volume II: Synopsis
Section IV: Divine Dynamics
page 30: The transfinite network
We can imagine any organisation as a filing system and a set of processes for updating the files. This businesslike idea is here expanded, using Cantor's theory of transfinite cardinal and ordinal numbers, into an infinite abstract structure we call the transfinite network. We assemble a selection of the mathematical, physical and philosophical ideas developed so far into a model large enough to talk about God, but with sufficient finesse to deal with every detail of the world, no matter how small.
We are looking for a language or space to describe God. By God we mean here the Universe as a whole. We can only imagine two constraints on the nature of God. The first is consistency. God must be consistent with itself. The second is size, not just in terms of distance and time, but in terms of complexity. The universe is very big, but a human body is an exquisite arrangement of some ten thousand trillion trillion atoms.
Cantor's theorem shows us that a consistent symbolic system as large as the natural numbers can grow without limit into the transfinite cardinal and ordinal numbers. This space, we postulate, is large enough to begin to model all the details of the Universe. Every quantum of action in it may be represented by a unique number, that is, a unique ordered set, or ordered set of ordered sets, of whatever symbols we choose to use to correspond to the natural numbers. Cantor's theorem - Wikipedia
Alan Turing envisaged two types of machine which he called a (for automatic) machines and c (for choice) machines. a-machines are deterministic, in the sense that once they are started they follow a definite course until they either halt (if their initial configuration is computable) or not (if the initial configuration is not computable). Alan Turing: On Computable Numbers, with an application to the Entscheidungsproblem
c-machines, on the other hand, go through a certain number of steps until they come to a point where the next step is indeterminate . They then an outside agent ('oracle') to choose their next move. In modern terms, we might call c-machines network machines. One would expect an c-machine to be more powerful than an a-machine since it has outside help. The machines populating the transfinite network are c-machines. Most real computers are c-machines. My computer spends most of its time working for BOINC, waiting for me to input another keystroke. I make the choices. At the fundamental level of embodied logical operators all computer are choice machines, since none do anything until they receive inputs from elsewhere. Andrew Tanenbaum: Computer Networks,, BOINC: Berkeley Open Infrastructure for Network Computing
Two ideas form the foundation for a universal computer network. The first is that we can build any classical computer out of a network of memories, connections and nand or Sheffer stroke logical operators. We understand these to be the atomic operations in computation, just as quanta of action are atomic operations in physics. Sheffer stroke - Wikipedia
The second is that there are a countably infinite number of different Turing machines, which means that we can establish a one-to-one correspondence between Turing machines and the natural numbers. Cantor established the existence of the transfinite network by showing that operations such as combinations and permutations on the set of natural numbers created bigger sets. We imagine performing similar operations on the computers corresponding to the natural numbers, connecting them together in different ways in order to build transfinite networks of computers. Collectively we can imagine the operations of all these computers being equinumerous with all the quanta of action in the lifetime of the universe, since there is no upper bound on the transfinite numbers.
Since the transfinite network is taken as a model of the whole Universe, every machine within it has, in principle, the ability to consult all the other machines, and can thus tap the power of the whole network. This is the first step in a recursive process, since we can imagine that the power of each machine grows as it learns what others have to teach it, so that a new generation of more powerful machines can emerge to begin another round of consultation. Networking increases the entropy and power of the information processing system. We imagine that this phenomenon is not unique to humans, but occurs at every level in the universal network.
Since there are only a countable number of different machines computable functions are a relatively scarce commodity in the transfinite network, and like the limitation of resources which causes natural selection in the biological sphere, we might imagine there is competition for computable functions in the transfinite network. Nevertheless, although there are only a countable number of computable functions, there does not appear to be any constraint on the number of instances of these functions spread over space and time. Natural selection - Wikipedia
Although the variety of computable functions is limited, these functions can nevertheless manipulate transfinite entities through the establishment of correspondences or meanings between these entities (eg people) and the natural numbers. This is the way bureaucracies, national and corporate, manipulate the people they control. All entities manipulate their environment to some extent to ensure their existence. Jeff Speaks (Stanford Encyclopedia of Philosophy): Theories of Meaning
The effective power of a machine depends on its place in the recursive hierarchy of the network. The transfinite net has an infinite hierarchy, and so allows for the existence of machines of unlimited power, governing larger and larger subsets of the network. This abstract network provides us a huge space for modelling reality. Like all networks, it is layered, its lowest physical layer comprising instances of the initial singularity. From this structureless entity, we imagine the whole Universe being created by the network process of copying and differentiation. Computer network - Wikipedia
(revised 5 April 2020)
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Further readingBooks
Beale, R, and T Jackson, Neural Computing: An Introduction, Adam Hilger 1991 Jacket: '. . . starts from basics and goes on to cover all the most important approaches to the subject. . . . The capabilities, advantages and disadvantages of each model are discussed as are possible applications of each. The relationship of the models developed to the brain and its functions are also explored.'
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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 1895, 1897, 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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Davis, Martin, Computability and Unsolvability, Dover 1982 Preface: 'This book is an introduction to the theory of computability and non-computability ususally referred to as the theory of recursive functions. The subject is concerned with the existence of purely mechanical procedures for solving problems. . . . The existence of absolutely unsolvable problems and the Goedel incompleteness theorem are among the results in the theory of computability that have philosophical significance.'
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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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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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Links
Alan Turing, On Computable Numbers, with an application to the Entscheidungsproblem, 'The “computable” numbers may be described briefly as the real numbers whose expression: s as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable or a real or computable variable, computable predicates, and so forth. The fundamental problems involved are, however, the same in each case, and I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique.' back |
BOINC, Berkeley Open Infrastructure for Network Computing, ' BOINC lets you help cutting-edge science research using your computer (Windows, Mac, Linux) or Android device. BOINC downloads scientific computing jobs to your computer and runs them invisibly in the background. It's easy and safe.' back |
Cantor's theorem - Wikipedia, Cantor's theorem - Wikipedia, the free encyclopedia, ' In elementary set theory, Cantor's theorem is a fundamental result which states that, for any set A, the set of all subsets of A (the power set of A, denoted by P(A) ) has a strictly greater cardinality than A itself. For finite sets, Cantor's theorem can be seen to be true by simple enumeration of the number of subsets. Counting the empty set as a subset, a set with n members has a total of 2n subsets, so that if card (A) = n, then card (P(A)) = 2 n , and the theorem holds because 2n > n for all non-negative integers. ' back |
Computer network - Wikipedia, Computer network - Wikipedia the free encyclopedia, 'A computer network, or simply a network, is a collection of computers and network hardware interconnected by communication channels that allow sharing of resources and information. . . . The best known computer network is the Internet. . . . Computer networking can be considered a branch of electrical engineering, telecommunications, computer science, information technology or computer engineering, since it relies upon the theoretical and practical application of the related disciplines.' back |
Jeff Speaks (Stanford Encyclopedia of Philosophy), Theories of Meaning , 'Here I focus on two sorts of “theory of meaning.” The first sort of theory—a semantic theory—is a theory which assigns semantic contents to expressions of a language. Approaches to semantics may be divided according to whether they assign propositions as the meanings of sentences and, if they do, what view they take of the nature of these propositions.
The second sort of theory—a foundational theory of meaning—is a theory which states the facts in virtue of which expressions have the semantic contents that they have. Approaches to the foundational theory of meaning may be divided into theories which do, and theories which do not, explain the meanings of expressions of a language used by a group in terms of the contents of the mental states of members of that group.' back |
Kurt Gödel - Wikipedia, Kurt Gödel - Wikipedia, the free encyclopedia, 'Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.' back |
Natural selection - Wikipedia, Natural selection - Wikipedia, the free encyclopedia, 'Natural selection is the differential survival and reproduction of individuals due to differences in phenotype; it is a key mechanism of evolution. The term "natural selection" was popularised by Charles Darwin, who intended it to be compared with artificial selection, now more commonly referred to as selective breeding. . . . Natural selection is one of the cornerstones of modern biology. The concept was published by Darwin and Alfred Russel Wallace in a joint presentation of papers in 1858, and set out in Darwin's influential 1859 book On the Origin of Species,[3] in which natural selection was described as analogous to artificial selection, a process by which animals and plants with traits considered desirable by human breeders are systematically favoured for reproduction.' back |
Sheffer stroke - Wikipedia, Sheffer stroke - Wikipedia, the free encyclopedia, 'In Boolean functions and propositional calculus, the Sheffer stroke, named after Henry M. Sheffer, written "|" . . . denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called nand ("not and") or the alternative denial, since it says in effect that at least one of its operands is false.' back |
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