Volume 1: About
The working hypothesis here is that
religion is the force that binds unrelated people into a peaceful
society. The plan is to apply the methods of science to the study of
religion, yielding scientific theology. Theology is the science
corresponding to religion, that is the science of the creation and
annihilation of human groups.
We begin by discussing the conditions for certain knowledge: Epistemology and then turn to the development of a hypothetical model of the Universe: Model.
We turn next to the bedrock of science,
which deals with relationships between the smallest particles and the
Universe as a whole. Physics confronts a set of problems that have
bedevilled European thought since Ancient Greece: how can the
Universe be both one and many? moving and still? certain and
uncertain? The modern physicists' answer to these questions is quantum field theory. This theory is remarkably successful, but has as yet failed to give a satisfactory account of gravitation. Quantum field theory - Wikipedia
The heart of the plan is to expand physical ideas from the world
of particles and forces to the human domain. This expansion relies on
the fact that many properties of the world are invariant with respect
to complexity. Subatomic particles are simple compared to vast
ordered sets of particles like a person, a planet or a galaxy. Yet
everything in the Universe has something in common: everything is a
part of the whole, interacting with other parts. The uniting element
Abstract thought is much assisted by
making models. The notion of symmetry with respect to complexity is
made concrete by constructing a model in which we can see it at work.
The playground we have chosen is the transfinite layered network of
communication described under the heading
model. The foundations of this model are the transfinite mathematical universe discovered by Georg Cantor at the end of the nineteenth century and the theory of computation pioneered by Alan Turing early in the twentieth century. Cantor, Alan Turing.
Both the model and the world become clearer when we fit them
together. We proceed step by step moving from physics to
To emphasize that model is bigger than any particular science we then
look at in the context of
From there we move to talk about the emergence of
matter. The final step in this speculative process is to use the
model as a foundation for
We then turn from pure science to applied science, that is art, technology or engineering. We begin with the art of Love and then add more and more detail working through Culture, Politics, Religion, Economics, Design and Work to the most concrete feature of life, Experience.
This plan is in effect the file structure of the site. The
dynamics are rather more sculptural. The site is evolving and will
never be finished, but it will probably converge to some ideal which
is not yet clear. So, apart from the historical texts, all is fluid.
This is one of the beauties of web publication.
All this may seem a roundabout way to approach an ancient
discipline like theology. The problem is that in the west, theology
has been dominated by Christian thought for nearly two thousand
years. There is very little non-Christian theological infrastructure
around. To build a new theology and a new religion, it is necessary
to build a new symbolic ecosystem for it to live in.
The output of religion is algorithms for living. Most of the
algorithms we study are already deeply entrenched in human societies,
but expressed in a multitude of different languages, customs and
views of the world. The hope here is to build a consistent and
transparent environment in which such algorithms can be expressed in
a common language that embraces all the activities of life.
Algorithms are static structures that
guide dynamic processes. Their advantage is that they are usually
more succinct and intelligible than the dynamics they describe. Christianity
compresses all the constraints on life into the simple phrase love
God, love your neighbour. Algorithm - WikipediaMark 12:30-31
The root of the peaceful life seems to reside in tolerance. How
far can tolerance go before the system breaks down? The
fundamentalist temperament tends to hold us to the letter of the law.
But when we look at the Universe as a whole, we see that formal
restrictions on behaviour operate with a very light rein. Nowhere is
this more obvious than in the realm of quantum mechanics, where a
given formal arrangement can lead to an infinity of different
The same lightness, we believe, can operate in human affairs. We
live in a Universe of divine possibilities. The only constraint on
our activities is that we do no harm. We must not destroy the
infrastructure (physical, biological and spiritual) upon which our
lives depend. With this proviso, the possibilities for human
development are literally transfinite.
(revised 7 August 2014)
You may copy this material freely provided only that you quote fairly and provide a link (or reference) to your source.
Click on the "Amazon" link below each book entry to see details of a book (and possibly buy it!)
|Cantor, Georg, Contributions to the FoundinCantorg 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.'
|Casti, John L, Five Golden Rules: Great Theories of 20th-Century Mathematics - and Why They Matter, John Wiley and Sons 1996 Preface: '[this book] is intended to tell the general reader about mathematics by showcasing five of the finest achievements of the mathematician's art in this [20th] century.' p ix. Treats the Minimax theorem (game theory), the Brouwer Fixed-Point theorem (topology), Morse's theorem (singularity theory), the Halting theorem (theory of computation) and the Simplex method (optimisation theory).
|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.'
|Galilei, Galileo, and Stillman Drake (translator), Discoveries and Opinions of Galileo: Including the Starry Messenger (1610 Letter to the Grand Duchess Christina), Doubleday Anchor 1957 Amazon: 'Although the introductory sections are a bit dated, this book contains some of the best translations available of Galileo's works in English. It includes a broad range of his theories (both those we recognize as "correct" and those in which he was "in error"). Both types indicate his creativity. The reproductions of his sketches of the moons of Jupiter (in "The Starry Messenger") are accurate enough to match to modern computer programs which show the positions of the moons for any date in history. The appendix with a chronological summary of Galileo's life is very useful in placing the readings in context.' A Reader.
|Goedel, Kurt, "On formally undecidable propositions of Principia Mathematica and related systems, I" in Solomon Fefferman et al (eds) Kurt Goedel: Collected Works Volume 1 Publications 1929-1936, Oxford UP 1986 Jacket: 'Kurt Goedel was the most outstanding logician of the twentieth century, famous for his work on the completeness of logic, the incompleteness of number theory and the consistency of the axiom of choice and the continuum hypotheses. ... The first volume of a comprehensive edition of Goedel's works, this book makes available for the first time in a single source all his publications from 1929 to 1936, including his dissertation. ...'
|Hulsman, Jef, Franz J Weissing, "Biodiversity of plankton by species oscillations and chaos", Nature, 402, 6760, 25 November 1999, page 407-410. Letters to Nature: 'In aquatic ecosystems the biodiversity puzzle is ... known as the 'paradox of the plankton'. Competition theory predicts that at equilibrium the number of coexisting species cannot exceed the number of limiting resources. ... Here we offer a solution to the plankton paradox. First, we show that resource competition models can generate oscillations and chaos when species compete for three or more resources. Second, we show that these oscillations and chaotic fluctuations in species abundances allow the coexistence of many species on a handful of resources.' . back |
|Nowak, Martin A, Joshua B Plotkin and Vincent A A Jansen, "The evolution of syntactic communication", Nature, 404, 6777, 30 March 2000, page 495-498. Letters to Nature: 'Animal communication is typically non-syntactic, which means that signals refer to whole situations. Human language is syntactic, and signals consist of discrete components that have their own meaning. Syntax is requisite for taking advantage of combinatorics, that is 'making infinite use of finite means'. ... Here we present a model for the population dynamics of language evolution, define the basic reproductive ratio of words and calculate the maximum size of a lexicon.'. 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 |
| 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 |
| Algorithm - Wikipedia, Algorithm - Wikipedia, the free encyclopedia, 'In mathematics, computing, linguistics, and related disciplines, an algorithm is a definite list of well-defined instructions for completing a task; that given an initial state, will proceed through a well-defined series of successive states, eventually terminating in an end-state.
The concept of an algorithm originated as a means of recording procedures for solving mathematical problems such as finding the common divisor of two numbers or multiplying two numbers. A partial formalization of the concept began with attempts to solve the Entscheidungsproblem (the "decision problem") that David Hilbert posed in 1928. Subsequent formalizations were framed as attempts to define "effective calculability" (cf Kleene 1943:274) or "effective method" (cf Rosser 1939:225); those formalizations included the Gödel-Herbrand-Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's "Formulation I" of 1936, and Alan Turing's Turing machines of 1936-7 and 1939.' back |
| 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 |
| Cantor's theorem - Wikipedia, Cantor's theorem - Wikipedia, the free encyclopedia, 'In elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A (the power set of A) has a strictly greater cardinality than A itself. For finite sets, Cantor's theorem can be seen to be true by a much simpler proof than that given below, since in addition to subsets of A with just one member, there are others as well, and since n < 2n for all natural numbers n. But the theorem is true of infinite sets as well. In particular, the power set of a countably infinite set is uncountably infinite. The theorem is named for German mathematician Georg Cantor, who first stated and proved it.' back |
| Claude E Shannon, A Mathematical Theory of Communication, 'The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages.' back |
| Claude Shannon - Wikipedia, Claude Shannon - Wikipedia, the free encyclopedia, 'Claude Elwood Shannon (April 30, 1916 – February 24, 2001), an American electrical engineer and mathematician, has been called "the father of information theory".
Shannon is famous for having founded information theory and both digital computer and digital circuit design theory when he was 21 years-old by way of a master's thesis published in 1937, wherein he articulated that electrical application of Boolean algebra could construct and resolve any logical, numerical relationship. It has been claimed that this was the most important master's thesis of all time.' back |
| Eugene Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, 'The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories.' back |
| Gödel's incompleteness theorems - Wikipedia, Gödel's incompleteness theorems - Wikipedia, 'Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all of mathematics is impossible, giving a negative answer to Hilbert's second problem.
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, a corollary of the first, shows that such a system cannot demonstrate its own consistency.' back |
| Hilbert's problems - Wikipedia, Hilbert's problems - Wikipedia, the free encyclopedia, 'ilbert's problems are a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21 and 22) at the Paris conference of the International Congress of Mathematicians, speaking on 8 August in the Sorbonne. The complete list of 23 problems was later published, most notably in English translation in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society.' back |
| Kurt Goedel I, On formally undecidable propositions of Principia Mathematica and related systems I, '1 Introduction
The development of mathematics towards greater exactness has, as is well-known, lead to formalization of large areas of it such that you can carry out proofs by following a few mechanical rules. The most comprehensive current formal systems are the system of Principia Mathematica (PM) on the one hand, the Zermelo-Fraenkelian axiom-system of set theory on the other hand. These two systems are so far developed that you can formalize in them all proof methods that are currently in use in mathematics, i.e. you can reduce these proof methods to a few axioms and deduction rules. Therefore, the conclusion seems plausible that these deduction rules are sufficient to decide all mathematical questions expressible in those systems. We will show that this is not true, but that there are even relatively easy problem in the theory of ordinary whole numbers that can not be decided from the axioms. This is not due to the nature of these systems, but it is true for a very wide class of formal systems, which in particular includes all those that you get by adding a finite number of axioms to the above mentioned systems, provided the additional axioms don’t make false theorems provable.' back |
| Mark 12:30-31, Love the Lord your God , '30 Love the Lord your God with all your heart and with all your soul and with all your mind and with all your strength.’ 31 The second is this: ‘Love your neighbor as yourself.’ There is no commandment greater than these.”' back |
| Productivity - Wikipedia, Productivity - Wikipedia, the free encyclopedia, 'Productivity is a measure of the efficiency of production. Productivity is a ratio of what is produced to what is required to produce it. Usually this ratio is in the form of an average, expressing the total output divided by the total input. Productivity is a measure of output from a production process, per unit of input.
At the national level, productivity growth raises living standards because more real income improves people's ability to purchase goods and services, enjoy leisure, improve housing and education and contribute to social and environmental programs. Productivity growth is important to the firm because it means that the firm can meet its (perhaps growing) obligations to customers, suppliers, workers, shareholders, and governments (taxes and regulation), and still remain competitive or even improve its competitiveness in the market place.' back |
| Quantum field theory - Wikipedia, Quantum field theory - Wikipedia, the free encyclopedia, 'Quantum field theory (QFT) provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or (especially in a condensed matter context) of many-body systems. . . . In QFT photons are not thought of as 'little billiard balls', they are considered to be field quanta - necessarily chunked ripples in a field that 'look like' particles. Fermions, like the electron, can also be described as ripples in a field, where each kind of fermion has its own field. In summary, the classical visualisation of "everything is particles and fields", in quantum field theory, resolves into "everything is particles", which then resolves into "everything is fields". In the end, particles are regarded as excited states of a field (field quanta). back |
| Set theory - Wikipedia, Set theory - Wikipedia, the free encyclopedia, 'Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.' back |
| The Assayer - Wikipedia, The Assayer - Wikipedia, the free encyclopedia, 'The Assayer (Il Saggiatore in Italian) was a book published in Rome by Galileo Galilei in October 1623. . . .
This is the book containing Galileo’s famous statement that mathematics is the language of God. . . . "Philosophy [i.e. physics] is written in this grand book — I mean the universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth."' back |
| Theorem - Wikipedia, Theorem - Wikipedia, the free encyclopedia, 'In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms.' back |
| Theory of computation - Wikipedia, Theory of computation - Wikipedia, the free encyclopedia, 'In theoretical computer science, the theory of computation is the branch that deals with whether and how efficiently problems can be solved on a model of computation, using an algorithm. The field is divided into three major branches: automata theory, computability theory and computational complexity theory.' back |
| Transfinite numbers - Wikipedia, Transfinite numbers - Wikipedia, the free encyclopedia, 'Transfinite numbers are cardinal numbers or ordinal numbers that are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were nevertheless not finite. Few contemporary workers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as "infinite". However, the term "transfinite" also remains in use.' back |
| Uncertainty principle - Wikipedia, Uncertainty principle - Wikipedia, the free encyclopedia, 'In quantum physics, the Heisenberg uncertainty principle states that the values of certain pairs of conjugate variables (position and momentum, for instance) cannot both be known with arbitrary precision. That is, the more precisely one variable is known, the less precisely the other is known. This is not a statement about the limitations of a researcher's ability to measure particular quantities of a system, but rather about the nature of the system itself.' back |