vol III Development:
Chapter 2: Model
Introduction
page 1: God
page 2: Fixed points
page 3: Immensity
page 4: Logical continuity
page 5: Computation
page 6: Invisibility
page 7 A transfinite computer network
Introduction
Here we set out to develop a model to help us understand the fixed points in the divine and plan successful action. The ancient religions teach that a successful life is the result of pleasing an invisible God. Locally, this God is often an abstract version of the reigning political power. Here, given our assumption that the Universe is divine, we seek a model which serves as a guide to all the physical and spiritual features of the the world.
page 1: God
A brief history of my personal God from ancient times until the present. My starting point is the traditional Christian model of God. My purpose is to develop and test a new model of God. The most important feature of this model is that it places us inside rather than outside God. If the Universe is divine, every experience in life is an revelation of God.
page 2: Fixed points
The God of the Hebrews and Christians is a living God. Aquinas, following Aristotle, sees it as pure activity, actus purus. Aquinas, following ancient mystical traditions, sees God as absolutely simple omnino simplex with no structure that the human mind can grasp. In particular, this God is utterly unlike our complex world. Mathematical fixed point theory, however, provides a way for us to understand fixed points of our world as part of the divine dynamics, inside rather than outside it.
page 3: Immensity
All agree that God is big. Physics has already taught us that natural language is too small to describe the Universe, and must be augmented with mathematics. To model the fixed point in God, therefore, we look for the biggest available mathematical structures. Here our starting point is the Cantor Universe, the space of transfinite numbers discovered by Georg Cantor in his efforts to
understand the relationship between continuous and discrete quantities.
page 4: Logical continuity
We distinguish two types of continuity, the physical continuity suggested by smooth motion through space, and the continuity of logical argument. We suggest that the Universe of logically consistent functions is much bigger than the Universe of continuous functions, and so more appropriate
for modelling the whole.
page 5: Computation
Hilbert thought there was no limit to the possibilities of formal mathematics. Gödel and Turing showed that this was not so. Regions of completeness and
computability in mathematics are relatively tiny. Computability is a
scarce and valuable resource in the mathematical realm.
page 6: Invisibility
A computer cannot both compute and explain its every move, since every explanation is also a computation, which will require explanation, and so on. The process can never halt. This is consistent with the idea that we cannot see all the processing that goes into the events we see. When we communicate over the internet, for instance, a large amount of invisible or transparent processing goes on in the communication link which we do not see.
page 7 A transfinite computer network
A communication network can model a permutation group. We interpret a Cantor Universe populated with Turing machines as a network, whose layers are measured by the transfinite numbers. The hardware level of this network, represented
by the natural numbers, has a high degree of symmetry, and is studied
by arithmetic and physics. Higher levels, which may represent things
like bacteria or politics, are much more complex, but still exhibit
useful symmetries which may be traced to the theory of communication
and the structure of the network.
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Copyright:
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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!)
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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Darwin, Charles, and Greg Suriano (editor), The Origin of Species, Gramercy 1998 Introduction: 'In considering the Origin of Species, it is quite conceivable that a naturalist, reflecting on the mutual affinities of organic beings, on their embryological relations, their geographical distribution, geological succession, and other such facts, might come to the conclusion that each species has not been independently created, but has descended, like varieties, from other species.'
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Lo, Hoi-Kwong, and Tim Spiller, Sandra Popescu, Introduction to Quantum Computation and Information, World Scientific 1998 Jacket: 'This book provides a pedagogical introduction to the subjects of quantum information and computation. Topics include non-locality of quantum mechanics, quantum computation, quantum cryptography, quantum error correction, fault tolerant quantum computation, as well as some experimental aspects of quantum computation and quantum cryptography. A knowledge of basic quantum mechanics is assumed.'
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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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Smart, Ninian, The World's Religions, Cambridge University Press 1992 Introduction: 'In undertaking a voyage into the world's religions, we should not define religion too narrowly. It is important for us to recognise secular ideologies as part of the story of human worldviews. ... Essentially this book is a history of ideas and practices that have moved human beings.'
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Links
| Brouwer fixed point theorem - Wikipedia, Brouwer fixed point theorem - Wikipedia, the free encyclopedia, 'Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f(x0) = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to itself. A more general form is for continuous functions from a convex compact subset K of Euclidean space to itself. back |
| Rolf Landauer, Information is a Physical Entity, 'Abstract: 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 |
| Thomas Aquinas, Summa Theologica 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. . . . ' back |
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