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... to restore theology to the mainstream of science
How much freedom should we trade for our security?
Essay submitted to the Shell and The Economist writing competition, August 2002.
Contents
1. Summary
2. Network
3. The physical world is a network ...
4. And the spirtual world also
5. The human condition
6. The principle of least action
1. Summary
Do we need to sacrifice freedom to gain security? The word 'trade' in the topic seems to suggest that we do: pay freedom to buy security. To the contrary is evidence flowing from the diversity and durability of the world's states. Hitler minimised freedom, and his thousand year Reich lasted little more than a decade. Free states, on the other hand, generally enjoy longer periods of stability.
To gain insight into the relationship of freedom and security, I assume a network model of the world and identify freedom with entropy and security with the ability of the network to survive damage and noise induced error. The mathematical theorems of communication tell us that entropy may be exploited to overcome error, so this model raises the possibility that increasing freedom may increase security.
Can we apply this result in the real world? My hunch is yes. I hope this rather abstract and impressionistic essay will move you in the same direction. The security of a community depends upon its ability to solve the problems that face it. If we increase community entropy by giving everybody an active voice in the search for solutions, we increase the experience and brainpower available for survival, increasing security. back
2. Network
The dotcoms and telcos may be a little tarnished, but the mathematical foundations of their technology remain forever brilliant, and can tell us much about freedom and security. Let us assume that all the world can be seen as a network. This is not implausible, as we will see below. In the abstract mathematical world, a network is a graph, a set of points (or nodes) connected by a set of lines (or edges) obeying certain axioms. When we apply the mathematics to real networks, the elements may be anything that obeys the axioms, atoms, people, nations ... , railways, phone lines, treaties ... .
There is a rich literature on networks and the internet is a handy practical model. The nodes are computers (and behind them, users), and the edges are communication channels. The most important feature of networks is their ability to work perfectly despite partial failure. The first computer networks were designed to preserve the defences of the United States even if many cities were destroyed by nuclear weapons. An insecure network may break down, a state than manifests as war, disaster, accident, loneliness or any of the other difficulties that beset us.
The security of a network is bought with redundancy, which we might think of as spare entropy, freedom or headroom. First, in a well designed network the nodes are connected to one another by multiple edges, so that if we lose a few, there is still a good chance that the remaining nodes can still communicate. Second, the mathematical theory of communication shows us how to exploit the redundancy in messages to guarantee error free transmission over noisy channels. Finally, security against unauthorised listeners can be achieved using special encodings. back
3. The physical world is a network ...
We study of history with an eye to predicting and controlling the future. Here we seek a cosmological foundation, placing human history in the context of the whole universe. Cosmology is founded on three disciplines: quantum mechanics, relativity and quantum field theory. These theories are now so trusted that they, like the theories of networks and communications, lie at the theoretical heart of national defence systems.
Quantum mechanics postulates an infinite world of states animated by a boundless zoo of operators. To see the quantum world as a network, we may make the operators nodes and the states edges. Quantum networks are subject to an important constraint: the operators transforming an isolated system must be unitary.
Unitarity corresponds to the common observation that physical resources are locally limited. The quantum formalism recognises that while the variety of possible events is without limit, the probability of something happening at any point in spacetime is normalised to one. The physical world is full. This has the consequence, recognised by Malthus, that every potential being must compete for the physical resources to realise itself. This constraint, plus the infinite combinatorial creativity of the world, has given us the tree of life in which we nest.
Newton assumed that information propagates through space instantaneously. In fact it takes time: space and time are intermingled. The geometrical structure of spacetime is described by relativity. Relativity is consistent with the network view, since networks take time to encode, transmit and decode messages. The space/time relationships in a network create a metric space which enables us to define and measure the geometry of networks (and physical spacetime, as Einstein did so brilliantly).
Quantum field theory lies at the intersection quantum mechanics and relativity. These two theories, plus natural selection, seem sufficient to explain the evolution of our universe. Quantum field theory holds that all communications involve the exchange of physical particles, a fact widely verified by experience. Many feel, nevertheless, that there exists a spiritual world in which communications (eg revelations from a god outside the universe) move by non-physical means. The needs of spirituality, however, might also be met in the network model. back
4. And the spiritual world also ...
For historical reasons, the term 'wave function' is often applied to the mathematical mechanism of quantum mechanics. This arises because a central construct of mechanics is the 'harmonic oscillator' which explains such things as musical instruments and light. Quantum mechanics works in Hilbert space, which is rather like ordinary space, but may have any number of dimensions. The quantum harmonic oscillator (revealed by Planck in 1900) has as many states as there are integers, 1, 2, 3, ... , a 'countably infinite' number. To construct the wave function of the universe, we use the fact that as networks can be joined to form bigger networks, quantum systems can be joined to form bigger systems working in bigger spaces.
We construct and measure such spaces using set theory. We call the cardinal number of the set of natural numbers 'aleph zero' (aleph(0)), the first transfinite number. The natural numbers have a natural order, 1, 2, 3, ... , but we can rearrange them in many different orders, like 1, 3, 2; 2, 3, 1; ... called permutations. The cardinal number of the set of all permutations of the natural numbers is a transfinite number (called aleph(1)) strictly greater than aleph(0). By considering all permutations of aleph(1) numbers, we arrive at aleph(2), and so on without end. This construction is often called the Cantor universe.
When we unite quantum systems, we find that the corresponding Hilbert spaces grow like the transfinite numbers. The product of two aleph(0)-dimensional spaces is aleph(1)-dimensional, the product of two aleph(1)-dimensional spaces an aleph(2)-dimensional space, and so on. For quantum mechanics, the physical universe looks very like the Cantor universe.
Is the real universe this big? It looks to be everywhere finite, and the unitarity constraint seems to explain why. A century of quantum theory, however, has forced us to accept that there are transfinite processes going on behind the scenes. Physicists have constructed a quantum model of the mind of the universe just as we, reading someone's words, try to picture their invisible mind. Like human activities, quantum events are beset by uncertainties because the huge space of mental possibilities can never be fully represented by observables.
Quantum information theory makes the picture intuitively clearer because it interprets wave functions in terms of information rather than physical motion. Vast amounts of information may be stored and manipulated with quantum systems. Many hope to exploit this quantum resource to construct very powerful computers. Quantum information theory, like its classical antecedent, relies on redundancy to overcome error. back
5. The human condition
How do we link such physics (and metaphysics) to human freedom and security? Our connection between Hilbert spaces and the Cantor universe is consistent with the observation that the world is a layered network, fundamental particles being assembled into atoms, atoms into molecules, molecules into cells, organisms, communities and so on through solar systems to galaxies and the universe as a whole. Each layer is populated by entities of the same power, sometimes called peers.
Somewhere in all this we find networks whose nodes are human and whose edges are human communication. Nodes come and go as people enter or leave communities. We notice, in most human networks, that all nodes are not equal. We are not all peers. Some, like newborn children, are still learning their network protocols and have limited functionality. Others, at the far end of their life cycle, are fading out.
Beside these natural processes, there are vast differences between people in their economic and political power, and so in their ability to access the goods of the world. Further, many weak spots in the human network are caused by geographical, religious and political barriers. To maximise the security of the human network, we must eliminate these weak spots. Forging human networks it not easy however, since it is hard to reach agreement about communication protocols. Babel complicates connectivity. This difficulty is compounded because some (despots, warlords) seek freedom at the expense of others (slaves, cannon fodder). There is, however, a further feature of the world which may simplify the task of making us all peers. back
6. The principle of least action
The self-similarity of networks tells us that what we learn about one we may apply to others. This suggests that apart from questions of scale, the intractably complex human world echoes the simpler world of physics. The mathematical wormhole which joins classical to quantum physics is Hamiltonian dynamics, at whose heart we find Hamilton's principle of least action. Action entered physics with Newton's second axiom of change: to every action there is an equal and opposite reaction. Classical action carries over to quantum physics where it is quantised (Planck's quantum of action) and conserved. Hamilton's principle tells us that physical systems that obey the second law evolve so as to minimise action.
Capitalism creates wealth at an unprecedented rate, but stability requires an effective distribution system as well. Newton's second axiom holds in the ideal of free trade between equals, but disparities in power lead to its frequent violation in other cases. A may bomb B into the stone age without B being able to react, or even make its plight public.
The beauty of the second axiom is that it is local. We do not need to take on the whole world. We can generate freedom in the world by generating freedom in our local environment. This move from local to global is reminiscent of Einstein's extrapolation of his local special theory of relativity to the global general theory, which constrains the overall structure of the physical universe. Globally we can strengthen the human network by using wealth to heal the weak spots.
We want a free and secure world, a dream. To get there we must implement the human equivalent of Newton's second axiom at every point: a combination of an eye for an eye with universal love. We live in a system stabilised by mutual assured destruction. Here I see a path to a world stabilised by mutually assured construction, the physical expression of the spirit of love. back
Books
| Cantor, Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 1955 Jacket: 'One of the greatest mathematical classics of all time, this work established a new field of mathematics which was to be of incalculable importance in topology, number theory, analysis, theory of functions, etc, as well as the entire field of modern logic.' Amazon back |
| de Witt, Bryce S and Neill Graham (eds) , and Hugh Everett III, J A Wheeler, B S DeWitt, L N Cooper, D van Vechten, N Graham (contributors), The Many-Worlds Interpretation of Quantum Mechanics, Princeton UP 1973 Jacket: 'A novel interpretation of quantum mechanics, first proposed in brief form by Hugh Everett in 1957, forms the nucleus around which this book is developed. The volume contains Dr Everett's short paper from 1957, "'Relative State' Formulation of Quantum Mechanics", and a far longer exposition of his interpretation, entitled "The Theory of the Universal Wave Function", never before published. In addition, other papers by De Witt, Graham and Cooper and van Vechtem provide further dicussion of the same theme. Together they constitute virtually the entire world output of scholarly commentary on the Everett interpretation.' Amazon back |
| Hawking, Steven W, and G F R Ellis, The Large Scale Structure of Space-Time , Cambridge UP 1975 Preface: Einstein's General Theory of Relativity ... leads to two remarkable predictions about the universe: first that the final fate of massive stars is to collapse behind an event horizon to form a 'black hole' which will contain a singularity; and secondly that there is a singularity in our past which constitutes, in some sense, a beginning to our universe. Our discussion is principally aimed at developing these two results.' Amazon back |
| Jech, Thomas, Set Theory, Springer 1997 Jacket: 'This book covers major areas of modern set theory: cardinal arithmetic, constructible sets, forcing and Boolean-valued models, large cardinals and descriptive set theory. ... It can be used as a textbook for a graduate course in set theory and can serve as a reference book.' Amazon back |
| Khinchin, A I, Mathematical Foundations of Information Theory (translated by P A Silvermann and M D Friedman), Dover 1957 Jacket: 'The first comprehensive introduction to information theory, this book places the work begun by Shannon and continued by McMillan, Feinstein and Khinchin on a rigorous mathematical basis. For the first time, mathematicians, statisticians, physicists, cyberneticists and communications engineers are offered a lucid, comprehensive introduction to this rapidly growing field.' Amazon back |
| Klein, Richard G, The Human Career : Human Biological and Cultural Origins , University of Chicago Press 1999 Review: 'The Human Career describes one of the most spectacular changes to have occurred in our understanding of human evolution. The once-popular fresco showing a single file of marching hominids becoming ever more vertical, tall and hairless now appears to be a fiction. ... For most of the past four million years several species of hominids coexisted, sometimes in limited geographical areas. The eventual peopling of the planet with a single homogeneous species of hominid is shown to be exceptional on the geological timescale. ... If you could have only one book that deals with human evolution, this is definitely the one to choose. ' Jean-Jacques Hublins, Nature. 403:364 27 January 2000. Amazon back |
| Kreyszig, Erwin, Introductory Functional Analysis with Applications, John Wiley and Sons 1989 Amazon: 'Kreyszig's "Introductory Functional Analysis with Applications", provides a great introduction to topics in real and functional analysis. This book is part of the Wiley Classics Library and is extremely well written, with plenty of examples to illustrate important concepts. It can provide you with a solid base in these subjects, before one takes on the likes of Rudin and Royden. I had purchased a copy of this book, when I was taking a graduate course on real analysis and can only strongly recommend it to anyone else.' Krishnan S. Kartik Amazon back |
| Kuhn, Thomas S, Black-Body Theory and the Quantum Discontinuity 1894-1912, University of Chicago Press 1987 Jacket: '[This book] traces the emergence of discontinuous physics during the early years of this century. Breaking with historiographic tradition, Kuhn maintains that, though clearly due to Max Planck, the concept of discontinuous energy change does not originate in his work. Instead it was introduced by physicists trying to understand the success of his brilliant new theory of black-body radiation.' Amazon back |
| Misner, Charles W, and Kip S Thorne, John Archibald Wheeler, Gravitation, Freeman 1973 Jacket: 'Einstein's description of gravitation as curvature of spacetime led directly to that greatest of all predictions of his theory, that the universe itself is dynamic. Physics still has far to go to come to terms with this amazing fact and what it means for man and his relation to the universe. John Archibald Wheeler. ... this is a book on Einstein's theory of gravity (general relativity).' Amazon back |
| Nielsen, Michael A, and Isaac L Chuang, Quantum Computation and Quantum Information, Cambridge University Press 2000 Review: A rigorous, comprehensive text on quantum information is timely. The study of quantum information and computation represents a particularly direct route to understanding quantum mechanics. Unlike the traditional route to quantum mechanics via Schroedinger's equation and the hydrogen atom, the study of quantum information requires no calculus, merely a knowledge of complex numbers and matrix multiplication. In addition, quantum information processing gives direct access to the traditionally advanced topics of measurement of quantum systems and decoherence.' Seth Lloyd, Department of Quantum Mechanical Engineering, MIT, Nature 6876: vol 416 page 19, 7 March 2002. Amazon back |
| Tanenbaum, Andrew S, Computer Networks, Prenctice Hall International 1996 Preface: 'The key to designing a computer network was first enunciated by Julius Caesar: Divide and Conquer. The idea is to design a network as a sequence of layers, or abstract machines, each one based upon the previous one. ... This book uses a model in which networks are divided into seven layers. The structure of the book follows the structure of the model to a considerable extent.' Amazon back |
| von Neumann, John, and Robert T Beyer (translator), Mathematical Foundations of Quantum Mechanics, Princeton University Press 1983 Jacket: '... a revolutionary book that caused a sea change in theoretical physics. ... JvN begins by presenting the theory of Hermitean operators and Hilbert spaces. These provide the framework for transformation theory, which vN regards as the definitive form of quantum mechanics. ... Regarded as a tour de force at the time of its publication, this book is still indispensible for those interested in the fundamental issues of quantum mechanics.' Amazon back |
| Weinberg, Steven, The Quantum Theory of Fields Volume II: Modern Applications, Cambridge University Press 1996 Jacket: '[This volume] gives an up-to-date and self-contained account of the methods of quantum field theory, and how they have led to an understanding of the weak, strong and electromagnetic interactions of the elementary particles. The presentation of modern mathematical methods is throughout interwoven with accounts of the problemd of elemntary partricle physics and condensed matter physics to which they have been applied. Topics are incl;uded that are not usually found in books on quantum field theory, such as the Batalin-Vilkovsky formalism and its application to renomralisation and anomalies in gauge theories; the background field method; the effective field theory approach to symmetry breaking; critical phenomena; and superconductivity. The book contains original material and is peppered with examples and insights from the author's experience as a lkeader of elementary particle physics. Problems are inlcuided at the end of each chapter.' Amazon back |
| Weinberg, Steven, The Quantum Theory of Fields Volume I: Foundations, Cambridge University Press 1995 Jacket: 'After a brief historical outline, the book begins anew with the principles about which we are most certain, relativity and quantum mechanics, and then the properties of particles that follow from these principles. Quantum field theory then emerges from this as a natural consequence. The classic calculations of quantum electrodynamics are presented in a thoroughly modern way, showing the use of path integrals and dimensional regularization. The account of renormalization theory reflects the changes in our view of quantum field theory since the advent of effective field theories. The book's scope extends beyond quantum elelctrodynamics to elementary partricle physics and nuclear physics. It contains much original material, and is peppered with examples and insights drawn from the author's experience as a leader of elementary particle research. Problems are included at the end of each chapter. ' Amazon back |
| Weinberg, Steven, The Quantum Theory of Fields, Volume III: Supersymmetry, Cambridge University Press 2000 Jacket: '[This volume] presents a self-contained, up-to-date and comprehensive introduction to supersymmetry, a highly active area of theoretical physics that is likely to be the centre of future progress in the physics of elementary particles and gravitation. The text introduces and explains a broad range of topics, including supersymmetric algebras, supersymmetric field theories, extended supersymmetry, super-graphs, non-perturbative results, theories of supersymmetry in high dimensions, and supergravity. A thorough review is given of the phenomenological implications of sypersymmetry, including theories of both gauge and gravitationally mediated supersymmetry breaking. Also provided is an introduction to mathematical techniques, based on holomorphy and duality, that have proved so fruitful in recent developments. The book contains much material not found in other books on supersymmetry, some of it new. Problems are included at the end of each chapter.' Amazon back |
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