1: Introduction: a prolegomenon to scientific theology part II
1.2: No information without symbols
1.4: Requisite variety and control
1.5: The divine initial singularity
1.6: From trinity to transfinity Trinity
1.8: A network model of the universe
1.9: Creation, evolution and entropy
1.10: Determinism is not only not creative but logically impossible
Abstract
This introduction is a brief run through some of the ideas at the root of this project. Everything will be elaborated in due course.
The starting point of my thesis was an identification of the classical god of Christianity with the conclusion drawn from general relativity that the universe began from an initial singularity. This identification follows from the observations that both god and the singularity are believed to exist; that both are absolutly without structure; and that both are the source of the world. Hawking & Ellis: The Large Scale Structure of Space-Time
Aristotle pioneered the path from physics to theology, the name he gave to what we now call metaphysics. His starting point we the idea of matter and form used above to explain the knowledge of god. He developed the theory of matter and rom to explain change. His teacher, Plato, proposed that the structure of the world was controlled by an invisible heaven of eternal forms. Plato added the idea that this world is a very poor copy of its heavenly prototype, and that only philosophers are in a position to understand the true beauty of the heavenly world. The rest of us are in effect trapped in a cave where only the shadows of the true world are visible to us.
1.1: God is not the World?
Thomas Aquinas begins his Summa Theologiae by proving the existence of God using the famous five ways. For those who believe that God is the creator of the world, these proofs are somewhat redundant. Since the world obviously exists, the creator must exist. The five ways do not so much prove that God exists, as prove that God is other than the world. Each of them points to a supposed deficiency in the world, and concludes that a God is necessary to compensate for deficiency. Five Ways (Aquinas) - Wikipedia
Traditional Christian theology abounds in mysteries. The biggest mystery of all is God itself. The traditional reason provided for this is that our minds are so tiny compared to God that we are totally incapable of understanding it. The idea is expressed in the term apophatic theology ancient Greek for denial theology. After he has proved that God exists, Aquinas gives us a clear warning that we are cannot say anything positive about it:
Once we have learnt that something exists, it remains to seek the mode of its existence, so that we may better understand what it is. But because we cannot know what God is, but what it is not, we cannot ask how God is, but rather how it is not. . . . It is possible to show how God is not by removing from it those properties which are not appropriate, like composition, motions and similar properties. So we ask first about the simplicity of God, removing composition. St Thomas Aquinas: Summa: I 3 1: Prologue
He then goes on to deny a series of attributes that a God might have, concluding that it does not have parts of any description: God is altogether simple, rather as we imagine fundamental particles like photons and electrons to be. Aquinas, Summa, I, 3, 7: Is God altogether simple?
This question reinforces the conclusion reached by the proofs of God. There is no way we can identify our extraordinarily complex world with an absolutely simple god. From a political point of view, this conclusion is an excellent starting point for creative theologians. Because we have no evidence based information about God we are free to create God in own own image, the inverse of the idea in Genesis that we are created in the image of God (I:27). It is not surprising, therefore, that the world abounds in a Babel of divinities, theologies and religions.
1.2 No information without symbols
Simplicity creates another problem for those who would know God.You can read this text (and possibly make sense of it) because it is encoded in the letters of the English alphabet. An absolutely simple God is very like like a blank sheet of paper: nothing there to read.
This was not a problem for the ancients who consider knowledge and intelligence to be attributes of spirituality. The traditional picture is described by Aquinas in his discussion of the knowledge of god:
He asks: Is there knowledge in God? Yes, he replies, building an argument on Aristotle’s doctrine of matter and form:In God there exists the most perfect knowledge. To prove this, we must note that intelligent beings are distinguished from non-intelligent beings in that the latter possess only their own form; whereas the intelligent being is naturally adapted to have also the form of some other thing; for the idea of the thing known is in the knower.
Hence it is manifest that the nature of a non-intelligent being is more contracted and limited; whereas the nature of intelligent beings has a greater amplitude and extension; therefore the Philosopher says (De Anima iii:8, 431b20) that "the soul is in a sense all things." Eugene T. Gendlin: Line by Line Commentary on Aristotle’s De Anima
Now the contraction of the form comes from the matter. Hence, . . . forms according as they are the more immaterial, approach more nearly to a kind of infinity. Therefore it is clear that the immateriality of a thing is the reason why it is cognitive; and according to the mode of immateriality is the mode of knowledge. . . .
Since therefore God is in the highest degree of immateriality . . . it follows that He occupies the highest place in knowledge Aquinas, Summa: I, 14, 1: Is there knowledge in God?.
From a modern point of view, this discussion completely misses the point since it is a general metaphysical argument which does not touch on the mechanism of knowledge. In a way, Aristotle comes close when he defines the soul as the first actuality of natural body possessed of organs, that is of details which may encode information. Christopher Shields (Stanford Encyclopedia of Philosophy): The Active Mind of De Anima III 5
We understand that knowledge is a form of information, and we now understand that all information is represented by a physical marks of some sort, as we can see on the everyday macroscopic scale in all forms of sensory stimuli, visible, audible and tangible, including the present medium, writing. In a nutshell, information is a physical entity Rolf Landauer: The physical nature of information (1996).
The claim that God is absolutely simple and omniscient is therefore a difficulty with the traditional divinity which is solved if we identify the God and the universe. Like the information encoded in our nervous systems by the chemical state of our synapses, information is stored in the divine universe by its own structure which we may understand by analogy to this text. There is need for a separate copy in a god
1.3 Logic and omnipotence
The traditional God is omnipotent as well as omniscient able, we imagine, to do anything thinkable, that is anything logically consistent. Aquinas asks Is God omnipotent? and answers:
. . . God is called omnipotent because He can do all things that are possible absolutely; . . .. For a thing is said to be possible or impossible absolutely, according to the relation in which the very terms stand to one another, possible if the predicate is not incompatible with the subject, as that Socrates sits; and absolutely impossible when the predicate is altogether incompatible with the subject, as, for instance, that a man is a donkey. Aquinas, Summa I, 25, 3: Is God omnipotent
Here Aquinas anticipates Hilbert's formalist approach to mathematics. This approach was inspired by Cantor's development of transfinite numbers using set theory. Cantor's work greatly expanded the domain of mathematics, but led to logical paradoxes (contradictions) which had to be weeded out to preserve the heart of Cantor's work. Among the first of these was Cantor's paradox. Formalism (mathematics) - Wikipedia, Cantor's theorem - Wikipedia, Cantor's paradox - Wikipedia
Hilbert thought that the demand for formal consistency would enable the solution of all mathematical problems. Whitehead and Russell were inspired by this idea to look for a way to define mathematics as applied logic. Gödel applied Whitehead and Russell's ideas to show that there were problems that formally consistent mathematics could not solve. Turing followed soon after with the discovery that there were problems that were formally incomputable. From a Thomistic point of view, these discoveries place boundaries on the omnipotence of any god. Hilbert's program - Wikipedia, Whitehead & Russell: Principia Mathematica
1.4: Requisite variety and control
The cybernetic principle of requisite variety establishes that a complex system can only be controlled by a system of equal or greater entropy or complexity. We measure variety logarithmically in bits. A two state system like a light can be controlled by a two state system like an on/off switch, both having a variety of one bit. A red / orange/ green traffic signal has, in principle, six states, requiring three bits of control via three on/off switches. One state, with red and green on simultaneously must be forbidden lest it lead to confusion and collisions. In some systems orange and red or orange and green may be on simultaneously, but both these conditions are temporary so independent switching is still required. Ashby: An Introduction to Cybernetics
Gregory Chaitin has shown that this principle is closely related to Gödel’s incompleteness theorem:
Gödel's theorem may be demonstrated using arguments having an information-theoretic flavour. In such an approach it is possible to argue that if a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. In contrast with the traditional proof based on the paradox of the liar, this new viewpoint suggests that the incompleteness phenomenon discovered by Gödel is natural and widespread rather than pathological and unusual. Gregory J. Chaitin: Gödel's Theorem and Information
Traditional theology maintains that the combination of the divine omniscience and omnipotence gives God the ability to completely know and control every detail of the future. Requisite variety invalidates this claim. Insofar as God is understood to be absolutely simple, its variety is zero, and so its powers of knowledge and control are absent. It is simply not possible to reconcile the absolute simplicity of god with omniscience and omnipotence.
1.5: The divine initial singularity
The solution of the cybernetic problem posed by requisite variety is a fundamental issue in this essay. We accept as our starting point the modern analogue of the ancient absolutely simple god, the initial singularity established by Penrose and elaborated by Hawking and Ellis as a consequence of Einstein's general theory of relativity. Hawking and Ellis conclude their book with the sentence: '. . . the actual point of creation, the singularity, is outside the scope of the presently known laws of physics.' Given the law of conservation of energy, it may be tempting to assume that the singularity contains of the energy of the universe in a point of zero size and structure. This seems unlikely.
Modern cosmology imagines that the initial singularity is not only an extremely hot and dense state, but it is by definition outside space-time, so we cannot make much sense of it. Instead we turn here to the ancient notion of god developed by Aristotle and Aquinas who see god is as pure actuality, in a broad sense the realization of all possibility. Here we identify it with the quantum of action.
In the beginning, Planck realized that black bodies radiate energy in units of hω which is in the first place simply a mathematical symbol. Einstein, using a statistical argument, proposed that these units (quanta) are the particles of light we now call photons, linking Planck's mathematics to a physical reality. Here we make a similar move, guessing that the quantum of action is not just a symbol but a particle whose properties we will imagine to be open to becoming a universe. Albert Einstein: On a heuristic point of view concerning the production and transformation of light
In modern physics, the quantum of action is the smallest possible event, while the universe is the largest. It may seem difficult to see the smallest as the source of that largest, but we will avoid the question of size by postulating that the quantum of action is a logical event whose effect is the same as the logical not. In abstract generality, an act changes some p into not-p, here into there, alive into dead, unborn into born.
Every interaction at the fundamental level involves a quantum of action, but the energy associated with this event may be anywhere from zero to a countable infinity, since we understand energy to be the inverse of the time taken for the event to happen. So energy in the order of n corresponds to an event whose period is 1/n. Every fundamental event in the physical world involves precisely one quantum of action. The whole business of quantum physics is to measure and calculate how much energy is associated with the action in each particular event.
1.6: From Trinity to transfinity
The notion that the living gods can reproduce and multiply like other living creatures is very ancient. Although the Hebrews were rigidly monotheistic, their successors, the Christians, differentiated three divine personalities, the Father, the Son and the Spirit. these were given distinct roles in the Christian Salvation History. Trinity - Wikipedia, Salvation History - Wikipedia
Augustine and Aquinas developed a psychological model of the Trinity. The Father is modelled on the ancient Hebrew God. In humans our self image is accidental, but according to the notion that things which are accidental in created beings are substantial in god, the Father's self-image is also god, the Son, and the love of the Father for the Son is a third person, the Spirit.
Aquinas point out that the differentiation of the persons from one another within the divine unity is mediated by their relationships to on another. He makes the point that the divine relationships are real, unlike human relationships that he sees, following Aristotle, as merely mental constructs. Here we understand created relationships to be real, established by communication. Not only are they the foundation of human societies, but they are the mechanism through which communication networks establish complex relationships at all scales.
The Christian Trinity is limited to three persons for dogmatic reasons. Here we see no such limit, and understand that the universal network can expand into the transfinite realm. The cardinal of a separable Hilbert space is ℵ0. Through the operation of Cantor's theorem, this first transfinite cardinal can yield increasing transfinite cardinals without bound.
1.7: Before the big bang
We are assuming the identity of the initial singularity, the classical god defined by Aristotle and Aquinas as pure action, and the quantum of action. The ancient view distinguishes action and potential and accepts Aristotle's opinion that no potential can actualize itself. This is the key assumption in Aquinas' first way of proving that god is other than the universe. Modern physics rejects this axiom, at least in the realm of energy, accepting that energy appears in two forms, potential and kinetic, and that the symmetry inherent in the conservation of energy requires that both kinetic and potential energy be taken into account.
The large scale space-time structure of the present universe is based on the notion that the four vectors time-space and energy-momentum transform identically. We understand that both space-time and all that it contains was created by the big bang. We model the resulting structure with a quantum field theory that combines quantum theory and special relativity.
The formalism of quantum mechanics does not require the existence of space so that we can consider it as a one dimensional field theory based solely on energy and its dual time, connected through the quantum of action by the fundamental equation E = ℏω. Of particular interest then is the mechanism by which space-time emerged from energy-time, the driver of the big bang. Our working hypothesis is that can be explained using a communication network based on quantum communication, quantum computation and the creative property of quantum observation, explained by von Neumann, which increases entropy. The emergence of space-time within the initial singularity is one of the first steps in the creation of the universe. John von Neumann: Mathematical Foundations of Quantum Mechanics Chapter 5
1.8: A network model of the universe
Bohr and his followers see quantum measurement as the interaction between a quantum and a classical system but in reality all actions are quantum interactions. A quantum network is therefore a natural framework in which to describe the universe, every event in the universe being understood as a communication between two quantum systems. Although in accelerator experiments we try to simplify affairs by observing the interactions of discrete particles, reality is in effect a sea of interaction closely analogous to a human society.
In 2019 I wrote an honours thesis Prolegomenon in the department of philosophy, University of Adelaide, introducing the idea that it is reasonable to identify the universe and God. Here I wish to extend this thesis, which was based on classical network computation and communication, by showing that quantum computation and communication provides a more direct and convincing route to binding physics to theology. If the universe is divine, physics is a the study of god's body and if we follow Aristotle's path, we will come through physics to a study of god's mind which is in effect the creative imagination of the universe. Aristotle, Metaphysics XII: vii: The divine life of the prime mover, Jeffrey Nicholls (2019): Prolegomenon to scientific theology
We may say, nevertheless, that the initial singularity, as a quantum of action has within it the potential, rather than the actuality of omniscience and omnipotence. In the ancient terminlogy, its existence is identical to its existence.
A key point of this site is to argue that we can equate the universe to the mind of god, giving rise to what I will call cognitive cosmology. I take the view that the most characteristic attribute of god is omniscience, from which all else follows, and that the line in Genesis that we are created in the image of god is best construed to mean that our intelligence is an image of the universal intelligence that created us.
1.9 Creation, evolution and entropy
Our problem is to understand how our enormously complex universe emanated from the structureless initial singularity. Our hypothesis is that a process analogous to evolution by natural selection, based on quantum theory, will provide the answer. Since deterministic copying is not creative, randomness is an essential ingredient of creative evolution. The uncertainty that many attribute to quantum mechanics may be essential to our existence.
The principle feature of the application of quantum mechanics to theology is that it enables us to apply the idea of evolution by natural selection at the quantum mechanical level. The quantum fluctuations of the vacuum provide us with an infinite space of possibilties from which observables are selected by Hermitian operators which identify the possible space of survivors in the quantum mechanical ecosystem. These survivors are in effect the common eigenvalues of the operators of the interacting systems.Garrett Birkoff & John Von Neumann; The Logic of Quantum Mechanics,The nature of the quantum of action is to act, but in the beginning where the quantum has no structure this action is uncontrolled and so it must be random. Since energy is the product of action by frequency, that is universe time, we understand energy to be a measure of how quickly an action is completed. Given uncontrolled but perpetual action, we naturally create a "vacuum" with an unlimited continuous spectrum of energies
Both energy and entropy are conserved in the unobserved quantum world but the real action is in entropy and information, and this is a matter of interaction and communication, that is of quantum measurement. Here (I suspect) we see how natural selection works at the quantum mechanical level, selecting "observables" from all other possibilities. Through the mechanism modelled by the eigenvalue equation quantum mechanics provides a very low level model of natural selection (sometimes called the collapse of the wave function) which selects one of many possibilities to be realized in a quantum event.
In this essay I wish to express the general ideas of my thesis but this time constructing the transfinite network in Hilbert space and using the theories of quantum computation and quantum computation to explore the theological consequences of this construction. An important consequence of this exploration is an account of the quantum mechanical origin of the four dimensional spacetime of our common experience, and in particular the Minkowski metric that applies in quantum field theory and gravitation. Success in this endeavour might point to a quantum theory of gravitation, thus bringing all the four modes of natural communication represented by the electromagnetic, weak, strong and gravitational forces under the same umbrella and providing us with a path to mapping a physical theory of everything to a theological theory of everything. Nielsen & Chuang (2000): Quantum Computation and Quantum Information, Jeffrey Nicholls (2019): Prolegomenon to Scientific Theology
It seems reasonable to imagine that the transformation of reality from a structureless singularity to the current universe involves evolution, the unfolding of possibility. Charles Darwin put his finger on the dynamics of evolution without any knowledge of modern physics, genetics or molecular biology.
We may imagine that his immediate source of inspiration was the farming community which has for thousands of years controlled breeding and selection in the search for faster horses, fatter cattle, tastier vegetables and more beautiful flowers. He augmented this with paleoontological information and his own observations on a trip around the world. At the core of the process, beautifully demonstrated by the COVID virus, is random variation and selection. In living creatures the variation arises either from errors in the duplication of genetic material or the mixing of genes by sexual reproduction. Natural selection is the tendency for those better adapted to their local environment to have a higher probability of reproducing themselves. Kirsty Short: UK, South African, Brazilian: a virologist explains each COVID variant and what they mean for the pandemic
We have now been able to reconstruct a relatively complete history of the evolution of life since close to the beginning. How the first living creatures came to be is still an open question. A more difficult question to answer is how the universe could have grown to its present state from the structureless initial singularity proposed by many cosmologists. At first glance, this would appear to be miraculous, explaining why so many in the past have proposed an omniscient and omnipotent creator to explain our existence.
1.10 Determinism is not only not creative but logically impossible
Creative evolution is not possible in a deterministic system. The formal constraint on the omnipotence of god is the same as the constraint on formalist mathematics: any formal mathematical structure is permitted so long as it is not inherently self contradictory. David Hilbert proposed this purely formal approach to mathematics, divorcing it from any empirical connection to physics. He felt that this approach would leave no mathematical problem unsolved. This was not to be. First Kurt Gödel showed that some propositions are undecidable. Soon after Alan Turing found that some functions are incomputable. This failure of total control in a consistent system opens the way for variation and creation. Kurt Gödel - Wikipedia, Alan Turing - Wikipedia
The network model provides a mechanism for evolution. The principal role of a classical network is to communicate by copying structures from one point to another in space-time. Copying errors provide variation and in rare cases, as in biological evolution, such variations provide improvement in the efficiency of software, providing it with a selective advantage. So we find that faster algorithms replace slower ones. A further source of variation in a computer network is that computers interrupt and redirect one another at random times. I am sitting in front of a deterministic machine, but at any instant I may strike a key that sends it off on a new course of action.
Physicists, being empirical scientists, are bound to take the world as they find it and try to explain how it fits together, a quest which seems to have settled on symmetries and gauge theories. As a theologian more interested in origins, one seeks to learn how it came to be this way starting from a zero entropy state of divine action. By bringing the two approaches into contact, one might hope to perfect both physics and theology, arriving at cognitive cosmology, a picture of god's creative mind essentially identical to the universe we inhabit.
The fact apparently accepted by physicists, that we can create anything out of energy, which involves time but predates space, would seem to imply that action, which predates both time and space, puts no constraints at all on possibility. Like the traditional god, action has unbounded potential limited only by the fact that local inconsistency, revealed by local contact, cannot exist. It is outside the bounds of possibility, "outside" the universe.
What we are really wishing to show is that quantum mechanics is a lot closer to god than classical mechanics. How so we achieve this? In other words, what is the special point of quantum theory. The idea I started with was that I can fix up the infinity problems in quantum field theory by making Hilbert space rather than Minkowski space the basic domain of the universe seen as a computation network and get the result that Minkowski space is naturally pixellated by the quantum of action, which disposes of the infinities.
The real task, however, is much deeper and must be built on quantum computation and communication. In the long historical perspective, we see Aristotle's axiom — no potential can actualize itself — precludes the existence of harmonic oscillators like the pendulum. The classical manifestation of the failure of Aristotle's axiom is the identity of potential and kinetic energy that we see in the ideal classical pendulum and the associated conservative gravitational field in which it operates.
Here we begin with the idea that the actual processing in the universe does not occur in space-time but in Hilbert space. Spacetime acts as the medium for memory and communication, that is it is the operating system of the universe. This is true even of classical computers where the actual processing is done by quantum systems embedded in chips and wires embedded in spacetime. Space-time must be four dimensional in order to maintain orthogonality between the underlying processes. All computing is quantum computing and always was. What the co-called quantum computing business is doing is fiddling with the interface between Minkowski space and Hilbert space.
Very little has changed in Christian theology since Aquinas developed the 'standard model' of God more than seven hundred years ago. The Summa is the inspiration and starting point for this work. Very little of what Aquinas wrote needs to be changed. It simply needs to be reinterpreted to reflect the position taken here. God remains a simple dynamic unity. The difference is that we are no longer outside God, but inside. Thomas Aquinas: Summa Theologiae, Thomas Aquinas: Opera Omnia
Our new starting point (replacing Scholastic philosophy) is John Von Neumann's description of Hilbert space, the natural home of quantum mechanics and the quantum of action. Quantum mechanics is the natural explanation of creation driven by action.John von Neumann: The Mathematical Foundations of Quantum Mechanics
(revised 11 May 2021}
