii) it represents what I believe to be the most
relevant and important topics; (iii) it will hopefully give the reader a sense of what it is like to be at the “pit-face” in such a
challenging area of research; and (iv) it ensures that any reader can challenge me directly on any claims that I make, and can
demand an informed answer. To facilitate this process of public scrutiny, a complete list of the relevant scientific research reports
is presented in the latter part of the Appendix. I also encourage any readers who wish to email me with questions, to do so at
n.johnson@physics.ox.ac.uk

Napoleon's Buttons is the fascinating account of seventeen groups of molecules that have greatly influenced the course of history. These molecules provided the impetus for early exploration, and made possible the voyages of discovery that ensued. The molecules resulted in grand feats of engineering and spurred advances in medicine and law; they determined what we now eat, drink, and wear. A change as small as the position of an atom can lead to enormous alterations in the properties of a substance-which, in turn, can result in great historical shifts.
With lively prose and an eye for colorful and unusual details, Le Couteur and Burreson offer a novel way to understand the shaping of civilization and the workings of our contemporary world.

"More years ago than I care to reckon up, I met Richard Feynman." So begins Herman Wouk's trenchant and exhilarating book on navigating the divide between science and religion.
Told by Feynman in that first meeting that he must learn "the language God talks"-calculus-Wouk set in motion the lifelong inquiry that has culminated in this rich, compact volume. Wouk draws on stories from his own life, on key events from the twentieth century, and on encounters not just with Feynman but with other masters of science and religion to address the eternal questions of why we are here, what purpose faith serves, and how scientific facts fit into the picture.

Review
Virginia Quarterly Review : A fascinating study of the effects that the theories of the notorious Viennese physician, Franz Mesmer, had upon social and political thinkers during the two decades preceding the French Revolution. This book is a skillful exploration of the various psychological factors that made mesmerism a widely accepted attitude[The book] will interest literary scholars as well as historians since mesmerism is examined as a phenomenon that bequeathed an attitude that found its expression in the writings of the preromantics and the romantics.
Isis : This is an excellent book and one of singular interest both to the historian of science and to the French historian.
Science : [An] excellent and exemplary study in the history of ideas. Based on a thorough study of manuscripts, pamphlets, and journals, learned in its broad setting and persuasive in its internal logic, supported by richly relevant quotations and reproductions of contemporary engravings, Mesmerism and the End of the Enlightenment in France provides a commendable model for those interested in the way 'true' and 'false' ideas interact and broadly influence behavior.

Elucidates the difference knowledge makes to our existence as civilized people. Advancing detailed criticism of philosophy's usual approach to knowledge, Knowledge and Civilization offers a new, original way of framing philosophical questions about knowledge. It describes a redirection in the philosophy of knowledge, away from textbook problems of epistemology, toward an ecological philosophy of technology and civilization.