vol III: Development

chapter 4: Physics

page 10: Space-time

Introduction: from classical God to modern physics

This promotes the idea that the universe is divine: that is it plays all the roles traditionally given to God. The principal difficulty for this project is reconciling the immense dynamic complexity of the world we live in with the eternity and absolute simplicity traditionally attributed to God. The solution proposed here is that the observable features of our world are fixed points in the divine dynamics. The existence of such fixed points is predicted by mathematical fixed point theory. Fixed point theorem - Wikipedia

Fixed point theory provides a general answer to the relationship between a simple divinity and a complex universe. The next problem we have to face is the relationship between a spiritual eternal divinity and the world of space and time. Since ancient times insight, our ability to know and understand, has convinced many authors that we are spiritual beings. This problem has been largely overcome by our discovery that matter is extraordinary complex and energetic. As Pierre Teilhard de Chardin noted, the universe becomes more spiritual as it becomes more complex. Lonergan: Insight, Teilhard de Chardin, Pierre Teilhard de Chardin - Wikipedia

We are assuming that the emergence of fixed points in the divine dynamics occurs in some sort of orderly fashion. It is not easy to discern this order, but our heuristic approach suggests that simpler structures occur before more complex structures and serve as the foundation upon which more complex structures are built. The simplest structures in everyday experience appear to be time and space, initially devoid of any content. Time flows evenly through our lives, and even though different things may happen at different times, time itself seems to be unchanging. The same is true of empty space.

The classical Christian God is eternal, existing all at once outside space and time. The Universe, on the other hand, clearly moves in time, having both a history and a future, and contains a lot of space. If we are to maintain that the Universe is divine, we must explain how time and space fit a consistent model of God. To do this we understand time and space to be fixed points in the divine dynamics. Aquinas 47: Is God eternal?, Aquinas 20: Is God altogether simple?

Thomas argues that God is eternal because it is immutable. God is immutable because change requires potentiality, and there is no potentiality in a purely actual being. This argument is weakened by the modern understanding that Aristotle's axiom that no potential can actualize itself is false. As the ideal pendulum demonstrates, potential and kinetic energy are exactly equivalent. Aquinas 43: Is God immutable?, Potential energy - Wikipedia, Kinetic energy - Wikipedia

Further, Aquinas argues, space is not an attribute of God because God is not a body. A body, he says, is continuous, and therefore infinitely divisible, which shows that it is in potentiality, which is also inconsistent with the assumption that God is pure actuality. Aquinas 14: Is God a body?

The history of science suggests that new scientific developments often take a broader view of the world within which the old view is a special case. Newtonian celestial dyamics, for instance, can be seen as a special case of the general theory of relativity in regions of low energy. Newtonian physics works quite well in the solar system, but has little to tell us about black holes. Kuhn: The Structure of Scientific Revolutions, David K Sing: Einstein's Field Equation and the Newtonian Limit

Space-time

Intuitively, space-time is the blank screen upon which the events of the world are projected. This concept lies at the foundation of modern physics, which explains all events as the outputs of fields on space-time. Mathematical physics sees a field as a function that has a value at every point in space and time. Quantum field theory imagines the whole of space-time (except perhaps its earliest moments) to be permeated with fields corresponding to the sixty or so known elementary particles. The observable particles themselves are believed to be energetic excited states of the underlying fields. The creation and annihilation of particles corresponds to the conserved transfer of energy from one particle field to another. Quantum field theory - Wikipedia, Elementary particle - Wikipedia

Quantum field theory takes space and time as given, and does not seek to explain their existence. General relativity goes deeper, seeing gravitation not so much as a field whose domain is space-time, but as the structure of space-time itself. A serious constraint on this theory is the assumption that we describe space-time and the fields on it using the mathematics of continuous differentiable manifolds. These manifolds represent only a tiny part of the enormous space of mathematics, which is limited only by consistency. The strategy on this site is to understand the universe using not just the results of mathematics, but the methods of mathematics. Differentiable manifold - Wikipedia

We understand mathematics itself as the framework of fixed points (published theorems) established by the creative work of the mathematical community over many centuries. This is the product both of the network of mathematicians spread over space and time and of the neural network that creates the mind of each mathematician. Our hypothesis that the universe is divine, that is that it is all that there is, means that there are no external constraints. The boundaries on the universe are the edges of reality bordering on nothingness, that is inconsistency. Inconsistency is also the boundary of mathematics, although some people study inconsistent mathematics. Mathematics - Wikipedia, Leon Horsten (Standord Encyclopedia of Philosophy), Mortensen: Inconsistent Mathematics

The network model that we are using here models insight as the encoding and decoding of messages. When we are conversing to our friends in our natural language, this process often seems instantaneous. Sometimes it takes a while to understand what someone is saying or to put something into words, sometimes a long discussion is necessary to share a complex thought. The progress of science is analogous, but can be much slower. It has taken us thousands of years to decode some of the messages that we receive from nature. This site is an attempt to understand the universe in the light of a new paradigm, that the Universe is divine.

We might call this a psychological rather than an analytical approach to modelling the world. The state space for this approach is a transfinite computer network described in Chapter 2. This network comprises a set of processes represented by computers with the power of turing machines and a set of memories which are read and written to by the computers. Although there are only ℵ₀ different computable functions, we place no limit on the number of instances of each process to be found in this network.

A network is a space and time

If we see memory as analogous to space and digital processes as analogous to the passage of time we may imagine the transfinite computer network as a space-time. As Cantor noted, there is no formally consistent limit to the cardinal of a set as long as we do not introduce a set which is inconsistently large, such as the set of all sets. We imagine the initial singularity as a one point network which has since expanded into the enormously complex cosmic network we now observe. Hallett: Cantorian set theory and limitation of size

Space-time makes networks possible. Let us assume that every source A in an ideal network should be abe to communicate with any other source B without interference from any other sources C, D . . .. There should be no crossed wires. As the makers of printed circuit boards learned long ago, this is not possible in two dimensions. Two sources can communicate in one dimension, three sources can communicate in two dimensions, but four or more sources require three dimensions. Formally, three dimensions can accomodate any number of connections without crossed wires. (Practically, of course, we have to deal with the thickness of the wires.) From a network point of view, therefore, there is no need for further dimensions.

We need three dimensional space to establish 'hard wired' connections between an unlimited number of sources, but this is not the only way, as we can see with road traffic. There is also 'time division multiplexing' the discipline imposed by traffic lights and give way rules. Two vehicles cannot occupy the same point on an intersection, but they can share the same space if they are separated in time. Time-division multiplexing - Wikipedia

The emergence of space-time

These ideas suggest that a comprehensive network needs space and time, but we have yet to understand why the initial singularity, effectively a single source, should differentiate into multiple sources.

An explanation of this tendency may be provided by Cantor's theorem, which tells us that given any set, we may produce a larger set by ordering the elements of the original set. This theorem provided Cantor with a process for moving from the natural numbers to the transfinite numbers. We might see Cantor's theorem as the source of a 'Cantor force' which exerts pressure toward increasing complexity. Cantor's theorem - Wikipedia

Christianity introduced a radical new idea into Hebrew monotheism when it postulated that although there is only one God, this God is a group of three distinct persons, commonly known as the Father, the Son and the Holy Spirit. Augustine and Aquinas produced a simple model of the Trinity which attempted to explain the distinction of the persons while maintaing the unity of God. This model is psychological. It sees the Son as a God's image of itself and the Spirit as the love binding the Father to the Son. Here we propose a quantum mechanical analogue to this idea. Monotheism - Wikipedia

In quantum mechanical language, we might say that the Father produces the Son by interfering with itself. We see the same phenomenon occurring in the two slit experiment, and in Feynman's extrapolation of this idea to his 'path integral' formulation of quantum mechanics. Zee

In the two slit experiment we imagine single particles like electrons or photons passing through two slits in a screen and forming an interference pattern behind the screen. This pattern appears even when we allow only one particle at a time through the apparatus, so we must assume that each particle interferes with itself. Feynman's methodology imagines the space between two states being filled with an infinity of screens, each with an infinity of holes, and uses a mathematical expression of this idea to compute the probability of an initial state moving to a final state. Double-slit experiment - Wikipedia, Interference (wave propagation) - Wikipedia, Path integral formulation - Wikipedia

The path integral approach works on the assumption that every path from A to B is possible but a selection process picks out the actual path taken. Feynman developed this idea from the Lagrangian formulation of classical mechanics, led on by a hint from Paul Dirac. The selection process is based on Hamilton's principle.

Hamilton's principle and natural selection

It has long been believed that the physical world is in some way optimal. Many theological thinkers would attribute this to the power and goodwill of the creator. It may also be a consequence of natural selection. Those systems closer to perfection have a better chance of reproduction than those with significant failings. Yourgrau & Mandelstam: Variational Principles . . .

This line of thought led to Hamilton"s principle, that the action associated with any event is a fixed point either a maximum or a minimum. The action is defined as the time integral of the Lagrangian, L. The Lagrangian, in its simplest form, is the difference between between the kinetic energy T and potential energy V, so we write L = T - V. The action, S is then written S = ∫ _t1 ^t₂ L dt where t₁ and t₂ are the start and finish times of the process for which we are computing the action. Neuenschwander

Hamilton's principle, adapted to quantum mechanics, has since become one of the most important heuristic methods in physics. It has been found to reproduce the results of classical Newtonian mechanics and quantum mechanics. One possible stationary value of the action functional is 0, which results in situations where the kinetic energy T is equal to the potential energy V, so that T - V = 0. This suggests that kinetic and potential energy may be equal and opposite in an ideal universe.

Is the total energy of the universe zero?

Richard Feynman and many others have speculated that the total energy of the Universe is zero. This assumption saves us from trying to imagine an initial singularity of infinite energy density and infinite temperature. It also supports the idea that the divine universe is pure eternal act. Since energy is the time rate of action, a zero energy universe has a zero net rate of action, and so is, at the lowest layer, eternal. Feynman: Feynman Lectures of Gravitation, Marcelo Samuel Berman: On the Zero-Energy Universe

The zero energy universe is made possible by the bifurcation of action into potential and kinetic energy, where the potential energy is equal to minus the kinetic energy, so that potential energy + kinetic energy = 0. This approach also seems consistent with the idea of a dynamic god with fixed points. The fixed points are mathematically formal but physically endowed with potential energy equal and opposite to the kinetic energy of the process in which they are embedded.

Here we assume that the fundamental symmetry in the universe, something which applies to all processes, is local consistency. For instance, we assume that nothing can be both black and white at the same time at the same point. The quantum mechanical notion of superposition appears to violate this requirement, since it is assumed that all the solutions to a given differential equation are real, different, and exist in the same place. To avoid this inconsistency we must assume that these solutions are separated in space and time.

We find that every fundamental particle falls into one of two categories: it is either a boson or a fermion. We guess that separation in time or frequency are characteristic of bosons. Separations in space are characteristic of fermions. Bosons of the same energy have a tendency to flock together into the same quantum state. Fermions, on the other hand, obey the Pauli exclusion principle so that no two may be found in the same position except under the extreme levels of duress found in black holes. So we have two levels of differentiation, bosons are not fermions, and no two fermions can be found in the same state. Pauli exclusion principle - Wikipedia

Bosons are 'messenger' particles and fermions are structural particles. From a quantum mechanical point of view, it is the fermions, in their attempts to stay away from one another, that enable spatially extended structures to be built from fundamental fermions like electrons, which are believed to have no spatial extension.

Bosons have integral spin (measured in units of Planck's constant), and fermions have half integral spin. Particles with integral spin execute a quantum of action by making a full turn, through 2π radians or 360 degrees. Particles with half integral spin, on the other hand, must make two full turns to execute a quantum of action so that one full turn leaves then in an unobservable anti-commutative state. Spin (physics) - Wikipedia, Spin-statistics theorem - Wikipedia

Space, memory and creation

We may assume that the tension between competition and cooperation exists at all levels in the evolutionary process, and so the idea of symmetry with respect to complexity allows us to carry it back to the relationship between bosons and fermions and the creation and extension of networks.

In a layered network, higher layers need to curate lower layers in order to preserve their own existence, as my mind, for instance, must give some thought to my food supply if it is to remain embodied. We might therefore expect to see an explanation of space-time in the communication processes between the fundamental particles which constitute the material universe. We will return to this subject on the next page Quantum field theory

The network model sees the universe as an information processing system, and so like the human brain and other networks, we see an element of creativity in it. Our hypothesis that the Universe is divine implies that the Universe is intelligent, since God is commonly understood to be the epitome of intelligence. Misner, Thorne and Wheeler present an excellent short statement on scientific intelligence:

. . . that view is out of date which used to say, "Define your terms before you proceed." All the laws and theories of physics, including the Lorentz force law, have this deep and subtle character, that they both define the concepts that they use . . . and make statements about these concepts. Contrariwise, the absence of some body of theory, laws and principles deprives one of the means to properly define or even to use concepts. Any forward step in human knowledge is truly creative in this sense: that theory, concept, law and method of measurement—forever inseperable— are born into the world in union. Misner, Thorne & Wheeler: Gravitation

We imagine that a similar process happens as the universe diversifies from the initial singularity to the complex network we now inhabit.

Transformation: Coding and decoding

Here we see the network world growing by introducing dualities. In the beginning we have energy, which we model by the transformation between the real and complex axes of the unit vector rotating in the complex plane (page 4: Energy). The complex phase of this rotation is invisible to us but serves as a coupling between real particles carrying potential energy and the invisible processes that generate them using kinetic energy.

So far we have dealt with 'pure' quantum mechanics which we assume to be a fixed point in the Universe closely related to conservation of action. Pure quantum mechanics has no memory and exists prior to physical space. In the layered network model proposed on this site, the conservation of action is the symmetry bounding the lowest physical layer of the Universe.

The act of decoding is the inverse of the act of encoding, so that the action associated with a received message adds up to zero. Encoding turns kinetic energy into potential (a fixed point, message). Decoding does the opposite, releasing the potential energy as kinetic energy. We may see the 'collapse of the wavefunction' in this light. An excited field bearing kinetic energy may collapse to create a particle carrying potential energy. A particle carrying potential energy may be annihilated to create an excited field.

Observation shows that unlike time, which marches along imperturbably, carrying us with it, space began small and has grown to the huge Universe than we now inhabit. Here we identify space with memory. Without memory, the whole concept of evolution is meaningless since to an entity without memory, there is no past or future. Further, although we cannot move in time, we are free to move in space. Metric expansion of space - Wikipedia

The symmetries of space-time

Like a computer network, the Universe is layered. In the physical layer we find the fundamental particles which are generally assumed to be points with no internal structure. These particles then assemble into more complex structures like atoms, molecules, cells and so on. As in a network, the members of each layer use the services provided by the layers beneath them to communicate with one another. In engineered network, all data flows through the physical layer. We may see a similar structure in the Universe, all human communication, for instance, passing through our physical organs of sense and movement.

A symmetry exists where changes in a system produce no observable effect. We say a perfect snowflake is symmetrical, since we may rotate it 60 degrees without making any observable change. Similarly a perfect sphere has continuous symmetry, since it may rotate any amount around any axis with no observable change. Space has a symmetrical quality. One can 'move' in 'pure space' without perceiving any difference. This symmetry may account for Newton's first law: 'a body in motion continues in its state of motion unless it is acted upon by a force'. The layering of the universal network explains both symmetry and symmetry breaking. Elements of lower layers (like atoms) are common to all higher layers but are differentiated by the specific roles given to them in higher layers.

The fields corresponding to the fundamental particles add structure to space-time. These fields are believed to carry the energy and momentum which manifest as particles. From the network point of view, space-time is the layer beneath field.

Fields therefore break the symmetry of space, giving local characteristics to something which is globally featureless. We may consider fields as higher layers using the processes provided by space-time as subroutines to execute their own routines.

Symmetries are closely related to conservation laws. The three basic symmetries of spacetime are:

1. Rotation through a fixed angle — conservation of angular momentum, ie action
2. Translation in time — conservation of energy
3. Translation in space — conservation of linear momentum

Each of these tells us something that space-time does not do. Rotation through a fixed angle does not change angular momentum, moving to a different time does not change energy, moving to a different place does not change momentum. But in the absence of space-time we cannot talk about rotation, moving in time or moving in place.

We may think of a symmetry as computable algorithm, that is a logical process which connects an input to an output. In reality all symmetries are broken in the same way that instances of a given algorithm are differentiated by their inputs and outputs and their place in the universal process.

Here we imagine that quantum mechanical algorithms or symmetries are elements of a network layer that lies beneath the four dimensional spacetime in which we find particles, ranging from strutureless fundamental particles to ourselves and beyond.

Quantum mechanics has no memory. Each new state follows logically from the state before it, the state before it being forgotten. When spin down turns to up, there is no memory of down. The origin of space, we guess, is the origin of memory.

The identification of symmetries and algorithms suggests that we can use the symmetries of the world to identify the algorithms that underlie its observed behaviour. By exploring the order of the symmetries we hope to be able to order these algorithms in order of complexity, beginning with the simplest and working toward what we might call 'the algorithm of the universe'.

We exist and observe in 4D space-time. We accept that this space-time is real. Our explanations for what happens in space-time, however, are based on quantum mechanics, which operates in complex, often infinite dimensional, Hilbert spaces which have no immediate connection to space-time. The transition from Hilbert space to real space is imagined to occur through observations represented by self adjoint operators which pick out real fixed points in the quantum dynamics.

Past and future

We can move around in space, but we cannot move in time, although many would wish for time travel. From a formal point of view, reversal of time replaces matter with antimatter. In the early says of the universe, when there was no space and no memory, the direction of time could have no meaning, and so it was reasonable to expect equal quantities of matter and antimattter. The development of memory, however, broke this symmetry, giving a direction to time and removing the possibility of antimatter. Although we can create antimatter in high energy processes which mimic early phases of the universe, extensive observations suggest that there are no longer significant quantities of antimatter in the universe.

In our current epoch, we experience time as a flowing boundary between past and future. It is a one way flow. The past is determined and cannot be revisited, as much as we would sometimes like to go back and change things.

We guess that the past is complete in the mathematical sense developed by Gödel and Turing. Experience suggests that the past is eternal and cannot be changed. Gödel's incompleteness theorem suggests that no consistent recurvsive Universe can be complete. This sugests we identify potential energy with the completed, static past, and kinetic energy with the incomplete dynamic future. Completeness - Wikipedia, Gödel's incompleteness theorems - Wikipedia

(revised 8 June 2016)

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Further reading

Books

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

Feynman, Richard, Feynman Lectures on Gravitation, Westview Press 2002 Amazon Editorial Reviews Book Description 'The Feynman Lectures on Gravitation are based on notes prepared during a course on gravitational physics that Richard Feynman taught at Caltech during the 1962-63 academic year. For several years prior to these lectures, Feynman thought long and hard about the fundamental problems in gravitational physics, yet he published very little. These lectures represent a useful record of his viewpoints and some of his insights into gravity and its application to cosmology, superstars, wormholes, and gravitational waves at that particular time. The lectures also contain a number of fascinating digressions and asides on the foundations of physics and other issues. Characteristically, Feynman took an untraditional non-geometric approach to gravitation and general relativity based on the underlying quantum aspects of gravity. Hence, these lectures contain a unique pedagogical account of the development of Einstein's general theory of relativity as the inevitable result of the demand for a self-consistent theory of a massless spin-2 field (the graviton) coupled to the energy-momentum tensor of matter. This approach also demonstrates the intimate and fundamental connection between gauge invariance and the principle of equivalence.'  Amazon   back

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.  Amazon   back

Kuhn, Thomas S, The Structure of Scientific Revolutions, U of Chicago Press 1996 Introduction: 'a new theory, however special its range of application, is seldom just an increment to what is already known. Its assimilation requires the reconstruction of prior theory and the re-evaluation of prior fact, an intrinsically revolutionary process that is seldom completed by a single man, and never overnight.' [p 7]   Amazon   back

Lonergan, Bernard J F, Insight : A Study of Human Understanding (Collected Works of Bernard Lonergan : Volume 3), University of Toronto Press 1992 '. . . Bernard Lonergan's masterwork. Its aim is nothing less than insight into insight itself, an understanding of understanding'  Amazon   back

Misner, Charles W, and Kip S Thorne, John Archibald Wheeler, Gravitation, Freeman 1973 Jacket: 'Einstein's description of gravitation as curvature of spacetime led directly to that greatest of all predictions of his theory, that the universe itself is dynamic. Physics still has far to go to come to terms with this amazing fact and what it means for man and his relation to the universe. John Archibald Wheeler. . . . this is a book on Einstein's theory of gravity. . . . '  Amazon   back

Mortensen, Chris, Inconsistent Mathematics, Kluwer Academic 1995 'The argument from pure mathematics for studying inconsistency is the best of reasons: because it is there. . . . It is always dangerous to think that a physical use will never be found for a given piece of mathematics. Nor is present-day mathematical physics anomaly free: witness the singularities at the beginning of time or in black holes, delta functions in elementary quantum theory, or renormalisation in quantum field theory.' p 8-9.  Amazon   back

Neuenschwander, Dwight E, Emmy Noether's Wonderful Theorem, Johns Hopkins University Press 2011 Jacket: A beautiful piece of mathematics, Noether's therem touches on every aspect of physics. Emmy Noether proved her theorem in 1915 and published it in 1918. This profound concept demonstrates the connection between conservation laws and symmetries. For instance, the theorem shows that a system invariant under translations of time, space or rotation will obey the laws of conservation of energy, linear momentum or angular momentum respectively. This exciting result offers a rich unifying principle for all of physics.'  Amazon   back

Schilpp, Paul Arthur, and (editor), Albert Einstein: Philosopher-Scientist, Open Court Publishing Company 1949 'Contains Einstein's autobiographical notes in German and English, 25 descriptive and critical essays on the Work of Albert Einstein, Einstein's reply to these essays, and a bibliography of Einstein's writings to May 1951'  Amazon   back

Teilhard de Chardin, Pierre, The Divine Milieu, Harper Collins 1989 Jacket: 'Not a single thought in these pages is the result of computation; everything that is expressed is the fruit of the writer's inner life. In fact this extraordinary book can be read on different levels. There is here, as in all the writings of Father Teillhard, the expression of a scientist who takes delight in the descriptive method and the ultimate meaning of all physical exploration.' Karl Stern  Amazon   back

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.   Amazon   back

Zee, Anthony, Quantum Field Theory in a Nutshell, Princeton University Press 2003 Amazon book description: 'An esteemed researcher and acclaimed popular author takes up the challenge of providing a clear, relatively brief, and fully up-to-date introduction to one of the most vital but notoriously difficult subjects in theoretical physics. A quantum field theory text for the twenty-first century, this book makes the essential tool of modern theoretical physics available to any student who has completed a course on quantum mechanics and is eager to go on. Quantum field theory was invented to deal simultaneously with special relativity and quantum mechanics, the two greatest discoveries of early twentieth-century physics, but it has become increasingly important to many areas of physics. These days, physicists turn to quantum field theory to describe a multitude of phenomena. Stressing critical ideas and insights, Zee uses numerous examples to lead students to a true conceptual understanding of quantum field theory--what it means and what it can do. He covers an unusually diverse range of topics, including various contemporary developments,while guiding readers through thoughtfully designed problems. In contrast to previous texts, Zee incorporates gravity from the outset and discusses the innovative use of quantum field theory in modern condensed matter theory. Without a solid understanding of quantum field theory, no student can claim to have mastered contemporary theoretical physics. Offering a remarkably accessible conceptual introduction, this text will be widely welcomed and used.   Amazon   back

Links

Aquinas 14: Is God a body?, Summa: I 3 1: Is God a body? , 'I answer that, It is absolutely true that God is not a body; and this can be shown in three ways. First, because no body is in motion unless it be put in motion, as is evident from induction. Now it has been already proved (2, 3), that God is the First Mover, and is Himself unmoved. Therefore it is clear that God is not a body. .. .' back

Aquinas 20, Summa I, 3, 7: Whether God is altogether simple? , 'I answer that, The absolute simplicity of God may be shown in many ways. First, from the previous articles of this question. For there is neither composition of quantitative parts in God, since He is not a body; nor composition of matter and form; nor does His nature differ from His "suppositum"; nor His essence from His existence; neither is there in Him composition of genus and difference, nor of subject and accident. Therefore, it is clear that God is nowise composite, but is altogether simple. . . . ' back

Aquinas 43, Summa I, 9, 1: Whether god is altogether immutable, 'I answer that, From what precedes, it is shown that God is altogether immutable. First, because it was shown above that there is some first being, whom we call God; and that this first being must be pure act, without the admixture of any potentiality, for the reason that, absolutely, potentiality is posterior to act. Now everything which is in any way changed, is in some way in potentiality. Hence it is evident that it is impossible for God to be in any way changeable. ... ' back

Aquinas 47, Summa I, 10, 3: Does it belong to god to be eternal?, 'I answer that, Eternity truly and properly so called is in God alone, because eternity follows on immutability; as appears from the first article. But God alone is altogether immutable, as was shown above (9, 1). Accordingly, however, as some receive immutability from Him, they share in His eternity. ... ' back

Cantor's theorem - Wikipedia, Cantor's theorem - Wikipedia, the free encyclopedia, 'In elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A (the power set of A) has a strictly greater cardinality than A itself. For finite sets, Cantor's theorem can be seen to be true by a much simpler proof than that given below, since in addition to subsets of A with just one member, there are others as well, and since n < 2n for all natural numbers n. But the theorem is true of infinite sets as well. In particular, the power set of a countably infinite set is uncountably infinite. The theorem is named for German mathematician Georg Cantor, who first stated and proved it.' back

Completeness - Wikipedia, Completeness - Wikipedia, the free encyclopedia, 'In general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields.' back

Cosmic microwave background radiation - Wikipedia, Cosmic microwave background radiation - Wikipedia, the free encyclopedia, 'The CMB is a snapshot of the oldest light in our Universe, imprinted on the sky when the Universe was just 380,000 years old. It shows tiny temperature fluctuations that correspond to regions of slightly different densities, representing the seeds of all future structure: the stars and galaxies of today.' back

David K Sing, 12. Einstein's Field Equation & Newtonian Limit, 'So Einstein’s field equation reducing to the Newtonian Poisson equation completes the proof that Newtonian gravity is the limiting case of GR.' back

Differentiable manifold - Wikipedia, Differentiable manifold - Wikipedia, the free encyclopedia, 'In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. back

Dirac equation - Wikipedia, Dirac equation - Wikipedia, the free encyclopedia, 'In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1⁄2 massive particles such as electrons and quarks, for which parity is a symmetry, and is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It accounted for the fine details of the hydrogen spectrum in a completely rigorous way.' back

Edmund Bertschinger, Coordinates and Proper Time, Now it came to me: . . . the independence of the gravitational acceleration from the nature of the falling substance, may be expressed as follows: In a gravitational field (of small spatial extension) things behave as they do in a space free of gravitation. . . . This happened in 1908. Why were another seven years required for the construction of the general theory of relativity? The main reason lies in the fact that it is not so easy to free oneself from the idea that coordinates must have an immediate metric al meaning. | A. Einstein (quoted in Albert Einstein: Philosopher-Scientist , ed. P.A. Schilpp, 1949) back

Elementary particle - Wikipedia, Elementary particle - Wikipedia, the free encyclopedia, 'In particle physics, an elementary particle or fundamental particle is a particle whose substructure is unknown, thus it is unknown whether it is composed of other particles. Known elementary particles include the fundamental fermions (quarks, leptons, antiquarks, and antileptons), which generally are "matter particles" and "antimatter particles", as well as the fundamental bosons (gauge bosons and Higgs boson), which generally are "force particles" that mediate interactions among fermions. A particle containing two or more elementary particles is a composite particle.' back

Fixed point theorem - Wikipedia, Fixed point theorem - Wikipedia, the free encyclopedia, 'In mathematics, a fixed point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. Results of this kind are amongst the most generally useful in mathematics. The Banach fixed point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. By contrast, the Brouwer fixed point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma).' back

Gödel's incompleteness theorems - Wikipedia, Gödel's incompleteness theorems - Wikipedia, 'Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (i.e., any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.' back

Hawking radiation - Wikipedia, Hawking radiation - Wikipedia, the free encyclopedia, 'Hawking radiation is black-body radiation that is predicted to be released by black holes, due to quantum effects near the event horizon. It is named after the physicist Stephen Hawking, who provided a theoretical argument for its existence in 1974] and sometimes also after Jacob Bekenstein, who predicted that black holes should have a finite, non-zero temperature and entropy.' back

Inflation (cosmology) - Wikipedia, Inflation (cosmology) - Wikipedia, the free encyclopedia, 'In physical cosmology, cosmic inflation, cosmological inflation, or just inflation is a theory of exponential expansion of space in the early universe. The inflationary epoch lasted from 10−36 seconds after the Big Bang to sometime between 10−33 and 10−32 seconds. Following the inflationary period, the Universe continues to expand, but at a less rapid rate.' . . . The detailed particle physics mechanism responsible for inflation is not known. The basic inflationary paradigm is accepted by most scientists, who believe a number of predictions have been confirmed by observation; however, a substantial minority of scientists dissent from this position.' back

Interference (wave propagation) - Wikipedia, Interference (wave propagation) - Wikipedia, the free encyclopedia, 'In physics, interference is a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. Interference usually refers to the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency. Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves or matter waves.' back

Kinetic energy - Wikipedia, Kinetic energy - Wikipedia, the free encyclopedia, 'The kinetic energy of an object is the energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body in decelerating from its current speed to a state of rest.' back

Leon Horsten (Standord Encyclopedia of Philosophy), Philosophy of Mathematics, 'If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space in time, it is not at all obvious that this also the case of the objects that are studied in mathematics.' back

Marcelo Samuel Berman, On the Zero-Energy Universe, 'We consider the energy of the Universe, from the pseudo-tens or point of view(Berman,1981). We find zero values, when the calculations are well-done.The doubts concerning this subject are clarified, with the novel idea that the justification for the calculation lies in the association of the equivalence principle, with the nature of co-motional observers, as demanded in Cosmology. In Section 4, we give a novel calculation for the zero-total energy result.' back

Mathematics - Wikipedia, Mathematics - Wikipedia, the free encyclopedia, 'Mathematics is the abstract study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.' back

Metric expansion of space - Wikipedia, Metric expansion of space - Wikipedia, the free encyclopedia, 'The metric expansion of space is the increase of the distance between two distant parts of the universe with time. It is an intrinsic expansion whereby the scale of space itself changes. This is different from other examples of expansions and explosions in that, as far as observations can ascertain, it is a property of the entirety of the universe rather than a phenomenon that can be contained and observed from the outside.' back

Monotheism - Wikipedia, Monotheism - Wikipedia, the free encyclopedia, 'Monotheism (from Greek μόνος, monos, "single", and θεός, theos, "god") is the belief in the existence of one and only one god. Monotheism is characteristic of the Baha'i Faith, Christianity, Druzism, Hinduism Islam, Judaism, Samaritanism, Sikhism and Zoroastrianism.' back

Path integral formulation - Wikipedia, Path integral formulation - Wikipedia, the free encyclopedia, 'The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a quantum amplitude. . . . This formulation has proved crucial to the subsequent development of theoretical physics, since it provided the basis for the grand synthesis of the 1970s which unified quantum field theory with statistical mechanics. . . . ' back

Pauli exclusion principle - Wikipedia, Pauli exclusion principle - Wikipedia, the free encyclopedia, 'The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions (particles with half-integer spin) may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles. The principle was formulated by Austrian physicist Wolfgang Pauli in 1925.' back

Pierre Teilhard de Chardin - Wikipedia, Pierre Teilhard de Chardin - Wikipedia, the free encyclopedia, 'Pierre Teilhard de Chardin SJ (1 May 1881 – 10 April 1955) was a French idealist philosopher and Jesuit priest who trained as a paleontologist and geologist and took part in the discovery of Peking Man. He conceived the idea of the Omega Point (a maximum level of complexity and consciousness towards which he believed the universe was evolving) and developed Vladimir Vernadsky's concept of noosphere.' back

Potential energy - Wikipedia, Potential energy - Wikipedia, the free encyclopedia, 'In physics, potential energy is the energy of an object or a system due to the position of the body or the arrangement of the particles of the system. The SI unit for measuring work and energy is the joule (symbol J). The term potential energy was coined by the 19th century Scottish engineer and physicist William Rankine although it has links to Greek philosopher Aristotle's concept of potentiality. Potential energy is associated with a set of forces that act on a body in a way that depends only on the body's position in space.' back

Quantum field theory - Wikipedia, Quantum field theory - Wikipedia, the free encyclopedia, 'Quantum field theory (QFT) provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or (especially in a condensed matter context) of many-body systems. . . . In QFT photons are not thought of as 'little billiard balls', they are considered to be field quanta - necessarily chunked ripples in a field that 'look like' particles. Fermions, like the electron, can also be described as ripples in a field, where each kind of fermion has its own field. In summary, the classical visualisation of "everything is particles and fields", in quantum field theory, resolves into "everything is particles", which then resolves into "everything is fields". In the end, particles are regarded as excited states of a field (field quanta). back

Spin (physics) - Wikipedia, Spin (physics) - Wikipedia, the free encyclopedia, 'In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei. Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum. Orbital angular momentum is the quantum-mechanical counterpart to the classical notion of angular momentum: it arises when a particle executes a rotating or twisting trajectory (such as when an electron orbits a nucleus). The existence of spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which particles are observed to possess angular momentum that cannot be accounted for by orbital angular momentum alone.' back

Spin-statistics theorem - Wikipedia, Spin-statistics theorem - Wikipedia, the free encyclopedia, 'In quantum mechanics, the spin–statistics theorem relates the spin of a particle to the particle statistics it obeys. The spin of a particle is its intrinsic angular momentum (that is, the contribution to the total angular momentum that is not due to the orbital motion of the particle). All particles have either integer spin or half-integer spin (in units of the reduced Planck constant ħ). The theorem states that: The wave function of a system of identical integer-spin particles has the same value when the positions of any two particles are swapped. Particles with wave functions symmetric under exchange are called bosons. The wave function of a system of identical half-integer spin particles changes sign when two particles are swapped. Particles with wave functions antisymmetric under exchange are called fermions.' back

Standard model - Wikipedia, Standard model - Wikipedia, the free encyclopedia, 'The Standard Model of particle physics is a theory that describes three of the four known fundamental interactions between the elementary particles that make up all matter. It is a quantum field theory developed between 1970 and 1973 which is consistent with both quantum mechanics and special relativity. To date, almost all experimental tests of the three forces described by the Standard Model have agreed with its predictions. However, the Standard Model falls short of being a complete theory of fundamental interactions, primarily because of its lack of inclusion of gravity, the fourth known fundamental interaction, but also because of the large number of numerical parameters (such as masses and coupling constants) that must be put "by hand" into the theory (rather than being derived from first principles) . . . ' back

Stellar evolution - Wikipedia, Stellar evolution - Wikipedia, the free encyclopedia, 'Stellar evolution is the process by which a star changes during its lifetime. Depending on the mass of the star, this lifetime ranges from a few million years for the most massive to trillions of years for the least massive, which is considerably longer than the age of the universe.' back

Time-division multiplexing - Wikipedia, Time-division multiplexing - Wikipedia, the free encyclopedia, 'Time-division multiplexing (TDM) is a method of transmitting and receiving independent signals over a common signal path by means of synchronized switches at each end of the transmission line so that each signal appears on the line only a fraction of time in an alternating pattern.' back