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vol 6: Essays

Is the Universe divine? (June 2015)

Table of contents

0: Abstract

1: Introduction
   1.1: Modelling God 1: Antiquity and the Patristic era
   1.2: Modelling God 2: The Middle Ages
   1.3: Modelling God 3: Recent developments

2: Fixed point theory

3: Quantum mechanics and fixed points

4: Mathematical theology

5: Logical continuity: the mathematical theory of computation

6: The transfinite digital network as a model of revelation

7: Evolution

8: Theological implications:
   8.1 Theology becomes a science, embracing all the sciences
   8.2 God is our judge
   8.. In the divine Universe, all experience is experience of God
   8.4 The Universe is one, that is a consistent whole
   8.5 Religion, following theology may become univeralized by human symmetry
   8.6 Digitizing the Universe points to an unbounded future

9: Practical implications
   9.1 The Roman Catholic Church
   9.2 The global environment
   9.3 The structure of nations
   9.4 Education and indoctrination

0. Abstract

It remains a fundamental and widely held belief in the Western Christian tradition that God is completely other than the visible Universe. Christian theologians defend this belief on two general grounds. The first is faith. From a scientific point of view a belief held solely on faith may be an hypothesis in need of testing that gains momentum only insofar as it is found consistent with observation. The second is proofs for the existence of God, such as the 'five ways' presented by Thomas Aquinas. These proofs are model dependent. Aquinas used contemporary understandings of the nature of God and the world to demonstrate that they could not be identified.

Here I propose that our modern scientific understanding of the Universe enables us to accept that the Universe is, for all practical purposes, divine. If this is so we exist within God and it is probable that all our experiences are experiences of God. Given an observable divinity, theology, the science of God, can become a real science.

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1: Introduction

1.1: Modelling God: Antiquity and the Patristic Era

At some point in the distant past, people became aware of their ability to control the world and so increase their probability of survival. This became possible because although life is pretty chaotic, there are many predictable events in our lives like the sunrise, the germination of seeds and the behaviour of animals.

This observation may have led to an idea of God, the supreme creator and controller of our destinies. Many of the first Gods were modelled on the warriors, warlords, monarchs and emperors who controlled much of ancient life. Of these, two sets of Gods have had a big impact in the Western world: the Hebrew Gods of the ancient near east culminating in Yahweh, and the Gods of Greek mythology who eventually coalesced into Plato's One and Aristotle's heavenly First Mover. Miles: God, A Biography

The Evangelists and Fathers of the Church who developed Christian theology tended to combine these ancient ideas to produce a God who is both a personality in the human sense, and a powerful transcendent being responsible for all that happens. One of the first known thinkers to move from a personal to a metaphysical view of God was Xenophanes (c. 570 - 480 bce), who wrote:

Homer and Hesiod have attributed to the gods
all sorts of things which are matters of reproach and censure among men:
theft, adultery and mutual deceit. James Lesher: Xenophanes (Stanford Encyclopedia of Philosophy)

The next recorded step in the scientific direction was taken by Parmenides (c. 515 - ? bce) who identified a central scientific problem: how can we write true texts about a world which is continually changing? We have fragments of a text in which Parmenides explains the solution to this problem revealed to him by a Goddess who distinguished between 'the way of truth' and 'the way of opinion'. The way of truth is grounded in 'what is', and what is is now, complete all at once, one and continuous . . . Nor is it divisible, since it is all alike; nor is there any more or less of it in one place which might prevent it from holding together, but all is full of what is. James Lesher; Parmenides (Stanford Encyclopedia of Philosophy)

This notion was taken up by Plato and eventually became part of the Christian model of God. This model was developed within the Church at various councils and became the foundation for the Christian creeds like the Nicene Creed which remain the foundations of Christian belief.

1.2: Modelling God: The Middle Ages

The metaphysical foundation for the early development of Christianity came principally from Plato and his followers and little was heard of Aristotle. The situation changed in the early middle ages when the works of Aristotle, translated from Greek and Arabic into Latin, began to enter European universities.

The most profound influence of Aristotelian thought on Christian theology came through the work of Thomas of Aquino (1225 - 1274), particularly his Summa Theologiae. Thomas probably first encountered Aristotle at university in Naples.

Later he was able to use the complete translation of Aristotle's writings made by William of Moerbecke, a fellow Dominican. Thomas also benefitted from a period as a student of Albert the Great, the foremost medieval authority on Aristotle. Using Aristotle, Thomas constructed the model of God that is still standard in the Catholic Church and enjoined in Canon Law. Code of Canon Law, 252 §3

Aristotle's doctrine of potency and act plays the central role in Thomas' theology. At the beginning of the Summa, Thomas provides five ways of proving the existence of God. The first is taken directly from Aristotle's discussion of the unmoved mover in the Metaphysics: Aristotle: Metaphysics, 1073b3 sqq

Does God exist?

I answer that the existence of God can be proved in five ways.

The first and more manifest way is the argument from motion. It is certain, and evident to our senses, that in the world some things are in motion. Now whatever is in motion is put in motion by another, for nothing can be in motion except it is in potentiality to that towards which it is in motion; whereas a thing moves inasmuch as it is in act. For motion is nothing else than the reduction of something from potentiality to actuality. But nothing can be reduced from potentiality to actuality, except by something in a state of actuality. Thus that which is actually hot, as fire, makes wood, which is potentially hot, to be actually hot, and thereby moves and changes it. Now it is not possible that the same thing should be at once in actuality and potentiality in the same respect, but only in different respects. For what is actually hot cannot simultaneously be potentially hot; but it is simultaneously potentially cold. It is therefore impossible that in the same respect and in the same way a thing should be both mover and moved, i.e. that it should move itself. Therefore, whatever is in motion must be put in motion by another. If that by which it is put in motion be itself put in motion, then this also must needs be put in motion by another, and that by another again. But this cannot go on to infinity, because then there would be no first mover, and, consequently, no other mover; seeing that subsequent movers move only inasmuch as they are put in motion by the first mover; as the staff moves only because it is put in motion by the hand. Therefore it is necessary to arrive at a first mover, put in motion by no other; and this everyone understands to be God. Aquinas Summa I, 2, 3: Does God exist?

Essentially Aquinas, proves that God is not the World because God, like the First mover, is pure actuality, and the World contains potential. Here he differs from Aristotle in that Aristotle seemed to think that the First Mover is part of the world rather than other than the World. Using the fact that God is pure actuality, Thomas goes on to establish the standard properties of God, absolute simplicity, goodness, perfection, infinity, omnipresence, immutability, eternity, unity, omnipotence, life, omniscience and so on (Aquinas op. cit. qq 3-26).

Thomas' God is consistent with the properties of the divinity developed by the early Christians and documented in their creeds. From a theological point of view, the principal difference between Yahweh and the Christian God is the belief that there are three distinct persons in the one God, the Trinity of Father, Son and Holy Spirit (Aquinas op. cit. qq 27-43).

Augustine imagined that he could see vestigia (footprints) of the Trinity in the World, particularly in the structure of our thought: mind, corresponding to the Father, intellect, corresponding to the Son and will, corresponding to the Spirit. Thomas developed this psychological model using the notion that the persons of the Trinity were differentiated by the relationships between them.

1.3: Modelling God: Recent developments

A fundamental axiom of Aristotelian metaphysics is that no potential can actualize itself. The World therefore needs the Unmoved Mover, pure act, to actualize it. This leads to a difficulty in the discussion of the life of God. Thomas accepts Aristotle's definition of life as self motion. If God is alive, this would appear to imply the existence of potential in God, contradicting the claim that God is pure act. Thomas deals with this problem by distinguishing motion from potency to act from motion from act to act, which is proper to intelligent beings like God:

Objection 1. It seems that life is not properly attributed to God. For things are said to live inasmuch as they move themselves, as previously stated. But movement does not belong to God. Neither therefore does life

Reply to Objection 1. As stated in [Aristotle] Metaph. ix, 16, 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 [Aristotle] 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, 3Is life properly attributed to God?

An important development in nineteenth century physics was the discovery and testing of the principle of conservation of energy. An important step in this development was recognition of the distinction between potential (hidden) and kinetic (visible) energy. Thus motor fuel contains potential energy which can be released by the motor to become the kinetic energy of a moving vehicle. A clear demonstration of the relationship between potential and kinetic energy is a simple harmonic oscillator, like the pendulum which (given frictionless perfection) converts energy from potential to kinetic and back again forever without loss. In this modern context, potency and act are equivalent, both in fact active. Elkana: The Discovery of the Conservation of Energy

Although it may seem to be a rather long shot to see the modern physical relationship between potential and kinetic energy invalidating Thomas's first proof for the existence of God and supporting the notion that action in the World is effectively the same as action in God, we can take the hint and explore the alternative to the Christian hypothesis. Christian theology places an absolute difference between God and the World, one is not the other. Here we will proceed on the assumption that God and the World are identical and see where it leads.

While physics is moving into theology, traditional theology is suffering from a combination of chaos and fundamentalism. As Ford notes, the scope of twentieth century Christian theology is marked by 'diversity amounting often to fragmentation'. The reason for this seems clear. The theologians are all working from a very limited and ancient 'deposit of faith'. This deposit simply does not contain enough data to decide modern questions, and there is no promise of new data, because God's revelation is considered to be 'once for all.' Davies: The Mind of God: Science and the Search for Ultimate Meaning, Ford: The Modern Theologians : An Introduction to Christian Theology in the Twentieth Century, Second Vatican Council: Dogmatic Constitution on Divine Revelation - Dei verbum

The hypothesis proposed here solves this problem by opening theology to all the public and private data in the whole Universe. There can be no question that a contemporary observable fact is at least as trustworthy as an ancient text. The real question is: how do we interpret the things we observe?

If the universe is divine it seems reasonable to assume that all our experience is experience of God. Theology can therefore become a real evidence based science drawing strength from every event in the universe.

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2: Fixed point theory

Let us, like Aquinas, accept Aristotle's opinion that the first mover is pure action. From a mathematical point of view, we may see God's action or the life as God mapping itself onto itself, since it is all that there is.

Mathematically we may understand such a mapping as the execution of a function, let us say y = f(x) where both y and x are both elements of the system we call God.

Aquinas uses the hypothesis that God is pure act (actus purus) to argue for the traditional properties of God, eternity, omnipotence, omniscience and so forth. The first point that he establishes, consistent with long mystical tradition, is that God is absolutely simple, with no constituent parts of any sort. This property renders God invisible and mysterious to us and serves to differentiate God from the world, which is full of parts. Underhill: Mysticism: A Study in the Nature and Development of Man's Spiritual Consciousness

If we are to assert that the Universe is divine, we need to deal with this problem. Since God has no internal structure at all, how can we distinguish x and y? This brings us to a species of chicken and egg situation. Mathematical fixed point theorems tell us that under certain conditions any dynamic system that maps onto itself has at least one point which is unchanged by the mapping, that is f(x) = x. A consequence of a fixed point theorem, for instance, is that no matter how much you stir your coffee, when it slows again to rest, there will be one point in the coffee in the same place as it was before.

If there are, necessarily, fixed points in the dynamics of God, then we have some ground for distinguishing x from y. In other words, if we can apply fixed point theory to God, then we can apply fixed point theory to God. Otherwise not. We will proceed here on the assumption that such theory does apply to God and leave further discussion until later.

An argument in favour of this position is that a typical fixed point theorem, that of Brouwer, has a non constructive proof. As Thomas notes in the Summa, the non-constructive proof, sometimes known as the via negativa or apophatic theology, is at home in Christian theology:

Once we know that something may exist, it remains to be inquired how it may exist so that we may know what it is. But because we cannot know what God may be, but what it may not be, we cannot talk about how God may exist, but rather how God is not. Therefore the first thing to be studied is how God may not be.

How God may not be can be shown by removing from it things that are not appropriate, like composition, motion and other similar features. Consequently, the first feature to be studied is the simplicity of God, equivalent to the removal from God of composition . . . Thomas Aquinas, Summa I, 3: God's simplicity, Brouwer fixed point theorem - Wikipedia

Brouwer proved that any continuous function mapping a convex compact set onto itself has a fixed point f(x) = x. The first proofs of Brouwer's theorem were non constructive and it was only later than a constructive proof was found. Casti: Five Golden Rules

Brouwer's theorem guarantees the existence of a fixed point in many functions. Thus, if God implements the axioms underlying Brouwer's theorem, we can expect to find at least one fixed point in the one God, a result that is at least consistent with our expectations, that point being God itself.

There are more complex fixed points on spaces larger than the infinite dimensional Euclidian space in which Brouwer's theorem operates. The beauty of infinite dimensional spaces is that a single point may represent a very complex structure. The most general forms of fixed point theory work in the space defined by logic and set theory and may predict any number of fixed points.

From a theological point of view, fixed point theory gives us a logically consistent path to reconciling the simplicity of God with the complexity of the observable Universe. In Parmenides' day the eternal being came to be contrasted with observable ephemeral being, which was deprecated. This deprecation of the World, the flesh and the devil reached its apogee in Christianity and still widely believed.

What fixed point theory tells us that motion and stillness are not two different things. Stillness is simply the fixed point of a motion: it is in the motion, not outside it. We may therefore consider the complex processes of the Universe as sequences of fixed points of the divine motion, and need no longer place God outside the world. As we learn from quantum mechanics the unmoved mover is part of the Universe, as Aristotle thought, not apart from it, as Christianity maintains.

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3: Quantum mechanics and fixed points

How do fixed points appear in reality? Although it has been often been assumed since ancient times that the Universe is continuous, it is a matter of fact that all our observations are of discrete events. This is true not only at the quantum level, but at all scales where we observe discrete objects like people, leaves or grains of sand. The principal argument for continuity is the apparent continuity of motion.

From a practical point of view, quantum mechanics is a very successful theory. We feel strongly that it touches on the truth. Zurek writes:

The only "failure" of quantum theory is its inability to provide a natural framework that can accommodate our prejudices about the workings of the universe. . . . at the root of our unease with quantum mechanics is the clash between the principle of superposition -- the consequence of the linearity of ψ and the everyday classical reality in which the principle appears to be violated.' Wojciech H Zurek: Decoherence and the Transition from quantum to classical - Revisited

The underlying mathematical theory suggests that the continuous superposition of solutions to the energy equation evolves deterministically and that each element of the superposition is in perpetual motion at a rate proportional to its energy, given by the equation E = hf. The wave equation is normalized so that the sum of all the frequencies to be found in the superposition is equal to the total energy of the system modeled. Dirac: The Principles of Quantum Mechanics, von Neumann: Mathematical Foundations of Quantum Mechanics

An isolated quantum system is observed or measured by coming into contact with another quantum system. An observation is represented by an ‘observable’ or measurement operator, M, and we find that the only states that we see are eigenfunctions of M. These states are the fixed points under the operation of M given by the ‘eigenvalue equation’ Mψ = mψ. The scalar parameter m is the eigenvalue corresponding to the eigenfunction ψ. The eigenfunctions of a measurement operator M are determinate functions or vectors which can be computed from M. The eigenvalue equation gives us insight into how the system generates fixed points at a fundamental physical level. Eigenfunction - Wikipedia

Zurek has found that the correspondence of observable quantities to orthogonal eigenfunctions of the measurement operator is necessary if we are to obtain information from a quantum system. This suggests that the quantization of observation and the requirements of mathematical communication theory are consistent with one another. Error free communication requires a deterministic process which is realized by an orthogonal set of symbols, symbols which cannot be confused with one another because they have nothing in common. Wojciech Hubert Zurek: Quantum origin of quantum jumps, Claude E Shannon: A Mathematical Theory of Communication

Although the continuous wave function is believed to evolve deterministically, and the eigenfunctions of a measurement operator can in principle be computed exactly,we can only compute the probability distribution of the eigenvalues revealed by the repetition of a given measurement.

These frequencies are predicted by the Born rule: pk = |< mk |ψ >|2 where ψ is the unknown pre-existing state of the system to be measured and pk is the probability of observing the eigenvalue corresponding to the kth eigenfunction mk of M. Provided the measurement process is properly normalized, the sum of the probabilities pk is 1. When we observe the spectrum of a system, the eigenfunctions determine the frequencies of the lines we observe and the eigenvalues the line weights.

From the point of view of communication theory, a quantum measurement is a source, that is it emits a set of symbols ai (eigenvalues) each with probability pi, such that the sum of the pi is 1. This enables us to think of the universal process as a network of quantum systems measuring one another, that is communicating with one another. Quantum mechanics, and its relativistic extension, quantum field theory, serve as methods for modeling both the nature and frequency of communication on this network. They are our means of physical network traffic analysis.

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4: Mathematical theology

The purpose of theology, 'talk about God' is to explain God. In modern terms this means to search for models of God. Following Aquinas, and long tradition, we accept that we cannot model the dynamic simplicity of God. Nevertheless, the hypothesis that there are fixed points in the divine motion provide an opening to model divine revelation. From this point of view, we understand every fixed point we observe to be a message from the dynamic divinity.

We assume that there is no limit on the number of fixed points in the divine Universe. The first step in mathematical modeling is to establish a one-to-one correspondence between a set of mathematical symbols and observable elements of the system to be modeled. The mathematical space in which the modeling takes place is sometimes called a phase space. Here we assume that an appropriate phase space to model the fixed points of God is the Cantor space of transfinite numbers. Cantor: Contributions to the Founding of the Theory of Transfinite Numbers

It has been known since antiquity that there are more points in a geometric line than there are natural numbers, 0, 1, 2 . . .. Cantor wished to produce a mathematical symbol which represented this number of points, the cardinal of the continuum. To do this he invented the theory of sets. For him, a set is 'the collection into a whole of definite and separate objects of our intuition or our thought'.

Cantor concentrated on two abstract properties of sets: their cardinal number, that is the number of elements in a set; and their ordinal number, the arrangement of the elements of the set among themselves. The set of natural numbers, for instance, has a 'natural' order, 1, 2, 3, . . . but we can imagine different orderings like 1, 3, 2, . . . 2, 3, 1, and so on. The total number of orderings of a set of n elements is n! , ie n × (n-1) × (n-2) × . . . × 2 × 1, a number which grows exponentially with n.

Cantor wrote:

The concept of 'ordinal type' developed here, when it is transferred in like manner to 'multiply ordered aggregates' embraces, in conjunction with the concept of 'cardinal number' or 'power' . . . , everything capable of being numbered (Anzahlmassige) that is thinkable, and in this sense cannot be further generalized. Cantor, page 117

In other words we can use the transfinite numbers not only count things, but also to represent their forms by mapping them to an appropriate ordered set. This makes the transfinite ordinal numbers the ideal structures to map the fixed points in the divinity.

From a mathematical point of view, an entity can be said to exist if it is self consistent. Traditionally, self-consistency is the only constraint that theology places on God since God is subject to no outside constraint. The assumption that the Universe is divine implies that the Universe has no outside constraint. Insofar as it is created, it creates itself. This methodological parallel between mathematics and theology suggests that they can be of mutual benefit.

Hallett writes:

It is clear that Cantor understands pure set theory as a quite general foundational theory which prepares the way for any theory which uses or relies on sets or numbers. But now we come back to theology and God, for this foundation, this understanding of what numbers are, or what sets etc. exist, is for Cantor intimately connected with the attempt to understand God's whole abstract creation and the nature of God himself. Hallett: Cantorian Set Theory and Limitation of Size, page 10

Here we propose to identify 'abstract creation' with 'the fixed points in the divine dynamics'.

Cantor saw that there is no contradiction involved in imagining the existence of the set of all the natural numbers N = {1, 2, 3, . . . }. Since there is no greatest natural number, the cardinal of the set of natural numbers cannot be a natural number but a new number which Cantor called 0, the first transfinite cardinal. He also saw that by ordering the set of 0 natural numbers in different ways, it is possible to make another number, hugely greater, the second transfinite cardinal 1. The process that generates 1 out of 0 is permutation.

Since permutation is indifferent to the cardinal of the set permuted, this recursive generation of transfinite cardinals could continue without end, like the recursive generation of natural numbers by adding one. In the transfinite case, however, the one that was added at each step was a new structure comprising everything that could be constructed from the elements of the prior set.

We may see permutation as a mathematical model of imagination, taking any set of elements (words, building blocks) and trying different arrangements in a search for something new and interesting.

Cantor surmised that the cardinal of the continuum is the 1. Cohen later showed that one cannot use set theory to determine the cardinal of the continuum. This is because the concept of set is independent of (orthogonal to) cardinality, ie sets are symmetrical with respect to size: they are essentially the same no matter how many elements they contain. Cohen: Set Theory and the Continuum Hypothesis

The formalism of quantum mechanics enjoys a similar symmetry. David Hilbert realized that Cantor's transfinite numbers are in effect a hierarchy of function spaces. 1 is the cardinal of the set of permutations of the natural numbers, that is all the functions from the natural numbers to themselves. Hilbert used Cantor's idea to develop Hilbert space a function space analogous of Euclidian space which is the home of the state vectors ψ which carry all the information in quantum mechanics.

Like the theory of sets, quantum mechanics is independent of the number of components in its state vectors, which correspond to the number of dimensions of the relevant Hilbert space. Quantum mechanics accepts systems from one dimension up to the cardinal of the continuum and beyond. This property of symmetry with respect to complexity serves as bridge to connect Hilbert spaces with any number of dimensions.

The unbounded size of the transfinite numbers means that no matter how how many fixed points there are in any system of interest, we are guaranteed a transfinite number of sufficient size to be placed into one to one correspondence with these fixed points. This correspondence gives us a mathematical grip on the system to be modeled.

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5: Logical continuity: the mathematical theory of computation

Some fixed points in the Universe appear to last forever, others, like ourselves and atomic states, have shorter lifetimes. In physics we describe these changes at two levels, kinematic, which simply records what happens, and dynamic, which seeks to explain the kinematic. We might call a system natural if its kinematics is driven directly by its dynamics. Otherwise it is artificial. The motion picture industry specializes in making real looking kinematics using tricks to avoid the expense of constructing the real dynamics, like an actual sinking of the Titanic.

At the fundamental level, physicists observe the kinematics of the world but must speculate about the dynamics, which is invisible. Experimental physicists use article accelerators like the Large Hadron Collider to build up kinematic pictures of the behavior of these particles. We test our dynamical theories by seeing how well they imitate the kinematic behavior actually observed.

Modern physics provides two somewhat incompatible models for the universal dynamics. The Standard Model, built with quantum field theory, describes the smaller scale structure of the Universe down to particles like electrons and quarks which are believed to have no size. The large scale structure of the Universe is described by Einstein' general theory of relativity. Between them, these theories have allowed us to develop quite a comprehensive history of the expanding observable Universe since it was very small. A quantum field theory of gravitation on the other hand, remains elusive.

Heisenberg helped us come to terms with the weirdness of quantum theory by stating that we can only be certain about what is observable and must speculate about what is invisible. Quantum theory posits a world characterized by the deterministic continuous evolution of wave functions which occasionally 'collapse' when they are observed. Werner Heisenberg: Quantum-theoretical re-interpretation of kinematic and mechanical relations

Because it is continuous motion, we model it with sets whose cardinal is the cardinal of the continuum, which we assume to be 1. The mechanism that makes all these points behave as they do is mysterious but mathematically consistent. Continuous mathematics has been developed over millennia and is thought to be logically sound. This we call logical continuity. A logical continuum is identical to a halting Turing machine, which joins a final state to an initial state through a series of deterministic logically sound steps. Alan Turing: On Computable Numbers, with an application to the Entscheidungsproblem

Here we propose an alternative model of the universal dynamics based on the notion that the that the invisible dynamics is logically rather than geometrically continuous. This approach is suggested by the hypothesis that the Universe is divine. If this is the case, we would expect the observable kinematics, that is the observable fixed points in the divine dynamics, to be the outcome of logically consistent processes reflecting the internal consistency and intelligence of divine action.

So let us assume that the eigenfunctions of quantum mechanical observables are computable functions. Instead of there being a continuous spectrum of 1 eigenfunctions, there are only 0 of them, that is the number of Turing machines whose algorithm is expressed by a finite ordered set of symbols. From this point of view, a quantum mechanical observation is equivalent to the exchange of a message between two quantum systems. The computations encoding and decoding such a message are accomplished by Turing machines.

Turing machines are deterministic. Many features of the world also bear deterministic relationships to one another, like the spectral frequencies of an atom. It seems natural to attribute this determinism to a deterministic natural process.

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6: The transfinite digital network as a model of revelation

We assume, if the Universe is divine, that all human experience is experience of God. In physical terms we can say that every event, that is every fixed point, is divine revelation.

We have already noted that the Cantor universe of transfinite numbers gives us mathematical space large enough to map the fixed points of the divinity, since there is no limit to the size of transfinite cardinal and ordinal numbers. We now add computers to the Cantor universe to construct a transfinite digital network to model the behavior of the observable world. We can model that transfinite computer network on finite networks, like the internet, built by engineers. Tanenbaum: Computer Networks

Engineered networks are layered, a structure necessary to make them easy to construct, expand and troubleshoot. We transform this idea into a transfinite network by mapping the layers of the Universal network onto the sequence of transfinite ordinal numbers, beginning by letting the natural numbers correspond to the physical layer of the Universe. The eigenfunctions of this physical layer are the countably infinite set of Turing machines.

Each subsequent software layer uses the layer beneath it as an alphabet of operations to achieve its ends. The topmost layer, in engineered networks, comprises human users. These people may be a part of a corporate network, reporting through further layers of management to the board of an organization. By analogy to this layered hierarchy, we may consider the Universe as a whole as the ultimate user of the universal network. Since the higher layers depend on the lower layers for their existence, we can expect an evolutionary tendency for higher layers to curate their alphabets to maintain its own stability.

Processes in corresponding layers (‘peers’) in a network may communicate if they share a suitable protocol. All such communication uses the services of all layers between the peers and the physical layer. These services are generally invisible or transparent to the peers unless they fail. Thus two people in conversation are generally unaware of the huge psychological, physiological and physical complexity of the systems that make their communication possible

Let us imagine that the actual work of permutation in the symmetric universe (ie its dynamics) is executed by Turing machines. As formal structures these Turing machines are themselves ordered sets, and are to be found among the ordered strings contained in the Universe.

The installation of these Turing machines turns the transfinite universe into the transfinite network. This network is a set of independent memories able to communicate with and change one another via Turing machines. The internet is a finite example of such a network, the memories of servers, routers, clients and users changing each other’s states through communication.

It seems clear that the transfinite network has sufficient variety to be placed in one-to-one correspondence with any structure or process in the Universe. In a case where a given layer of the network universe is found to be too small to accommodate the system of interest, we have only to move up through the layers until we find a level whose cardinal is adequate for the task.

Permutations can be divided into subsets or cycles of smaller closed permutations. This process means that no matter what the cardinal of a permutation, we can find finite local permutations whose action nevertheless permutes the whole Universe. Moving my pen from a to b (and moving an equivalent volume of air from b to a ) is such an action .

Although there are i mappings of the 0 natural numbers onto themselves, there are only 0 different Turing machines. As a consequence, almost all mappings are incomputable, and so cannot be generated by a deterministic process. This limitation on determinism explains the randomness that we experience in life and makes evolution possible.

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7. Evolution

We have introduced the notion of logical continuity as a superior form of continuity. A logical continuum is a Turing machine or computer, a mechanism that given a certain suitable starting state, proceeds inexorably to a certain fixed ending or halting state. David Hilbert speculated that there is a deterministic mechanical process available for every starting state, but Alan Turing was able to demonstrate the existence of incomputable functions, that is starting states for a computer which would never halt.

The Universe is believed to have evolved from a structureless initial singularity, formally identical to the traditional Christian God, to its current state The evolutionary process is a product of creation, variation and selection of fixed points in the divine dynamics. Creation and variation are made possible by the existence of non-deterministic (ie non-computable) processes. The duplication of genetic material during cell division, for instance, is subject to a certain small error rate which may ultimately affect the fate of the daughter cells.

Selection culls the variations. The net effect of variation and selection is to optimize systems for survival, that is for stability or the occupation of a fixed point (which may be in a space of transfinite dimension). Before the explicit modeling of evolution, however, writers like de Maupertuis and others speculated that the processes of the world were as perfect as possible. Yourgrau & Mandelstam: Variational Principles in Dynamics and Quantum Theory

Turing's work on incomputability built upon Gödel's work on incompleteness which established that there are mathematical propositions that may be true or false but cannot be proved to be so. In our model of the fixed points of God, this is equivalent to saying that there are fixed points in the divine dynamics which cannot be reached by deterministic processes. Feferman et al: Kurt Goedel: Collected Works Volume 1 Publications 1929-1936

Nevertheless, an incomputable mapping, once discovered, may be tested by a computable process. In the context of biological evolution, a random genetic change may give rise to an individual that survives and reproduces, that it successfully passes its random genetic variation on to future generations.

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8: Theological implications:

8.1: Theology becomes a science, embracing all the sciences

Theology is the traditional theory of everything, projecting a broad image of the physical and psychological worlds we inhabit and providing us with assistance to navigate the often stormy seas of life. From a critical point of view, the scriptures of the ancient religions are secondary source, the authors recoding their personal views of God.

We lay the foundation for theology as a science by assuming that the observables of physics are fixed points in the divine dynamics. We imagine an infinity of these points, but accept that they are not isolated, but consistently connected by the underlying divine dynamics. I, for instance, am a huge network of quantum events controlling my structure and metabolism at the atomic level. At a rough calculation, my lifetime is the product of approximately 1053 quanta of action.

Our current description of the divine dynamics is quantum theory. This theory explains the occurrence of fixed points in the Universe and gives us a practically useful but theoretically incomplete account of the processes that lead to the fixed points that we observe.

The transfinite computer network outlined above provides a logical phase space in which to model both the processes that lie behind the observations and the permanently or temporarily fixed structures arising from these processes. The most powerful feature of this model is its symmetry with respect to complexity or Cantor symmetry. The basic idea is that the world creates itself by recursive processes some of which stay in the same groove forever and others of which explore the whole space of possibility. Mathematically this idea is explored by the theory of computation, often called recursive function theory. Hopcroft, Motwani & Ullman: Introduction to Automata Theory, Languages and Computation.

8.2 God is our judge

It seems widely agreed that our actions are judged by a divinity and some sort of account kept which will be paid out (or called in) in the end. This view was shaken by the story of Job, who became a pawn in a game between God and Satan. Before that the Hebrews had often been disappointed when their God Yahweh showed himself to be a slow payer. Job (KJV)

The picture changes slightly if we assume that the Universe is divine and that all our actions are events in the overall event we call God. In the divine milieu, every action at every scale is judged by its its fit to the divine reality: some things succeed, some things fail. Things work, or they don't. The art of living is to understand the local divinity well enough to devise actions that work, like learning to shear a sheep.

8.3. In the divine Universe, all experience is experience of God

Our working hypothesis is that all the observable phenomena in the Universe are fixed points in the divine dynamics. All our experiences call into the category 'observable phenomena' even though I am the only one who can immediately observe my own experiences. Nevertheless, all personal experience is part of the universal process, and we therefore assume that our experiences are fixed points in the divine dynamics, that is experiences of God.

8.4 The Universe is one, that is it is a consistent whole

This is just what we would expect if the fixed points that we observe are points in the divine dynamics. We can only observe fixed points, not moving points. Because the world is one, theology, scientifically tracking the world, will itself tend toward unity like the other evidence based sciences

8.5 Religion, following theology, may become universalized by human symmetry

Much of the current strife in the world traces to different views of the nature and will of God. Since the Enlightenment many people have thought that theology and religion, coupled with theocratic governments, have hindered rather than helped human development. It has been found helpful, for instance, to separate religion and politics. The political aspirations of modern educated people have long outstripped the traditional religious offerings. It is time for a theological and religious spring to put the ancient religions on a new divine foundation.

It is clear that we have to overcome many of our differences if we are to work together to fix the damage we have done to the planet before it brings too much more suffering.

8.6 Digitizing the Universe points to an unbounded future

The history of the Universe contains clues to its future. This is a consequence of the conservation of energy, or symmetry with respect to time. The Christian God is in no way subject to action by us. We are powerless before it, since it is the absolute, omnipotent, omniscient monarch. In a divine Universe this is not the case, since our action is part of divine action, and we can either fit in with the desires of the divinity and prosper, or act blindly and suffer.

The salvific feature of the Universe is that it has fixed points guaranteed by the underlying divine dynamics which we can use as foundations for building the technology necessary for peaceful sustained existence on Earth. These foundations are digital or quantized, definite points in the universal dynamics. A little reflection will show that all our technology to date depends on such points which define things like the strengths of materials and the behavior of semiconductors. The structures that we can make on these fixed points is limited only by our imagination and the properties of God.

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9: Practical implications

9.1: The Roman Catholic Church

The hypothesis outlined in this essay has some pretty high hurdles to clear. The theory proposed here suggests that the Roman Catholic Church currently depends on a false model of God.

This Church has taken a dogmatic stand on the proposition that God is the mysterious being described in the first part of Thomas's Summa. It holds the position that God is absolutely other than the world, totally outside our ken. It holds that it is the sole reliable channel of communication between this God and humanity.

This position is inconsistent with empirical science, as was dramatically illustrated in the Galileo affair. In modern times we can see its equivalent in those vested interests which assert that human activities are having no effect on global climate. Galileo lost his battle against the political power of the Church. To avoid worse punishment the Church made him publicly deny the consequences phenomena he had seen with his own eyes. Despite this, and although he was an old man, the Church put him under house arrest for the rest of his life.

The Church remains an absolute monarchy ruled by the Pope who enjoys absolute legislative, executive and judicial power:

'Can. 333 §1. By virtue of his office, the Roman Pontiff not only possesses power offer the universal Church but also obtains the primacy of ordinary power over all particular churches and groups of them. Moreover, this primacy strengthens and protects the proper, ordinary, and immediate power which bishops possess in the particular churches entrusted to their care. §2. In fulfilling the office of supreme pastor of the Church, the Roman Pontiff is always joined in communion with the other bishops and with the universal Church. He nevertheless has the right, according to the needs of the Church, to determine the manner, whether personal or collegial, of exercising this office. §3. No appeal or recourse is permitted against a sentence or decree of the Roman Pontiff.'

Furthermore, the Pope, when he so decides, is held to be infallible:

Chapter IV: On the infallibility of the Roman Pontiff . . . 9. . . . we teach and define as a divinely revealed dogma that when the Roman Pontiff speaks ex cathedra, that is, when, in the exercise of his office as shepherd and teacher of all Christians, in virtue of his supreme apostolic authority, he defines a doctrine concerning faith or morals to be held by the whole Church, he possesses, by the divine assistance promised to him in blessed Peter, that infallibility which the divine Redeemer willed his Church to enjoy in defining doctrine concerning faith or morals. Therefore, such definitions of the Roman Pontiff are of themselves, and not by the consent of the Church, irreformable. So then, should anyone, which God forbid, have the temerity to reject this definition of ours: let him be anathema. Manning: The Vatican Council and its Definitions: A Pastoral Letter to the Clergy

The Pope claims to hold power by divine right given in the Gospel of Matthew 16:17-19 (NIV) and parallels:

Jesus replied, “Blessed are you, Simon son of Jonah, for this was not revealed to you by flesh and blood, but by my Father in heaven. And I tell you that you are Peter, and on this rock I will build my church, and the gates of Hades will not overcome it. I will give you the keys of the kingdom of heaven; whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven.”

Here we work from the definitive exposition of the nature of God given by Thomas Aquinas in his Summa Theologiae. Aquinas' work is specially recommended by the Church in Canon Law: Can. 252 . . . §3. There are to be classes in dogmatic theology, always grounded in the written word of God together with sacred tradition; through these, students are to learn to penetrate more intimately the mysteries of salvation, especially with St. Thomas as a teacher. . . . '

In other words, the Church, as an institution is totally committed to the position that God is not the Universe. Given the enormous social momentum of the Church, it will take a long time for the alternative view to become dominant. It is important, however, that we base our future on reality rather than ancient myth.

9.2 The global environment

Christianity believes that we own the Earth: "Be fruitful and and increase in number, and fill the earth, and subdue it" (Genesis 1:28, NIV). This is not the case. Our existence depends upon how well we fit the global system upon which we are completely dependent. In general, human survival requires that we turn resources of the Earth that might be used by other species toward ourselves. We are predators on the Earth.

We have now become aware that we are overloading the Earth and must being to lift off by 'reducing our footprint'. The doctrine is no longer 'subdue the Earth' (since it was made for us) but 'treat the Earth as God' (because we are absolutely dependent upon it). Ecological footprint - Wikipedia

9.3. The structure of nations

A nation is a set of people, which we can study at various levels of resolution. The most basic is population, the cardinal of the national set. Next we turn to the ordinal number, the structures developed in the national population through their relationships to one another. Historically, we can see two extremes in national structure, monarchy and democracy. We have characterized the Roman Catholic Church as a monarchy above. Here we turn to democracy.

Let us define a democracy as a network of peers. We can define peers in terms of game theory. In any game peers are equal, so that if they play often we will see random numbers of wins and losses. This is opposed to monarchy, where the monarch wins every time.

Monarchies are unstable. 'Uneasy lies the head that wears a crown' (Shakespeare Henry The Fourth, Part 2 Act 3, scene 1, 31). The monarch is just one person, who can easily fail. Democracy, is stable, because it has no failure modes as long as peers remain peers. A democracy is rendered more liable to failure the more often the peer constraint is broken by the accumulation by one of power over others.

From this formal point of view, we interpret the Christian injunction to love one's neighbour as the algorithm 'maintain peerhood'.

9.4. Education and indoctrination

On the one hand I resent having been bought up a Catholic; Catholic constraints significantly reduced the pleasure of my life in my early years. On the other hand, this imposition, once I saw it was unnecessary, motivated me to develop an alternative first for myself and then for anybody who find it useful.

One can imagine that we begin to absorb our native culture from the moment we are born, and that much of this information is accepted uncritically. We quite naturally learn the languages and customs of our environment and gradually find and develop a niche of our own in which to survive.

Navigation through life, like navigation in space, required a fixed frame of reference like the stars to determine our position in the world. If we are to survive, this reference frame must be God, the way things are.

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Further reading

Books

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Canon Law Society of America, Holy See, Code of Canon Law: Latin-English Edition, Canon Law Society of America 1984 Pope John Paul XXXIII announced his decision to reform the existing corpus of canonical legislation on 25 January 1959. Pope John Paul II ordered the promulgation of the revised Code of Canon law on the same day in 1983. The latin text is definitive. This English translation has been approved by the Canonical Affairs Committee of the [US] National Conference of Catholic Bishops in October 1983. 
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Cantor, Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 1955 Jacket: 'One of the greatest mathematical classics of all time, this work established a new field of mathematics which was to be of incalculable importance in topology, number theory, analysis, theory of functions, etc, as well as the entire field of modern logic.' 
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Casti, John L, Five Golden Rules: Great Theories of 20th-Century Mathematics - and Why They Matter, John Wiley and Sons 1996 Preface: '[this book] is intended to tell the general reader about mathematics by showcasing five of the finest achievements of the mathematician's art in this [20th] century.' p ix. Treats the Minimax theorem (game theory), the Brouwer Fixed-Point theorem (topology), Morse's theorem (singularity theory), the Halting theorem (theory of computation) and the Simplex method (optimisation theory). 
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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. . . .'  
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Davies, Paul, The Mind of God: Science and the Search for Ultimate Meaning, Penguin Books 1992 'Paul Davies' "The Mind of God: Science and the Search for Ultimate Meaning" explores how modern science is beginning to shed light on the mysteries of our existence. Is the universe - and our place in it - the result of random chance, or is there an ultimate meaning to existence? Where did the laws of nature come from? Were they created by a higher force, or can they be explained in some other way? How, for example, could a mechanism as complex as an eye have evolved without a creator? Paul Davies argues that the achievement of science and mathematics in unlocking the secrets of nature mean that there must be a deep and significant link between the human mind and the organization of the physical world. . . . ' 
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Dirac, P A M, The Principles of Quantum Mechanics (4th ed), Oxford UP/Clarendon 1983 Jacket: '[this] is the standard work in the fundamental principles of quantum mechanics, indispensible both to the advanced student and the mature research worker, who will always find it a fresh source of knowledge and stimulation.' (Nature)  
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Elkana, Yehuda, The Discovery of the Conservation of Energy, Hutchinson Educational 1974 Jacket: 'This book chronicles historically and in a philosophical context the discovery and gradual develoment of the concept of energy ... Metaphysical beliefs in the principle of 'conservation of something' in nature resulted finally in the statement of the physical laws of the conservaiton of energy in the work of Hermann von Helmholtz.' 
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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. ...' 
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Ford, David, The Modern Theologians : An Introduction to Christian Theology in the Twentieth Century, Blackwell 1997 Preface: 'The main aim of this volume is to introduce the theology of most leading twentieth-century Christian theologians and movements in theology. . . . The contributors are mostly based in Europe of North America and come from a wide range of institutions, denominational backgrounds, and countries. Most are themselves constructively engaged in modern theology, and their purpose has been to produce a scholarly account of their subject and also carry further the theological dialogue in each case.'  
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Hallett, Michael, Cantorian Set Theory and Limitation of Size, Oxford UP 1984 Jacket: 'This book will be of use to a wide audience, from beginning students of set theory (who can gain from it a sense of how the subject reached its present form), to mathematical set theorists (who will find an expert guide to the early literature), and for anyone concerned with the philosophy of mathematics (who will be interested by the extensive and perceptive discussion of the set concept).' Daniel Isaacson. 
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Hopcroft, John E, and Rajeev Motwani and Jeffrey D Ullman, Introduction to Automata Theory, Languages and Computation (Second edition), Addison Wesley 2001 Amazon Product Description 'This book is a rigorous exposition of formal languages and models of computation, with an introduction to computational complexity. The authors present the theory in a concise and straightforward manner, with an eye out for the practical applications. Exercises at the end of each chapter, including some that have been solved, help readers confirm and enhance their understanding of the material. This book is appropriate for upper-level computer science undergraduates who are comfortable with mathematical arguments.' 
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Manning, Henry Edward, The Vatican Council and its Definitions: A Pastoral Letter to the Clergy, Excelsior Catholic Publishing House 1905 Latin original of quoted passage: '... docemus et divinitus revelatum dogma esse definimus; Romanum Pontificem, cum ex Cathedra loquitur, id est, cum omnium Christianorum Pastoris et Doctoris munere fungens, pro suprema sua Apostolica auctoritate, doctrinam de fide vel moribus ab universa Ecclesia tenendam definit, per assistentiam divinam, ipsi in beato Petro promissam, ea infallibilitate pollere, qua divinus Redemptor Ecclesiam suam in definienda docrtina de fide vel moribus instructam esse voluit; ideoque eiusmodi Romani Pontificis definitiones ex sese, non autum ex consensu Ecclesia irreformabiles esse. Si quis autem huic Nostrae definitioni contradicere, quod Deus avertat, praesumpserit; anathema sit.'back
Miles, Jack, God : A Biography, Vintage Books 1996 Jacket: 'Jack Miles's remarkable work examines the hero of the Old Testament . . . from his first appearance as Creator to his last as Ancient of Days. . . . We see God torn by conflicting urges. To his own sorrow, he is by turns destructive and creative, vain and modest, subtle and naive, ruthless and tender, lawful and lawless, powerful yet powerless, omniscient and blind.' 
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Tanenbaum, Andrew S, Computer Networks, Prentice Hall International 1996 Preface: 'The key to designing a computer network was first enunciated by Julius Caesar: Divide and Conquer. The idea is to design a network as a sequence of layers, or abstract machines, each one based upon the previous one. . . . This book uses a model in which networks are divided into seven layers. The structure of the book follows the structure of the model to a considerable extent.'  
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Underhill, Evelyn, Mysticism: A Study in the Nature and Development of Man's Spiritual Consciousness, E P Dutton and Co 1911-  
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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.' 
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Yourgrau, Wolfgang, and Stanley Mandelstam, Variational Principles in Dynamics and Quantum Theory, Dover 1979 Variational principles serve as filters for parititioning the set of dynamic possibilities of a system into a high probability and a low probability set. The method derives from De Maupertuis (1698-1759) who formulated the principle of least action, which states that physical laws include a rule of economy, the principle of least action. This principle states that in a mathematically described dynamic system will move so as to minimise action. Yourgrau and andelstam explains the application of this principle to a variety of physical systems.  
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Links
Alan Turing, On Computable Numbers, with an application to the Entscheidungsproblem, 'The "computable" numbers may be described briefly as the real numbers whose 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
Aquinas Summa I, 2, 3, Does God exist?, 'I answer that, The existence of God can be proved in five ways. The first and more manifest way is the argument from motion. . . . ' 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
Brouwer fixed point theorem - Wikipedia, Brouwer fixed point theorem - Wikipedia, the free encyclopedia, 'Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f(x0) = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to itself. A more general form is for continuous functions from a convex compact subset K of Euclidean space to itself. back
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
Code of Canon Law, 252, The formation of clerics, 'Can. 252 §1. Theological instruction is to be imparted in the light of faith and under the leadership of the magisterium in such a way that the students understand the entire Catholic doctrine grounded in divine revelation, gain nourishment for their own spiritual life, and are able properly to announce and safeguard it in the exercise of the ministry. §2. Students are to be instructed in sacred scripture with special diligence in such a way that they acquire a comprehensive view of the whole of sacred scripture. §3. There are to be classes in dogmatic theology, always grounded in the written word of God together with sacred tradition; through these, students are to learn to penetrate more intimately the mysteries of salvation, especially with St. Thomas as a teacher. There are also to be classes in moral and pastoral theology, canon law, liturgy, ecclesiastical history, and other auxiliary and special disciplines, according to the norm of the prescripts of the program of priestly formation.' back
Ecological footprint - Wikipedia, Ecological footprint - Wikipedia, the free encyclopedia, 'An ecological footprint is a measure of human impact on Earth's ecosystems. It's typically measured in area of wilderness or amount of natural capital consumed each year. A common way of estimating footprint is, the area of wilderness of both land and sea needed to supply resources to a human population; This includes the area of wilderness needed to assimilate human waste. At a global scale, it is used to estimate how rapidly we are depleting natural capital. The Global Footprint Network calculates the global ecological footprint from UN and other data. They estimate that as of 2007 our planet has been using natural capital 1.6 times as fast as nature can renew it.]' back
Eigenfunction - Wikipedia, Eigenfunction - Wikipedia, the free encyclopedia, 'In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has Af = λf for some scalar, λ, the corresponding eigenvalue.' back
James Lesher, Xenophanes (Stanford Encyclopedia of Philosophy), 'Xenophanes of Colophon was a philosophically-minded poet who lived in various parts of the ancient Greek world during the late 6th and early 5th centuries BCE He is best remembered for a novel critique of anthropomorphism in religion, a partial advance toward monotheism, and some pioneering reflections on the conditions of knowledge. Many later writers, perhaps influenced by two brief characterizations of Xenophanes by Plato (Sophist 242c–d) and Aristotle (Metaphysics 986b18-27), identified him as the founder of Eleatic philosophy (the view that, despite appearances, what there is is a changeless, motionless, and eternal ‘One’). In fact, the Xenophanes who emerges from the surviving fragments defies simple classification. He was a travelling rhapsode who criticised the stories about the gods told by the poets, and he defended a novel conception of the divine nature. But he was also a reflective observer of the human condition, a practitioner of the special form of ‘inquiry’ (historiê) introduced by the Milesian philosopher-scientists, and a civic counselor who encouraged his fellow citizens to respect the gods and work to safeguard the well-being of their city.' back
Job, Job: King James Version, '8 And the Lord said unto Satan, Hast thou considered my servant Job, that there is none like him in the earth, a perfect and an upright man, one that feareth God, and escheweth evil? 9 Then Satan answered the Lord, and said, Doth Job fear God for nought? 10 Hast not thou made an hedge about him, and about his house, and about all that he hath on every side? thou hast blessed the work of his hands, and his substance is increased in the land. 11 But put forth thine hand now, and touch all that he hath, and he will curse thee to thy face.' 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
Second Vatican Council, Dogmatic Constitution on Divine Revelation - Dei verbum, Dogmatic Constitution on Divine Revelation Dei Verbum solemnly promulgated by his Holiness Pope Paul VI on November 18, 1965. 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
Thomas Aquinas, Summa I, 3, God's Simplicity, 'Once we have ascertained that a given thing exists, we then have to inquire into its mode of being in order to come to know its real definition (quid est). However, in the case of God we cannot know His real definition, but can know only what He is not; and so we are unable to examine God’s mode of being, but instead can examine only what His mode of being is not.
Therefore, we have to consider, first, what His mode of being is not; . . . By excluding from God certain things that do not befit Him, e.g., composition, change, and other things of this sort, it is possible to show what His mode of being is not.' 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
Wojciech H Zurek, Decoherence and the Transition from quantum to classical - Revisited, 'Quantum mechanics works exceedingly well in all practical applications. No example of conflict between its predictions and experiment is known. Without quantum physics, we could not explain the behavior of the solids, the structure and function of DNA, the color of the stars, the action of lasers, or the properties of superfluids. Yet nearly a century after its inception, the debate about the relation of quantum physics to the familiar physical world continues. Why is a theory that seems to account with precision for everything we can measure still deemed lacking? The only “failure” of quantum theory is its inability to provide a natural framework for our prejudices about the workings of the Universe.' 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

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