vol III Development:
Chapter 2: Model
page 7: A transfinite computer network
Introduction
We are working on the hypothesis that the Universe is divine and that divinity, as understood by the ancients, is pure activity. Our problem is to reconcile this ancient model of God with the world of our experience.
The first step is made for us by the mathematical theory of fixed points, which tells us that logically, closed dynamic systems must have fixed points. These fixed points are not 'outside' the dynamics, as the ancients thought, but simply elements of the dynamics that do not move. We can see this relationship in all our engineered systems with moving parts. The fixed parts guide the moving parts. The desired motion of the moving parts guides engineers in their design of the fixed parts. Fixed point theorem - Wikipedia
The next step is to bring the vast hierarchical function space created by Cantor's set theory into correspondence with the fixed points of the Universe. This space is the 'paradise' in which mathematics exists, and because there is no limit to its transfinite size we are guaranteed to find a place to model any part of the Universe, no matter how complex the set of fixed points manifested by the local dynamics. Cantor's paradise - Wikipedia
The structure of mathematics is built by proof. Proofs connect different symbolic points in the mathematical structure to one another by logical argument. A proof is equivalent to a Turing machine, that is a computer or logical continuum leading deterministically from an initial state to a final state in which the computer halts. Turing machine - Wikipedia
As Alan Turing discovered, however, proofs have their limitations. The first transfinite number, ℵ0 is the cardinal of the set of natural numbers, N. The second transfinite number, ℵ1 is the cardinal of the set of permutations of N, so we can write ℵ1 = ℵ0!.
Permutation - Wikipedia
There are, however only a countable infinity, ℵ0, of Turing machines. so that the majority of the mappings of N onto itself are incomputable or unprovable. We wish to use computers to describe the dynamics of the Universe. How then can this relatively small set of computers be used to manipulate the vast array of transfinite numbers, and so imitate the workings of the Universe? The answers lies in meaning or correspondence.
Meaning
Cantor based his transfinite numbers on the notion of a set, which he defined when he wrote: By an "aggregate" [set] we are to understand any collection into a whole M of definite and separate objects of our intuition or our thought. These objects are called the "elements" of M. We can use the idea of a set to model natural objects. A rock, for instance, is a set of atoms bonded together into a whole. This may not be mathematically exact, because atoms are so small that at any moment a number of them will be joining or leaving the rock. Cantor, page 85
Cantor captured the notion of meaning in his set theory using the notion of 'correspondence'. Dictionaries, for instance, set up correspondences between individual words and extended definitions of these words, usually using other words from the same language. We count a mob of sheep by establishing a one to one correspondence between each sheep and a natural number and say that the number we reach when we have labelled each sheep just once is the cardinal of the mob of sheep.
We too may be modelled as a sets of atoms. Not so well defined, because we eat and breathe and sweat, etc, so we am more of a moving population than a rock. Nevertheless we are a relatively durable units with names. Even though we aare transfinitely complex, the conservation laws of the Universe dictate that there are only a countable number of systems like ourselves in the local Universe. We are all unique and can be labelled in a way that makes us amenable to computation. A process can be imagined (in a government office, for instance) where certain variables refer to certain people and calculations yield, for instance, our pension payments.
The relativity of transfinity
In the Platonic world of mathematics we can imagine and deal with the whole transfinite set of natural numbers. In reality, however, because information is physical and every unit must be represented by a physical object whose size will be at least as great as Plack's constant, real transfinite numbers cannot be Platonically large.
Instead, in a location where there are three distinct physical symbols, we may set the first transfinite number ℵ0 to 3. ℵ1 will then be 6 (3!), ℵ2 720 (6!) and so on.
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. The relativity of transfinity helps to make this process logically palatable. Hallett, page 7
A
computer network comprises a hardware layer and a series of software
layers culminating in the user layer. The hardware layer comprises the physical elements of the network, wires, optic fibres, integrated circuits, disk drives and so on. Tanenbaum, About Inc
The software layers transform the physical input into a form acceptable to the users, and conversely, convert the user's input into a form suitable for transmission over the physical links. We may think of each layer as the user of the layer below it, the lower layers providing more and more complex 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.
To construct the formal 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 ℵ0 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 are deterministic.
In addition to deterministic machines, Turing envisaged oracle
machines which proceed with a computation
to a given point and then halt to wait for external input from an 'oracle' which
determines which of a set of possible next steps to execute.
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
ℵ0 possible next steps. Communication, therefore,
increases the possible number of different computational paths from
one permutation to the next from ℵ0 to somewhere between
2ℵ0and ℵ0ℵ0, both of which are equal toℵ1 (a peculiarity of transfinite arithmetic Ordinal
arithmetic - Wikipedia).
Now each of these ℵ1 computational paths (or
processes) may may send an instruction to another process. Since we
have ℵ1 processes, this gives us ℵ1ℵ1 = ℵ2 different permutations of the computational paths.
The networking process both introduces a network of causal
linkages between our computers, and also guarantees that paths of
ℵn computers each performing a computation with a
variety of ℵn can communicate to create
ℵn+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 also seems to be the way imagination grows, beginning with one set of ideas and permuting them to get new pictures, a multi-dimensional kaleidoscope.
This model resembles the ancient concept
of mental growth, the recursive creation of more complex processes.
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. Hussey,
350.
(revised 27 February 2015)
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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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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.'
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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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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.
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Hussey, E L, " Heraclitus of Ephesus" in Ted Honderich The Oxford Companion to Philosophy, Oxford University Press 1995
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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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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.)'
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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
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Papers
| 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 |
| 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:
• 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.
• 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:
• the title is changed to ``The mathematical theory of communication'' and some sections have new headings,
• Appendix 4 is rewritten,
• 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 |
| 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 |
Links
| A Mathematical Theory of Communication - Wikipedia, A Mathematical Theory of Communication - Wikipedia, the free encyclopedia, 'The article entitled "A Mathematical Theory of Communication", published in 1948 by mathematician Claude E. Shannon, was one of the founding works of the field of information theory. Shannon's paper laid out the basic elements of any digital communication:
• An information source which produces a message
• A transmitter which operates on the message to create a signal which can be sent through a channel
• A channel, which is the medium over which the signal, carrying the information that composes the message, is sent
• A receiver, which transforms the signal back into the message intended for delivery
• A destination, which can be a person or a machine, for whom or which the message is intended
It also developed the concepts of information entropy and redundancy.' back |
| 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 |
| Cantor's paradise - Wikipedia, Cantor's paradise - Wikipedia, the fre encyclopedia, 'Cantor's paradise is an expression used by David Hilbert . . . in describing set theory and infinite cardinal numbers developed by Georg Cantor. The context of Hilbert's comment was his opposition to what he saw as L. E. J. Brouwer's reductive attempts to circumscribe what kind of mathematics is acceptable; see Brouwer–Hilbert controversy.' back |
| Emil L Post, Finite Combinatoru 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 |
| Fixed point theorem - Wikipedia, Fixed point theorem - Wikipedia, the free encyclopedia, 'In mathematics, a fixed point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. Results of this kind are amongst the most generally useful in mathematics.
The Banach fixed point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point.
By contrast, the Brouwer fixed point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma).' 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 |
| Permutation - Wikipedia, Permutation - Wikipedia, the free encyclopedia, 'In mathematics, the notion of permutation relates to the act of permuting, or rearranging, members of a set into a particular sequence or order (unlike combinations, which are selections that disregard order). For example, there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). As another example, an anagram of a word, all of whose letters are different, is a permutation of its letters. The study of permutations of finite sets is a topic in the field of combinatorics.' back |
| Set (mathematics) - Wikipedia, Set (mathematics) - Wikipedia, the free encyclopedia, 'In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived.' back |
| Sheffer stroke - Wikipedia, Sheffer stroke - Wikipedia, the fre encyclopedia, 'The Sheffer stroke, written "|" or "?", in the subject matter of boolean functions or propositional calculus, 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 the alternative denial, since it says in effect that at least one of its operands is false. In Boolean algebra and digital electronics it is known as the NAND operation ("not and").' back |
| Turing machine - Wikipedia, Turing machine - Wikipedia, the free encyclopedia, 'Turing machines are extremely basic abstract symbol-manipulating devices which, despite their simplicity, can be adapted to simulate the logic of any computer algorithm (as we understand them). They were described in 1936 by Alan Turing. Though they were intended to be technically feasible, Turing machines were not meant to be a practical computing technology, but a thought experiment about the limits of mechanical computation; thus they were not actually constructed. Studying their abstract properties yields many insights into computer science and complexity theory.' back |
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