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vol 6: Essays

Physical Theology, 2006

Contents

Abstract
The standard model of God
A new model of God
A symmetric Universe
Relativity
Quantum mechanics
Creation
Evolution
Conclusion

Abstract

This essay proposes a transfinite network to link physics and theology. It follows the time-worn scientific path of uniting apparently disparate elements of the world by creating a mathematical space large enough to hold them both. By redefining continuity in terms of logical inference and infinity in terms of permutation, we hope to construct a model of reality which unifies our notions of 'God' and 'the world'. The approach is 'formalist' Cohen, Hofstadter, back

Introduction

I address a fundamental issue for humanity: how do we construct and validate our models of God? 'God' here means the whole of reality. Since we must survive within God so defined, knowledge of God is of supreme practical value.

Traditional western theology divides reality into 'God' and 'The World' and holds that God is essentially invisible. Our knowledge of God comes through ancient texts. Modern scientific epistemology, on the other hand, takes Einstein's view that we can trust information only when it is obtained by contact with the entity we wish to know. A trustworthy theology must therefore be scientific, that is devoted to producing models based on direct experience of God. Scientific theology is only possible if God is observable, that is if, in the spirit of Occam, we ignore thet distinction between God and the world.

The scientific method is empirical. Because we are all studying the same Universe our results are compatible, comparable and communicable. One Universe plus scientific method will eventually lead us to one theology, as it does in the other sciences. In this theology, God is truly creator and judge: all our activity is divine activity, and our schemes work or do not work depending on whether they are judged fit or unfit by reality.

For a traditional believer (which I once was) the identification of God and the Universe seems a priori impossible. While God is perfect, eternal and unchanging the World is imperfect, temporary and subject to change. A little reflection shows that this impossibility is model dependent, and need not appear in a revised model of God. We begin with a brief account of the Western model. back

The standard model of God

The standard model of God has a very long history which culminated in the work of the medieval theologian Thomas Aquinas (1224 - 1274). Aquinas built on the work of Aristotle (384 - 322 bc). Aristotle's model of reality distinguished two elements, potential (the ability to be) and act (actual being), and proposed an axiom of actualization: no potential being can actualize itself.

Aristotle used his model to establish to existence of an unmoved mover. To move is to realize the potential to move. Since no potential can actualize itself, any motion implies a chain of moved movers which must have a beginning if anything is to move at all. Since things do move, there must be an unmoved mover Aristotle, Physics,, Book VIII Chapter vi

Aquinas provides five ways to prove the existence of God, the first of which is identical to Aristotle's argument for the unmoved mover, renamed 'God'. Although called 'proofs for the existence of God', these texts are rather arguments for the partition of reality into God and not-God. As the proofs make clear, we and our world are part of not-God. Aquinas 13

In each case, this partition is based on the premiss that the world cannot explain some aspect itself, which must therefore be explained by God. Since the traditional God is invisible however, we do not have access to this explanation, but must 'take it on faith'. Science, on the other hand, believes that there is an accessible and reasonable explanation for all phenomena, that is all messages received from our environment, which includes our hearts.

For Aquinas, God is pure act, the realization of all possibility. From this axiom he derives all the traditional properties of God, simplicity, perfection, goodness, infinity, eternity, unity, omniscience, life omnipotence, justice, mercy, providence and beatitude Aquinas 14, Summa, qq 3-26

In addition to the monotheism inherited from the Hebrews, Christianity embraced a divine Trinity, asserting the existence of three distinct divine Persons in the one God. Aquinas modeled the Trinity on the properties of relationship, knowledge and desire, also derived from Aristotle . The Trinity takes Christianity some distance in the direction of religions which attribute many personalities or incarnations to the power that controls human life. We follow this course below, unfolding a transfinite array of 'personalities' (independent sources of information) from an initial single source which shares many of the attributes of the traditional God. Aquinas 160, Summa, qq 27-43

The Christian God is a living God. Since Aquinas accepted the traditional definition of life as self-motion he faced the difficulty that motion implies potential and God is pure act. Following Aristotle, Aquinas distinguished between transeunt and immanent action. Transeunt action actualizes something other than the agent while immanent action, like the life of God, perfects the agent itself. Aristotle 1 Metaphysics Book IX chapter viii, 1050a22

On the present hypothesis we are part of God, and so all our actions may be considered part of God's life. The modern descendants of Aristotle potency and act are potential and kinetic energy. Here there is no axiom of actualization: the two forms of energy are 'peers' freely interchangeable and equally real, as illustrated by the harmonic oscillator. back

A new model of God

Theology creates and studies models of God. The hypothesis here is that the terms God and the Universe refer to the same thing, whose defining property is that there is nothing outside it and so no external constraints upon it. Any constraints we find must arise from self-consistency.

To unite God and the Universe, we need a new model of God. In effect, I want to replace classical continua, which are traditionally a characteristic physical bodies, with logical continua (represented by Turing machines), which is are a characteristic of networks.

Whatever God is, it is big. At least as big physically as the observed Universe. The use of mathematical function spaces in quantum mechanics suggests that the Universe is even bigger behind the scenes. We estimate big here in terms of complexity (measured by information or entropy) rather than by spatial extent.

The mathematical archetype of big is 'Cantor's Paradise', the transfinite Universe of sets and correspondences invented by Georg Cantor. Although our engineered realizations of computation and communication theory are finite, the relevant mathematics often holds in transfinite domains as well. It is in this respect invariant with respect to complexity. It derives this feature from set theory itself, which deals indifferently with aggregates of any size. We therefore propose to model God with a transfinite network whose basic properties are complexity invariant. Cantor, Dauben

Symmetry with respect to complexity is the foundation of abstract knowledge. It allows us to map the transfinite scale of complexity to finite models, talking (as we do) with equal facility about atoms, people, planets, and feelings. The essential features of communication in the transfinite network are the same regardless of the complexity of the sources and of the messages exchanged. Since physics is the most general and abstract theory of the Universe, we might expect the foundations of physics to lie in the nature of communication alone, with no further specialization.

Soon after the development of set theory, Cantor and others realized that any attempt to work with the set of all sets, (which might serve as a model of God) leads to contradiction Formally, the set of all sets does not exist. This result is in accord with the ancient notion that we cannot comprehend God. Nevertheless, set theory provides a means to talk consistently about subsets of some larger aggregate. Modelling and communication of God must therefore be local. This is also a feature of our Universe imposed by the finite velocity of communication and explicated in detail by Einstein's theory of relativity. Mendelson, back

A symmetric Universe

Let us construct a 'symmetric Universe', modelled on the Cantor Universe. We begin with the set N of natural numbers 1, 2, 3, . . . . N is said to be infinite because given any natural number n, we can always form the next one, n+1. Cantor chose the symbol 0 to represent the cardinal of N. 0 is the first transfinite number.

In modern set theory the axiom of the power set provides that given any set S, there exists a power set P(S) which contains all the subsets of S. Cantor's theorem states that the cardinal of P(S) is strictly greater than the cardinal of S, even when S is transfinite. This theorem gives rise to the second transfinite number, 1, the third 2 and so on without end. The axiom of the power set is blind to the size of any set we choose, and so acts in a uniform (or symmetric) way on all sets. Cantor's theorem is based on the hypothesis that counting by one to one correspondence is also blind to the size of the sets compared. Jech

Cardinal number ignores the nature of the elements of a set. Set theory relies heavily on a second, more concrete abstraction, the ordinal number or ordinal type of a set. The 0 natural numbers have a natural order, but we may permute this order to obtain the 1 permutations of the 0 natural numbers. These 1 permutations may be ordered alphabetically based on the order of the natural numbers. These symbols may themselves be permuted to create 1 distinct symbols, and so on without end. Let us call the all permutations associated with each cardinal a peer group, analogous to the peer levels in computer network engineering.Tanenbaum

The set of permutations of a set S of cardinal n form the symmetric group on n objects. Group theory finds that every group of order n is a subgroup of the symmetric group. We call the Cantor Universe interpreted as a transfinite hierarchy of groups of permutations the symmetric Universe. The symmetric Universe is unlimited in size and complexity; every element of it is unique; and it contains, as a subgroup, every possible abstract group. As Cantor stated, it is large enough to represent anything thinkable, which we assume to include all local manifestations of God.

We use permutation rather than combination to generate our model because we are ultimately modelling a concrete set of symbols (the Universe) and accept Landauer's thesis that all information is encoded physically. We assume that nothing is different (ie no message is different) unless it is physically different and that every physical event is unique and can be placed into correspondence with a unique sequence of symbols. The address of an element arises partly from within itself and partly from its place in the overall system. Landauer

The symmetric Universe serves as a configuration space for the transfinite network. We can map this space onto our Universe by assuming that every event in our Universe corresponds to a transition between permutations or sets of permutations in the symmetric Universe.

We gain insight into the functioning of the transfinite network by comparing it to an ordinary computer with multiple processors. The symmetric Universe corresponds to the memory. We assume the logical equivalent of Newton's first law, that the state of a memory location is only changed when it is written to by a process. We model processes with Turing machines. Each execution of each computable function results in change of state of the entire system.

Simple logical machine operations are 'not', 'and' and 'or'. We may imagine the operation of a conventional computer as a time evolving network. Connections are continually being made and broken, each cycle transferring a certain amount of information. The making and breaking of connections corresponds to the physical creation and annihilation of particles.

Such atomic operations may be organized into more complex processes. This embedding confers meaning on physical events. A multiplication operation may itself be part of the execution of a climate model and the execution of the climate model part of a more complex scientific, social and political process of deciding how to manage our interface with the Universe.

Conflict between processes and failure can result if a deterministic processor is faced with memory changed by processes outside its control. Change in the environment of a deterministic processor is the source of indeterminism in the model.

We can continue the development of this model by fitting it to quantum mechanics and relativity. back

Relativity

At the heart Einstein's theory of relativity is an epistemological hypothesis: that information comes from contact. Here we propose a vast network of 'personalities' talking to one another, a transfinite extension of the triune threesome developed by Platonic theologians in the Patristic age. Everywhere we look we experience things talking to us and we only experience those that do talk to us. We begin with the conversation which holds us on the Earth and models the large scale structure of our Universe. Kelly

The network model suggests that gravitation is a protocol that constrains the way the whole system fits together. General relativity is founded on two propositions. The first, in Einstein's words 'All Gaussian coordinate systems are essentially equivalent in for the formulation of the general laws of nature'. The second is that all communication takes place by contact. These propositions constitute the 'principle of general covariance'. Einstein

Gaussian coordinates generalize Cartesian coordinates, the principal requirement being that there is a continuous relationship between Gaussian coordinates and the space they measure. Intervals between Gaussian coordinates are defined with the help of a metric tensor. Two events are in contact (ie become one event) if the interval between them is zero.

Einstein developed relativity in two stages. Newton worked in a rigid Euclidean space endowed with a universal time and instantaneous communication at a distance. The special theory follows as soon as we introduce delay into communication. The result is a four dimensional space-time with a Minkowski metric and a group of Lorentz transformations which tell us what uniformly moving observers look like to each other when they communicate at light speed. The only special thing about the velocity of light, c, is that it is fixed and finite. Horse drawn communications also induce relativistic structure on the networks they serve, where c = horse speed. Newton

By his own account, the happiest thought in Einstein's life was the realization that an observer in free fall does not feel his own weight . Although gravitation can be transformed away locally, this cannot be done globally. Each observer has its own path of free fall (geodesic), and the web of geodesics marks out the curvature of space. Pais

General covariance accepts that every event is unique and goes its own way. The complexity of transformations required to transform a randomly chosen event into my local frame of reference must be equal to the variety of relative motions possible in the Universe. The set of transformations allowed under general relativity is much larger that the set of Lorentz transformations. Here we assume that relativity operates at every scale in the Universe, wherever there is communication. In the human sphere, general covariance allows us to put ourselves in one another's shoes and see the world from one another's point of view.

In his 1915 paper. Einstein represented general covariance in the language of Riemann's differential geometry. Riemann represented n dimensional space by an n dimensional Gaussian coordinate system and found that the whole structure of the resulting curved or dynamic space could be encoded as a field of metric tensors, gmn, He also found that 'The basis of metrical determination must be sought outside the manifold in the binding forces which act on it'. Einstein 1, Jammer

This determination for the actual space we inhabit was provided by Einstein, who found that the metrical structure of space-time, represented by the Einstein tensor G. is a function of the energy tensor T: G = 8 π T. The curvature in a region of space-time is a function of the energy in that region.

The symmetric Universe is naturally covariant, since each event in a peer group is represented by a unique permutation of the relevant alphabet and group transformations exists to convert every element of the peer group into every other. Of these transformations, a subset are computable. The computable subset of the transfinite network we will call a computable manifold, the logical analogue of the continuous differentiable manifold of general relativity.

We define computable transformations as a transformations whose symbolic expressions can be represented as a Turing machine. Turing found that the set of computable functions is countably infinite, ie its cardinal is 0.

Let us identify every execution of Turing machine in the computable manifold with an event in the world, measured by one or more 'quanta of action'. The rate of communication f between two points is therefore (from a quantum mechanical point of view) measured by energy according to the relationship E= hf.

In general relativity, the shape of space is determined by the distribution of energy. Energy, that it traffic, attracts. At this level of abstraction, the accretion of stars and planets and the accretion of human groups and cities are driven by the same force, the attraction of the 'bright lights' of a network. Bandwidth attracts bandwidth.

In addition to its position in the physical layer of the transfinite network, each event has has a symbolic role in the higher layers of the network. As in a computer network, symbols are processed through multiple layers of software before they are presented to the user. So we may model God (that is the whole system) as the ultimate user of the transfinite network. Every particle is a user at its own peer level.

It may seem at first sight that since the transformations in Riemann space are continuous and differentiable, the cardinal of the set of permissible transformations must be the cardinal of the continuum, that is 1 or greater. To explore this question, we turn to quantum mechanics. back

Quantum mechanics

Since its birth in 1900, quantum mechanics has enjoyed a reputation for profound obscurity. The clouds began to part in the eighties, when Feynman , Deutsch and others began to interpret the quantum formalism in terms of communication and computation. Feynman 1, Deutsch

The foundation of quantum mechanics, first seen by Planck, is that everything that we see in the Universe is a discrete (quantum) event. Here we suppose that there are many formally different quanta of action as there are Turing machines. This observation connects the logical model to the physical Universe of experience.

Why is all communication quantized and countable? The answer to this question may lie in the theory of communication. Assume that stable structures in this Universe, like ourselves and atoms, are maintained by some sort of dynamic control. Assume further that effective control requires a sufficiently low error rate in communication. The theory of communication discovered by Shannon tells us that we can avoid error if the messages we use to communicate are so far apart (in some abstract space) that the probability of their being confused approaches zero. Error free communication requires quantization. Shannon

Shannon's scheme for the defeat of error also introduces delay into communications, since the encoder must wait until the source has emitted a certain number of symbols before it can encode them into an error resistant packet. Since the velocity of light is the maximum attainable, we might assume that the code used to transmit information by massless particles is the shortest possible.

Quantum mechanics began as 'wave mechanics' and gave rise to talk of 'particle-wave duality'. This duality is epistemologically asymmetrical. We observe particles. We postulate waves. By waves we mean periodic functions, traditionally continuous functions like sines, cosines and complex exponentials. There is also digital periodicity, as the name recursive function theory applied to the theory of computation suggests. Much of the power of a digital computer arises from its ability to repeat simple cycles of operations very quickly.

The waves are part of the abstract model, and provide a means of visualizing quantum processes. No-one can deny the utility of wave models as a bridge between the classical notion of continuum and the particulate nature of experience, but the classical continuous formalism of wave mechanics may not have the power to tell the whole story. Brandt

Quantum mechanics is a natural fit to the symmetric Universe because the function space, Hilbert space, which it inhabits is a direct descendant of the Cantor Universe. Quantum mechanics represents states of a physical system by functions or vectors in spaces of finite, countably infinite or transfinite dimensions. Observations or measurements of a physical system are represented by operators in this space. The laws of quantum mechanics are expressed as constraints on these vectors and operators. von Neumann

In the century of its existence, quantum mechanics has shown itself to be perfect, as far as it goes. For some, particularly Einstein, this is not far enough. Quantum mechanics is incomplete in that it does not predict the precise outcome of individual events, but it does predict (to apparently unlimited precision) both the 'stationary states' that represent permanent structures in the physical world, and the probability of transitions between various pairs of stationary states.

This probabilistic interpretation is reflected in quantum algebra. All vectors are normalized to 1, which insures (in the Born interpretation of the formalism) that the sum of the probabilities of the different possible outcomes of a particular quantum history is 1. This feature makes a source of quantum events formally identical to an information source as defined in the mathematical theory of communication. The alphabet of the source is the set of eigenvalues of the operator used to observe it. Khinchin

The Hilbert space required to represent the joint state of two particles is the tensor product of the Hilbert spaces representing the individuals. The state space of a quantum system thus grows exponentially with the number of particles involved. From this we conclude that the cardinal of the joint state of 0 particles is 1, and so on. This rate of growth allows a bijective mapping between the Cantor Universe and quantum mechanics.

An important feature of the construction of joint states is entanglement. The states of entangled particles are correlated in a non-classical manner which enables information about the state of one particle to be inferred from measurements of the state of another. The particles in entangled states are thus 'logically bound' to one another and can be used to transmit information about quantum states by 'teleportation'.Nielsen

The quantum states are not visible, but part of the model. In this respect quantum mechanics resembles all other explanations of the world which postulate a hidden process to explain a visible one. What we see are events, modelled as the operations that transform one state into another.

Let us think of the four-space of general relativity as the physical user interface of the Universe. All events are observed in this space. Behind this user interface is the process which gives meaning to the observed structure. In a computer the pixels on the screen are controlled by software and user input to give an image whose behaviour is explained by the software.

The software of the universal process is described by quantum mechanics. In quantum mechanics, one looks with an operator, an observable. What one sees after repeated observations is the frequency with which the observed system makes transitions between various pairs of eigenstates of the observable. From this point of view, quantum mechanics is equivalent to network traffic analysis, predicting, for instance, how often an atom will emit or absorb a photon in a given environment.

In more anthropomorphic (human friendly) terms, to observe is to ask a question. To ask a question is to set up a frame of reference (the 'eigenvectors of the question') which the respondent may use to frame an answer meaningful to the questioner. back

Creation

The union of quantum mechanics and relativity is quantum field theory. One consequence of the special relativity is that mass and energy are equivalent. Physical particles may be created and annihilated provided that energy, momentum and action are conserved. Quantum field theory sees every event as the coupled creation and annihilation of particles, and computes the frequency of these events. Zee

The most successful quantum field theory is quantum electrodynamics, described by Feynman whose approach is called the path integral method. One integrates the action along every possible path from a to b. Action is measured by phase, a number readily available from the formalism. The most probable path is that whose action is stationary: phases in the vicinity of this path are relatively constant and so add to give a large probability amplitude for the path. Feynman 2

A logical version of this scenario may be constructed using the ancient principle of bonum ex integro ('good comes form the whole'). An engine will not run (or anything else happen for that matter) unless all systems are go. Let us suppose that the inner product ('overlap integral') of quantum mechanics tells us how often a logical continuum forms between a and b. In logical terms the transition from a to b only happens when it is proven. This is equivalent to 100% overlap between input and output, that is logical contact. Each of these contacts is modelled by a Turing machine in the computable manifold.

We assume that we what observe is computable. The frequency with which various computable functions are executed thus corresponds to the frequencies of certain events. In neural network terms, we may think of Turing machines or computable functions as synapses in the cosmic nervous system. Synapse - Wikipedia

Much difficulty in quantum field theory comes from the infinities arising from division by zero in the continuous formalism. These infinities, and the large (unobserved) energy of the vacuum predicted by quantum field theory, might be overcome by using the countable formalism of the computable network. We are only required to explain the phenomena, not the vagaries of inappropriate mathematical models.

A Turing machine is a deterministic entity, implementing predicate calculus which was shown by Gödel to be completeThe principle of requisite variety in cybernetics tells us that simple systems cannot control more complex ones. Every quantum event has a history (or as a physicist would say, a preparation). This preparation is effected by communicating with the system to be studied. If that system is more complex than the preparation, the outcome will not be determined. The relationship between relatively finite communication and relatively infinite possibility may thus explain the incompleteness in quantum mechanics. Gödel, Ashby

Although Einstein felt the incompleteness of quantum mechanics to be a defect, it is seen here as a clear manifestation of the openness of the Universe. It also suggests that no local manifestation of God is capable of intelligent design, that is deterministic progress into the future along a preordained path. It is always at risk of 'decoherence' arising from communications with its environment. back

Evolution

Quantum mechanics and relativity describe the basic network structure of the Universe, and can be applied a any level of complexity, but what about the detail, the actual messages that pass over the network?

It is of the essence of a network that it be able to convey any possible message. In practical terms, all possible (ie 'well formed') strings are transmissible. The designation 'well formed' is relative to a protocol or grammar which partitions the full set of permutations of an alphabet into grammatical and ungrammatical. How is this partition implemented? The answer seems to be evolution by natural selection. This occurs because there are more possibilities in most situations than can be expressed in physical messages.

Modern solutions of Einstein's equation coupled with astronomical observations give unequivocal support for the big bang hypothesis, that in the last thirteen billion years, the Universe has grown from an initial singularity to its current size.

The metaphysical properties of the initial singularity were analyzed by ancient theologians. like Aquinas. During its growth, the Universe has remained forever one because there is nothing outside to fragment it. From a formal point of view we might ascribe this expansion to Cantor's theorem: for every set there is set of greater cardinal number. A countable string of actions may be permuted to give an uncountable variety of such strings. Cantor's proof is non-constructive: it is held to be formally true because its negation implies a formal contradiction.

If the hypotheses of the proof are fulfilled, we must expect the conclusion to follow. God, communicating with itself using distinct countable symbols, can, by permutation, create uncountably infinite structures. Since this happens at all scales, we too are part of the creation and communication process. We model this as a network, because looked at as a whole, a network communicates with itself.

We thus have a force for unity and a force for diversity. Between them, these forces exert selective pressure on the Turing machines that inhabit the computable network. In the competition for limited computing resources, inefficient algorithms and strings of algorithms are selected against.

We may gain further insight into the evolutionary process by considering the 'P—NP' question, which explores the relationship between foresight and hindsight. Turing established that there are truths inaccessible to finitely defined deterministic processes. We take this to mean that in certain circumstances the future cannot be determined by any process executing in 'polynomial time', a member of the class P. Many steps into the future must therefore be classified as NP, requiring 'exponential time' or chance to find a solution, which may then be checked in polynomial time. Applying this to evolution, we see natural selection as a P process which tests the solutions reached by indeterminate processes classifies them as fit or unfit. back

Conclusion

The beautiful fit between the large scale structure of the Universe and Einstein's general theory of relativity and between quantum mechanics and the detailed structure of the Universe are examples of the 'unreasonable effectiveness of mathematics in the natural sciences' noted by Wigner .Wigner

The effectiveness of mathematics seems more reasonable if we note that both mathematics and the observed Universe are symbolic systems. An event, as understood by quantum mechanics, is a symbol, that is a definite and separate addressable thing which may be used with a suitable code to convey meaning, like these letters. Logically, there is no difference between the way elementary particles talk and people talk. The conversation of elementary particles is the physical layer of human conversation.

The 'religions of the book' hold that God communicated with us once for all. The source of this notion would seem to be epistemological ideas attributed to Parmenides We can only have reliable communication with (ie truly know) our environment if it does not change. Fundamental reality must therefore be, eternal or rigid. Since God does not change, there is no need for us to receive updates from God. Burnet: Parmenides

Here we take a dynamic view. We are always communicating with God. Both we and God can change and still maintain true communication as long as our frequency of interaction with God matches God's frequency of change.

Uncontrolled power tends to corrupt communication because it can prevent negative reviews of its actions. Traditionally, God is so great that He judges the powerful and the weak with complete indifference. Historically, dictators have overcome this problem by taking control of God, claiming to rule by divine right.

We can reassert the power of God by recognizing that it exists independently of any subset of itself, and is open to observation and verification by all. Given a scientific theology, we can sift spurious claims about God from genuine ones, and so place the governance of our societies on a sound realistic footing.

The best feature of this model, to my mind, is its community. It sees a Universe structured by conversation. The totality of this conversation is measured by the space-time size of the Universe, and its local intensity by local energy density.

Conversation is natural to us. We know its properties intuitively, The network paradigm allows us to transform the subject of our inquiry into our own frame of reference and then ask how we would behave in the circumstances. With this model, we can use our personal experience to explore all aspects of the life of God. back

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Further reading

Books

Click on the "Amazon" link below each book entry to see details of a book (and possibly buy it!)

Aristotle, and P H Wickstead and F M Cornford, translators, Physics books V-VIII, Harvard University Press,William Heinemann 1980 Introduction: 'Simplicius tells us that Books I - IV of the Physics were referred to as the books Concerning the Principles, while Books V - VIII were called On Movement. The earlier books have, in fact, defined the things which are subject to movement (the contents of the physical world) and analyzed certain concepts - Time, Place and so forth - which are involved in the occurrence of movement.' Book V is a further introduction to the detailed analysis in Books VI - VIII. Book VI deals with continuity, Book VII is an introductory study for Book VIII, which brings us to the conclusion that all change and motion in the universe are ultimately caused by a Prime Mover which is itself unchanging and unmoved and which has neither magnitude nor parts, but is spiritual and not in space.' 
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Aristotle 1, and H Tedennick (translator), Metaphysics I-IX , Harvard University Press, William Heinemann 1980 Introduction: "[Aristotle] felt that there must be a regular system of sciences, each concerned with a different aspect of reality. At the same time it was only reasonable to suppose that there was a supreme science which was more ultimate, more exact, more truly Wisdom than the others. The discussion of ths science - Wisdom, Primary Philosophy or Theology, as it is variously called - and of its scope, forms the subject of the Metaphysics' page xxv. 
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Ashby, W Ross, An Introduction to Cybernetics, Methuen 1964 'This book is intended to provide [an introduction to cybernetics]. It starts from common-place and well understood concepts, and proceeds step by step to show how these concepts can be made exact, and how they can be developed until they lead into such subjects as feedback, stability, regulation, ultrastability, information, coding, noise and other cybernetic topics' 
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Brandt, Siegmund, and Hans Dieter Dahmen, The Picture Book of Quantum Mechanics, Springer-Verlag 1995 Jacket: 'This book is an introduction to the basic concepts and phenomena of quantum mechanics. Computer-generated illustrations are used extensively throughout the text, helping to establish the relation between quantum mechanics on one side and classical physics ... on the other side. Even more by studying the pictures in parallel with the text, readers develop an intuition for notoriously abstract quantum phenomena ...' 
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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.' 
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Cohen, Paul J, Set Theory and the Continuum Hypothesis, Benjamin/Cummings 1966-1980 Preface: 'The notes that follow are based on a course given at Harvard University, Spring 1965. The main objective was to give the proof of the independence of the continuum hypothesis [from the Zermelo-Fraenkel axioms for set theory with the axiom of choice included]. To keep the course as self contained as possible we included background materials in logic and axiomatic set theory as well as an account of Gödel's proof of the consistency of the continuum hypothesis. . . .'  
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Dauben, Joseph Warren, Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton University Press 1990 Jacket: 'One of the greatest revolutions in mathematics occurred when Georg Cantor (1843-1918) promulgated his theory of transfinite sets. ... Set theory has been widely adopted in mathematics and philosophy, but the controversy surrounding it at the turn of the century remains of great interest. Cantor's own faith in his theory was partly theological. His religious beliefs led him to expect paradox in any concept of the infinite, and he always retained his belief in the utter veracity of transfinite set theory. Later in his life, he was troubled by attacks of severe depression. Dauben shows that these played an integral part in his understanding and defense of set theory.' 
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Einstein, Albert, and Robert W Lawson (translator) Roger Penrose (Introduction), Robert Geroch (Commentary), David C Cassidy (Historical Essay) , Relativity: The Special and General Theory, Pi Press 2005 Preface: 'The present book is intended, as far as possible, to give an exact insight into the theory of relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. ... The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated.' page 3  
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Hofstadter, Douglas R, Goedel Escher Bach: An Eternal Golden Braid, Basic/Harvester 1979 An illustrated essay on the philosophy of mathematics. Formal systems, recursion, self reference and meaning explored with a dazzling array of examples in music, dialogue, text and graphics. 
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Jammer, Max, Concepts of Space: The History of Theories of Space in Physics, Dover 1994 Jacket: 'Although the concept of space is of fundamental importance in both physics and philosophy, until the publication of this book, the idea of space had never been treated in terms of its historical development. ... Following an introductory chapter on the concept of space in antiguity, subsequent chapters consider Judeaeo-Christian ideas about space, the emancipation of the space concept from Aristotelianism, Newton's concept of absolute space and the concept of space from the 18th century to the present. ... It is essential reading for philosphers, physicists and mathematicians, but even the nonprofessional reader will find it accessible, for the author has kept the technical language and mathematical details to a minimum.' 
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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.' 
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Kelly, J N D, Early Christian Creeds, Longman 1972 Jacket: "Dr Kelly's famous book - a comprehensive study of the rise, development and use of creadal formularies in the creative centuries of the Church's history - was immediately acclaimed ... as the standard work on the subject. "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.' 
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Mendelson, Elliott, Introduction to Mathematical Logic, van Nostrand 1987 Preface: '... a compact introduction to some of the principal topics of mathematical logic. . . . In the belief that beginners should be exposed to the most natural and easiest proofs, free swinging set-theoretical methods have been used."  
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Newton, Isaac, and Julia Budenz, I. Bernard Cohen, Anne Whitman (Translators), The Principia : Mathematical Principles of Natural Philosophy, University of California Press 1999 This completely new translation, the first in 270 years, is based on the third (1726) edition, the final revised version approved by Newton; it includes extracts from the earlier editions, corrects errors found in earlier versions, and replaces archaic English with contemporary prose and up-to-date mathematical forms. ... The illuminating Guide to the Principia by I. Bernard Cohen, along with his and Anne Whitman's translation, will make this preeminent work truly accessible for today's scientists, scholars, and students. 
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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. 
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Pais, Abraham, 'Subtle is the Lord...': The Science and Life of Albert Einstein, Oxford UP 1982 Jacket: In this ... major work Abraham Pais, himself an eminent physicist who worked alongside Einstein in the post-war years, traces the development of Einstein's entire ouvre. ... Running through the book is a completely non-scientific biography ... including many letters which appear in English for the first time, as well as other information not published before.' 
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Shannon, Claude, and Warren Weaver, The Mathematical Theory of Communication, University of Illinois Press 1949 'Before this there was no universal way of measuring the complexities of messages or the capabilities of circuits to transmit them. Shannon gave us a mathematical way . . . invaluable . . . to scientists and engineers the world over.' Scientific American 
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Tanenbaum, Andrew S, Computer Networks, Prentice 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.'  
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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 JvN regards as the definitive form of quantum mechanics. . . . Regarded as a tour de force at the time of its publication, this book is still indispensable for those interested in the fundamental issues of quantum mechanics.' 
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Wigner, Eugene, Symmetries and Reflections: Scientific Essays , MIT Press 1970 Jacket: 'This volume contains some of Professor Wigner's more popular papers which, in their diversity of subject and clarity of style, reflect the author's deep analytical powers and the remarkable scope of his interests. Included are articles on the nature of physical symmetry, invariance and conservation principles, the structure of solid bodies and of the compound nucleus, the theory of nuclear fission, the effects of radiation on solids, and the epistemological problems of quantum mechanics. Other articles deal with the story of the first man-made nuclear chain reaction, the long term prospects of nuclear energy, the problems of Big Science, and the role of mathematics in the natural sciences. In addition, the book contains statements of Wigner's convictions and beliefs as well as memoirs of his friends Enrico Fermi and John von Neumann. Eugene P. Wigner is one of the architects of the atomic age. He worked with Enrco Fermi at the Metallurgical Laboratory of the University of Chicago at the beginning of the Manhattan Project, and he has gone on to receive the highest honours that science and his country can bestow, including the Nobel Prize for physics, the Max Planck Medal, the Enrico Fermi Award and the Atoms for Peace Award. '. 
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Zee, Anthony, Quantum Field Theory in a Nutshell, Princeton University Press 2003 Amazon book description: 'An esteemed researcher and acclaimed popular author takes up the challenge of providing a clear, relatively brief, and fully up-to-date introduction to one of the most vital but notoriously difficult subjects in theoretical physics. A quantum field theory text for the twenty-first century, this book makes the essential tool of modern theoretical physics available to any student who has completed a course on quantum mechanics and is eager to go on. Quantum field theory was invented to deal simultaneously with special relativity and quantum mechanics, the two greatest discoveries of early twentieth-century physics, but it has become increasingly important to many areas of physics. These days, physicists turn to quantum field theory to describe a multitude of phenomena. Stressing critical ideas and insights, Zee uses numerous examples to lead students to a true conceptual understanding of quantum field theory--what it means and what it can do. He covers an unusually diverse range of topics, including various contemporary developments,while guiding readers through thoughtfully designed problems. In contrast to previous texts, Zee incorporates gravity from the outset and discusses the innovative use of quantum field theory in modern condensed matter theory. Without a solid understanding of quantum field theory, no student can claim to have mastered contemporary theoretical physics. Offering a remarkably accessible conceptual introduction, this text will be widely welcomed and used.  
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Papers
Deutsch, David, "Quantum theory, the Church-Turing principle and the universal quantum computer", Proceedings of the Royal Society of London, , A 400, 1985, page 97-117. 'It is argued that underlying the Church-Turing hypothesis there is an implicit physical assertion. Here this assertion is presented explicitly as a physical principle: 'every finitely realizible physical system can be perfectly simulated by a universal model computing machine operating by finite means'.'. back
Einstein 1, Albert, "Die Feldgleichungun der Gravitation", Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin, , , 25 November 1915, page 844-847. back
Feynman 1, Richard P, "Simulating Physics with Computers", International Journal of Theoretical Physics, 21, 6/7, 1982, page 467. back
Goedel, Kurt, "On the completeness of the calculus of logic", in Solomon Fefferman et al (eds), Kurt Goedel: Collected Works Volume 1 Publications 1929-1936, New York, OUP 1986, , , , , page 61-101. '1. Introduction The main object of the following is the proof of the completeness of the axiom system for what is clled the restricted functional calculus, namely the system given in Whitehead and Russel 1910 part 1 *1 ns *10 ... Here 'completeness' is to mean that every valid formula expressible in the restricted functional calculus ... can be derived from the axioms by means of a finite sequence of formal inferences. ...'. back
Landauer, Rolf, "Information is a physical entity", Physica A, 263, 1, 1 February 1999, page 63-7. 'This paper, associated with a broader conference talk on the fundamental physical limits of information handling, emphasizes the aspects still least appreciated. Information is not an abstract entity but exists only through a physical representation, thus tying it to all the restrictions and possibilities of our real physical universe. The mathematician's vision of an unlimited sequence of totally reliable operations is unlikely to be implementable in this real universe. Speculative remarks about the possible impact of that, on the ultimate nature of the laws of physics are included.'. back
Links
Aquinas 13, Summa: I 2 3: Whether God exists?, I answer that the existence of God can be proved in five ways. The first and more manifest way is the argument from motion. . . . The second way is from the nature of the efficient cause. . . . The third way is taken from possibility and necessity . . . The fourth way is taken from the gradation to be found in things. . . . The fifth way is taken from the governance of the world. back
Aquinas 14, Summa: I 3 1: Is God a body? , 'I answer that, It is absolutely true that God is not a body; and this can be shown in three ways. First, because no body is in motion unless it be put in motion, as is evident from induction. Now it has been already proved (2, 3), that God is the First Mover, and is Himself unmoved. Therefore it is clear that God is not a body. .. .' back
Aquinas 140, The procession of the divine persons, 'As God is above all things, we should understand what is said of God, not according to the mode of the lowest creatures, namely bodies, but from the similitude of the highest creatures, the intellectual substances; while even the similitudes derived from these fall short in the representation of divine objects. Procession, therefore, is not to be understood from what it is in bodies, either according to local movement or by way of a cause proceeding forth to its exterior effect, as, for instance, like heat from the agent to the thing made hot. Rather it is to be understood by way of an intelligible emanation, for example, of the intelligible word which proceeds from the speaker, yet remains in him. In that sense the Catholic Faith understands procession as existing in God.' back
John Burnet, John Burnet's Early Greek Philosophy: chapter IV, Parmenides of Elea: 85: The Poem, back
Synapse - Wikipedia, Synapse - Wikipedia, the free encyclopedia, 'In the nervous system, a synapse is a structure that permits a neuron (or nerve cell) to pass an electrical or chemical signal to another cell (neural or otherwise). Santiago Ramón y Cajal proposed that neurons are not continuous throughout the body, yet still communicate with each other, an idea known as the neuron doctrine The word "synapse" (from Greek synapsis "conjunction," from synaptein "to clasp," from syn- "together" and haptein "to fasten") was introduced in 1897 by English physiologist Michael Foster at the suggestion of English classical scholar Arthur Woollgar Verrall.' back

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