vol 6: Essays
Essay 15: Is the Universe divine? (2015)
Table of contents
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
3: Quantum mechanics and fixed points
5: Logical continuity: the mathematical theory of computation
6: The transfinite digital network as a model of revelation
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.
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.
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. Back to toc3: 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.
Back to toc4: 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.
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 ℵ0, the first transfinite cardinal. He also saw that by ordering the set of ℵ0 natural numbers in different ways, it is possible to make another number, hugely greater, the second transfinite cardinal ℵ1. The process that generates ℵ1 out of ℵ0 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 ℵ1. 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. ℵ1 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.
Back to toc5: 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 ℵ1. 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 ℵ1 eigenfunctions, there are only ℵ0 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.
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 ℵi mappings of the ℵ0 natural numbers onto themselves, there are only ℵ0 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.
Back to toc7. 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.
Back to toc8: 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.
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. Acemoglu & Robinson: Why Nations Fail: The Origins of Power, Prosperity and Poverty
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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