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A transfinite computer network
Introduction
The Cantor universe described on page 2 is an enormous static structure, a transfinite array of sets (= named pigeonholes). We surmise that Cantor universe will prove useful for mapping the structure that guides our universe of experience.
We need to model function as well as structure. To do this, we use the formal equivalent of an agent or actor, a computer. We have imagined the universe as a tree, or set of trees. We may imagine the computers as the birds singing to one another in these trees.
Computer
An imaginative and logical paper model of a computer was written by Alan Turing in 1937. Turing, Davis. Others had similar ideas about the same time. Church, Post. A 'Turing machine' is a deterministic automaton which follows a certain written algorithm to convert a certain text (the initial state) into another text (the final state). Such a machine is capable of doing anything which many people would consider to be a computation.
A Network
Following Cantor's 'principle of finitism' we use the ubiquitous finite computer networks in our world as a model for a transfinite computer network. Hallett page 7
A computer network comprises a hardware layer and a series of software layers culminating in the user layer. Tanenbaum, About Inc The hardware layer comprises the physical elements of the network, wires, optic fibres, integrated circuits, disk drives and so on.
The software layers transform the physical input into a form acceptable to the user, and conversely, convert the user's input into a form suitable for transmission over the physical link. We may think of each layer as the user of the layer below it, the lower layers providing services to the layers above them.
Since the physical layer of most networks is subject to a certain amount of error, the first software layer is general assigned to error detection and correction, so that the layers above it receive only correct data.
The transmission of a message is error free if the receiver receives exactly string of symbols sent by the transmitter, The mathematical theory of communication was first developed by Shannon and has been improved ever since. The basic idea is that by utilizing the spare entropy to be found in almost all messages, one can make the message space larger and so distribute possible messages further apart. Being further apart, they are much less likely to be confused with one another. The result is that all messages are discrete points in a space, like the natural numbers.
To construct the transfinite network, we model the physical layer with the natural numbers and the software layers with the increasing hierarchy of transfinite numbers.
Network computers
The execution of a Turing machine is a deterministic process leading without interruption from some computable initial state to the final state that is logically implied by the initial state. There are aleph(0) such processes, which we may consider to be the alphabet of network computing.
If we arrange that the outcome of one Turing machine may be the input of another, we may create strings of deterministic computations Which may lead from a given input through a string of deterministic processes to a certain output.
Turing machines are recursive. We can construct a complex Turing using a string of Turing machines which operate as subroutines of the master process. Turing himself used subroutines to build a Turing machine complex enough to prove incomputability. Such systems fulfill the definition of a Turing machine as long as they re deterministic.
In addition to deterministic machines, Turing envisaged oracle machines or o-machines which proceeded with a computation to a given point and then halted to wait for external input which determines which of a set of possible next steps to choose.
We can imagine an o-machine as a concatenation of two ordinary Turing machines whose ordering is determined by an outside source, the oracle. Such machines can be connected into a designed so that the machines all act as oracles for one another through commnication.
How many different states can we find in such a network? Each message can direct an oracle machine to choose between a minimum of two and aleph(0) possible next steps. Communication, therefore, increases the possible number of different computational paths from one permutation to the next from aleph(0) to somewhere between 2aleph(0) and aleph(0)aleph(0), both of which are equal to aleph(1) (a peculiarity of transfinite arithmetic Ordinal arithmetic - Wikipedia).
Now each of these aleph(1) computational paths (or processes) may may send an instruction to another process. Since we have aleph(1) processes, this gives us aleph(1)aleph(1) = aleph(2) different permutations of the computational paths.
The networking process both introduces a network of causal linkages between our computers, and also guarantees that aleph(n) computers each performing a computation with a variety of aleph(n) can communicate to create aleph(n+1) processes.
What we are thinking is that by making the Turing machines communicate, we both multiply the computations and bind the system together at the same time. Our computer network grows in the same way as the Cantor universe grows, permutations feeding off permutations.
This model resembles the ancient concept of mental growth, the recursive creation of more complex processes. Hussey, 350. The beauty of the present picture is that it may be the beginning of a road to mathematical rigour in our understanding of our environment, particularly the ancient questions of the relationship between mind and matter, or more abstractly, between physics and theology.
Further reading
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 |
| Church, Alonzo, Introduction to Mathematical Logic, Princeton UP 1996 Jacket: 'One of the pioneers of mathematical logic in the twentieth century was Alobzo Church, He introduced such concepts as the lambda calculus, now an essential tool of computer science, and was the founder of the Journal of Symbolic Logic. In Introduction to Mathematical Logic, Church presents a masterful overview of the subject - one which should be read by every researcher and student of logic.' Amazon back |
| 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.' Amazon back |
| Hallett, Michael, Cantorian set theory and limitation of size, Oxford UP 1984 Jacket: 'This book will be of use to a wide audience, from beginning students of set theory (who can gain from it a sense of how the subject reached its present form), to mathematical set theorists (who will find an expert guide to the early literature), and for anyone concerned with the philosophy of mathematics (who will be interested by the extensive and perceptive discussion of the set concept).' Daniel Isaacson. Amazon back |
| Hussey, E L, "Parmenides" in Ted Honderich (editor) The Oxford Companion to Philosophy, Oxford University Press 1995 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 |
| Zemanian, Armen H, Transfiniteness for Graphs, Electrical Newtorks and Random Walks, Springer Verlag 1996 'A substantial introduction is followed by chapters covering transfinite graphs; connectedness problems; finitely structured transfinite graphs; transfinite electrical networks; permissively finitely structured networks; and a theory for random walks on a finitely structured transfinite network. Appendices present brief surveys of ordinal and cardinal numbers; summable series; and irreducible and reversible Markov chains. Accessible to those familiar with basic ideas about graphs, Hilbert spaces, and resistive electrical networks. (Annotation copyright Book News, Inc. Portland, Or.)' Amazon back |
| Zemanian, Armen H, Graphs and Networks: Transfinite and Nonstandard, Birkhäuser 2004 Amazon editorial review :'For about thirty years Zemanian has been developing a theory of infinite electrical networks. This book is the latest in a series of books...on the subject. The subject is necessarily abstract and sophisticated because infinite objects are the main objects of discourse.... The first few chapters are important not only to remind the reader of the terms, but also to give an improved or alternate treatment of some earlier results. There does not yet seem to be a large following of researchers in this area, but it seems very attractive and ripe for investigation. Its intriguing to see the connections between set theory and electrical network problems.... To understand these concepts fully the reader must consult the book under review. The reviewer highly recommends devoting the effort needed to understand these original and surprising concepts.' SIAM Review Amazon back |
Papers
| Turing, Alan, "On Computable Numbers, with an application to the Entscheidungsproblem", Proceedings of the London Mathematical Society, 2, 42, 12 November 1937, page 230-265. 'The "computable" numbers maybe described briefly as the real numbers whose expressions as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost as easy to define and investigate computable functions of an integrable 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. I hope shortly to give an account of the rewlations of the computable numbers, functions and so forth to one another. This will include a development of the theory of functions of a real variable expressed in terms of computable numbers. According to my definition, a number is computable if its decimal can be written down by a machine'. back |
| Shannon, Claude E, "The mathematical theory of communication", Bell System Technical Journal, 27, , July and October, 1948, page 379-423, 623-656. 'A Note on the Edition Claude Shannon's ``A mathematical theory of communication'' was first published in two parts in the July and October 1948 editions of the Bell System Technical Journal [1]. The paper has appeared in a number of republications since: o The original 1948 version was reproduced in the collection Key Papers in the Development of Information Theory [2]. The paper also appears in Claude Elwood Shannon: Collected Papers [3]. The text of the latter is a reproduction from the Bell Telephone System Technical Publications, a series of monographs by engineers and scientists of the Bell System published in the BSTJ and elsewhere. This version has correct section numbering (the BSTJ version has two sections numbered 21), and as far as we can tell, this is the only difference from the BSTJ version. o Prefaced by Warren Weaver's introduction, ``Recent contributions to the mathematical theory of communication,'' the paper was included in The Mathematical Theory of Communication, published by the University of Illinois Press in 1949 [4]. The text in this book differs from the original mainly in the following points: o the title is changed to ``The mathematical theory of communication'' and some sections have new headings, o Appendix 4 is rewritten, o the references to unpublished material have been updated to refer to the published material. The text we present here is based on the BSTJ version with a number of corrections.. back |
| Post, E L, "Finite combinatory processes - formulation 1.", The Journal of Symbolic Logic, 1, 42, 1936, page 103-105. 'The present formulation should prove significant in the development of symbolic logic along the lines of Godel's theorem on the incompleteness of symbolic logics1 and Church's results concerning absolutely unsolvable problems.' We have in mind a general problem consisting of a class of specific problems. A solution of the general problem will then be one which furnishes an answer to each specific problem. In the following formulation of such a solution two concepts are involved: that of a symbol space in which the work leading from problem to answer is to be carried out,' and a fixed unalterable set of directions which will both direct operations in the symbol space and determine the order in which those directions are to be applied.'. back |
| Post, Emil L, "Recursively Enumerable Sets of Positive Integers and their Decision Problems ", Bulletin of the American Mathematical Society, 50, , February 26 1944, page 284-316.. back |
Links
| About Inc Networking Basics - Essential Concepts in Computer Networking 'Networking Basics - Key Concepts in Computer Networking Begin your study of computer networking basics by exploring these key concepts and essential technologies.' back |
| Alan Turing On Computable Numbers, with an application to the Entscheidungsproblem 'The "computable" numbers may be described briefly as the real numbers whose expressions 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 |
| Armen H Zemanian Research Reports on Transfinite Graphs and Networks back |
| Emil L Post Finite Combinaroty Processes - Formulation 1 'The present formulation should prove significant in the development of symbolic logic along the lines of Godel's theorem on the incompleteness of symbolic logics1 and Church's results concerning absolutely unsolvable problems.' We have in mind a general problem consisting of a class of specific problems. A solution of the general problem will then be one which furnishes an answer to each specific problem. In the following formulation of such a solution two concepts are involved: that of a symbol space in which the work leading from problem to answer is to be carried out,' and a fixed unalterable set of directions which will both direct operations in the symbol space and determine the order in which those directions are to be applied.' back |
| Mike Stannet Hypercomputation Research Network 'Online resources for download: All files listed here remain the intellectual property of their respective owners, and are made available here for personal research and classroom use only. ' back |
| Ordinal arithmetic - Wikipedia Ordinal arithmetic - Wikipedia, the free encyclopedia 'In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set which represents the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. The so-called "natural" arithmetical operations retain commutativity at the expense of continuity.' back |
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