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vol 6: Essays

Scientific Theology (June 2017)


0: Abstract
1: My theological history
2: Sources of knowledge
3: Creation: Fixed point theory
4: Formal mathematics: the transfinite network
5: The transfinite network reaches the limits of computability and completeness
6: Quantum mechanics: interfacing formalism to reality
7. Why is the Universe quantized?
8: Relativity
9: Quantum field theory: applied symmetry
10: Natural selection
11: From Trinity to transfinite complexity
12: Human symmetry and social equilibrium
13: Conclusion: Scientific method and justice

In scientific investigations . . . it is permitted to invent any hypothesis, and if it explains various large and independent classes of facts, it rises to the rank of a well grounded theory. Darwin: The Variation of Animals and Plants Under Domestication
Perhaps now is the time to widen the quest for understanding still further, to expand the intellectual effort beyond conventional science—to engage the larger non-scientific communities of philosophers, theologians and others who often resonate with the cosmic-evolutionary theme even if not in name, all in an ambitious effort to construct a millennial world view of who we are, where we came from, and how we fit into the cosmic scheme of things as wise, ethical, human beings. Chaisson: Cosmic Evolution, page 211
0: Abstract

I explore the possibility and consequences of replacing the proposition ‘God is not identical to the Universe’ with its contradiction ‘God is identical with the Universe’. I explain that this change may be made possible by the mathematical theory of fixed points, and draw the conclusion that if this is the case, the way is open for theology to become a science in the modern sense of the word.

The Catholic Church maintains that it is the sole trustworthy channel of personal communication between God and us. This channel is necessary, it claims, because there is an absolute divide between God and the World. God is totally beyond our ken and can be known only insofar as it chooses to reveal itself. This revelation, the Church’s raison’detre, is principally embodied in the Bible. The Church claims the right to produce definitive and even infallible interpretations of the Bible.

Here we explore the alternative, replacing the hypothesis God is not the Universe with the hypothesis God is the Universe.

This second hypothesis faces two problems. First, it flies in the face of thousands of years of tradition; and second, it would seem to be impossible to identify the perfect, infinite, omnipotent, omniscient and absolutely simple God of tradition with the complex activity at every scale visible in the Universe. From a scientific point of view, we can dismiss the first problem. As Werner Heisenberg pointed out in the midst of the epistemological problems raised by quantum mechanics, all we have to do is find a way to understand the data. Werner Heisenberg: Quantum-theoretical re-interpretation of kinematic and mechanical relations

History has set the scene, but the answer may require a new interpretation of older ideas. In the case of quantum mechanics, the Lagrangian approach in classical mechanics provided a bridge and taught us to understand physics not as a continuous infinitesimal process, but as the optimum route between two states in a computable space. Lagrangian - Wikipedia

The overall task here is to suggest that the world is divine by fitting the world of experience to the ancient models of God and showing that the ideas of a completely simple God and a world of growing complexity are consistent with one another. The key to this identification is the mathematical theory of fixed points.

Keywords: theology, science, cosmology, mathematics, motion, creation, quantum mechanics, natural selection

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1. My theological history

I was brought up in a Catholic milieu, went to Catholic schools and ultimately entered the Dominican Order and was solemnly professed. Soon afterwards I was asked to leave for openly exploring the option that God and the Universe are identical rather than the distinct elements of reality embedded in Catholic tradition.

I came to this position with the help of Bernard Lonergan’s book Insight3. Following the tradition of Plato, Aristotle and Aquinas, Lonergan developed his metaphysics from a theory of knowledge.

We understand insight to be the act by which the mind apprehends the meaning of data presented to it by the senses. Insight moves a person from the state of possessing potentially intelligible information to actually understanding the information. This transformation makes no difference to the data, but is a change in the mental state of the knower. If the insight is true it will correspond to the process that renders the data.

In addition to insight, Lonergan asserts the existence of inverse insight, the insight that there is nothing to be understood. There are data that do not arise from any reality and are meaningless. Lonergan calls such meaningless data empirical residue4. He feels that the existence of the empirical residue tells us that the Universe is not fully intelligible. Since he holds that God must be fully intelligible, he concludes that the world is not God.

I had studied science at school. Using the evolutionary paradigm, we can see the world as a tree whose leaves, for us, are all our local events events and whose root is the initial singularity. Every branching in the tree is an event in the past. Consequently every event, that is every datum, has a pedigree going back to the beginning. It is this genetic history that gives every event a unique meaning within the overall structure of the Universe.

My idea was found to be in breach of 4 of the 24 theses published by Pope Pius X in his struggle against Modernism. Thus ended the first part of my theological education. I learnt that classical theology, because it is not empirical, has no observable foundation and can only be maintained by an authority like the Papacy.

If the Dominican Order had been a scientific organization, my dismissal would have been wrongful. As we have become aware, the major lasting injury to abused children is the psychological damage that may last a lifetime. Teaching people false doctrines may also result in psychological injury.

Much of the data for the physical and biological sciences is public, open to be observed by anybody. When we turn to psychology and theology however, many of the sources of data become private and personal experiences which we reveal to one another by actions and words. Among these communications is the written literature historically the subject and product of theological scholarship.

Talar writes:

Modernity's—and Modernism's—commitments to the historicity and subjectivity of thought come up against the speculative outlook, deductive method and objective notion of religious truth embraced by neo-Thomism. This system has been described as a "supernatural rationalism"—supernatural in that is derived its data from supernatural revelation, not from autonomous reason; rationalist in its marginalization of experience as a valid theological category. . . . . Talar: "The Synthesis of All Heresies", page 506

It can fairly be said that, in imposing renewed commitment to Scholasticism as a solution, Pascendi significantly misdiagnosed the problem. . . . ibid page 511

The Modernists] personalize the dilemma of the scholar (indeed of the Catholic) who finds the conclusions that emerge from experience at variance with the dicta of ecclesiastical authorities. When fidelity to ones own integrity and fidelity to one religious tradition conflict, what is to be done? ibid page 513

Jean Calvert: If you ever deal with the modernist crisis, do not forget to tell how much we suffered. Boland: La crise moderniste hier et aujourdhui, page 90

This essay is at attempt to restate my position as it has developed since I left the Order. Every intellectual step forward provides a new interpretation of old data. This interpretation erases previous difficulties and is often followed by an explosion of new understanding. Einstein’s theories of invariance, for instance, totally changed our vision of the Universe. My feeling is that the mathematical theory of fixed points enables us to overcome to the perennial problem, identified by Parmenides, of the relationship of motion to stillness. This opens the way to a new view of God. John Palmer - Parmenides

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2. Sources of knowledge

Here I assume the theory of evolution, that is creation of new forms by variation and selection. Evolution is well attested by cosmological, paleontological and archeological evidence. We also have fragmentary documentary evidence of the evolution of human ideas since people first began recoding their experiences of the world, starting with accountancy and tending toward poetry.

Human survival depends upon learning. The ability to learn enables each new child to adapt its genetically controlled phenotype to the details of the environment into which it is born. The languages and cultures thus acquired are often very local, and it is not surprising that local world views experience tension when they come into contact with more distant neighbours.

Science is catholic. It sees that since we are all one species living on one planet in one Universe there exists a foundation for universal agreement about our place in the world. From this we can work toward agreement on prudent means to secure our own health and safety.

The explosion in science and technology which has brought us to our current situation began in Galileo's time. This transition, which remains a work in process, was from the establishment of credibility by belief in an authority to the establishment of credibility by studying the world.

Aquinas understood science to be the logical deduction of conclusions from principles which were either known in themselves (per se nota) or derived from some higher knowledge. He saw theology as being derived from the higher knowledge revealed to us in the scriptures. Thomas Aquinas Summa Theologiae I, 1, 2: Is sacred doctrine a science

This approach to truth is essentially political. An established political power defines what is to be held as true, ultimately under pain of death. Such enforcement of belief is widespread in the world today.

This ancient view of science slowly changed as channels of communication opened between practical tradespersons and the intelligentsia. Galileo, the instrument maker and mathematician, embodied this transition.

Scientific faith has two articles. First, that the Universe is consistent. If we perceive an inconsistency, we assume that we have not understood what is happening. The second is that the conclusions of science must be consistent with data obtained by observing the actual world.

Anselm saw theology as fides quaerens intellectum, faith seeking insight, a view common among scientists. From the modern point of view, the source of faith is experiment and experience, and the understanding we seek takes the form of models, like quantum mechanics, that appear to fit the data.

Augustine sought understanding using Platonic philosophy. Aquinas based much of his theology on the work of Aristotle. Aristotle had already devised the Unmoved Mover which was easy to transform into the omnipotent, omniscient, and eternally blessed Catholic God. Aristotle: Metaphysics, Book XII, 1072b3 sqq

Aristotle’s unmoved mover was part of the cosmos. Aquinas, a Christian, distinguishes God from the cosmos. This has the effect of making God invisible to us. Since this God is invisible, its creators are free to design it as they wish. Their ideas cannot be checked against experience.

Science proceeds by conjecture and refutation, as Popper explained, or by muddling through, according to Fortun and Bernstein. Scientists collect data, imagine possible explanations of the data, and test their explanations against further data. Eventually, after enough conjecture and testing, a consensus develops in the relevant community and becomes part of the established foundation for further research. Occasionally, new evidence requires a complete revision of old ideas, as in physics when quantum mechanics became necessary. Popper: Conjectures and Refutations, Fortun & Bernstein: Muddling Through

Scientific method is not confined to professional scientists: we do it informally in our everyday search for the truth.

The passage from the classical God to the divine Universe took me three major insights. First I saw that Bernard Lonergan’s approach to God is open to question; second that the Cantor universe of transfinite numbers is big enough to approach a description of God; and third, that mathematical fixed point theory unites the classical actus purus, omnino simplex God to the complex world of action we observe.

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3. Creation: Fixed point theory

The decision to place God outside the world was forced upon the ancients by their inability to reconcile dynamic and static systems. Since the world is clearly dynamic, and true and permanent knowledge requires stasis, they concluded that God and the world are different entities. Plato built on Parmenides’ work, and the idea has since become the foundation of Christian theology. The Christian God is eternal, and time and space are understood to be inconsistent with the divine nature. Plato: Parmenides

We represent motion mathematically using functions which map the domain of the function onto its range. From a formal point of view, we see no reason not to approximate the life of God by functions, finite and transfinite. Luitzen Brouwer and others have found that any mapping f of a continuous, compact, convex set onto itself must contain a fixed point, that is a point x for which f(x) = x. William K. Allard: Brouwer's fixed point therem

Following this hint, we may guess that the although the divine Universe is pure action, the invariances we observe come not from outside the dynamics, but are part of it. Fixed point theory shows us that the concept of motion includes not-motion. We find a similar idea in mathematics, where the concept of number includes 0.

We may understand fixed point theory to have made Parmenides’ dichotomy unnecessary. Even though the Universe is purely dynamic, if it fulfills the hypotheses of fixed point theorems, we may expect to find fixed points which are not outside the dynamics, but within it.

Brouwer’s theorem is subject to three conditions, continuity, convexity and compactness.

Does the Universe meet these criteria? Does a Universe of pure action have fixed points? Observation and quantum theory suggest that it does. As far as we can tell, electrons and photons have had the same properties now as they had very soon after the Universe began to differentiate.

It also seems reasonable to expect that the Universe is convex and compact. By definition there is nothing outside it so that it cannot have holes and must contain its own boundaries. Aristotle imagined a duality whose elements he named potential (dunamis) and act (entelecheia). He also introduced one axiom: no potential can actualize itself. The actualization of any potential required the existence of an efficient cause already in act. From this he concluded that there must be an unmoved first mover which is the source of all motion in the world.

Aquinas concludes that God, like Aristotle's first mover, is pure actuality. This God is a is a living God. Aquinas following Aristotle, defines life as self-motion. Further, motion is the passage potency to act. This would seem to imply that the living God contains potential, which contradicts the assertion that it is actus purus.

The received solution is that this definition of motion applies only to physical being, not intellectual being. Aquinas writes:

. . . action is twofold. Actions of one kind pass out to external matter, as to heat or to cut; whilst actions of the other kind remain in the agent, as to understand, to sense and to will. The difference between them is this, that the former action is the perfection not of the agent that moves, but of the thing moved; whereas the latter action is the perfection of the agent.

Hence, because movement is an act of the thing in movement, the latter action, in so far as it is the act of the operator, is called its movement, by this similitude, that as movement is an act of the thing moved, so an act of this kind is the act of the agent, although movement is an act of the imperfect, that is, of what is in potentiality; while this kind of act is an act of the perfect, that is to say, of what is in act as stated in De Anima iii, 28. In the sense, therefore, in which understanding is movement, that which understands itself is said to move itself.

It is in this sense that Plato also taught that God moves Himself; not in the sense in which movement is an act of the imperfect. Thomas Aquinas Summa Theologiae I, 18, 3: Is life properly attributed to God?

Since we are here supposing that the Universe is divine, we may consider all actions of the Universe as falling into the second class. This is consistent with our modern understanding of energy. Kinetic and potential energy are precisely equivalent, freely interconvertable, as we see in the pendulum. In modern physics potential is an active force, so we may think of the action of a pendulum moving from potential to kinetic energy and back again as a transition from act to act of the type Aquinas attributes to immanent actions.

If we assume that fixed point theory is indifferent to the complexity of the sets to which it is applied we might say that the mathematical literature is a (multi-dimensional) fixed point in the human mathematical community. We thus arrive at a sort of bootstrap: the existence of fixed points in dynamical systems explains how the active living mathematical community can arrive at proofs that establish fixed relationships between certain propositions, in particular the proof that under certain conditions there are fixed points in dynamic systems.

We can now go on to model the relationship of the fixed points to one another. By studying these relationships we gain insight into the underlying divine dynamics. We may imagine establishing a correspondence between quanta of action in the Universe, each of which has unique identity, and the natural numbers. The 0 quanta of action can be ordered in 1 different ways. This suggests how a transfinite computer network might apply to the fixed points of the world.

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4. Formal mathematics: the transfinite network

As Parmenides realized, we cannot make a permanent record of motion per se, but we can write down the invariant features of a motion. Like Newton, we often represent invariants with algorithms called differential equations, formal mathematical texts that capture the essence of a motion in symbols and enable us to reproduce it by computation.

Calculus reopened old questions about continuity, points and infinitesimals. By the nineteenth century these questions had been solved to almost everybody’s satisfaction with the theory of limits. There are no atoms in classical continuous mathematics. Every interval can be subdivided ad infinitum.

The ancients knew that one cannot label all the points on the natural line with integral numbers, or ratios of integers. Instead we need the real numbers. Just as in a line there are points between any two points, there are real numbers between any two real numbers.

Georg Cantor, working on the assumption that a line is made of points, set out to find the cardinal of the continuum: how many points does it take to make a line? As the points become smaller, their number becomes larger, so Cantor created a new world of numbers, the transfinite numbers, which, he hoped, would be big enough to number all the points in a line, no matter how small the points or long the line. Cantor: Contributions to the Theory of Transfinite Numbers

Like Newton, Cantor invented a new branch of mathematics, now known as set theory, to study his problem. Set theory and logic have become foundations for mathematics. A set is a collection of elements with individual identities so that they can be counted and ordered. The principal operation in set theory is establishing one-to-one correspondences between elements of sets. So we can imagine the set of all the natural numbers. The cardinal of this set cannot be a particular natural number, because we can always add 1 and get a larger number. Instead Cantor named the cardinal of the set 0, the first transfinite number, using the first letter of the Hebrew alphabet.

He then exploited the power of order: he showed that the cardinal of the set of all orderings or permutations of the natural numbers, 1, is strictly greater than 0. The set of orderings of 1 items produced the next transfinite cardinal, 2, and so on without end. Cantor hoped that 1 would turn out to be the cardinal of the continuum. Cohen showed, nearly seventy years later, that the continuum hypothesis is independent of the standard axioms of set theory. Nevertheless, set theory remains a principal tool for the development of mathematics. Cohen: Set Theory and the Continuum Hypothesis

Cantor's theory upset some theologians who felt that infinity is the unique attribute of God and there can be no 'created' infinity. This problem was solved by Cantor's tacit use of formalism, subsequently made explicit by Hilbert. Formalism is the process of manipulating symbols under the sole constraint of consistency. Although in reality all information is represented physically, mathematicians are still free to imagine that the symbol x may stand for anything, such as the infinite set of natural numbers, or the set of all permutations of the natural numbers.

Much of Cantor's work was theologically motivated, and he imagined an absolute infinity which was characteristic of God. He also found that the existence of such an infinity was not self consistent, something now known as Cantor's paradox. Cantor's proof for the existence of the transfinite numbers tells us that given a set of plausible assumptions, every set, no matter how big, can generate a bigger set. This would hold for the absolutely infinite set as well, which is therefore no longer absolutely infinite. Formally the absolutely infinite set cannot exist because it is inconsistent. Dauben: Georg Cantor, Hallett: Cantorian Set Theory and Limitation of Size

From this we may conclude that transfinite mathematics does not have an element corresponding to God. Instead it has a series of elements which may approach but never reach the immensity of God. We can talk about these subsets of the whole (sometimes called universes of discourse) without contradiction. Further, Cantor's theorem guarantees that no how matter large a universe of discourse we decide to study, it will remain always a subset of the strictly greater set arising from the Cantor expansion of our chosen universe.

Formally, a network is a set of addresses and a set of processors capable of reading and writing information stored at the addresses. If the processors are trucks, reading means loading the message at the source and writing means unloading it at the destination. In our world all information is physically embodied. Rolf Landauer: Information is a Physical Entity

The addresses in the transfinite network are the transfinite numbers. The transport agents are universal computers, that is machines capable of doing anything that can reasonably considered computation.

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5. The transfinite network reaches the limits of computability and completeness

Formal mathematics is bounded by consistency. David Hilbert thought consistent mathematics would be able to answer all questions, but this is not the case. In 1931 Kurt Gödel proved that large enough consistent formal systems must be incomplete. In other words they contain true propositions that cannot be proved nor disproved Richard Zach: Hilbert's Program, Feferman: Kurt Gödel: Collected Works vol 1.

Gregory Chaitin recast Gödel’s idea in information theoretical terms which makes its implications clearer. Chaitin writes:

Gödel's theorem may be demonstrated using arguments having an information theoretic flavor. In such an approach it is possible to argue that if a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. In contrast with the traditional proof based on the paradox of the liar, this new viewpoint suggests that the incompleteness phenomenon discovered by Gödel is natural and widespread rather than pathological and unusual. Chaitin: Gödel's Theorem and Information

Gödel’s theorem, as understood by Chaitin, comes very close to the cybernetic principle of requisite variety which tells us that one system can control another only if it has equal or greater complexity. Ashby: An Introduction to Cybernetics

Mathematical proof is a deterministic logical process that begins with a set of hypotheses and arrives at a conclusion. It is, in other words, a computation process. Hilbert thought that mathematics, as well as being complete, would be computable. Alan Turing complemented Gödel's work by showing that this is not so. Turing devised (formally) a machine, now known as a Turing machine, which can execute any algorithm that might reasonably be considered computable, and showed that there are problems this machine can not solve. Alan Turing: On Computable Numbers

A function maps from its domain to its range. A function space is a set of functions on the same domain. A function space of interest here is the space of all the mappings of the natural numbers to themselves. This space has as many dimensions as there are natural numbers, and each point in it represents one permutation of the natural numbers.

Turing found that there are only 0 Turing machines, so that only 0 of the 1 functions on the natural numbers onto themselves can be computable. 0 computers can only control 0 deterministic processes. Consistency, through the work of Gödel and Turing places bounds on mathematics.

Cantor's enlargement of the number system to the transfinite cardinal and ordinal numbers provides us with a system large enough to put into correspondence with any structure in the Universe. If a system with transfinite cardinal n is too small to do the job, we need only move to some larger cardinal n+m .

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6. Quantum mechanics: interfacing formalism to reality

We now turn to mapping the transfinite network to the observable world. We express our inner thoughts and feelings in body language. In the scholarly world this is mostly speech and writing. Here we assume, following Landauer, that information is a physical entity.

The Christian history of salvation is recorded in the physical words of a Bible. We know now that the information in our minds is stored in the physical states of our central nervous systems and the genes that define us are encoded as long chemical strings.

Cantor, discussing the transfinite numbers, introduced a principle of finitism: The transfinite is on a par with the finite and mathematically is to be treated as far as possible like the finite. We improve our picture of the formal transfinite network by studying the properties of real physical networks like the internet and the network described by quantum mechanics. Hallett, op cit page 7

A network is essentially a system of processors communicating through a set of memories. To communicate, one computer will write something in the memory of another. The recipient will read that memory to receive the message. Some memory may be isolated to a single stand-alone machine. Other memories have physical network connections which enable them to read from and write to each other.

The software that operates a computer network is built in layers. These layers perform the sequence of transformations necessary to interface human users with the underlying physical processes that implement logical computation.

The most significant difference between classical and quantum theory is, as the name suggests, quantization. Every discrete operation in the Universe comprises one or more sub-operations which are also discrete. There is a smallest discrete operation, the quantum or atom of action, measured in conventional units by Planck’s constant. It therefore becomes possible to establish correspondences between the discrete symbols of mathematics and discrete events. Various functional relationships between symbols have been found to model the behaviour of the real world. Eugene Wigner: The Unreasonable Effectiveness of Mathematics in the Natural Sciences

Physicists see two major regions of mathematical correspondence to the world, relativity and quantum mechanics.

Quantum mechanics is the study of events at the quantum scale. This scale is very small. The quantum of action is about 6.6 x 10-34 Joule second. A Joule is approximately the amount of energy required to lift an egg one metre. The blink of an eye requires trillions of quanta of action. On this scale, we are huge and complex organisms.

Quantum mechanics is built in complex Hilbert space. This space is a function space developed from Cantor’s system of transfinite cardinal and ordinal numbers. Physical states are represented by state vectors, ψ. All the information we have about a physical states is encoded in the direction of ψ in its Hilbert space. Complex numbers are periodic, and so ideal for dealing with periodic phenomena like waves and computation. From a quantum mechanical point of view the physical world is in perpetual motion with an unbounded range of frequencies. von Neumann: Mathematical Foundations of Quantum Mechanics

Physical motions are described by the energy equation: dψ/dt = Hψ. H is a Hamiltonian or energy operator, which encodes how the elements of the vector ψ transform with time. In quantum mechanics, frequency is directly related to energy by the relationship f = E/h, where h is Planck’s constant. Quantum mechanics yields results through two further equations, the eigenvalue equation and the Born rule. These equations serve to pick observable results out of the infinity of possibilities offered by the energy equation.

The eigenvalue equation defines the fixed points of the energy equation. In the terminology above, we read Hψ = kψ, where we now think of H as a measurement operator or observable. This equation picks out the scalar values k which correspond to eigenfunctions, that is the fixed algorithms, of the operator H. An observer using the operator H will see only those eigenvalues k which correspond to eigenfunctions of the operator H. Such eigenfunctions are represented by state vectors whose direction is unchanged by H.

A classical experimenter expects to see just one result (within errors) for each experimental setup. Quantum mechanics, on the other hand, can yield many different outputs from the same input.

There is no way to predict the exact outcome from a given input, but the Born Rule predicts the probability that we will observe a particular eigenvalue k. Experimental physicists must repeat identical initial conditions many times to estimate the probabilities of various possible outcomes.

Here we come to the basic conundrum of quantum mechanics, one that baffled Einstein. He remained convinced till the end of his life that a true model of the world would be causal, giving definite outputs for definite inputs. Quantum mechanics does not do this. Although the wave equation is a deterministic continuous equation, it does not give deterministic results. There is a degree of freedom between the equation and the results of observation.

This degree of freedom is sometimes explained by the reduction or collapse of the wave function. An isolated system is believed to embody all the solutions of the wave equation, usually an infinite number. When it is observed, however, this complex system ‘collapses’ and we see only one eigenvalue. Often it is tacitly assumed that such observation must involve physicists, but we can also imagine the elements of the Universe observing one another with the same result.

Although its mathematical expression is a bit difficult, there is quite a simple interpretation of quantum mechanics. It describes a communication network operating between two or more sources. Eigenvalues are the content of the messages transmitted on this network, which we can observe with our own senses or suitable machinery. The eigenfunctions are the algorithms used by the system to encode and decode these messages. This coding is necessary to prevent error in the network, as we see next. The particles we observe, defined by the eigenvalue equation, are messages on this network. The frequency spectrum of these messages is computed using the Born rule.

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7. Why is the Universe quantized?

Quantization surprised nineteenth century physicists steeped in the ancient tradition of a continuous world, yet it is to be seen everywhere. We are all individuals, surrounded by individual trees, flowers and blades of grass. Although we are inclined to think of these things as static objects, in a dynamic Universe they are better conceived as operations. From this point of view my life is one long complex action or event comprising a huge number of quanta of action, that is atomic events.

It seems reasonable to assume that error free communication is necessary if we are to have a stable Universe. We can defeat error by making different signals so far apart in the space of possible signals that the probability of confusion is minimized. In practical networks, messages are quantized into packets, enabling us to send gigabytes of information over noisy channels without error. This idea is the foundation of the mathematical theory of communication devised by Claude Shannon. Claude E Shannon: A Mathematical Theory of Communication

Before the advent of digital computers some of Shannon’s strategy could be implemented by analogue methods, but the full power of his insight requires computable digital encoding and decoding algorithms. This suggests that the set of eigenfunction of quantum mechanics may be the same as the set of computable functions defined by a universal Turing machine.

These functions provide an independent (orthogonal) set of signals from which the processes that create the observed Universe is built. As Turing discovered, there are only a countable infinity of computable functions. As we have seen above, the number of functions of the natural numbers to themselves is the second transfinite number, 1, which means that only an infinitesimal proportion of these functions is computable. One might therefore expect all the possible functions to be equiprobable, leaving us with a Universe of purely random structure. Nevertheless, computable functions have an advantage over the rest in that they may be able to predictably copy themselves.

Quantum mechanics is reversible, which means that in the quantum mechanical world time has no direction. Instead we have an eternal motion which we may characterize as pure act. Relativity introduces a direction to time and provides an ordering of events with respect to causality.

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8. Relativity

According to our hypothesis, the creation of the world is the emergence of fixed points in the divine dynamics. We understand these fixed points to be carriers of information in the universal network.

Quantum mechanics describes the lowest physical layer of the universal network. Although the quantum formalism is usually applied in space-time, all its essential features can be realized in the domain of energy and time. Quantum mechanics identifies energy and time through the Planck-Einstein equation E = hf, where f is frequency, inverse time. Quantum mechanics does not see absolute energies, only energy differences, that is frequency differences, which determine the outcome when the wave functions are added according to the Born rule.

Classically, God is pure action. In modern physics energy is the time rate of action, so that we can imagine the first act of creation was the emergence of two orthogonal entities, energy and time, from action.

Relativity is concerned with the emergence of the next layer of structure in the Universe, space-time. This introduces a new duality, space and time, with two properties: space is not time; and it takes time to move in space. Einstein developed the special theory of relativity when he realized that Maxwell’s equations are not consistent with the Galilean transformations used in Newtonian physics. The velocity of light, c, is fixed locally by Maxwell’s equations and is completely independent of the velocity of either the source or the observer of the light. This leads to the relation c + c = c, a counterintuitive result that is nevertheless consistent with observed reality. It is explained by the Lorentz transformation. This transformation is the fixed point or algorithm which governs the local structure of space-time. Albert Einstein: On the Elecrodynamics of Moving Bodies, Lorentz transformation - Wikipedia

Relativity inherits the conservation of energy from quantum mechanics. The Universe as a whole, we believe, contains a constant amount of energy, which may in fact be zero, the physical equivalent of eternity. This is because in space-time we can distinguish potential energy and kinetic energy. Conservation holds if we count kinetic energy as positive and potential energy us negative. A frictionless pendulum would swing forever. Feynman: Lectures on Gravitation

The special theory deals with inertial motion, that is systems in free fall. In this system, Newton’s first law of motion holds: a body at rest remains at rest and a body in motion continues to move in a straight line unless it is acted upon by a force. The general theory takes force (like gravitation) into account. It corresponds to Newton’s second law: that a body with mass m acted upon by force F accelerates at a rate a, ie a = F/m.

Einstein began his study of gravitation with the insight that a person in free fall does not feel their own weight. This led to the equivalence principle, that there is no way to distinguish between weight and acceleration, between gravitational mass and inertial mass. Gravitation sees only energy regardless of its form. Einstein: Relativity

Two constraints bind the general theory of relativity. First, that we live in a four dimensional space-time. And second, that energy is conserved. We live in a world with three spatial dimensions. This is the simplest space in which a network can be completely connected without ‘crossed wires’, that is messages interfering with one another and becoming confused.

The mathematical implementation of the general theory in four dimensional spacetime is quite complex. General relativity is an application of differential geometry. The domain of differentiation is a differential manifold. A manifold is a space the points of which are addressed by numbers or sets of numbers. A non-Euclidean manifold is constructed from a set of overlapping ‘charts’. Different addressing systems may be used in different charts, but we require that where the charts overlap we can transform smoothly from one addressing system to the other. These transformations are expressed as connection coefficients that relate the charts. The result is a flexible dynamic topological space made from little segments like chain mail.

General relativity describes a patchwork of inertial spaces. These inertial frames may accelerate toward or away from one another while remaining inertial. In free fall they do not ‘feel’ the curvature that controls their motion. We may look upon this as a network structure, the charts being sources and the connection coefficients between the charts as messages between the sources.

General relativity is a classical theory which, after a century of testing appears to describe the large scale dynamics of space-time perfectly. The special theory is linear first order and not creative. The general theory is second order and creative: the gravitational field carries energy and gravitation responds to energy.

Formally, general relativity describes a dynamic space-time which may expand or contract. At present we see an expanding Universe, and can extrapolate this expansion back to a singular point which we see as the source of the ‘big bang’ which created the current Universe. The inverse process, which occurs in regions of concentrated energy, is the formation of black holes. General relativity may be the only consistent implementation of conservation of energy in 4D space-time, having been selected from all those possibilities that entail inconsistency. Hawking & Ellis: The Large-Scale Structure of Space-Time

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9. Quantum field theory: applied symmetry

Historically we have talked about laws of nature and natural law. Currently the preferred term is symmetry. A symmetry is something that stays the same when other things change, as laws do. We are all equal before the law: the response of the law is (theoretically) the same, no matter who it is dealing with. One of the key ingredients of modern physics is Emmy Noethers’ explanation of the close relationship between symmetries and conservation laws like the conservation of energy. Neuenschwander: Emmy Noether's Wonderful Theorem.

The next step in the emergence of the observable universe is the creation of particles. We model this process using quantum field theory (QFT) which lies at the intersection of quantum mechanics and special relativity. Quantum field theory has been applied to observations of the most fundamental particles in the Universe to give us the standard model. Particle Data Group. Lawrence Berkeley National Laboratory

A key relationship arising from the special theory is the equation of mass and energy: E = mc2. Massive particles can be created from energy and energy is released when they are annihilated. One of the most successful applications of quantum field theory is quantum electrodynamics, the theory of the interactions between photons and electrically charged particles like electrons. Most of the macroscopic phenomena we observe are explained by quantum electrodynamics, with the particular exception of gravitation, which has so far eluded incorporation into the standard model. Feynman: QED

In a layered network, the symmetries of a given layer are provided by the layer beneath it: conservation of energy is provided to relativity by quantum mechanics. Relativity introduces the new layer of space, mass and momentum which provides the symmetries of particle physics, that is applied QFT.

Particles fall into two classes, bosons and fermions. Bosons serve as messenger particles, fermions as structural particles. Identical bosons tend to congregate together. Identical fermions, on the other hand, obey the Pauli exclusion principle: only one fermion is allowed in any quantum state. This leads to the spatial extension of structures like atoms and everything made from atoms. Quantum field theory understands this differentiation by the spin-statistics theorem. Streater & Wightman: PCT, Spin, Statistics and All That

Classical physicists thought that information could be transmitted instantaneously through space. The special theory of relativity tell us that it takes time to move a distance. This fact leads to the time ordering of causality and the effective isolation of events that are outside one another’s ‘light cones’ and cannot communicate with one another.

Particles are the observable fixed points in the divine dynamics. On the whole, they are observed by one another rather than by physicists. We use the term ‘particle’ to apply to everything from fundamental particles to ourselves any other discrete entities.

Particles interact with one another through the underlying, largely invisible processes which we represent by quantum wave functions, analogous to computer software. These processes are invisible, since to be visible, they must communicate. Since communication is itself a process, the overhead of communication would bring every process to a standstill if it tried to communicate a real time history of every step it took. For the purposes of our model, these fixed points correspond to transfinite numbers and the underlying processes are modelled by Turing machines.

Quantum mechanics becomes creative when we introduce 'measurement', ie systems communicating with one another. The act of communication takes place in a product of the Hilbert spaces of the communicating particles This product space is in effect a new creation. Zurek explains that communication can only take place if the two particles share an orthogonal basis, that is a common code, as we should expect from Shannon’s theory. Networks extend themselves by creating new network addresses, that is new space. Quantum communication or measurement has a similar effect. Wojciech Hubert Zurek: The QuantumOrigin of Quantum Jumps

Fermions bind to one another by exchanging bosons. We can see layer after layer of this structure, beginning with fundamental particles and moving up through atoms, molecules, cells, organisms, ourselves, ecosystems, planets and galaxies to the Universe as a whole. This process is governed by natural selection which is, if the universe is divine, tantamount to divine judgement.

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10. Natural selection

We understand each mapping of the natural numbers onto themselves to be the representation of a string of events, each of which is executed by a sequence Turing machines. We imagine these sequences being built from the atomic Turing machine which is simply a quantum of action, formally identical to God understood as actus purus.

Let us assume that the generation of permutations of the Turing machines is random, so we might expect all of the permutations to be equiprobable. Since there are 1 permutations of the 0 computers, this assigns a probability of 1/1 to each one. It is this vast pool of improbable possibility (which continues on up the transfinite scale) which may be the source of both creativity and error in the observed Universe.

Successful reproduction requires deterministic processes and deterministic processing requires the error free transmission of information to different points in the computing network. Random motion enables the Universe to explore states which are forbidden to a deterministic universe. So we imagine that the evolving cosmos is a a possible sequence of states of the divine dynamics.

Birth is the the result of an error free instance of reproduction. Death is the result of the cumulation of errors in vital processes which ultimately lead to their cessation. Death, so defined, is a common occurrence in computing machinery, and the first available remedy is to turn the machine off and reboot it so that it starts in an error free state. Complex biological systems like ourselves cannot be rebooted when they are properly dead because their initial state cannot be reloaded from some other medium.

The symmetry of equiprobable processes is broken by processes that are able to reproduce themselves. This achieved biologically by organism carrying itself in two forms which we call genotype and phenotype. The genotype is a detailed specification of the organism usually encoded in strands of DNA or RNA. It serves as a clean copy of the organism which is used to specify new phenotypes which are free of the errors which will eventually kill their parents. Often these copies are not perfect, introducing variation into the offspring.

Charles Darwin knew that centuries of selective breeding created a wide variety of domesticated animals and plants. He imagined that natural selection could do the same in the wild world. In the domestic realm the breeder seeks to mate individuals that are tending in the desired direction. This selection is guided with an eye on the desired end result. Darwin: The Origin of Species

In the wild mating is rather more random, since in the wild there is often room for choice by the participants themselves. The more severe selection acts upon the children, who must survive infancy, grow, prosper, mate and reproduce if they are to pass on the traits they received from their parents

In a divine world, natural selection is effectively a judgement from God, and it operates at all scales from the quantum of action to the Universe as a whole. Every time I hit a nail, my action is judged. If I do not hit it squarely, it may bend, an error which must be repaired. In each case, the individual is judged by its interaction with its environment.

Our scientific understanding of the tree of life exhibits the work of natural selection. Darwin had far less information to go on, but was able to discern the general pattern and publish it in The Origin of Species. His work stirred even greater controversy when he applied his theoryto The Descent of Man. This work was understood to directly contradict the Catholic doctrine that God creates a new immortal soul for every child born. Darwin; The Descent of Man, Holy See: Catechism of the Catholic Church, § 382

Variation and selection enable organisms to adapt to environmental change. If the change is so fast that species cannot keep up it will probably become extinct through failure to reproduce. Human anthropocene activity, by making swift and massive changes in the planetary environment, has been responsible for tens thousands of extinctions since our global footprint began to explode.

In the theological language suggested here, adaptation means listening to God and acting on what we hear. We listen through the sciences that tell us how God works, and the arts that communicate these ideas in images, symbols and machinery.

Variation and selection apply in all environments where competition is induced by a shortage of resources. In the transfinite network, the limiting resource is processing power. In general survivors are the ones who can gather abundant resources or learn to live with less. Creatures adapt through the slow process of genetic evolution, the faster process of the epigenetic emergence of instincts, and the fastest process, cultural development.

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11. From Trinity to transfinite complexity

The notion of fixed points in the divine dynamics is not new. It is implicit in the Catholic doctrine of the Trinity. The transfinite network expands this idea from trinity to transfinity.

The first work in this direction was done by the Fathers of the Church in their attempts to reconcile the Christian notion that God is three Persons, Father, Son and Spirit, with the unitary God embodied in Jewish culture.

The first person to attempt an understanding of the relationship between the Father and the Son was the evangelist John, who wrote: In the beginning was the Word, and the Word was with God, and the Word was God (John 1:1). This idea was developed by Augustine, Aquinas and Lonergan to produce what we might call the standard model of the Trinity. Lonergan: Verbum, Augustine: The Trinity, Aquinas, Summa, I, 27, 1: Is there procession in God?, Lonergan: The Triune God, Doctrines, Lonergan: The Triune God, Systematics

The Son proceeds from the Father rather as the mental word proceeds from the mind. From an information processing point of view, the procession of the Word is an act of copying, producing an identical but distinct entity. In Aquinas' model the persons of the Trinity are differentiated by the real relationships between them.

Here we view the Trinity as an instance of the simplest element of a communication network. The unit of communication is two sources connected by communication channel between them. In the Trinity the sources are Father and Son, and the channel between them is the Spirit, their love for one another. Or we might say that Father and Son are two fixed points in God, and the Spirit is their dynamical relationship, or that the Father and Son are fermions, and the Spirit is the boson through which they communicate.

In software terms we have a parent, a child and a communication protocol. Such units can communicate with one another to form more complex networks. We model these networks as computer networks, because there is plenty of clear mathematical theory available in this field and it makes intuitive sense because we are natural communicators. The network technology we have developed in the last century provides us with concrete examples of the abstract ideas in question 50. The explosive growth of the internet gives us a slow motion glimpse of the big bang, the initial stages in the development of a network of fixed points in the divine dynamics. Tanenbaum: Computer Networks

A network is a structure of processors and memory. The processors read data from memory, possibly transform it in some way and write it back into memory. Here we imagine that the world is revealed through the emergence of fixed points in the divine dynamics so that there is no real distinction between God and the world. The natural numbers are infinite because we can always add 1. The transfinite numbers grow without limit because we can always create the set of permutations of the biggest one we have in hand. This network can therefore be imagined big enough to be put into correspondence with any physical Universe, no matter how large. The transfinite space expands as the Universe expands in both size and complexity.

Like everyday computer networks, the transfinite computer network is layered. A message passing between two users goes from the sender down through many layers of software to the physical layer, is transmitted through the physical layer to the recipient machine, and then moves up through software layers to the receiving user. This process is transparent to the users. From this point of view, all communication in the world is rooted in God.

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12. Human symmetry and social equilibrium

The Catholic history of salvation claims that we are fallen species, the harmony between body and spirit broken by an ancestral sin. We are now universally inclined toward evil, and must be controlled, by violence if necessary; spare the rod . . . .

There is no evidence for such an original sin. What we do know is that we adapt to the environment in which we find ourselves. If we are treated badly we will probably react badly. We are one of the billions of species selected by the divine Universe to enjoy a few million generations in the Sun, and we are as perfect as possible given the constraints of consistency.

We expect a new scientific theory to adequately explain the existing data and serve as a foundation to look for more results. General relativity and quantum mechanics both operate in enormous and complex mathematical spaces which we have just begun to explore. Here we understand these spaces to be subsets of a transfinite network.

Cosmological history describes the complexification of the world from the initial singularity to its current complexity. We measure the complexity of the system by counting its states, to yield a number we call entropy. The second law of thermodynamics tells us that on the whole entropy tends to increase. This increase is a numerical expression of the creative power of the Universe which we understand to be the development of more complex fixed points in the divine dynamics as we move into higher transfinite spaces.

We are among these fixed points. I was created about 70 years ago and will be annihilated within the next thirty years or so. In the interim I have had a life experience, the whole set of external influences that have shaped me. On this hypothesis, all these messages from my environment are revelations of God and my actions responses to these messages.

We understand the world to be governed by divine law, which is in effect the essence of God. Here, as noted above, we think in terms of symmetry. Symmetry is built into the layered structure of networks. The lower layers appear as symmetries in the higher layers. The most basic symmetry, present in every layer above quantum mechanics, is conservation of energy.

Symmetry is closely related to abstraction and meaning. A species is a symmetry, and we can take advantage of this to talk about all them at once: ‘all Homo sapiens are mortal.’ We can point to different systems in our environment and classify them into human and not-human.

Symmetry also gives us control. Many societies are built on the shared understanding that we are all equal before the law so that legislators can bind everybody with a simple statement of law.

Computing processes are bound to the real world by meanings or correspondences. So the social security system processes each of us under a unique social security identity which connects the data in the system to the individual. Manipulations of this finite set of data effectively manipulates the transfinite state of each individual referenced.

The key to this control is the global ‘symmetry with respect to complexity’ exhibited by set theory. As Cantor found, the concept of set embraces sets of all finite, infinite and transfinite cardinal and ordinal numbers. Insofar as we can develop both quantum mechanics and the dynamics of mathematics from set theory, we should expect to find some similarities.

In particular, this leads to the idea that the human psychological phenomenon of insight and the quantum mechanical phenomena of measurement are instances of the same universal process. From this point of view, human intelligence is simply an instance of the creative process that makes the world what it is. This identification is made possible by the symmetry of networks with respect to complexity.

A computer is a deterministic process with a beginning and sometimes an end. We may see a computer as a logical continuum, every step being determined by the outcome of the step before. Like a continuum the dynamics of a computer is invisible. We see only the output and we can only ever see a source when it emits a signal. Feynman was first to conjecture that quantum mechanics represents a computational process, so initiating a large areas of contemporary research in mathematics, physics and logic. Nielsen & Chuang: Quantum Computation and Quantum Information

How do these ideas couple to the political tasks of human governance? What we are looking for is an implementation of divine providence designed to run in the human layer of the transfinite network. This process is already well under way. We have all the necessary technology to live sustainably on Earth, but the deployment of this technology is greatly hindered by political fragmentation of our species which can be traced back to religious and theological fragmentation.

The scientific method was first applied to politics by Aristotle, who is said to have collected the constitutions of hundreds of city states. This work has continued so that we have a relatively clear understanding of how to manage nations with respect to human rights, the rule of law, and sustainable exploitation of the resources necessary for our existence. In particular we are fully aware of the speed with which uncontrolled political, military and financial power and cults of personality can destroy a nation. Acemoglu & Robinson: Why Nations Fail

The problem is to work out ways to make a set of self interested agents work for their common good. The short answer is love one another, look out for one another, work for the collective well-being. The long answer is to implement this love in a way that respects human symmetry.

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13. Conclusion: Scientific method and justice

And ye shall know the truth, and the truth shall make you free (John 8:32, KJV). Conversely, error imprisons us. Scientists use the natural world as their touchstone, as Christian theologians use the Bible. Christians believe the Bible, and the task of theologians is to decode this information using the tools approved by their institutional masters. The Biblical ‘deposit of faith’ is less than a million words. Although Christian theologians allow that world teaches us something about God, this knowledge is severely limited and definitely not the full story of our existence.

This essay proposes an alternative path to God, that God is open to scientific exploration. This alternative is much richer: all experience, not just the words of the Bible, is experience of God.

The Church made a dangerous move during the "Modernist crisis" when it decided to deprecate the credibility of science. As it has done more recently in the child sexual abuse crisis, it took refuge in its power, allowing political expediency to replace evidence based information. Talar quotes Bernard Gaudeau to give us insight into the Church's fear of scientifically revealed reality:

Indeed the facts demonstrate that the doctrine of Modernism leads logically and fatally not only to the destruction of the element specific to Catholicism as opposed to Protestantism (dogmatic infallibility of the Church and of the Pope); not only to the destruction of what may be call the generic element of Christianity itself (real divinity of Jesus Christ, biblical inspiration, miracle and other supernatural realities admitted by primitive and orthodox Protestantism) but the facts demonstrate that the doctrine of Modernism leads logically and fatally to the destruction in humanity of the very conception of a real God distinct from the world, personal and transcendent, that is the very foundation of all religious belief, every religious idea, all religious meaning, of all religions. Riley Stuart: Royal commission into sexual abuse: Catholic Church 'sacrificed the welfare of children', Talar, op cit page 494

A sense of betrayal pervades Pascendi, surfacing at intervals in the portrayal of the character and motivation of Modernists. Their views are called "a delirium,", "insanity," and "audacious sacrilege." Modernism is a "monstrosity," and its proponents characterized as guilty of "pride and obstinacy," as having cast off all "sense of modesty," as being the "most pernicious of all adversaries of the Church" because they "lay the axe to the root, not the branches" while working from within. Talar op cit page 404

This essay suggests that there is a consistent alternative view to the Catholic dogma that God is a mysterious other known only to the Church. The data of theology are human experience, that is they lie in human consciousness and we have access to these data through the words and actions of their possessors. At present, many people on the planet are having negative experiences, and in most cases this negativity is the result of the actions of the others, predominantly intimate partners, warlords, and all those others who get their pleasures by inflicting pain on others.

I feel that the Church is a major player in this task, disenfranchising the female half of the population, maintaining that we are sinners living in a broken world, denying current reality infavour of some future world and a host of other errors. At present the Church wishes to dictate our beliefs. Instead it might do better to base its theology on dialogue with individual people in order to elicit their experiences. Humanity is the layer upon which the Church is built. If the properties of a lower layer are not respected, the layer built upon them will eventually fail.

The alternative to an effective parliament, however, is civil war. At present religious forces are verging on civil war, and it is imperative that the Catholic Church, the wealthiest and most powerful religion on the planet, open itself to a reform which will unify the theological and religious forces on the Earth for the welfare of the planet.

The lasting consequences child sexual assault are psychological and experience has shown that such injury can have lifelong effects. Teaching unverified materials as fact, particularly pernicious doctrines that we are all sinners, can also cause psychological damage. In my own case, I experienced about 20 years of pain until I reached my forties. At that point I had overcome much of my Catholic indoctrination and begun to replace it with something representing a scientific theory. My coming out is recorded at A theory of peace. Jeffrey Nicholls

From the above discussion, I conclude that the Church must listen more closely to science. Pope Francis has already made a very good start by recognizing the reality of human influence on global climate. Pope Francis: On the Care of Our Common Home


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Further reading


Click on the "Amazon" link below each book entry to see details of a book (and possibly buy it!)

Acemoglu, Daron, and James Robinson, Why Nations Fail: The Origins of Power, Prosperity and Poverty, Crown Business 2012 "Some time ago a little-known Scottish philosopher wrote a book on what makes nations succeed and what makes them fail. The Wealth of Nations is still being read today. With the same perspicacity and with the same broad historical perspective, Daron Acemoglu and James Robinson have retackled this same question for our own times. Two centuries from now our great-great- . . . -great grandchildren will be, similarly, reading Why Nations Fail." —George Akerlof, Nobel laureate in economics, 2001  
Ashby, W Ross, An Introduction to Cybernetics, Methuen 1964 'This book is intended to provide [an introduction to cybernetics]. It starts from common-place and well understood concepts, and proceeds step by step to show how these concepts can be made exact, and how they can be developed until they lead into such subjects as feedback, stability, regulation, ultrastability, information, coding, noise and other cybernetic topics' 
Augustine, Saint, and Edmond Hill (Introduction, translation and notes), and John E Rotelle (editor), The Trinity, New City Press 1991 Written 399 - 419: De Trinitate is a radical restatement, defence and development of the Christian doctrine of the Trinity. Augistine's book has served as a foundation for most subsequent work, particularly that of Thomas Aquinas.  
Boland, Andre, La crise moderniste hier et aujourdhui, Beauchesne 1980  
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.' 
Chaisson, Eric J., Cosmic Evolution: The Rise of Complexity in Nsture, Harvard University Press 2002 'In Cosmic Evolution Chaisson addresses some of the most basic issues we can contemplate: the origin of matter and the origin of life, and the ways matter, life, and radiation interact and change with time. Guided by notions of beauty and symmetry, by the search for simplicity and elegance, by the ambition to explain the widest range of phenomena with the fewest possible principles, Chaisson designs for us an expansive yet intricate model depicting the origin and evolution of all material structures. He shows us that neither new science nor appeals to nonscience are needed to understand the impressive hierarchy of the cosmic evolutionary story, from quark to quasar, from microbe to mind.' 
Cohen, Paul J, Set Theory and the Continuum Hypothesis, Benjamin/Cummings 1966-1980 Preface: 'The notes that follow are based on a course given at Harvard University, Spring 1965. The main objective was to give the proof of the independence of the continuum hypothesis [from the Zermelo-Fraenkel axioms for set theory with the axiom of choice included]. To keep the course as self contained as possible we included background materials in logic and axiomatic set theory as well as an account of Gödel's proof of the consistency of the continuum hypothesis. . . .'  
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.' 
Darwin, Charles, and Harriet Ritvo (Introduction), The Variation of Animals and Plants Under Domestication (Foundations of Natural History) , Johns Hopkins University Press 1998  
Darwin, Charles, The Descent of Man, and Selection in Relation to Sex, Penguin Classics 1871, 2004 'No book made a greater impact on the intellectual world of its first Victorian readers nor has had such an enduring influence on our thinking on science, literature, theology and philosophy. In The Descent of Man, Darwin addresses the crucial question of the origins, evolution and racial divergence of mankind, that he had deliberately left out of On the Origin of Species. And the evidence he presents forces us to question what it is that makes us uniquely human.' 
Dauben, Joseph Warren, Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton University Press 1990 Jacket: 'One of the greatest revolutions in mathematics occurred when Georg Cantor (1843-1918) promulgated his theory of transfinite sets. . . . Set theory has been widely adopted in mathematics and philosophy, but the controversy surrounding it at the turn of the century remains of great interest. Cantor's own faith in his theory was partly theological. His religious beliefs led him to expect paradox in any concept of the infinite, and he always retained his belief in the utter veracity of transfinite set theory. Later in his life, he was troubled by attacks of severe depression. Dauben shows that these played an integral part in his understanding and defense of set theory.' 
Einstein, Albert, and Robert W Lawson (translator) Roger Penrose (Introduction), Robert Geroch (Commentary), David C Cassidy (Historical Essay) , Relativity: The Special and General Theory, Pi Press 2005 Preface: 'The present book is intended, as far as possible, to give an exact insight into the theory of relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. ... The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated.' page 3  
Feferman, Solomon, and John W Dawson, Stephen C Kleene, Gregory H Moore, Robert M Solovay, Jean van Heijenoort (editors), 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. ...' 
Feynman, Richard, QED: The Strange Story of Light and Matter, Princeton UP 1988 Jacket: 'Quantum electrodynamics - or QED for short - is the 'strange theory' that explains how light and electrons interact. Thanks to Richard Feynmann and his colleagues, it is also one of the rare parts of physics that is known for sure, a theory that has stood the test of time. . . . In this beautifully lucid set of lectures he provides a definitive introduction to QED.' 
Feynman, Richard, Feynman Lectures on Gravitation, Westview Press 2002 Amazon Editorial Reviews Book Description 'The Feynman Lectures on Gravitation are based on notes prepared during a course on gravitational physics that Richard Feynman taught at Caltech during the 1962-63 academic year. For several years prior to these lectures, Feynman thought long and hard about the fundamental problems in gravitational physics, yet he published very little. These lectures represent a useful record of his viewpoints and some of his insights into gravity and its application to cosmology, superstars, wormholes, and gravitational waves at that particular time. The lectures also contain a number of fascinating digressions and asides on the foundations of physics and other issues. Characteristically, Feynman took an untraditional non-geometric approach to gravitation and general relativity based on the underlying quantum aspects of gravity. Hence, these lectures contain a unique pedagogical account of the development of Einstein's general theory of relativity as the inevitable result of the demand for a self-consistent theory of a massless spin-2 field (the graviton) coupled to the energy-momentum tensor of matter. This approach also demonstrates the intimate and fundamental connection between gauge invariance and the principle of equivalence.' 
Fortun, Mike, and Herbert J Bernstein, Muddling Through: Pursuing Science and Truths in the Twenty-First Century, Counterpoint 1998 Amazon editorial review: 'Does science discover truths or create them? Does dioxin cause cancer or not? Is corporate-sponsored research valid or not? Although these questions reflect the way we're used to thinking, maybe they're not the best way to approach science and its place in our culture. Physicist Herbert J. Bernstein and science historian Mike Fortun, both of the Institute for Science and Interdisciplinary Studies (ISIS), suggest a third way of seeing, beyond taking one side or another, in Muddling Through: Pursuing Science and Truths in the 21st Century. While they deal with weighty issues and encourage us to completely rethink our beliefs about science and truth, they do so with such grace and humor that we follow with ease discussions of toxic-waste disposal, the Human Genome Project, and retooling our language to better fit the way science is actually done.' 
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. 
Hawking, Steven W, and G F R Ellis, The Large Scale Structure of Space-Time , Cambridge UP 1975 Preface: Einstein's General Theory of Relativity . . . leads to two remarkable predictions about the universe: first that the final fate of massive stars is to collapse behind an event horizon to form a 'black hole' which will contain a singularity; and secondly that there is a singularity in our past which constitutes, in some sense, a beginning to our universe. Our discussion is principally aimed at developing these two results.' 
Lonergan, Bernard J F, and Robert M. Doran, Frederick E. Crowe (eds), Verbum : Word and Idea in Aquinas (Collected Works of Bernard Lonergan volume 2), University of Toronto Press 1997 Jacket: 'Verbum is a product of Lonergan's eleven years of study of the thought of Thomas Aquinas. The work is considered by many to be a breakthrough in the history of Lonergan's theology ... . Here he interprets aspects in the writing of Aquinas relevant to trinitarian theory and, as in most of Lonergan's work, one of the principal aims is to assist the reader in the search to understand the workings of the human mind.' 
Lonergan, Bernard J F, and Michael G Shields (translator), Robert M Doran & H Daniel Monsour (editors), The Triune God: Systematics, University of Toronto press 2007 Translated from De Deo Trino: Pars systematica (1964) by Michael G Shields. Amazon Product Description 'Buried for more than forty years in a Latin text written for seminarian students at the Gregorian University in Rome, Bernard Lonergan's 1964 masterpiece of systematic-theological writing, De Deo trino: Pars systematica, is only now being published in an edition that includes the original Latin along with an exact and literal translation. De Deo trino, or The Triune God, is the third great installment on one particular strand in trinitarian theology, namely, the tradition that appeals to a psychological analogy for understanding trinitarian processions and relations. The analogy dates back to St Augustine but was significantly developed by St Thomas Aquinas. Lonergan advances it to a new level of sophistication by rooting it in his own highly nuanced cognitional theory and in his early position on decision and love. Suggestions for a further development of the analogy appear in Lonergan's late work, but these cannot be understood and implemented without working through this volume. This is truly one of the great masterpieces in the history of systematic theology, perhaps even the greatest of all time.' 
Lonergan, Bernard J F, and Robert M Doran and H Daniel Monsour (eds), The Triune God: Doctrines (Volume 11 of Collected Works), University of Toronto Press 2009 Bernard Lonergan (1904-1984), a professor of theology, taught at Regis College, Harvard University, and Boston College. An established author known for his Insight and Method in Theology, Lonergan received numerous honorary doctorates, was a Companion of the Order of Canada in 1971 and was named as an original members of the International Theological Commission by Pope Paul VI. 
Neuenschwander, Dwight E, Emmy Noether's Wonderful Theorem, Johns Hopkins University Press 2011 Jacket: A beautiful piece of mathematics, Noether's therem touches on every aspect of physics. Emmy Noether proved her theorem in 1915 and published it in 1918. This profound concept demonstrates the connection between conservation laws and symmetries. For instance, the theorem shows that a system invariant under translations of time, space or rotation will obey the laws of conservation of energy, linear momentum or angular momentum respectively. This exciting result offers a rich unifying principle for all of physics.' 
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. 
Popper, Karl Raimund, Conjectures and Refutations: The Growth of Scientific Knowledge, Routledge and Kegan Paul 1972 Preface: 'The way in which knowledge progresses, and expecially our scientific knowledge, is by unjustified (and unjustifiable) anticipations, by guesses, by tentative solutions to our problems, by conjectures. These conjectures are controlled by criticism; that is, by attempted refutations, which include severely critical tests.' [p viii]  
Streater, Raymond F, and Arthur S Wightman, PCT, Spin, Statistics and All That, Princeton University Press 2000 Amazon product description: 'PCT, Spin and Statistics, and All That is the classic summary of and introduction to the achievements of Axiomatic Quantum Field Theory. This theory gives precise mathematical responses to questions like: What is a quantized field? What are the physically indispensable attributes of a quantized field? Furthermore, Axiomatic Field Theory shows that a number of physically important predictions of quantum field theory are mathematical consequences of the axioms. Here Raymond Streater and Arthur Wightman treat only results that can be rigorously proved, and these are presented in an elegant style that makes them available to a broad range of physics and theoretical mathematics.' 
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.'  
von Neumann, John, and Robert T Beyer (translator), Mathematical Foundations of Quantum Mechanics, Princeton University Press 1983 Jacket: '. . . a revolutionary book that caused a sea change in theoretical physics. . . . JvN begins by presenting the theory of Hermitean operators and Hilbert spaces. These provide the framework for transformation theory, which JvN regards as the definitive form of quantum mechanics. . . . Regarded as a tour de force at the time of its publication, this book is still indispensable for those interested in the fundamental issues of quantum mechanics.' 
Chaitin, Gregory J, "Godel's Theorem and Information", International Journal of Theoretical Physics,, 22, , 1982, page 941. back
Talar, Charles J. T., ""The Synthesis of All Heresies" - 100 years on", Theological Studies, 68, 3, September 2007, page 491-514 . 'The condemnation of Roman Catholic Modernism in 1907 was a traumatic event—in the dual sense that it reflected the traumatic impact of intellectual and political modernity on the Church, and in the induced climate of repressive reaction that affected Catholic scholarship for decades thereafter. The issues raised by the Modernists form an integral part of the trajectory of 20th-century theology.'. 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 some 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 of a real or computable variable, computable predicates and so forth. . . . ' back
Albert Einstein, On the Electrodynamics of Moving Bodies, An english translation of the paper that founded Special relativity. 'Examples of this sort, [in the contemporary application of Maxwell's electrodynamics to moving bodies] together with the unsuccessful attempts to discover any motion of the earth relatively to the ``light medium,'' suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.' back
Aquinas , Summa Theologiae I, 1, 2: Whether sacred doctrine is a science?, 'I answer that, Sacred doctrine is a science. We must bear in mind that there are two kinds of sciences. There are some which proceed from a principle known by the natural light of intelligence, such as arithmetic and geometry and the like. There are some which proceed from principles known by the light of a higher science: thus the science of perspective proceeds from principles established by geometry, and music from principles established by arithmetic. So it is that sacred doctrine is a science because it proceeds from principles established by the light of a higher science, namely, the science of God and the blessed.' back
Aquinas, Summa, I, 27, 1, Is there procession in God?, 'As God is above all things, we should understand what is said of God, not according to the mode of the lowest creatures, namely bodies, but from the similitude of the highest creatures, the intellectual substances; while even the similitudes derived from these fall short in the representation of divine objects. Procession, therefore, is not to be understood from what it is in bodies, either according to local movement or by way of a cause proceeding forth to its exterior effect, as, for instance, like heat from the agent to the thing made hot. Rather it is to be understood by way of an intelligible emanation, for example, of the intelligible word which proceeds from the speaker, yet remains in him. In that sense the Catholic Faith understands procession as existing in God.' back
Aristotle, Metaphysics, Book XII, vii, 'But since there is something which moves while itself unmoved, existing actually, this can in no way be otherwise than as it is. For motion in space is the first of the kinds of change, and motion in a circle the first kind of spatial motion; and this the first mover produces. The first mover, then, exists of necessity; and in so far as it exists by necessity, its mode of being is good, and it is in this sense a first principle.' 1072b3 sqq back
Aristotle_2, Metaphysics, Book XII, vii, 'But since there is something which moves while itself unmoved, existing actually, this can in no way be otherwise than as it is. For motion in space is the first of the kinds of change, and motion in a circle the first kind of spatial motion; and this the first mover produces. The first mover, then, exists of necessity; and in so far as it exists by necessity, its mode of being is good, and it is in this sense a first principle.' 1072b3 sqq 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
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
Holy See, Catechism of the Catholic Church, Paragraph 382: ' "Man, though made of body and soul, is a unity" (GS 14 § 1). The doctrine of the faith affirms that the spiritual and immortal soul is created immediately by God.' back
Jeffrey Nicholls, A Theory of Peace, ' It is the task of religion to reconcile the apparent contradiction between good and evil. It is that reconciliation, deeply and thoroughly believed, that gives us peace. The power of Christianity, the power of any religion, is measured by the serenity of its martyrs. I became deeply immersed in this Christian peace within the walls of a Catholic monastery. All my beliefs and training pointed to the fact that I was on my way to heaven by the most heavenly possible route. But I was deluded. There came a time when the order could no longer tolerate the way I was thinking. I was ejected from paradise.' back
John Palmer - Parmenides, Parmenides (Stanford Encyclopedia of Philosophy), First published Fri Feb 8, 2008 'Immediately after welcoming Parmenides to her abode, the goddess describes as follows the content of the revelation he is about to receive:
You must needs learn all things,/ both the unshaken heart of well-rounded reality/ and the notions of mortals, in which there is no genuine trustworthiness./ Nonetheless these things too will you learn, how what they resolved/ had actually to be, all through all pervading. (Fr. 1.28b-32) ' back
Lagrangian - Wikipedia, Lagrangian - Wikipedia, the free encyclopedia, 'The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as Lagrangian mechanics. In classical mechanics, the Lagrangian is defined as the kinetic energy, T, of the system minus its potential energy, V. In symbols, L = T - V. ' back
Lorentz transformation - Wikipedia, Lorentz transformation - Wikipedia, the free encyclopedia, 'In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik Lorentz. It reflects the surprising fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events.' back
Particle Data Group. Lawrence Berkeley National Laboratory, The Particle Adventure, The Particle Data Group of Lawrence Berkeley National Laboratory presents an award winning interactive tour of quarks, neutrinos, antimatter, extra dimensions, dark matter, accelerators and particle detectors. back
Plato, Parmenides, 'Parmenides By Plato Written 370 B.C.E Translated by Benjamin Jowett Persons of the Dialogue CEPHALUS ADEIMANTUS GLAUCON ANTIPHON PYTHODORUS SOCRATES ZENO PARMENIDES ARISTOTELES Scene Cephalus rehearses a dialogue which is supposed to have been narrated in his presence by Antiphon, the half-brother of Adeimantus and Glaucon, to certain Clazomenians. back
Pope Francis, Laudato Si': On care of our common home, '1. “LAUDATO SI’, mi’ Signore” – “Praise be to you, my Lord”. In the words of this beautiful canticle, Saint Francis of Assisi reminds us that our common home is like a sister with whom we share our life and a beautiful mother who opens her arms to embrace us. “Praise be to you, my Lord, through our Sister, Mother Earth, who sustains and governs us, and who produces various fruit with coloured flowers and herbs”. 2. This sister now cries out to us because of the harm we have inflicted on her by our irresponsible use and abuse of the goods with which God has endowed her. We have come to see ourselves as her lords and masters, entitled to plunder her at will. The violence present in our hearts, wounded by sin, is also reflected in the symptoms of sickness evident in the soil, in the water, in the air and in all forms of life. This is why the earth herself, burdened and laid waste, is among the most abandoned and maltreated of our poor; she “groans in travail” (Rom 8:22). We have forgotten that we ourselves are dust of the earth (cf. Gen 2:7); our very bodies are made up of her elements, we breathe her air and we receive life and refreshment from her waters.' back
Richard Zach, Hilbert's Program (Stanford Encyclopedia of Philosophy), 'In the early 1920s, the German mathematician David Hilbert (1862-1943) put forward a new proposal for the foundation of classical mathematics which has come to be known as Hilbert's Program. It calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof itself was to be carried out using only what Hilbert called "finitary" methods. The special epistemological character of finitary reasoning then yields the required justification of classical mathematics.' back
Riley Stuart, Royal commission into sexual abuse: Catholic Church 'sacrificed the welfare of children', 'Dr Thomas Doyle, a United States canon lawyer and expert in areas of sexual abuse by clergy, has been giving evidence at the Royal Commission into Institutional Responses to Child Sexual Abuse in Sydney. He claimed Pope John Paul II knew children were being sexually abused by priests and the Catholic Church attempted to cover up allegations. Dr Doyle said victims and their families were coerced into keeping quiet about the abuse, even threatened with ex-communication.' 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 Theologiae I, 1, 2, Is sacred doctrine is a science?, 'I answer that, Sacred doctrine is a science. We must bear in mind that there are two kinds of sciences. There are some which proceed from a principle known by the natural light of intelligence, such as arithmetic and geometry and the like. There are some which proceed from principles known by the light of a higher science: thus the science of perspective proceeds from principles established by geometry, and music from principles established by arithmetic. So it is that sacred doctrine is a science because it proceeds from principles established by the light of a higher science, namely, the science of God and the blessed.' back
Thomas Aquinas Summa Theologiae I, 18, 3, Is life properly attributed to God?, Life is in the highest degree properly in God. In proof of which it must be considered that since a thing is said to live in so far as it operates of itself and not as moved by another, the more perfectly this power is found in anything, the more perfect is the life of that thing. ' back
Werner Heisenberg, Quantum-theoretical re-interpretation of kinematic and mechanical relations, 'The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable.' back
William K. Allard, The Brouwer Fixed Point Theorem, 'High on the list of my priorities is Stokes' Theorem. So we will be sure to cover that material which is necessary to prove Stokes' Theorem. Using Stokes' theorem we will prove the Brouwer Fixed Point Theorem.' back
Wojciech Hubert Zurek, Quantum origin of quantum jumps: breaking of unitary symmetry induced by information transfer and the transition from quantum to classical, 'Submitted on 17 Mar 2007 (v1), last revised 18 Mar 2008 (this version, v3)) "Measurements transfer information about a system to the apparatus, and then further on -- to observers and (often inadvertently) to the environment. I show that even imperfect copying essential in such situations restricts possible unperturbed outcomes to an orthogonal subset of all possible states of the system, thus breaking the unitary symmetry of its Hilbert space implied by the quantum superposition principle. Preferred outcome states emerge as a result. They provide framework for the ``wavepacket collapse'', designating terminal points of quantum jumps, and defining the measured observable by specifying its eigenstates. In quantum Darwinism, they are the progenitors of multiple copies spread throughout the environment -- the fittest quantum states that not only survive decoherence, but subvert it into carrying information about them -- into becoming a witness.' back is maintained by The Theology Company Proprietary Limited ACN 097 887 075 ABN 74 097 887 075 Copyright 2000-2017 © Jeffrey Nicholls